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@c This file is generated automatically
@c Do not edit
@chapter Examples
@section MDSSystem
Robust control of a mass-damper-spring system.
Type @code{which MDSSystem} to locate,
@code{edit MDSSystem} to open and simply
@code{MDSSystem} to run the example file.
@section optiPID
Numerical optimization of a PID controller using an objective function.
The objective function is located in the file @command{optiPIDfun}.
Type @code{which optiPID} to locate, @code{edit optiPID} to open
and simply @code{optiPID} to run the example file.
@section Anderson
Frequency-weighted coprime factorization controller reduction.
@section Madievski
Frequency-weighted controller reduction.
@chapter Linear Time Invariant Models
@section dss
@deftypefn {Function File} {@var{sys} =} dss (@var{sys})
@deftypefnx {Function File} {@var{sys} =} dss (@var{d})
@deftypefnx {Function File} {@var{sys} =} dss (@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @dots{})
@deftypefnx {Function File} {@var{sys} =} dss (@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @var{tsam}, @dots{})
Create or convert to descriptor state-space model.
@strong{Inputs}
@table @var
@item sys
LTI model to be converted to state-space.
@item a
State transition matrix (n-by-n).
@item b
Input matrix (n-by-m).
@item c
Measurement matrix (p-by-n).
@item d
Feedthrough matrix (p-by-m).
@item e
Descriptor matrix (n-by-n).
@item tsam
Sampling time in seconds. If @var{tsam} is not specified,
a continuous-time model is assumed.
@item @dots{}
Optional pairs of properties and values.
Type @command{set (dss)} for more information.
@end table
@strong{Outputs}
@table @var
@item sys
Descriptor state-space model.
@end table
@strong{Equations}
@example
@group
.
E x = A x + B u
y = C x + D u
@end group
@end example
@seealso{ss, tf}
@end deftypefn
@section filt
@deftypefn {Function File} {@var{sys} =} filt (@var{num}, @var{den}, @dots{})
@deftypefnx {Function File} {@var{sys} =} filt (@var{num}, @var{den}, @var{tsam}, @dots{})
Create discrete-time transfer function model from data in DSP format.
@strong{Inputs}
@table @var
@item num
Numerator or cell of numerators. Each numerator must be a row vector
containing the coefficients of the polynomial in ascending powers of z^-1.
num@{i,j@} contains the numerator polynomial from input j to output i.
In the SISO case, a single vector is accepted as well.
@item den
Denominator or cell of denominators. Each denominator must be a row vector
containing the coefficients of the polynomial in ascending powers of z^-1.
den@{i,j@} contains the denominator polynomial from input j to output i.
In the SISO case, a single vector is accepted as well.
@item tsam
Sampling time in seconds. If @var{tsam} is not specified,
default value -1 (unspecified) is taken.
@item @dots{}
Optional pairs of properties and values.
Type @command{set (filt)} for more information.
@end table
@strong{Outputs}
@table @var
@item sys
Discrete-time transfer function model.
@end table
@strong{Example}
@example
@group
3 z^-1
H(z^-1) = -------------------
1 + 4 z^-1 + 2 z^-2
octave:1> H = filt ([0, 3], [1, 4, 2])
Transfer function 'H' from input 'u1' to output ...
3 z^-1
y1: -------------------
1 + 4 z^-1 + 2 z^-2
Sampling time: unspecified
Discrete-time model.
@end group
@end example
@seealso{tf}
@end deftypefn
@section frd
@deftypefn {Function File} {@var{sys} =} frd (@var{sys})
@deftypefnx {Function File} {@var{sys} =} frd (@var{sys}, @var{w})
@deftypefnx {Function File} {@var{sys} =} frd (@var{H}, @var{w}, @dots{})
@deftypefnx {Function File} {@var{sys} =} frd (@var{H}, @var{w}, @var{tsam}, @dots{})
Create or convert to frequency response data.
@strong{Inputs}
@table @var
@item sys
LTI model to be converted to frequency response data.
If second argument @var{w} is omitted, the interesting
frequency range is calculated by the zeros and poles of @var{sys}.
@item H
Frequency response array (p-by-m-by-lw). H(i,j,k) contains the
response from input j to output i at frequency k. In the SISO case,
a vector (lw-by-1) or (1-by-lw) is accepted as well.
@item w
Frequency vector (lw-by-1) in radian per second [rad/s].
Frequencies must be in ascending order.
@item tsam
Sampling time in seconds. If @var{tsam} is not specified,
a continuous-time model is assumed.
@item @dots{}
Optional pairs of properties and values.
Type @command{set (frd)} for more information.
@end table
@strong{Outputs}
@table @var
@item sys
Frequency response data object.
@end table
@seealso{dss, ss, tf}
@end deftypefn
@section ss
@deftypefn {Function File} {@var{sys} =} ss (@var{sys})
@deftypefnx {Function File} {@var{sys} =} ss (@var{d})
@deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b})
@deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}, @var{c})
@deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}, @var{c}, @var{d}, @dots{})
@deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}, @var{c}, @var{d}, @var{tsam}, @dots{})
Create or convert to state-space model.
@strong{Inputs}
@table @var
@item sys
LTI model to be converted to state-space.
@item a
State transition matrix (n-by-n).
@item b
Input matrix (n-by-m).
@item c
Measurement matrix (p-by-n).
If @var{c} is empty @code{[]} or not specified, an identity matrix is assumed.
@item d
Feedthrough matrix (p-by-m).
If @var{d} is empty @code{[]} or not specified, a zero matrix is assumed.
@item tsam
Sampling time in seconds. If @var{tsam} is not specified, a continuous-time model is assumed.
@item @dots{}
Optional pairs of properties and values.
Type @command{set (ss)} for more information.
@end table
@strong{Outputs}
@table @var
@item sys
State-space model.
@end table
@strong{Example}
@example
@group
octave:1> a = [1 2 3; 4 5 6; 7 8 9];
octave:2> b = [10; 11; 12];
octave:3> stname = @{"V", "A", "kJ"@};
octave:4> sys = ss (a, b, [], [], "stname", stname)
sys.a =
V A kJ
V 1 2 3
A 4 5 6
kJ 7 8 9
sys.b =
u1
V 10
A 11
kJ 12
sys.c =
V A kJ
y1 1 0 0
y2 0 1 0
y3 0 0 1
sys.d =
u1
y1 0
y2 0
y3 0
Continuous-time model.
octave:5>
@end group
@end example
@seealso{tf, dss}
@end deftypefn
@section tf
@deftypefn {Function File} {@var{s} =} tf (@var{"s"})
@deftypefnx {Function File} {@var{z} =} tf (@var{"z"}, @var{tsam})
@deftypefnx {Function File} {@var{sys} =} tf (@var{sys})
@deftypefnx {Function File} {@var{sys} =} tf (@var{num}, @var{den}, @dots{})
@deftypefnx {Function File} {@var{sys} =} tf (@var{num}, @var{den}, @var{tsam}, @dots{})
Create or convert to transfer function model.
@strong{Inputs}
@table @var
@item sys
LTI model to be converted to transfer function.
@item num
Numerator or cell of numerators. Each numerator must be a row vector
containing the coefficients of the polynomial in descending powers of
the transfer function variable.
num@{i,j@} contains the numerator polynomial from input j to output i.
In the SISO case, a single vector is accepted as well.
@item den
Denominator or cell of denominators. Each denominator must be a row vector
containing the coefficients of the polynomial in descending powers of
the transfer function variable.
den@{i,j@} contains the denominator polynomial from input j to output i.
In the SISO case, a single vector is accepted as well.
@item tsam
Sampling time in seconds. If @var{tsam} is not specified, a continuous-time
model is assumed.
@item @dots{}
Optional pairs of properties and values.
Type @command{set (tf)} for more information.
@end table
@strong{Outputs}
@table @var
@item sys
Transfer function model.
@end table
@strong{Example}
@example
@group
octave:1> s = tf ("s");
octave:2> G = 1/(s+1)
Transfer function "G" from input "u1" to output ...
1
y1: -----
s + 1
Continuous-time model.
@end group
@end example
@example
@group
octave:3> z = tf ("z", 0.2);
octave:4> H = 0.095/(z-0.9)
Transfer function "H" from input "u1" to output ...
0.095
y1: -------
z - 0.9
Sampling time: 0.2 s
Discrete-time model.
@end group
@end example
@example
@group
octave:5> num = @{[1, 5, 7], [1]; [1, 7], [1, 5, 5]@};
octave:6> den = @{[1, 5, 6], [1, 2]; [1, 8, 6], [1, 3, 2]@};
octave:7> sys = tf (num, den)
Transfer function "sys" from input "u1" to output ...
s^2 + 5 s + 7
y1: -------------
s^2 + 5 s + 6
s + 7
y2: -------------
s^2 + 8 s + 6
Transfer function "sys" from input "u2" to output ...
1
y1: -----
s + 2
s^2 + 5 s + 5
y2: -------------
s^2 + 3 s + 2
Continuous-time model.
octave:8>
@end group
@end example
@seealso{ss, dss}
@end deftypefn
@section zpk
@deftypefn {Function File} {@var{s} =} zpk (@var{"s"})
@deftypefnx {Function File} {@var{z} =} zpk (@var{"z"}, @var{tsam})
@deftypefnx {Function File} {@var{sys} =} zpk (@var{sys})
@deftypefnx {Function File} {@var{sys} =} zpk (@var{k})
@deftypefnx {Function File} {@var{sys} =} zpk (@var{z}, @var{p}, @var{k}, @dots{})
@deftypefnx {Function File} {@var{sys} =} zpk (@var{z}, @var{p}, @var{k}, @var{tsam}, @dots{})
@deftypefnx {Function File} {@var{sys} =} zpk (@var{z}, @var{p}, @var{k}, @var{tsam}, @dots{})
Create transfer function model from zero-pole-gain data.
This is just a stop-gap compatibility wrapper since zpk
models are not yet implemented.
@strong{Inputs}
@table @var
@item sys
LTI model to be converted to transfer function.
@item z
Cell of vectors containing the zeros for each channel.
z@{i,j@} contains the zeros from input j to output i.
In the SISO case, a single vector is accepted as well.
@item p
Cell of vectors containing the poles for each channel.
p@{i,j@} contains the poles from input j to output i.
In the SISO case, a single vector is accepted as well.
@item k
Matrix containing the gains for each channel.
k(i,j) contains the gain from input j to output i.
@item tsam
Sampling time in seconds. If @var{tsam} is not specified,
a continuous-time model is assumed.
@item @dots{}
Optional pairs of properties and values.
Type @command{set (tf)} for more information.
@end table
@strong{Outputs}
@table @var
@item sys
Transfer function model.
@end table
@seealso{tf, ss, dss, frd}
@end deftypefn
@chapter Model Data Access
@section @@lti/dssdata
@deftypefn {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @var{tsam}] =} dssdata (@var{sys})
@deftypefnx {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @var{tsam}] =} dssdata (@var{sys}, @var{[]})
Access descriptor state-space model data.
Argument @var{sys} is not limited to descriptor state-space models.
If @var{sys} is not a descriptor state-space model, it is converted automatically.
@strong{Inputs}
@table @var
@item sys
Any type of LTI model.
@item []
In case @var{sys} is not a dss model (descriptor matrix @var{e} empty),
@code{dssdata (sys, [])} returns the empty element @code{e = []} whereas
@code{dssdata (sys)} returns the identity matrix @code{e = eye (size (a))}.
@end table
@strong{Outputs}
@table @var
@item a
State transition matrix (n-by-n).
@item b
Input matrix (n-by-m).
@item c
Measurement matrix (p-by-n).
@item d
Feedthrough matrix (p-by-m).
@item e
Descriptor matrix (n-by-n).
@item tsam
Sampling time in seconds. If @var{sys} is a continuous-time model,
a zero is returned.
@end table
@end deftypefn
@section @@lti/filtdata
@deftypefn {Function File} {[@var{num}, @var{den}, @var{tsam}] =} filtdata (@var{sys})
@deftypefnx {Function File} {[@var{num}, @var{den}, @var{tsam}] =} filtdata (@var{sys}, @var{"vector"})
Access discrete-time transfer function data in DSP format.
Argument @var{sys} is not limited to transfer function models.
If @var{sys} is not a transfer function, it is converted automatically.
@strong{Inputs}
@table @var
@item sys
Any type of discrete-time LTI model.
@item "v", "vector"
For SISO models, return @var{num} and @var{den} directly as column vectors
instead of cells containing a single column vector.
@end table
@strong{Outputs}
@table @var
@item num
Cell of numerator(s). Each numerator is a row vector
containing the coefficients of the polynomial in ascending powers of z^-1.
num@{i,j@} contains the numerator polynomial from input j to output i.
In the SISO case, a single vector is possible as well.
@item den
Cell of denominator(s). Each denominator is a row vector
containing the coefficients of the polynomial in ascending powers of z^-1.
den@{i,j@} contains the denominator polynomial from input j to output i.
In the SISO case, a single vector is possible as well.
@item tsam
Sampling time in seconds. If @var{tsam} is not specified, -1 is returned.
@end table
@end deftypefn
@section @@lti/frdata
@deftypefn {Function File} {[@var{H}, @var{w}, @var{tsam}] =} frdata (@var{sys})
@deftypefnx {Function File} {[@var{H}, @var{w}, @var{tsam}] =} frdata (@var{sys}, @var{"vector"})
Access frequency response data.
Argument @var{sys} is not limited to frequency response data objects.
If @var{sys} is not a frd object, it is converted automatically.
@strong{Inputs}
@table @var
@item sys
Any type of LTI model.
@item "v", "vector"
In case @var{sys} is a SISO model, this option returns the frequency response
as a column vector (lw-by-1) instead of an array (p-by-m-by-lw).
@end table
@strong{Outputs}
@table @var
@item H
Frequency response array (p-by-m-by-lw). H(i,j,k) contains the
response from input j to output i at frequency k. In the SISO case,
a vector (lw-by-1) is possible as well.
@item w
Frequency vector (lw-by-1) in radian per second [rad/s].
Frequencies are in ascending order.
@item tsam
Sampling time in seconds. If @var{sys} is a continuous-time model,
a zero is returned.
@end table
@end deftypefn
@section @@lti/get
@deftypefn {Function File} get (@var{sys})
@deftypefnx {Function File} {@var{value} =} get (@var{sys}, @var{"property"})
Access property values of LTI objects.
@end deftypefn
@section @@lti/set
@deftypefn {Function File} set (@var{sys})
@deftypefnx {Function File} set (@var{sys}, @var{"property"}, @var{value}, @dots{})
@deftypefnx {Function File} {@var{retsys} =} set (@var{sys}, @var{"property"}, @var{value}, @dots{})
Set or modify properties of LTI objects.
If no return argument @var{retsys} is specified, the modified LTI object is stored
in input argument @var{sys}. @command{set} can handle multiple properties in one call:
@code{set (sys, 'prop1', val1, 'prop2', val2, 'prop3', val3)}.
@code{set (sys)} prints a list of the object's property names.
@end deftypefn
@section @@lti/ssdata
@deftypefn {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}, @var{tsam}] =} ssdata (@var{sys})
Access state-space model data.
Argument @var{sys} is not limited to state-space models.
If @var{sys} is not a state-space model, it is converted automatically.
@strong{Inputs}
@table @var
@item sys
Any type of LTI model.
@end table
@strong{Outputs}
@table @var
@item a
State transition matrix (n-by-n).
@item b
Input matrix (n-by-m).
@item c
Measurement matrix (p-by-n).
@item d
Feedthrough matrix (p-by-m).
@item tsam
Sampling time in seconds. If @var{sys} is a continuous-time model,
a zero is returned.
@end table
@end deftypefn
@section @@lti/tfdata
@deftypefn {Function File} {[@var{num}, @var{den}, @var{tsam}] =} tfdata (@var{sys})
@deftypefnx {Function File} {[@var{num}, @var{den}, @var{tsam}] =} tfdata (@var{sys}, @var{"vector"})
@deftypefnx {Function File} {[@var{num}, @var{den}, @var{tsam}] =} tfdata (@var{sys}, @var{"tfpoly"})
Access transfer function data.
Argument @var{sys} is not limited to transfer function models.
If @var{sys} is not a transfer function, it is converted automatically.
@strong{Inputs}
@table @var
@item sys
Any type of LTI model.
@item "v", "vector"
For SISO models, return @var{num} and @var{den} directly as column vectors
instead of cells containing a single column vector.
@end table
@strong{Outputs}
@table @var
@item num
Cell of numerator(s). Each numerator is a row vector
containing the coefficients of the polynomial in descending powers of
the transfer function variable.
num@{i,j@} contains the numerator polynomial from input j to output i.
In the SISO case, a single vector is possible as well.
@item den
Cell of denominator(s). Each denominator is a row vector
containing the coefficients of the polynomial in descending powers of
the transfer function variable.
den@{i,j@} contains the denominator polynomial from input j to output i.
In the SISO case, a single vector is possible as well.
@item tsam
Sampling time in seconds. If @var{sys} is a continuous-time model,
a zero is returned.
@end table
@end deftypefn
@section @@lti/zpkdata
@deftypefn {Function File} {[@var{z}, @var{p}, @var{k}, @var{tsam}] =} zpkdata (@var{sys})
@deftypefnx {Function File} {[@var{z}, @var{p}, @var{k}, @var{tsam}] =} zpkdata (@var{sys}, @var{"v"})
Access zero-pole-gain data.
@strong{Inputs}
@table @var
@item sys
Any type of LTI model.
@item "v", "vector"
For SISO models, return @var{z} and @var{p} directly as column vectors
instead of cells containing a single column vector.
@end table
@strong{Outputs}
@table @var
@item z
Cell of column vectors containing the zeros for each channel.
z@{i,j@} contains the zeros from input j to output i.
@item p
Cell of column vectors containing the poles for each channel.
p@{i,j@} contains the poles from input j to output i.
@item k
Matrix containing the gains for each channel.
k(i,j) contains the gain from input j to output i.
@item tsam
Sampling time in seconds. If @var{sys} is a continuous-time model,
a zero is returned.
@end table
@end deftypefn
@chapter Model Conversions
@section @@lti/c2d
@deftypefn {Function File} {@var{sys} =} c2d (@var{sys}, @var{tsam})
@deftypefnx {Function File} {@var{sys} =} c2d (@var{sys}, @var{tsam}, @var{method})
@deftypefnx {Function File} {@var{sys} =} c2d (@var{sys}, @var{tsam}, @var{"prewarp"}, @var{w0})
Convert the continuous lti model into its discrete-time equivalent.
@strong{Inputs}
@table @var
@item sys
Continuous-time LTI model.
@item tsam
Sampling time in seconds.
@item method
Optional conversion method. If not specified, default method @var{"zoh"}
is taken.
@table @var
@item "zoh"
Zero-order hold or matrix exponential.
@item "tustin", "bilin"
Bilinear transformation or Tustin approximation.
@item "prewarp"
Bilinear transformation with pre-warping at frequency @var{w0}.
@end table
@end table
@strong{Outputs}
@table @var
@item sys
Discrete-time LTI model.
@end table
@end deftypefn
@section @@lti/d2c
@deftypefn {Function File} {@var{sys} =} d2c (@var{sys})
@deftypefnx {Function File} {@var{sys} =} d2c (@var{sys}, @var{method})
@deftypefnx {Function File} {@var{sys} =} d2c (@var{sys}, @var{"prewarp"}, @var{w0})
Convert the discrete lti model into its continuous-time equivalent.
@strong{Inputs}
@table @var
@item sys
Discrete-time LTI model.
@item method
Optional conversion method. If not specified, default method @var{"zoh"}
is taken.
@table @var
@item "zoh"
Zero-order hold or matrix logarithm.
@item "tustin", "bilin"
Bilinear transformation or Tustin approximation.
@item "prewarp"
Bilinear transformation with pre-warping at frequency @var{w0}.
@end table
@end table
@strong{Outputs}
@table @var
@item sys
Continuous-time LTI model.
@end table
@end deftypefn
@section @@lti/prescale
@deftypefn {Function File} {[@var{scaledsys}, @var{info}] =} prescale (@var{sys})
Prescale state-space model.
Frequency response commands perform automatic scaling unless model property
@var{scaled} is set to @var{true}.
@strong{Inputs}
@table @var
@item sys
LTI model.
@end table
@strong{Outputs}
@table @var
@item scaledsys
Scaled state-space model.
@item info
Structure containing additional information.
@item info.SL
Left scaling factors. @code{Tl = diag (info.SL)}.
@item info.SR
Right scaling factors. @code{Tr = diag (info.SR)}.
@end table
@strong{Equations}
@example
@group
Es = Tl * E * Tr
As = Tl * A * Tr
Bs = Tl * B
Cs = C * Tr
Ds = D
@end group
@end example
For proper state-space models, @var{Tl} and @var{Tr} are inverse of each other.
@strong{Algorithm}@*
Uses SLICOT TB01ID and TG01AD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}.
@end deftypefn
@section @@lti/xperm
@deftypefn {Function File} {@var{sys} =} xperm (@var{sys}, @var{st_idx})
Reorder states in state-space models.
@end deftypefn
@chapter Model Interconnections
@section @@lti/append
@deftypefn {Function File} {@var{sys} =} append (@var{sys1}, @var{sys2})
Group LTI models by appending their inputs and outputs.
@end deftypefn
@section @@lti/blkdiag
@deftypefn {Function File} {@var{sys} =} blkdiag (@var{sys1}, @var{sys2})
Block-diagonal concatenation of LTI models.
@end deftypefn
@section @@lti/connect
@deftypefn {Function File} {@var{sys} =} connect (@var{sys}, @var{cm}, @var{inputs}, @var{outputs})
Arbitrary interconnections between the inputs and outputs of an LTI model.
@end deftypefn
@section @@lti/feedback
@deftypefn {Function File} {@var{sys} =} feedback (@var{sys1})
@deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{"+"})
@deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2})
@deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}, @var{"+"})
@deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}, @var{feedin}, @var{feedout})
@deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}, @var{feedin}, @var{feedout}, @var{"+"})
Feedback connection of two LTI models.
@strong{Inputs}
@table @var
@item sys1
LTI model of forward transmission. @code{[p1, m1] = size (sys1)}.
@item sys2
LTI model of backward transmission.
If not specified, an identity matrix of appropriate size is taken.
@item feedin
Vector containing indices of inputs to @var{sys1} which are involved in the feedback loop.
The number of @var{feedin} indices and outputs of @var{sys2} must be equal.
If not specified, @code{1:m1} is taken.
@item feedout
Vector containing indices of outputs from @var{sys1} which are to be connected to @var{sys2}.
The number of @var{feedout} indices and inputs of @var{sys2} must be equal.
If not specified, @code{1:p1} is taken.
@item "+"
Positive feedback sign. If not specified, @var{"-"} for a negative feedback interconnection
is assumed. @var{+1} and @var{-1} are possible as well, but only from the third argument
onward due to ambiguity.
@end table
@strong{Outputs}
@table @var
@item sys
Resulting LTI model.
@end table
@strong{Block Diagram}
@example
@group
u + +--------+ y
------>(+)----->| sys1 |-------+------->
^ - +--------+ |
| |
| +--------+ |
+-------| sys2 |<------+
+--------+
@end group
@end example
@end deftypefn
@section @@lti/lft
@deftypefn {Function File} {@var{sys} =} lft (@var{sys1}, @var{sys2})
@deftypefnx {Function File} {@var{sys} =} lft (@var{sys1}, @var{sys2}, @var{nu}, @var{ny})
Linear fractional tranformation, also known as Redheffer star product.
@strong{Inputs}
@table @var
@item sys1
Upper LTI model.
@item sys2
Lower LTI model.
@item nu
The last nu inputs of @var{sys1} are connected with the first nu outputs of @var{sys2}.
If not specified, @code{min (m1, p2)} is taken.
@item ny
The last ny outputs of @var{sys1} are connected with the first ny inputs of @var{sys2}.
If not specified, @code{min (p1, m2)} is taken.
@end table
@strong{Outputs}
@table @var
@item sys
Resulting LTI model.
@end table
@strong{Block Diagram}
@example
@group
.............sys..............
: +--------+ :
w1 ------------>| |------------> z1
: | sys1 | :
: u +---->| |-----+ y :
: | +--------+ | : Lower LFT
: | | :
: | +--------+ | : lft (sys1, sys2)
: +-----| sys2 |<----+ :
: +--------+ :
:............................:
@end group
@end example
@example
@group
.............sys..............
: +--------+ :
: u +---->| sys1 |-----+ y :
: | +--------+ | : Upper LFT
: | | :
: | +--------+ | : lft (sys1, sys2)
: +-----| |<----+ :
: | sys2 | :
z2 <------------| |<------------ w2
: +--------+ :
:............................:
@end group
@end example
@example
@group
.............sys..............
: +--------+ :
w1 ------------>| |------------> z1
: | sys1 | :
: u +---->| |-----+ y :
: | +--------+ | :
: | | : lft (sys1, sys2, nu, ny)
: | +--------+ | :
: +-----| |<----+ :
: | sys2 | :
z2 <------------| |<------------ w2
: +--------+ :
:............................:
@end group
@end example
@end deftypefn
@section @@lti/mconnect
@deftypefn {Function File} {@var{sys} =} mconnect (@var{sys}, @var{m})
@deftypefnx {Function File} {@var{sys} =} mconnect (@var{sys}, @var{m}, @var{inputs}, @var{outputs})
Arbitrary interconnections between the inputs and outputs of an LTI model.
@strong{Inputs}
@table @var
@item sys
LTI system.
@item m
Connection matrix. Each row belongs to an input and each column represents an output.
@item inputs
Vector of indices of those inputs which are retained. If not specified, all inputs are kept.
@item outputs
Vector of indices of those outputs which are retained. If not specified, all outputs are kept.
@end table
@strong{Outputs}
@table @var
@item sys
Interconnected system.
@end table
@strong{Example}
@example
@group
Solve the system equations of
y(t) = G e(t)
e(t) = u(t) + M y(t)
in order to build
y(t) = H u(t)
The matrix M for a (p-by-m) system G
has m rows and p columns (m-by-p).
Example for a 3x2 system:
u1 = -1*y1 + 5*y2 + 0*y3
u2 = pi*y1 + 0*y2 - 7*y3
| -1 5 0 |
M = | pi 0 7 |
@end group
@end example
@end deftypefn
@section @@lti/parallel
@deftypefn{Function File} {@var{sys} =} parallel (@var{sys1}, @var{sys2})
Parallel connection of two LTI systems.
@strong{Block Diagram}
@example
@group
..........................
: +--------+ :
: +-->| sys1 |---+ :
u : | +--------+ | + : y
-------+ O--------->
: | +--------+ | + :
: +-->| sys2 |---+ :
: +--------+ :
:.........sys............:
sys = parallel (sys1, sys2)
@end group
@end example
@end deftypefn
@section @@lti/series
@deftypefn {Function File} {@var{sys} =} series (@var{sys1}, @var{sys2})
@deftypefnx {Function File} {@var{sys} =} series (@var{sys1}, @var{sys2}, @var{outputs1}, @var{inputs2})
Series connection of two LTI models.
@strong{Block Diagram}
@example
@group
.....................................
u : +--------+ y1 u2 +--------+ : y
------>| sys1 |---------->| sys2 |------->
: +--------+ +--------+ :
:................sys.................
sys = series (sys1, sys2)
@end group
@end example
@example
@group
.....................................
: v2 +--------+ :
: ---------->| | : y
: +--------+ y1 u2 | sys2 |------->
u : | |---------->| | :
------>| sys1 | z1 +--------+ :
: | |----------> :
: +--------+ :
:................sys.................
outputs1 = [1]
inputs2 = [2]
sys = series (sys1, sys2, outputs1, inputs2)
@end group
@end example
@end deftypefn
@chapter Model Characteristics
@section ctrb
@deftypefn {Function File} {@var{co} =} ctrb (@var{sys})
@deftypefnx {Function File} {@var{co} =} ctrb (@var{a}, @var{b})
Return controllability matrix.
@strong{Inputs}
@table @var
@item sys
LTI model.
@item a
State transition matrix (n-by-n).
@item b
Input matrix (n-by-m).
@end table
@strong{Outputs}
@table @var
@item co
Controllability matrix.
@end table
@strong{Equation}
@iftex
@tex
$$ C_o = [ B \ \ AB \ \ A^2B \ \ldots \ A^{n-1}B ] $$
@end tex
@end iftex
@ifnottex
@example
2 n-1
Co = [ B AB A B ... A B ]
@end example
@end ifnottex
@end deftypefn
@section ctrbf
@deftypefn{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} ctrbf (@var{sys})
@deftypefnx{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} ctrbf (@var{sys}, @var{tol})
@deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} ctrbf (@var{A}, @var{B}, @var{C})
@deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} ctrbf (@var{A}, @var{B}, @var{C}, @var{TOL})
If Co=ctrb(A,B) has rank r <= n = SIZE(A,1), then there is a
similarity transformation Tc such that Tc = [t1 t2] where t1
is the controllable subspace and t2 is orthogonal to t1
@example
@group
Abar = Tc A * Tc , Bbar = Tc B , Cbar = C * Tc
@end group
@end example
and the transformed system has the form
@example
@group
| Ac A12| | Bc |
Abar = |----------|, Bbar = | ---|, Cbar = [Cc | Cnc].
| 0 Anc| | 0 |
@end group
@end example
where (Ac,Bc) is controllable, and Cc(sI-Ac)^(-1)Bc = C(sI-A)^(-1)B.
and the system is stabilizable if Anc has no eigenvalues in
the right half plane. The last output K is a vector of length n
containing the number of controllable states.
@end deftypefn
@section @@lti/dcgain
@deftypefn {Function File} {@var{k} =} dcgain (@var{sys})
DC gain of LTI model.
@strong{Inputs}
@table @var
@item sys
LTI system.
@end table
@strong{Outputs}
@table @var
@item k
DC gain matrice. For a system with m inputs and p outputs, the array @var{k}
has dimensions [p, m].
@end table
@seealso{freqresp}
@end deftypefn
@section gram
@deftypefn {Function File} {@var{W} =} gram (@var{sys}, @var{mode})
@deftypefnx {Function File} {@var{Wc} =} gram (@var{a}, @var{b})
@code{gram (@var{sys}, "c")} returns the controllability gramian of
the (continuous- or discrete-time) system @var{sys}.
@code{gram (@var{sys}, "o")} returns the observability gramian of the
(continuous- or discrete-time) system @var{sys}.
@code{gram (@var{a}, @var{b})} returns the controllability gramian
@var{Wc} of the continuous-time system @math{dx/dt = a x + b u};
i.e., @var{Wc} satisfies @math{a Wc + m Wc' + b b' = 0}.
@end deftypefn
@section hsvd
@deftypefn{Function File} {@var{hsv} =} hsvd (@var{sys})
@deftypefnx{Function File} {@var{hsv} =} hsvd (@var{sys}, @var{"offset"}, @var{offset})
@deftypefnx{Function File} {@var{hsv} =} hsvd (@var{sys}, @var{"alpha"}, @var{alpha})
Hankel singular values of the stable part of an LTI model. If no output arguments are
given, the Hankel singular values are displayed in a plot.
@strong{Algorithm}@*
Uses SLICOT AB13AD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section @@lti/isct
@deftypefn {Function File} {@var{bool} =} isct (@var{sys})
Determine whether LTI model is a continuous-time system.
@strong{Inputs}
@table @var
@item sys
LTI system.
@end table
@strong{Outputs}
@table @var
@item bool = 0
@var{sys} is a discrete-time system.
@item bool = 1
@var{sys} is a continuous-time system or a static gain.
@end table
@end deftypefn
@section isctrb
@deftypefn {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{sys})
@deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{sys}, @var{tol})
@deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b})
@deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}, @var{e})
@deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}, @var{[]}, @var{tol})
@deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}, @var{e}, @var{tol})
Logical check for system controllability.
For numerical reasons, @code{isctrb (sys)}
should be used instead of @code{rank (ctrb (sys))}.
@strong{Inputs}
@table @var
@item sys
LTI model. Descriptor state-space models are possible.
@item a
State transition matrix.
@item b
Input matrix.
@item e
Descriptor matrix.
If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed.
@item tol
Optional roundoff parameter. Default value is 0.
@end table
@strong{Outputs}
@table @var
@item bool = 0
System is not controllable.
@item bool = 1
System is controllable.
@item ncon
Number of controllable states.
@end table
@strong{Algorithm}@*
Uses SLICOT AB01OD and TG01HD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{isobsv}
@end deftypefn
@section isdetectable
@deftypefn {Function File} {@var{bool} =} isdetectable (@var{sys})
@deftypefnx {Function File} {@var{bool} =} isdetectable (@var{sys}, @var{tol})
@deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c})
@deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e})
@deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{[]}, @var{tol})
@deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}, @var{tol})
@deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{[]}, @var{[]}, @var{dflg})
@deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}, @var{[]}, @var{dflg})
@deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{[]}, @var{tol}, @var{dflg})
@deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}, @var{tol}, @var{dflg})
Logical test for system detectability.
All unstable modes must be observable or all unobservable states must be stable.
@strong{Inputs}
@table @var
@item sys
LTI system.
@item a
State transition matrix.
@item c
Measurement matrix.
@item e
Descriptor matrix.
If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed.
@item tol
Optional tolerance for stability. Default value is 0.
@item dflg = 0
Matrices (@var{a}, @var{c}) are part of a continuous-time system. Default Value.
@item dflg = 1
Matrices (@var{a}, @var{c}) are part of a discrete-time system.
@end table
@strong{Outputs}
@table @var
@item bool = 0
System is not detectable.
@item bool = 1
System is detectable.
@end table
@strong{Algorithm}@*
Uses SLICOT AB01OD and TG01HD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
See @command{isstabilizable} for description of computational method.
@seealso{isstabilizable, isstable, isctrb, isobsv}
@end deftypefn
@section @@lti/isdt
@deftypefn {Function File} {@var{bool} =} isdt (@var{sys})
Determine whether LTI model is a discrete-time system.
@strong{Inputs}
@table @var
@item sys
LTI system.
@end table
@strong{Outputs}
@table @var
@item bool = 0
@var{sys} is a continuous-time system.
@item bool = 1
@var{sys} is a discrete-time system or a static gain.
@end table
@end deftypefn
@section @@lti/isminimumphase
@deftypefn {Function File} {@var{bool} =} isminimumphase (@var{sys})
@deftypefnx {Function File} {@var{bool} =} isminimumphase (@var{sys}, @var{tol})
Determine whether LTI system is minimum phase.
The zeros must lie in the left complex half-plane.
The name minimum-phase refers to the fact that such a system has the
minimum possible phase lag for the given magnitude response |sys(jw)|.
@strong{Inputs}
@table @var
@item sys
LTI system.
@item tol
Optional tolerance. Default value is 0.
@end table
@strong{Outputs}
@table @var
@item bool = 0
System is not minimum phase.
@item bool = 1
System is minimum phase.
@end table
@example
@group
real (z) < -tol*(1 + abs (z)) continuous-time
abs (z) < 1 - tol discrete-time
@end group
@end example
@end deftypefn
@section isobsv
@deftypefn {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{sys})
@deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{sys}, @var{tol})
@deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c})
@deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}, @var{e})
@deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}, @var{[]}, @var{tol})
@deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}, @var{e}, @var{tol})
Logical check for system observability.
For numerical reasons, @code{isobsv (sys)}
should be used instead of @code{rank (obsv (sys))}.
@strong{Inputs}
@table @var
@item sys
LTI model. Descriptor state-space models are possible.
@item a
State transition matrix.
@item c
Measurement matrix.
@item e
Descriptor matrix.
If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed.
@item tol
Optional roundoff parameter. Default value is 0.
@end table
@strong{Outputs}
@table @var
@item bool = 0
System is not observable.
@item bool = 1
System is observable.
@item nobs
Number of observable states.
@end table
@strong{Algorithm}@*
Uses SLICOT AB01OD and TG01HD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{isctrb}
@end deftypefn
@section @@lti/issiso
@deftypefn {Function File} {@var{bool} =} issiso (@var{sys})
Determine whether LTI model is single-input/single-output (SISO).
@end deftypefn
@section isstabilizable
@deftypefn {Function File} {@var{bool} =} isstabilizable (@var{sys})
@deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{sys}, @var{tol})
@deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b})
@deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e})
@deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{[]}, @var{tol})
@deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}, @var{tol})
@deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{[]}, @var{[]}, @var{dflg})
@deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}, @var{[]}, @var{dflg})
@deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{[]}, @var{tol}, @var{dflg})
@deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}, @var{tol}, @var{dflg})
Logical check for system stabilizability.
All unstable modes must be controllable or all uncontrollable states must be stable.
@strong{Inputs}
@table @var
@item sys
LTI system.
@item a
State transition matrix.
@item b
Input matrix.
@item e
Descriptor matrix.
If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed.
@item tol
Optional tolerance for stability. Default value is 0.
@item dflg = 0
Matrices (@var{a}, @var{b}) are part of a continuous-time system. Default Value.
@item dflg = 1
Matrices (@var{a}, @var{b}) are part of a discrete-time system.
@end table
@strong{Outputs}
@table @var
@item bool = 0
System is not stabilizable.
@item bool = 1
System is stabilizable.
@end table
@strong{Algorithm}@*
Uses SLICOT AB01OD and TG01HD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@example
@group
* Calculate staircase form (SLICOT AB01OD)
* Extract unobservable part of state transition matrix
* Calculate eigenvalues of unobservable part
* Check whether
real (ev) < -tol*(1 + abs (ev)) continuous-time
abs (ev) < 1 - tol discrete-time
@end group
@end example
@seealso{isdetectable, isstable, isctrb, isobsv}
@end deftypefn
@section @@lti/isstable
@deftypefn {Function File} {@var{bool} =} isstable (@var{sys})
@deftypefnx {Function File} {@var{bool} =} isstable (@var{sys}, @var{tol})
Determine whether LTI system is stable.
@strong{Inputs}
@table @var
@item sys
LTI system.
@item tol
Optional tolerance for stability. Default value is 0.
@end table
@strong{Outputs}
@table @var
@item bool = 0
System is not stable.
@item bool = 1
System is stable.
@end table
@example
@group
real (p) < -tol*(1 + abs (p)) continuous-time
abs (p) < 1 - tol discrete-time
@end group
@end example
@end deftypefn
@section @@lti/norm
@deftypefn {Function File} {@var{gain} =} norm (@var{sys}, @var{2})
@deftypefnx {Function File} {[@var{gain}, @var{wpeak}] =} norm (@var{sys}, @var{inf})
@deftypefnx {Function File} {[@var{gain}, @var{wpeak}] =} norm (@var{sys}, @var{inf}, @var{tol})
Return H-2 or L-inf norm of LTI model.
@strong{Algorithm}@*
Uses SLICOT AB13BD and AB13DD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section obsv
@deftypefn {Function File} {@var{ob} =} obsv (@var{sys})
@deftypefnx {Function File} {@var{ob} =} obsv (@var{a}, @var{c})
Return observability matrix.
@strong{Inputs}
@table @var
@item sys
LTI model.
@item a
State transition matrix (n-by-n).
@item c
Measurement matrix (p-by-n).
@end table
@strong{Outputs}
@table @var
@item ob
Observability matrix.
@end table
@strong{Equation}
@iftex
@tex
$$ O_b = \left[ \matrix{ C \cr
CA \cr
CA^2 \cr
\vdots \cr
CA^{n-1} } \right ] $$
@end tex
@end iftex
@ifnottex
@example
@group
| C |
| CA |
Ob = | CA^2 |
| ... |
| CA^(n-1) |
@end group
@end example
@end ifnottex
@end deftypefn
@section obsvf
@deftypefn{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} obsvf (@var{sys})
@deftypefnx{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} obsvf (@var{sys}, @var{tol})
@deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} obsvf (@var{A}, @var{B}, @var{C})
@deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} obsvf (@var{A}, @var{B}, @var{C}, @var{TOL})
If Ob=obsv(A,C) has rank r <= n = SIZE(A,1), then there is a
similarity transformation Tc such that To = [t1;t2] where t1 is c
and t2 is orthogonal to t1
@example
@group
Abar = To A * To , Bbar = To B , Cbar = C * To
@end group
@end example
and the transformed system has the form
@example
@group
| Ao 0 | | Bo |
Abar = |----------|, Bbar = | --- |, Cbar = [Co | 0 ].
| A21 Ano| | Bno |
@end group
@end example
where (Ao,Bo) is observable, and Co(sI-Ao)^(-1)Bo = C(sI-A)^(-1)B. And
system is detectable if Ano has no eigenvalues in the right
half plane. The last output K is a vector of length n containing the
number of observable states.
@end deftypefn
@section @@lti/pole
@deftypefn {Function File} {@var{p} =} pole (@var{sys})
Compute poles of LTI system.
@strong{Inputs}
@table @var
@item sys
LTI model.
@end table
@strong{Outputs}
@table @var
@item p
Poles of @var{sys}.
@end table
@end deftypefn
@section pzmap
@deftypefn {Function File} pzmap (@var{sys})
@deftypefnx {Function File} {[@var{p}, @var{z}] =} pzmap (@var{sys})
Plot the poles and zeros of an LTI system in the complex plane.
If no output arguments are given, the result is plotted on the screen.
Otherwise, the poles and zeros are computed and returned.
@strong{Inputs}
@table @var
@item sys
LTI model.
@end table
@strong{Outputs}
@table @var
@item p
Poles of @var{sys}.
@item z
Transmission zeros of @var{sys}.
@end table
@end deftypefn
@section @@lti/size
@deftypefn {Function File} {@var{nvec} =} size (@var{sys})
@deftypefnx {Function File} {@var{n} =} size (@var{sys}, @var{dim})
@deftypefnx {Function File} {[@var{p}, @var{m}] =} size (@var{sys})
LTI model size, i.e. number of outputs and inputs.
@strong{Inputs}
@table @var
@item sys
LTI system.
@item dim
If given a second argument, @command{size} will return the size of the
corresponding dimension.
@end table
@strong{Outputs}
@table @var
@item nvec
Row vector. The first element is the number of outputs (rows) and the second
element the number of inputs (columns).
@item n
Scalar value. The size of the dimension @var{dim}.
@item p
Number of outputs.
@item m
Number of inputs.
@end table
@end deftypefn
@section @@lti/zero
@deftypefn {Function File} {@var{z} =} zero (@var{sys})
@deftypefnx {Function File} {[@var{z}, @var{k}] =} zero (@var{sys})
Compute transmission zeros and gain of LTI model.
@strong{Inputs}
@table @var
@item sys
LTI model.
@end table
@strong{Outputs}
@table @var
@item z
Transmission zeros of @var{sys}.
@item k
Gain of @var{sys}.
@end table
@end deftypefn
@chapter Model Simplification
@section @@lti/minreal
@deftypefn {Function File} {@var{sys} =} minreal (@var{sys})
@deftypefnx {Function File} {@var{sys} =} minreal (@var{sys}, @var{tol})
Minimal realization or zero-pole cancellation of LTI models.
@end deftypefn
@section @@lti/sminreal
@deftypefn {Function File} {@var{sys} =} sminreal (@var{sys})
@deftypefnx {Function File} {@var{sys} =} sminreal (@var{sys}, @var{tol})
Perform state-space model reduction based on structure.
Remove states which have no influence on the input-output behaviour.
The physical meaning of the states is retained.
@strong{Inputs}
@table @var
@item sys
State-space model.
@item tol
Optional tolerance for controllability and observability.
Entries of the state-space matrices whose moduli are less or equal to @var{tol}
are assumed to be zero. Default value is 0.
@end table
@strong{Outputs}
@table @var
@item sys
Reduced state-space model.
@end table
@seealso{minreal}
@end deftypefn
@chapter Time Domain Analysis
@section covar
@deftypefn{Function File} {[@var{p}, @var{q}] =} covar (@var{sys}, @var{w})
Return the steady-state covariance.
@strong{Inputs}
@table @var
@item sys
LTI model.
@item w
Intensity of Gaussian white noise inputs which drive @var{sys}.
@end table
@strong{Outputs}
@table @var
@item p
Output covariance.
@item q
State covariance.
@end table
@seealso{lyap, dlyap}
@end deftypefn
@section gensig
@deftypefn{Function File} {[@var{u}, @var{t}] =} gensig (@var{sigtype}, @var{tau})
@deftypefnx{Function File} {[@var{u}, @var{t}] =} gensig (@var{sigtype}, @var{tau}, @var{tfinal})
@deftypefnx{Function File} {[@var{u}, @var{t}] =} gensig (@var{sigtype}, @var{tau}, @var{tfinal}, @var{tsam})
Generate periodic signal. Useful in combination with lsim.
@strong{Inputs}
@table @var
@item sigtype = "sin"
Sine wave.
@item sigtype = "cos"
Cosine wave.
@item sigtype = "square"
Square wave.
@item sigtype = "pulse"
Periodic pulse.
@item tau
Duration of one period in seconds.
@item tfinal
Optional duration of the signal in seconds. Default duration is 5 periods.
@item tsam
Optional sampling time in seconds. Default spacing is tau/64.
@end table
@strong{Outputs}
@table @var
@item u
Vector of signal values.
@item t
Time vector of the signal.
@end table
@seealso{lsim}
@end deftypefn
@section impulse
@deftypefn{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}, @var{t})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}, @var{tfinal})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}, @var{tfinal}, @var{dt})
Impulse response of LTI system.
If no output arguments are given, the response is printed on the screen.
@strong{Inputs}
@table @var
@item sys
LTI model.
@item t
Time vector. Should be evenly spaced. If not specified, it is calculated by
the poles of the system to reflect adequately the response transients.
@item tfinal
Optional simulation horizon. If not specified, it is calculated by
the poles of the system to reflect adequately the response transients.
@item dt
Optional sampling time. Be sure to choose it small enough to capture transient
phenomena. If not specified, it is calculated by the poles of the system.
@end table
@strong{Outputs}
@table @var
@item y
Output response array. Has as many rows as time samples (length of t)
and as many columns as outputs.
@item t
Time row vector.
@item x
State trajectories array. Has @code{length (t)} rows and as many columns as states.
@end table
@seealso{initial, lsim, step}
@end deftypefn
@section initial
@deftypefn{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}, @var{t})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}, @var{tfinal})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}, @var{tfinal}, @var{dt})
Initial condition response of state-space model.
If no output arguments are given, the response is printed on the screen.
@strong{Inputs}
@table @var
@item sys
State-space model.
@item x0
Vector of initial conditions for each state.
@item t
Optional time vector. Should be evenly spaced. If not specified, it is calculated
by the poles of the system to reflect adequately the response transients.
@item tfinal
Optional simulation horizon. If not specified, it is calculated by
the poles of the system to reflect adequately the response transients.
@item dt
Optional sampling time. Be sure to choose it small enough to capture transient
phenomena. If not specified, it is calculated by the poles of the system.
@end table
@strong{Outputs}
@table @var
@item y
Output response array. Has as many rows as time samples (length of t)
and as many columns as outputs.
@item t
Time row vector.
@item x
State trajectories array. Has @code{length (t)} rows and as many columns as states.
@end table
@strong{Example}
@example
@group
.
Continuous Time: x = A x , y = C x , x(0) = x0
Discrete Time: x[k+1] = A x[k] , y[k] = C x[k] , x[0] = x0
@end group
@end example
@seealso{impulse, lsim, step}
@end deftypefn
@section lsim
@deftypefn{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}, @var{x0})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}, @var{[]}, @var{method})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}, @var{x0}, @var{method})
Simulate LTI model response to arbitrary inputs. If no output arguments are given,
the system response is plotted on the screen.
@strong{Inputs}
@table @var
@item sys
LTI model. System must be proper, i.e. it must not have more zeros than poles.
@item u
Vector or array of input signal. Needs @code{length(t)} rows and as many columns
as there are inputs. If @var{sys} is a single-input system, row vectors @var{u}
of length @code{length(t)} are accepted as well.
@item t
Time vector. Should be evenly spaced. If @var{sys} is a continuous-time system
and @var{t} is a real scalar, @var{sys} is discretized with sampling time
@code{tsam = t/(rows(u)-1)}. If @var{sys} is a discrete-time system and @var{t}
is not specified, vector @var{t} is assumed to be @code{0 : tsam : tsam*(rows(u)-1)}.
@item x0
Vector of initial conditions for each state. If not specified, a zero vector is assumed.
@item method
Discretization method for continuous-time models. Default value is zoh
(zero-order hold). All methods from @code{c2d} are supported.
@end table
@strong{Outputs}
@table @var
@item y
Output response array. Has as many rows as time samples (length of t)
and as many columns as outputs.
@item t
Time row vector. It is always evenly spaced.
@item x
State trajectories array. Has @code{length (t)} rows and as many columns as states.
@end table
@seealso{impulse, initial, step}
@end deftypefn
@section step
@deftypefn{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}, @var{t})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}, @var{tfinal})
@deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}, @var{tfinal}, @var{dt})
Step response of LTI system.
If no output arguments are given, the response is printed on the screen.
@strong{Inputs}
@table @var
@item sys
LTI model.
@item t
Time vector. Should be evenly spaced. If not specified, it is calculated by
the poles of the system to reflect adequately the response transients.
@item tfinal
Optional simulation horizon. If not specified, it is calculated by
the poles of the system to reflect adequately the response transients.
@item dt
Optional sampling time. Be sure to choose it small enough to capture transient
phenomena. If not specified, it is calculated by the poles of the system.
@end table
@strong{Outputs}
@table @var
@item y
Output response array. Has as many rows as time samples (length of t)
and as many columns as outputs.
@item t
Time row vector.
@item x
State trajectories array. Has @code{length (t)} rows and as many columns as states.
@end table
@seealso{impulse, initial, lsim}
@end deftypefn
@chapter Frequency Domain Analysis
@section bode
@deftypefn {Function File} {[@var{mag}, @var{pha}, @var{w}] =} bode (@var{sys})
@deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} bode (@var{sys}, @var{w})
Bode diagram of frequency response. If no output arguments are given,
the response is printed on the screen.
@strong{Inputs}
@table @var
@item sys
LTI system. Must be a single-input and single-output (SISO) system.
@item w
Optional vector of frequency values. If @var{w} is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range,
where @var{wmin} and @var{wmax} denote minimum and maximum frequencies
in rad/s.
@end table
@strong{Outputs}
@table @var
@item mag
Vector of magnitude. Has length of frequency vector @var{w}.
@item pha
Vector of phase. Has length of frequency vector @var{w}.
@item w
Vector of frequency values used.
@end table
@seealso{nichols, nyquist, sigma}
@end deftypefn
@section bodemag
@deftypefn {Function File} {[@var{mag}, @var{w}] =} bodemag (@var{sys})
@deftypefnx {Function File} {[@var{mag}, @var{w}] =} bodemag (@var{sys}, @var{w})
Bode magnitude diagram of frequency response. If no output arguments are given,
the response is printed on the screen.
@strong{Inputs}
@table @var
@item sys
LTI system. Must be a single-input and single-output (SISO) system.
@item w
Optional vector of frequency values. If @var{w} is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range,
where @var{wmin} and @var{wmax} denote minimum and maximum frequencies
in rad/s.
@end table
@strong{Outputs}
@table @var
@item mag
Vector of magnitude. Has length of frequency vector @var{w}.
@item w
Vector of frequency values used.
@end table
@seealso{bode, nichols, nyquist, sigma}
@end deftypefn
@section @@lti/freqresp
@deftypefn{Function File} {@var{H} =} freqresp (@var{sys}, @var{w})
Evaluate frequency response at given frequencies.
@strong{Inputs}
@table @var
@item sys
LTI system.
@item w
Vector of frequency values.
@end table
@strong{Outputs}
@table @var
@item H
Array of frequency response. For a system with m inputs and p outputs, the array @var{H}
has dimensions [p, m, length (w)].
The frequency response at the frequency w(k) is given by H(:,:,k).
@end table
@seealso{dcgain}
@end deftypefn
@section margin
@deftypefn{Function File} {[@var{gamma}, @var{phi}, @var{w_gamma}, @var{w_phi}] =} margin (@var{sys})
@deftypefnx{Function File} {[@var{gamma}, @var{phi}, @var{w_gamma}, @var{w_phi}] =} margin (@var{sys}, @var{tol})
Gain and phase margin of a system. If no output arguments are given, both gain and phase margin
are plotted on a bode diagram. Otherwise, the margins and their corresponding frequencies are
computed and returned.
@strong{Inputs}
@table @var
@item sys
LTI model. Must be a single-input and single-output (SISO) system.
@item tol
Imaginary parts below @var{tol} are assumed to be zero.
If not specified, default value @code{sqrt (eps)} is taken.
@end table
@strong{Outputs}
@table @var
@item gamma
Gain margin (as gain, not dBs).
@item phi
Phase margin (in degrees).
@item w_gamma
Frequency for the gain margin (in rad/s).
@item w_phi
Frequency for the phase margin (in rad/s).
@end table
@strong{Equations}
@example
@group
CONTINUOUS SYSTEMS
Gain Margin
_ _
L(jw) = L(jw) BTW: L(jw) = L(-jw) = conj (L(jw))
num(jw) num(-jw)
------- = --------
den(jw) den(-jw)
num(jw) den(-jw) = num(-jw) den(jw)
imag (num(jw) den(-jw)) = 0
imag (num(-jw) den(jw)) = 0
@end group
@end example
@example
@group
Phase Margin
|num(jw)|
|L(jw)| = |-------| = 1
|den(jw)|
_ 2 2
z z = Re z + Im z
num(jw) num(-jw)
------- * -------- = 1
den(jw) den(-jw)
num(jw) num(-jw) - den(jw) den(-jw) = 0
real (num(jw) num(-jw) - den(jw) den(-jw)) = 0
@end group
@end example
@example
@group
DISCRETE SYSTEMS
Gain Margin
jwT log z
L(z) = L(1/z) BTW: z = e --> w = -----
j T
num(z) num(1/z)
------ = --------
den(z) den(1/z)
num(z) den(1/z) - num(1/z) den(z) = 0
@end group
@end example
@example
@group
Phase Margin
|num(z)|
|L(z)| = |------| = 1
|den(z)|
L(z) L(1/z) = 1
num(z) num(1/z)
------ * -------- = 1
den(z) den(1/z)
num(z) num(1/z) - den(z) den(1/z) = 0
@end group
@end example
@example
@group
PS: How to get L(1/z)
4 3 2
p(z) = a z + b z + c z + d z + e
-4 -3 -2 -1
p(1/z) = a z + b z + c z + d z + e
-4 2 3 4
= z ( a + b z + c z + d z + e z )
4 3 2 4
= ( e z + d z + c z + b z + a ) / ( z )
@end group
@end example
@seealso{roots}
@end deftypefn
@section nichols
@deftypefn {Function File} {[@var{mag}, @var{pha}, @var{w}] =} nichols (@var{sys})
@deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} nichols (@var{sys}, @var{w})
Nichols chart of frequency response. If no output arguments are given,
the response is printed on the screen.
@strong{Inputs}
@table @var
@item sys
LTI system. Must be a single-input and single-output (SISO) system.
@item w
Optional vector of frequency values. If @var{w} is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range,
where @var{wmin} and @var{wmax} denote minimum and maximum frequencies
in rad/s.
@end table
@strong{Outputs}
@table @var
@item mag
Vector of magnitude. Has length of frequency vector @var{w}.
@item pha
Vector of phase. Has length of frequency vector @var{w}.
@item w
Vector of frequency values used.
@end table
@seealso{bode, nyquist, sigma}
@end deftypefn
@section nyquist
@deftypefn {Function File} {[@var{re}, @var{im}, @var{w}] =} nyquist (@var{sys})
@deftypefnx {Function File} {[@var{re}, @var{im}, @var{w}] =} nyquist (@var{sys}, @var{w})
Nyquist diagram of frequency response. If no output arguments are given,
the response is printed on the screen.
@strong{Inputs}
@table @var
@item sys
LTI system. Must be a single-input and single-output (SISO) system.
@item w
Optional vector of frequency values. If @var{w} is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range,
where @var{wmin} and @var{wmax} denote minimum and maximum frequencies
in rad/s.
@end table
@strong{Outputs}
@table @var
@item re
Vector of real parts. Has length of frequency vector @var{w}.
@item im
Vector of imaginary parts. Has length of frequency vector @var{w}.
@item w
Vector of frequency values used.
@end table
@seealso{bode, nichols, sigma}
@end deftypefn
@section sigma
@deftypefn{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys})
@deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}, @var{w})
@deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}, @var{[]}, @var{ptype})
@deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}, @var{w}, @var{ptype})
Singular values of frequency response. If no output arguments are given,
the singular value plot is printed on the screen;
@strong{Inputs}
@table @var
@item sys
LTI system. Multiple inputs and/or outputs (MIMO systems) make practical sense.
@item w
Optional vector of frequency values. If @var{w} is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range,
where @var{wmin} and @var{wmax} denote minimum and maximum frequencies
in rad/s.
@item ptype = 0
Singular values of the frequency response @var{H} of system @var{sys}. Default Value.
@item ptype = 1
Singular values of the frequency response @code{inv(H)}; i.e. inversed system.
@item ptype = 2
Singular values of the frequency response @code{I + H}; i.e. inversed sensitivity
(or return difference) if @code{H = P * C}.
@item ptype = 3
Singular values of the frequency response @code{I + inv(H)}; i.e. inversed complementary
sensitivity if @code{H = P * C}.
@end table
@strong{Outputs}
@table @var
@item sv
Array of singular values. For a system with m inputs and p outputs, the array sv
has @code{min (m, p)} rows and as many columns as frequency points @code{length (w)}.
The singular values at the frequency @code{w(k)} are given by @code{sv(:,k)}.
@item w
Vector of frequency values used.
@end table
@seealso{bodemag, svd}
@end deftypefn
@chapter Pole Placement
@section place
@deftypefn {Function File} {@var{f} =} place (@var{sys}, @var{p})
@deftypefnx {Function File} {@var{f} =} place (@var{a}, @var{b}, @var{p})
@deftypefnx {Function File} {[@var{f}, @var{info}] =} place (@var{sys}, @var{p}, @var{alpha})
@deftypefnx {Function File} {[@var{f}, @var{info}] =} place (@var{a}, @var{b}, @var{p}, @var{alpha})
Pole assignment for a given matrix pair (@var{A},@var{B}) such that @code{p = eig (A-B*F)}.
If parameter @var{alpha} is specified, poles with real parts (continuous-time)
or moduli (discrete-time) below @var{alpha} are left untouched.
@strong{Inputs}
@table @var
@item sys
LTI system.
@item a
State transition matrix (n-by-n) of a continuous-time system.
@item b
Input matrix (n-by-m) of a continuous-time system.
@item p
Desired eigenvalues of the closed-loop system state-matrix @var{A-B*F}.
@code{length (p) <= rows (A)}.
@item alpha
Specifies the maximum admissible value, either for real
parts or for moduli, of the eigenvalues of @var{A} which will
not be modified by the eigenvalue assignment algorithm.
@code{alpha >= 0} for discrete-time systems.
@end table
@strong{Outputs}
@table @var
@item f
State feedback gain matrix.
@item info
Structure containing additional information.
@item info.nfp
The number of fixed poles, i.e. eigenvalues of @var{A} having
real parts less than @var{alpha}, or moduli less than @var{alpha}.
These eigenvalues are not modified by @command{place}.
@item info.nap
The number of assigned eigenvalues. @code{nap = n-nfp-nup}.
@item info.nup
The number of uncontrollable eigenvalues detected by the
eigenvalue assignment algorithm.
@item info.z
The orthogonal matrix @var{z} reduces the closed-loop
system state matrix @code{A + B*F} to upper real Schur form.
Note the positive sign in @code{A + B*F}.
@end table
@strong{Note}
@example
Place is also suitable to design estimator gains:
@group
L = place (A.', C.', p).'
L = place (sys.', p).' # useful for discrete-time systems
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB01BD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section rlocus
@deftypefn {Function File} rlocus (@var{sys})
@deftypefnx {Function File} {[@var{rldata}, @var{k}] =} rlocus (@var{sys}, @var{increment}, @var{min_k}, @var{max_k})
Display root locus plot of the specified @acronym{SISO} system.
@strong{Inputs}
@table @var
@item sys
LTI model. Must be a single-input and single-output (SISO) system.
@item min_k
Minimum value of @var{k}.
@item max_k
Maximum value of @var{k}.
@item increment
The increment used in computing gain values.
@end table
@strong{Outputs}
@table @var
@item rldata
Data points plotted: in column 1 real values, in column 2 the imaginary values.
@item k
Gains for real axis break points.
@end table
@strong{Block Diagram}
@example
@group
u + +---+ +------+ y
------>(+)----->| k |----->| SISO |-------+------->
^ - +---+ +------+ |
| |
+---------------------------------+
@end group
@end example
@end deftypefn
@chapter Linear-Quadratic Control
@section dlqe
@deftypefn {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}, @var{s})
Kalman filter for discrete-time systems.
@example
@group
x[k] = Ax[k] + Bu[k] + Gw[k] (State equation)
y[k] = Cx[k] + Du[k] + v[k] (Measurement Equation)
E(w) = 0, E(v) = 0, cov(w) = Q, cov(v) = R, cov(w,v) = S
@end group
@end example
@strong{Inputs}
@table @var
@item a
State transition matrix of discrete-time system (n-by-n).
@item g
Process noise matrix of discrete-time system (n-by-g).
If @var{g} is empty @code{[]}, an identity matrix is assumed.
@item c
Measurement matrix of discrete-time system (p-by-n).
@item q
Process noise covariance matrix (g-by-g).
@item r
Measurement noise covariance matrix (p-by-p).
@item s
Optional cross term covariance matrix (g-by-p), s = cov(w,v).
If @var{s} is empty @code{[]} or not specified, a zero matrix is assumed.
@end table
@strong{Outputs}
@table @var
@item m
Kalman filter gain matrix (n-by-p).
@item p
Unique stabilizing solution of the discrete-time Riccati equation (n-by-n).
Symmetric matrix.
@item z
Error covariance (n-by-n), cov(x(k|k)-x)
@item e
Closed-loop poles (n-by-1).
@end table
@strong{Equations}
@example
@group
x[k|k] = x[k|k-1] + M(y[k] - Cx[k|k-1] - Du[k])
x[k+1|k] = Ax[k|k] + Bu[k] for S=0
x[k+1|k] = Ax[k|k] + Bu[k] + G*S*(C*P*C' + R)^-1*(y[k] - C*x[k|k-1]) for non-zero S
E = eig(A - A*M*C) for S=0
E = eig(A - A*M*C - G*S*(C*P*C' + Rv)^-1*C) for non-zero S
@end group
@end example
@seealso{dare, care, dlqr, lqr, lqe}
@end deftypefn
@section dlqr
@deftypefn {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{sys}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{sys}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e})
Linear-quadratic regulator for discrete-time systems.
@strong{Inputs}
@table @var
@item sys
Continuous or discrete-time LTI model (p-by-m, n states).
@item a
State transition matrix of discrete-time system (n-by-n).
@item b
Input matrix of discrete-time system (n-by-m).
@item q
State weighting matrix (n-by-n).
@item r
Input weighting matrix (m-by-m).
@item s
Optional cross term matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed.
@item e
Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed.
@end table
@strong{Outputs}
@table @var
@item g
State feedback matrix (m-by-n).
@item x
Unique stabilizing solution of the discrete-time Riccati equation (n-by-n).
@item l
Closed-loop poles (n-by-1).
@end table
@strong{Equations}
@example
@group
x[k+1] = A x[k] + B u[k], x[0] = x0
inf
J(x0) = SUM (x' Q x + u' R u + 2 x' S u)
k=0
L = eig (A - B*G)
@end group
@end example
@seealso{dare, care, lqr}
@end deftypefn
@section estim
@deftypefn {Function File} {@var{est} =} estim (@var{sys}, @var{l})
@deftypefnx {Function File} {@var{est} =} estim (@var{sys}, @var{l}, @var{sensors}, @var{known})
Return state estimator for a given estimator gain.
@strong{Inputs}
@table @var
@item sys
LTI model.
@item l
State feedback matrix.
@item sensors
Indices of measured output signals y from @var{sys}. If omitted, all outputs are measured.
@item known
Indices of known input signals u (deterministic) to @var{sys}. All other inputs to @var{sys}
are assumed stochastic. If argument @var{known} is omitted, no inputs u are known.
@end table
@strong{Outputs}
@table @var
@item est
State-space model of estimator.
@end table
@seealso{kalman, place}
@end deftypefn
@section kalman
@deftypefn {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}, @var{[]}, @var{sensors}, @var{known})
@deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}, @var{s}, @var{sensors}, @var{known})
Design Kalman estimator for LTI systems.
@strong{Inputs}
@table @var
@item sys
Nominal plant model.
@item q
Covariance of white process noise.
@item r
Covariance of white measurement noise.
@item s
Optional cross term covariance. Default value is 0.
@item sensors
Indices of measured output signals y from @var{sys}. If omitted, all outputs are measured.
@item known
Indices of known input signals u (deterministic) to @var{sys}. All other inputs to @var{sys}
are assumed stochastic. If argument @var{known} is omitted, no inputs u are known.
@end table
@strong{Outputs}
@table @var
@item est
State-space model of the Kalman estimator.
@item g
Estimator gain.
@item x
Solution of the Riccati equation.
@end table
@strong{Block Diagram}
@example
@group
u +-------+ ^
+---------------------------->| |-------> y
| +-------+ + y | est | ^
u ----+--->| |----->(+)------>| |-------> x
| sys | ^ + +-------+
w -------->| | |
+-------+ | v
Q = cov (w, w') R = cov (v, v') S = cov (w, v')
@end group
@end example
@seealso{care, dare, estim, lqr}
@end deftypefn
@section lqe
@deftypefn {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{sys}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{sys}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}, @var{s})
Kalman filter for continuous-time systems.
@example
@group
.
x = Ax + Bu + Gw (State equation)
y = Cx + Du + v (Measurement Equation)
E(w) = 0, E(v) = 0, cov(w) = Q, cov(v) = R, cov(w,v) = S
@end group
@end example
@strong{Inputs}
@table @var
@item sys
Continuous or discrete-time LTI model (p-by-m, n states).
@item a
State transition matrix of continuous-time system (n-by-n).
@item g
Process noise matrix of continuous-time system (n-by-g).
If @var{g} is empty @code{[]}, an identity matrix is assumed.
@item c
Measurement matrix of continuous-time system (p-by-n).
@item q
Process noise covariance matrix (g-by-g).
@item r
Measurement noise covariance matrix (p-by-p).
@item s
Optional cross term covariance matrix (g-by-p), s = cov(w,v).
If @var{s} is empty @code{[]} or not specified, a zero matrix is assumed.
@end table
@strong{Outputs}
@table @var
@item l
Kalman filter gain matrix (n-by-p).
@item p
Unique stabilizing solution of the continuous-time Riccati equation (n-by-n).
Symmetric matrix. If @var{sys} is a discrete-time model, the solution of the
corresponding discrete-time Riccati equation is returned.
@item e
Closed-loop poles (n-by-1).
@end table
@strong{Equations}
@example
@group
.
x = Ax + Bu + L(y - Cx -Du)
E = eig(A - L*C)
@end group
@end example
@seealso{dare, care, dlqr, lqr, dlqe}
@end deftypefn
@section lqr
@deftypefn {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{sys}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{sys}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e})
@deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e})
Linear-quadratic regulator.
@strong{Inputs}
@table @var
@item sys
Continuous or discrete-time LTI model (p-by-m, n states).
@item a
State transition matrix of continuous-time system (n-by-n).
@item b
Input matrix of continuous-time system (n-by-m).
@item q
State weighting matrix (n-by-n).
@item r
Input weighting matrix (m-by-m).
@item s
Optional cross term matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed.
@item e
Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed.
@end table
@strong{Outputs}
@table @var
@item g
State feedback matrix (m-by-n).
@item x
Unique stabilizing solution of the continuous-time Riccati equation (n-by-n).
@item l
Closed-loop poles (n-by-1).
@end table
@strong{Equations}
@example
@group
.
x = A x + B u, x(0) = x0
inf
J(x0) = INT (x' Q x + u' R u + 2 x' S u) dt
0
L = eig (A - B*G)
@end group
@end example
@seealso{care, dare, dlqr}
@end deftypefn
@chapter Robust Control
@section augw
@deftypefn{Function File} {@var{P} =} augw (@var{G}, @var{W1}, @var{W2}, @var{W3})
Extend plant for stacked S/KS/T problem. Subsequently, the robust control problem
can be solved by h2syn or hinfsyn.
@strong{Inputs}
@table @var
@item G
LTI model of plant.
@item W1
LTI model of performance weight. Bounds the largest singular values of sensitivity @var{S}.
Model must be empty @code{[]}, SISO or of appropriate size.
@item W2
LTI model to penalize large control inputs. Bounds the largest singular values of @var{KS}.
Model must be empty @code{[]}, SISO or of appropriate size.
@item W3
LTI model of robustness and noise sensitivity weight. Bounds the largest singular values of
complementary sensitivity @var{T}. Model must be empty @code{[]}, SISO or of appropriate size.
@end table
All inputs must be proper/realizable.
Scalars, vectors and matrices are possible instead of LTI models.
@strong{Outputs}
@table @var
@item P
State-space model of augmented plant.
@end table
@strong{Block Diagram}
@example
@group
| W1 | -W1*G | z1 = W1 r - W1 G u
| 0 | W2 | z2 = W2 u
P = | 0 | W3*G | z3 = W3 G u
|----+-------|
| I | -G | e = r - G u
@end group
@end example
@example
@group
+------+ z1
+---------------------------------------->| W1 |----->
| +------+
| +------+ z2
| +---------------------->| W2 |----->
| | +------+
r + e | +--------+ u | +--------+ y +------+ z3
----->(+)---+-->| K(s) |----+-->| G(s) |----+---->| W3 |----->
^ - +--------+ +--------+ | +------+
| |
+----------------------------------------+
@end group
@end example
@example
@group
+--------+
| |-----> z1 (p1x1) z1 = W1 e
r (px1) ----->| P(s) |-----> z2 (p2x1) z2 = W2 u
| |-----> z3 (p3x1) z3 = W3 y
u (mx1) ----->| |-----> e (px1) e = r - y
+--------+
@end group
@end example
@example
@group
+--------+
r ----->| |-----> z
| P(s) |
u +---->| |-----+ e
| +--------+ |
| |
| +--------+ |
+-----| K(s) |<----+
+--------+
@end group
@end example
@example
@group
Reference:
Skogestad, S. and Postlethwaite I.
Multivariable Feedback Control: Analysis and Design
Second Edition
Wiley 2005
Chapter 3.8: General Control Problem Formulation
@end group
@end example
@seealso{h2syn, hinfsyn, mixsyn}
@end deftypefn
@section fitfrd
@deftypefn{Function File} {[@var{sys}, @var{n}] =} fitfrd (@var{dat}, @var{n})
@deftypefnx{Function File} {[@var{sys}, @var{n}] =} fitfrd (@var{dat}, @var{n}, @var{flag})
Fit frequency response data with a state-space system.
If requested, the returned system is stable and minimum-phase.
@strong{Inputs}
@table @var
@item dat
LTI model containing frequency response data of a SISO system.
@item n
The desired order of the system to be fitted. @code{n <= length(dat.w)}.
@item flag
The flag controls whether the returned system is stable and minimum-phase.
@table @var
@item 0
The system zeros and poles are not constrained. Default value.
@item 1
The system zeros and poles will have negative real parts in the
continuous-time case, or moduli less than 1 in the discrete-time case.
@end table
@end table
@strong{Outputs}
@table @var
@item sys
State-space model of order @var{n}, fitted to frequency response data @var{dat}.
@item n
The order of the obtained system. The value of @var{n}
could only be modified if inputs @code{n > 0} and @code{flag = 1}.
@end table
@strong{Algorithm}@*
Uses SLICOT SB10YD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section h2syn
@deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{rcond}] =} h2syn (@var{P}, @var{nmeas}, @var{ncon})
H-2 control synthesis for LTI plant.
@strong{Inputs}
@table @var
@item P
Generalized plant. Must be a proper/realizable LTI model.
@item nmeas
Number of measured outputs v. The last @var{nmeas} outputs of @var{P} are connected to the
inputs of controller @var{K}. The remaining outputs z (indices 1 to p-nmeas) are used
to calculate the H-2 norm.
@item ncon
Number of controlled inputs u. The last @var{ncon} inputs of @var{P} are connected to the
outputs of controller @var{K}. The remaining inputs w (indices 1 to m-ncon) are excited
by a harmonic test signal.
@end table
@strong{Outputs}
@table @var
@item K
State-space model of the H-2 optimal controller.
@item N
State-space model of the lower LFT of @var{P} and @var{K}.
@item gamma
H-2 norm of @var{N}.
@item rcond
Vector @var{rcond} contains estimates of the reciprocal condition
numbers of the matrices which are to be inverted and
estimates of the reciprocal condition numbers of the
Riccati equations which have to be solved during the
computation of the controller @var{K}. For details,
see the description of the corresponding SLICOT algorithm.
@end table
@strong{Block Diagram}
@example
@group
gamma = min||N(K)|| N = lft (P, K)
K 2
+--------+
w ----->| |-----> z
| P(s) |
u +---->| |-----+ v
| +--------+ |
| |
| +--------+ |
+-----| K(s) |<----+
+--------+
+--------+
w ----->| N(s) |-----> z
+--------+
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB10HD and SB10ED by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{augw, lqr, dlqr, kalman}
@end deftypefn
@section hinfsyn
@deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{rcond}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon})
@deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{rcond}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon}, @var{gmax})
H-infinity control synthesis for LTI plant.
@strong{Inputs}
@table @var
@item P
Generalized plant. Must be a proper/realizable LTI model.
@item nmeas
Number of measured outputs v. The last @var{nmeas} outputs of @var{P} are connected to the
inputs of controller @var{K}. The remaining outputs z (indices 1 to p-nmeas) are used
to calculate the H-infinity norm.
@item ncon
Number of controlled inputs u. The last @var{ncon} inputs of @var{P} are connected to the
outputs of controller @var{K}. The remaining inputs w (indices 1 to m-ncon) are excited
by a harmonic test signal.
@item gmax
The maximum value of the H-infinity norm of @var{N}. It is assumed that @var{gmax} is
sufficiently large so that the controller is admissible.
@end table
@strong{Outputs}
@table @var
@item K
State-space model of the H-infinity (sub-)optimal controller.
@item N
State-space model of the lower LFT of @var{P} and @var{K}.
@item gamma
L-infinity norm of @var{N}.
@item rcond
Vector @var{rcond} contains estimates of the reciprocal condition
numbers of the matrices which are to be inverted and
estimates of the reciprocal condition numbers of the
Riccati equations which have to be solved during the
computation of the controller @var{K}. For details,
see the description of the corresponding SLICOT algorithm.
@end table
@strong{Block Diagram}
@example
@group
gamma = min||N(K)|| N = lft (P, K)
K inf
+--------+
w ----->| |-----> z
| P(s) |
u +---->| |-----+ v
| +--------+ |
| |
| +--------+ |
+-----| K(s) |<----+
+--------+
+--------+
w ----->| N(s) |-----> z
+--------+
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB10FD and SB10DD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{augw, mixsyn}
@end deftypefn
@section mixsyn
@deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{rcond}] =} mixsyn (@var{G}, @var{W1}, @var{W2}, @var{W3}, @dots{})
Solve stacked S/KS/T H-infinity problem. Bound the largest singular values
of @var{S} (for performance), @var{K S} (to penalize large inputs) and
@var{T} (for robustness and to avoid sensitivity to noise).
In other words, the inputs r are excited by a harmonic test signal.
Then the algorithm tries to find a controller @var{K} which minimizes
the H-infinity norm calculated from the outputs z.
@strong{Inputs}
@table @var
@item G
LTI model of plant.
@item W1
LTI model of performance weight. Bounds the largest singular values of sensitivity @var{S}.
Model must be empty @code{[]}, SISO or of appropriate size.
@item W2
LTI model to penalize large control inputs. Bounds the largest singular values of @var{KS}.
Model must be empty @code{[]}, SISO or of appropriate size.
@item W3
LTI model of robustness and noise sensitivity weight. Bounds the largest singular values of
complementary sensitivity @var{T}. Model must be empty @code{[]}, SISO or of appropriate size.
@item @dots{}
Optional arguments of @command{hinfsyn}. Type @command{help hinfsyn} for more information.
@end table
All inputs must be proper/realizable.
Scalars, vectors and matrices are possible instead of LTI models.
@strong{Outputs}
@table @var
@item K
State-space model of the H-infinity (sub-)optimal controller.
@item N
State-space model of the lower LFT of @var{P} and @var{K}.
@item gamma
L-infinity norm of @var{N}.
@item rcond
Vector @var{rcond} contains estimates of the reciprocal condition
numbers of the matrices which are to be inverted and
estimates of the reciprocal condition numbers of the
Riccati equations which have to be solved during the
computation of the controller @var{K}. For details,
see the description of the corresponding SLICOT algorithm.
@end table
@strong{Block Diagram}
@example
@group
| W1 S |
gamma = min||N(K)|| N = | W2 K S | = lft (P, K)
K inf | W3 T |
@end group
@end example
@example
@group
+------+ z1
+---------------------------------------->| W1 |----->
| +------+
| +------+ z2
| +---------------------->| W2 |----->
| | +------+
r + e | +--------+ u | +--------+ y +------+ z3
----->(+)---+-->| K(s) |----+-->| G(s) |----+---->| W3 |----->
^ - +--------+ +--------+ | +------+
| |
+----------------------------------------+
@end group
@end example
@example
@group
+--------+
| |-----> z1 (p1x1) z1 = W1 e
r (px1) ----->| P(s) |-----> z2 (p2x1) z2 = W2 u
| |-----> z3 (p3x1) z3 = W3 y
u (mx1) ----->| |-----> e (px1) e = r - y
+--------+
@end group
@end example
@example
@group
+--------+
r ----->| |-----> z
| P(s) |
u +---->| |-----+ e
| +--------+ |
| |
| +--------+ |
+-----| K(s) |<----+
+--------+
@end group
@end example
@example
@group
+--------+
r ----->| N(s) |-----> z
+--------+
@end group
@end example
@example
@group
Extended Plant: P = augw (G, W1, W2, W3)
Controller: K = mixsyn (G, W1, W2, W3)
Entire System: N = lft (P, K)
Open Loop: L = G * K
Closed Loop: T = feedback (L)
@end group
@end example
@example
@group
Reference:
Skogestad, S. and Postlethwaite I.
Multivariable Feedback Control: Analysis and Design
Second Edition
Wiley 2005
Chapter 3.8: General Control Problem Formulation
@end group
@end example
@strong{Algorithm}@*
Relies on commands @command{augw} and @command{hinfsyn},
which use SLICOT SB10FD and SB10DD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{hinfsyn, augw}
@end deftypefn
@section ncfsyn
@deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} ncfsyn (@var{G}, @var{W1}, @var{W2}, @var{factor})
Loop shaping H-infinity synthesis. Compute positive feedback controller using
the McFarlane/Glover normalized coprime factor (NCF) loop shaping design procedure.
@strong{Inputs}
@table @var
@item G
LTI model of plant.
@item W1
LTI model of precompensator. Model must be SISO or of appropriate size.
An identity matrix is taken if @var{W1} is not specified or if an empty model
@code{[]} is passed.
@item W2
LTI model of postcompensator. Model must be SISO or of appropriate size.
An identity matrix is taken if @var{W2} is not specified or if an empty model
@code{[]} is passed.
@item factor
@code{factor = 1} implies that an optimal controller is required.
@code{factor > 1} implies that a suboptimal controller is required,
achieving a performance that is @var{factor} times less than optimal.
Default value is 1.
@end table
@strong{Outputs}
@table @var
@item K
State-space model of the H-infinity loop-shaping controller.
@item N
State-space model of the closed loop depicted below.
@item gamma
L-infinity norm of @var{N}. @code{gamma = norm (N, inf)}.
@item info
Structure containing additional information.
@item info.emax
Nugap robustness. @code{emax = inv (gamma)}.
@item info.Gs
Shaped plant. @code{Gs = W2 * G * W1}.
@item info.Ks
Controller for shaped plant. @code{Ks = ncfsyn (Gs)}.
@item info.rcond
Estimates of the reciprocal condition numbers of the Riccati equations
and a few other things. For details, see the description of the
corresponding SLICOT algorithm.
@end table
@strong{Block Diagram of N}
@example
@group
^ z1 ^ z2
| |
w1 + | +--------+ | +--------+
----->(+)---+-->| Ks |----+--->(+)---->| Gs |----+
^ + +--------+ ^ +--------+ |
| w2 | |
| |
+-------------------------------------------------+
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB10ID, SB10KD and SB10ZD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@chapter Matrix Equation Solvers
@section care
@deftypefn {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e})
@deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e})
Solve continuous-time algebraic Riccati equation (ARE).
@strong{Inputs}
@table @var
@item a
Real matrix (n-by-n).
@item b
Real matrix (n-by-m).
@item q
Real matrix (n-by-n).
@item r
Real matrix (m-by-m).
@item s
Optional real matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed.
@item e
Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed.
@end table
@strong{Outputs}
@table @var
@item x
Unique stabilizing solution of the continuous-time Riccati equation (n-by-n).
@item l
Closed-loop poles (n-by-1).
@item g
Corresponding gain matrix (m-by-n).
@end table
@strong{Equations}
@example
@group
-1
A'X + XA - XB R B'X + Q = 0
-1
A'X + XA - (XB + S) R (B'X + S') + Q = 0
-1
G = R B'X
-1
G = R (B'X + S')
L = eig (A - B*G)
@end group
@end example
@example
@group
-1
A'XE + E'XA - E'XB R B'XE + Q = 0
-1
A'XE + E'XA - (E'XB + S) R (B'XE + S') + Q = 0
-1
G = R B'XE
-1
G = R (B'XE + S)
L = eig (A - B*G, E)
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB02OD and SG02AD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{dare, lqr, dlqr, kalman}
@end deftypefn
@section dare
@deftypefn {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r})
@deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{s})
@deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e})
@deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e})
Solve discrete-time algebraic Riccati equation (ARE).
@strong{Inputs}
@table @var
@item a
Real matrix (n-by-n).
@item b
Real matrix (n-by-m).
@item q
Real matrix (n-by-n).
@item r
Real matrix (m-by-m).
@item s
Optional real matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed.
@item e
Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed.
@end table
@strong{Outputs}
@table @var
@item x
Unique stabilizing solution of the discrete-time Riccati equation (n-by-n).
@item l
Closed-loop poles (n-by-1).
@item g
Corresponding gain matrix (m-by-n).
@end table
@strong{Equations}
@example
@group
-1
A'XA - X - A'XB (B'XB + R) B'XA + Q = 0
-1
A'XA - X - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0
-1
G = (B'XB + R) B'XA
-1
G = (B'XB + R) (B'XA + S')
L = eig (A - B*G)
@end group
@end example
@example
@group
-1
A'XA - E'XE - A'XB (B'XB + R) B'XA + Q = 0
-1
A'XA - E'XE - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0
-1
G = (B'XB + R) B'XA
-1
G = (B'XB + R) (B'XA + S')
L = eig (A - B*G, E)
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB02OD and SG02AD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{care, lqr, dlqr, kalman}
@end deftypefn
@section dlyap
@deftypefn{Function File} {@var{x} =} dlyap (@var{a}, @var{b})
@deftypefnx{Function File} {@var{x} =} dlyap (@var{a}, @var{b}, @var{c})
@deftypefnx{Function File} {@var{x} =} dlyap (@var{a}, @var{b}, @var{[]}, @var{e})
Solve discrete-time Lyapunov or Sylvester equations.
@strong{Equations}
@example
@group
AXA' - X + B = 0 (Lyapunov Equation)
AXB' - X + C = 0 (Sylvester Equation)
AXA' - EXE' + B = 0 (Generalized Lyapunov Equation)
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB03MD, SB04QD and SG03AD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{dlyapchol, lyap, lyapchol}
@end deftypefn
@section dlyapchol
@deftypefn{Function File} {@var{u} =} dlyapchol (@var{a}, @var{b})
@deftypefnx{Function File} {@var{u} =} dlyapchol (@var{a}, @var{b}, @var{e})
Compute Cholesky factor of discrete-time Lyapunov equations.
@strong{Equations}
@example
@group
A U' U A' - U' U + B B' = 0 (Lyapunov Equation)
A U' U A' - E U' U E' + B B' = 0 (Generalized Lyapunov Equation)
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB03OD and SG03BD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{dlyap, lyap, lyapchol}
@end deftypefn
@section lyap
@deftypefn{Function File} {@var{x} =} lyap (@var{a}, @var{b})
@deftypefnx{Function File} {@var{x} =} lyap (@var{a}, @var{b}, @var{c})
@deftypefnx{Function File} {@var{x} =} lyap (@var{a}, @var{b}, @var{[]}, @var{e})
Solve continuous-time Lyapunov or Sylvester equations.
@strong{Equations}
@example
@group
AX + XA' + B = 0 (Lyapunov Equation)
AX + XB + C = 0 (Sylvester Equation)
AXE' + EXA' + B = 0 (Generalized Lyapunov Equation)
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB03MD, SB04MD and SG03AD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{lyapchol, dlyap, dlyapchol}
@end deftypefn
@section lyapchol
@deftypefn{Function File} {@var{u} =} lyapchol (@var{a}, @var{b})
@deftypefnx{Function File} {@var{u} =} lyapchol (@var{a}, @var{b}, @var{e})
Compute Cholesky factor of continuous-time Lyapunov equations.
@strong{Equations}
@example
@group
A U' U + U' U A' + B B' = 0 (Lyapunov Equation)
A U' U E' + E U' U A' + B B' = 0 (Generalized Lyapunov Equation)
@end group
@end example
@strong{Algorithm}@*
Uses SLICOT SB03OD and SG03BD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@seealso{lyap, dlyap, dlyapchol}
@end deftypefn
@chapter Model Reduction
@section bstmodred
@deftypefn{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{nr}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{opt}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{nr}, @var{opt}, @dots{})
Model order reduction by Balanced Stochastic Truncation (BST) method.
The aim of model reduction is to find an LTI system @var{Gr} of order
@var{nr} (nr < n) such that the input-output behaviour of @var{Gr}
approximates the one from original system @var{G}.
BST is a relative error method which tries to minimize
@iftex
@tex
$$ || G^{-1} (G-G_r) ||_{\infty} = min $$
@end tex
@end iftex
@ifnottex
@example
-1
||G (G-Gr)|| = min
inf
@end example
@end ifnottex
@strong{Inputs}
@table @var
@item G
LTI model to be reduced.
@item nr
The desired order of the resulting reduced order system @var{Gr}.
If not specified, @var{nr} is chosen automatically according
to the description of key @var{'order'}.
@item @dots{}
Optional pairs of keys and values. @code{"key1", value1, "key2", value2}.
@item opt
Optional struct with keys as field names.
Struct @var{opt} can be created directly or
by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
@end table
@strong{Outputs}
@table @var
@item Gr
Reduced order state-space model.
@item info
Struct containing additional information.
@table @var
@item info.n
The order of the original system @var{G}.
@item info.ns
The order of the @var{alpha}-stable subsystem of the original system @var{G}.
@item info.hsv
The Hankel singular values of the phase system corresponding
to the @var{alpha}-stable part of the original system @var{G}.
The @var{ns} Hankel singular values are ordered decreasingly.
@item info.nu
The order of the @var{alpha}-unstable subsystem of both the original
system @var{G} and the reduced-order system @var{Gr}.
@item info.nr
The order of the obtained reduced order system @var{Gr}.
@end table
@end table
@strong{Option Keys and Values}
@table @var
@item 'order', 'nr'
The desired order of the resulting reduced order system @var{Gr}.
If not specified, @var{nr} is the sum of NU and the number of
Hankel singular values greater than @code{MAX(TOL1,NS*EPS)};
@var{nr} can be further reduced to ensure that
@code{HSV(NR-NU) > HSV(NR+1-NU)}.
@item 'method'
Approximation method for the H-infinity norm.
Valid values corresponding to this key are:
@table @var
@item 'sr-bta', 'b'
Use the square-root Balance & Truncate method.
@item 'bfsr-bta', 'f'
Use the balancing-free square-root Balance & Truncate method. Default method.
@item 'sr-spa', 's'
Use the square-root Singular Perturbation Approximation method.
@item 'bfsr-spa', 'p'
Use the balancing-free square-root Singular Perturbation Approximation method.
@end table
@item 'alpha'
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix @var{G.A}. For a continuous-time
system, ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system, 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time systems and
1 for discrete-time systems.
@item 'beta'
Use @code{[G, beta*I]} as new system @var{G} to combine
absolute and relative error methods.
BETA > 0 specifies the absolute/relative error weighting
parameter. A large positive value of BETA favours the
minimization of the absolute approximation error, while a
small value of BETA is appropriate for the minimization
of the relative error.
BETA = 0 means a pure relative error method and can be
used only if rank(G.D) = rows(G.D) which means that
the feedthrough matrice must not be rank-deficient.
Default value is 0.
@item 'tol1'
If @var{'order'} is not specified, @var{tol1} contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value of @var{tol1} lies
in the interval [0.00001, 0.001]. @var{tol1} < 1.
If @var{tol1} <= 0 on entry, the used default value is
@var{tol1} = NS*EPS, where NS is the number of
ALPHA-stable eigenvalues of A and EPS is the machine
precision.
If @var{'order'} is specified, the value of @var{tol1} is ignored.
@item 'tol2'
The tolerance for determining the order of a minimal
realization of the phase system (see METHOD) corresponding
to the ALPHA-stable part of the given system.
The recommended value is TOL2 = NS*EPS. TOL2 <= TOL1 < 1.
This value is used by default if @var{'tol2'} is not specified
or if TOL2 <= 0 on entry.
@item 'equil', 'scale'
Boolean indicating whether equilibration (scaling) should be
performed on system @var{G} prior to order reduction.
Default value is true if @code{G.scaled == false} and
false if @code{G.scaled == true}.
Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs
is of utmost importance. The input and output scaling can @strong{not}
be done by the equilibration option or the @command{prescale} command
because these functions perform state transformations only.
Furthermore, signals should not be scaled simply to a certain range.
For all inputs (or outputs), a certain change should be of the same
importance for the model.
@end table
BST is often suitable to perform model reduction in order to obtain
low order design models for controller synthesis.
Approximation Properties:
@itemize @bullet
@item
Guaranteed stability of reduced models
@item
Approximates simultaneously gain and phase
@item
Preserves non-minimum phase zeros
@item
Guaranteed a priori error bound
@iftex
@tex
$$ || G^{-1} (G-G_r) ||_{\infty} \leq 2 \sum_{j=r+1}^{n} \frac{1+\sigma_j}{1-\sigma_j} - 1 $$
@end tex
@end iftex
@end itemize
@strong{Algorithm}@*
Uses SLICOT AB09HD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section btamodred
@deftypefn{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{nr}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{opt}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{nr}, @var{opt}, @dots{})
Model order reduction by frequency weighted Balanced Truncation Approximation (BTA) method.
The aim of model reduction is to find an LTI system @var{Gr} of order
@var{nr} (nr < n) such that the input-output behaviour of @var{Gr}
approximates the one from original system @var{G}.
BTA is an absolute error method which tries to minimize
@iftex
@tex
$$ || G - G_r ||_{\infty} = min $$
$$ || V \ (G - G_r) \ W ||_{\infty} = min $$
@end tex
@end iftex
@ifnottex
@example
||G-Gr|| = min
inf
||V (G-Gr) W|| = min
inf
@end example
@end ifnottex
where @var{V} and @var{W} denote output and input weightings.
@strong{Inputs}
@table @var
@item G
LTI model to be reduced.
@item nr
The desired order of the resulting reduced order system @var{Gr}.
If not specified, @var{nr} is chosen automatically according
to the description of key @var{'order'}.
@item @dots{}
Optional pairs of keys and values. @code{"key1", value1, "key2", value2}.
@item opt
Optional struct with keys as field names.
Struct @var{opt} can be created directly or
by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
@end table
@strong{Outputs}
@table @var
@item Gr
Reduced order state-space model.
@item info
Struct containing additional information.
@table @var
@item info.n
The order of the original system @var{G}.
@item info.ns
The order of the @var{alpha}-stable subsystem of the original system @var{G}.
@item info.hsv
The Hankel singular values of the @var{alpha}-stable part of
the original system @var{G}, ordered decreasingly.
@item info.nu
The order of the @var{alpha}-unstable subsystem of both the original
system @var{G} and the reduced-order system @var{Gr}.
@item info.nr
The order of the obtained reduced order system @var{Gr}.
@end table
@end table
@strong{Option Keys and Values}
@table @var
@item 'order', 'nr'
The desired order of the resulting reduced order system @var{Gr}.
If not specified, @var{nr} is chosen automatically such that states with
Hankel singular values @var{info.hsv} > @var{tol1} are retained.
@item 'left', 'output'
LTI model of the left/output frequency weighting @var{V}.
Default value is an identity matrix.
@item 'right', 'input'
LTI model of the right/input frequency weighting @var{W}.
Default value is an identity matrix.
@item 'method'
Approximation method for the L-infinity norm to be used as follows:
@table @var
@item 'sr', 'b'
Use the square-root Balance & Truncate method.
@item 'bfsr', 'f'
Use the balancing-free square-root Balance & Truncate method. Default method.
@end table
@item 'alpha'
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix @var{G.A}. For a continuous-time
system, ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system, 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time systems and
1 for discrete-time systems.
@item 'tol1'
If @var{'order'} is not specified, @var{tol1} contains the tolerance for
determining the order of the reduced model.
For model reduction, the recommended value of @var{tol1} is
c*info.hsv(1), where c lies in the interval [0.00001, 0.001].
Default value is info.ns*eps*info.hsv(1).
If @var{'order'} is specified, the value of @var{tol1} is ignored.
@item 'tol2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given
model. TOL2 <= TOL1.
If not specified, ns*eps*info.hsv(1) is chosen.
@item 'gram-ctrb'
Specifies the choice of frequency-weighted controllability
Grammian as follows:
@table @var
@item 'standard'
Choice corresponding to a combination method [4]
of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method.
@item 'enhanced'
Choice corresponding to the stability enhanced
modified combination method of [4].
@end table
@item 'gram-obsv'
Specifies the choice of frequency-weighted observability
Grammian as follows:
@table @var
@item 'standard'
Choice corresponding to a combination method [4]
of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method.
@item 'enhanced'
Choice corresponding to the stability enhanced
modified combination method of [4].
@end table
@item 'alpha-ctrb'
Combination method parameter for defining the
frequency-weighted controllability Grammian.
abs(alphac) <= 1.
If alphac = 0, the choice of
Grammian corresponds to the method of Enns [1], while if
alphac = 1, the choice of Grammian corresponds
to the method of Lin and Chiu [2,3].
Default value is 0.
@item 'alpha-obsv'
Combination method parameter for defining the
frequency-weighted observability Grammian.
abs(alphao) <= 1.
If alphao = 0, the choice of
Grammian corresponds to the method of Enns [1], while if
alphao = 1, the choice of Grammian corresponds
to the method of Lin and Chiu [2,3].
Default value is 0.
@item 'equil', 'scale'
Boolean indicating whether equilibration (scaling) should be
performed on system @var{G} prior to order reduction.
This is done by state transformations.
Default value is true if @code{G.scaled == false} and
false if @code{G.scaled == true}.
Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs
is of utmost importance. The input and output scaling can @strong{not}
be done by the equilibration option or the @command{prescale} command
because these functions perform state transformations only.
Furthermore, signals should not be scaled simply to a certain range.
For all inputs (or outputs), a certain change should be of the same
importance for the model.
@end table
Approximation Properties:
@itemize @bullet
@item
Guaranteed stability of reduced models
@item
Lower guaranteed error bound
@item
Guaranteed a priori error bound
@iftex
@tex
$$ \sigma_{r+1} \leq || (G-G_r) ||_{\infty} \leq 2 \sum_{j=r+1}^{n} \sigma_j $$
@end tex
@end iftex
@end itemize
@strong{References}@*
[1] Enns, D.
Model reduction with balanced realizations: An error bound
and a frequency weighted generalization.
Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984.
[2] Lin, C.-A. and Chiu, T.-Y.
Model reduction via frequency-weighted balanced realization.
Control Theory and Advanced Technology, vol. 8,
pp. 341-351, 1992.
[3] Sreeram, V., Anderson, B.D.O and Madievski, A.G.
New results on frequency weighted balanced reduction
technique.
Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995.
[4] Varga, A. and Anderson, B.D.O.
Square-root balancing-free methods for the frequency-weighted
balancing related model reduction.
(report in preparation)
@strong{Algorithm}@*
Uses SLICOT AB09ID by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section hnamodred
@deftypefn{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{nr}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{opt}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{nr}, @var{opt}, @dots{})
Model order reduction by frequency weighted optimal Hankel-norm (HNA) method.
The aim of model reduction is to find an LTI system @var{Gr} of order
@var{nr} (nr < n) such that the input-output behaviour of @var{Gr}
approximates the one from original system @var{G}.
HNA is an absolute error method which tries to minimize
@iftex
@tex
$$ || G - G_r ||_H = min $$
$$ || V \ (G - G_r) \ W ||_H = min $$
@end tex
@end iftex
@ifnottex
@example
||G-Gr|| = min
H
||V (G-Gr) W|| = min
H
@end example
@end ifnottex
where @var{V} and @var{W} denote output and input weightings.
@strong{Inputs}
@table @var
@item G
LTI model to be reduced.
@item nr
The desired order of the resulting reduced order system @var{Gr}.
If not specified, @var{nr} is chosen automatically according
to the description of key @var{"order"}.
@item @dots{}
Optional pairs of keys and values. @code{"key1", value1, "key2", value2}.
@item opt
Optional struct with keys as field names.
Struct @var{opt} can be created directly or
by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
@end table
@strong{Outputs}
@table @var
@item Gr
Reduced order state-space model.
@item info
Struct containing additional information.
@table @var
@item info.n
The order of the original system @var{G}.
@item info.ns
The order of the @var{alpha}-stable subsystem of the original system @var{G}.
@item info.hsv
The Hankel singular values corresponding to the projection @code{op(V)*G1*op(W)},
where G1 denotes the @var{alpha}-stable part of the original system @var{G}.
The @var{ns} Hankel singular values are ordered decreasingly.
@item info.nu
The order of the @var{alpha}-unstable subsystem of both the original
system @var{G} and the reduced-order system @var{Gr}.
@item info.nr
The order of the obtained reduced order system @var{Gr}.
@end table
@end table
@strong{Option Keys and Values}
@table @var
@item 'order', 'nr'
The desired order of the resulting reduced order system @var{Gr}.
If not specified, @var{nr} is the sum of @var{info.nu} and the number of
Hankel singular values greater than @code{max(tol1, ns*eps*info.hsv(1)};
@item 'method'
Specifies the computational approach to be used.
Valid values corresponding to this key are:
@table @var
@item 'descriptor'
Use the inverse free descriptor system approach.
@item 'standard'
Use the inversion based standard approach.
@item 'auto'
Switch automatically to the inverse free
descriptor approach in case of badly conditioned
feedthrough matrices in V or W. Default method.
@end table
@item 'left', 'v'
LTI model of the left/output frequency weighting.
The weighting must be antistable.
@iftex
@math{|| V \ (G-G_r) \dots ||_H = min}
@end iftex
@ifnottex
@example
|| V (G-Gr) . || = min
H
@end example
@end ifnottex
@item 'right', 'w'
LTI model of the right/input frequency weighting.
The weighting must be antistable.
@iftex
@math{|| \dots (G-G_r) \ W ||_H = min}
@end iftex
@ifnottex
@example
|| . (G-Gr) W || = min
H
@end example
@end ifnottex
@item 'left-inv', 'inv-v'
LTI model of the left/output frequency weighting.
The weighting must have only antistable zeros.
@iftex
@math{|| inv(V) \ (G-G_r) \dots ||_H = min}
@end iftex
@ifnottex
@example
|| inv(V) (G-Gr) . || = min
H
@end example
@end ifnottex
@item 'right-inv', 'inv-w'
LTI model of the right/input frequency weighting.
The weighting must have only antistable zeros.
@iftex
@math{|| \dots (G-G_r) \ inv(W) ||_H = min}
@end iftex
@ifnottex
@example
|| . (G-Gr) inv(W) || = min
H
@end example
@end ifnottex
@item 'left-conj', 'conj-v'
LTI model of the left/output frequency weighting.
The weighting must be stable.
@iftex
@math{|| conj(V) \ (G-G_r) \dots ||_H = min}
@end iftex
@ifnottex
@example
|| V (G-Gr) . || = min
H
@end example
@end ifnottex
@item 'right-conj', 'conj-w'
LTI model of the right/input frequency weighting.
The weighting must be stable.
@iftex
@math{|| \dots (G-G_r) \ conj(W) ||_H = min}
@end iftex
@ifnottex
@example
|| . (G-Gr) W || = min
H
@end example
@end ifnottex
@item 'left-conj-inv', 'conj-inv-v'
LTI model of the left/output frequency weighting.
The weighting must be minimum-phase.
@iftex
@math{|| conj(inv(V)) \ (G-G_r) \dots ||_H = min}
@end iftex
@ifnottex
@example
|| V (G-Gr) . || = min
H
@end example
@end ifnottex
@item 'right-conj-inv', 'conj-inv-w'
LTI model of the right/input frequency weighting.
The weighting must be minimum-phase.
@iftex
@math{|| \dots (G-G_r) \ conj(inv(W)) ||_H = min}
@end iftex
@ifnottex
@example
|| . (G-Gr) W || = min
H
@end example
@end ifnottex
@item 'alpha'
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix @var{G.A}. For a continuous-time
system, ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system, 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time systems and
1 for discrete-time systems.
@item 'tol1'
If @var{'order'} is not specified, @var{tol1} contains the tolerance for
determining the order of the reduced model.
For model reduction, the recommended value of @var{tol1} is
c*info.hsv(1), where c lies in the interval [0.00001, 0.001].
@var{tol1} < 1.
If @var{'order'} is specified, the value of @var{tol1} is ignored.
@item 'tol2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given
model. @var{tol2} <= @var{tol1} < 1.
If not specified, ns*eps*info.hsv(1) is chosen.
@item 'equil', 'scale'
Boolean indicating whether equilibration (scaling) should be
performed on system @var{G} prior to order reduction.
Default value is true if @code{G.scaled == false} and
false if @code{G.scaled == true}.
Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs
is of utmost importance. The input and output scaling can @strong{not}
be done by the equilibration option or the @command{prescale} command
because these functions perform state transformations only.
Furthermore, signals should not be scaled simply to a certain range.
For all inputs (or outputs), a certain change should be of the same
importance for the model.
@end table
Approximation Properties:
@itemize @bullet
@item
Guaranteed stability of reduced models
@item
Lower guaranteed error bound
@item
Guaranteed a priori error bound
@iftex
@tex
$$ \sigma_{r+1} \leq || (G-G_r) ||_{\infty} \leq 2 \sum_{j=r+1}^{n} \sigma_j $$
@end tex
@end iftex
@end itemize
@strong{Algorithm}@*
Uses SLICOT AB09JD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section spamodred
@deftypefn{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{opt}, @dots{})
@deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @var{opt}, @dots{})
Model order reduction by frequency weighted Singular Perturbation Approximation (SPA).
The aim of model reduction is to find an LTI system @var{Gr} of order
@var{nr} (nr < n) such that the input-output behaviour of @var{Gr}
approximates the one from original system @var{G}.
SPA is an absolute error method which tries to minimize
@iftex
@tex
$$ || G - G_r ||_{\infty} = min $$
$$ || V \ (G - G_r) \ W ||_{\infty} = min $$
@end tex
@end iftex
@ifnottex
@example
||G-Gr|| = min
inf
||V (G-Gr) W|| = min
inf
@end example
@end ifnottex
where @var{V} and @var{W} denote output and input weightings.
@strong{Inputs}
@table @var
@item G
LTI model to be reduced.
@item nr
The desired order of the resulting reduced order system @var{Gr}.
If not specified, @var{nr} is chosen automatically according
to the description of key @var{'order'}.
@item @dots{}
Optional pairs of keys and values. @code{"key1", value1, "key2", value2}.
@item opt
Optional struct with keys as field names.
Struct @var{opt} can be created directly or
by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
@end table
@strong{Outputs}
@table @var
@item Gr
Reduced order state-space model.
@item info
Struct containing additional information.
@table @var
@item info.n
The order of the original system @var{G}.
@item info.ns
The order of the @var{alpha}-stable subsystem of the original system @var{G}.
@item info.hsv
The Hankel singular values of the @var{alpha}-stable part of
the original system @var{G}, ordered decreasingly.
@item info.nu
The order of the @var{alpha}-unstable subsystem of both the original
system @var{G} and the reduced-order system @var{Gr}.
@item info.nr
The order of the obtained reduced order system @var{Gr}.
@end table
@end table
@strong{Option Keys and Values}
@table @var
@item 'order', 'nr'
The desired order of the resulting reduced order system @var{Gr}.
If not specified, @var{nr} is chosen automatically such that states with
Hankel singular values @var{info.hsv} > @var{tol1} are retained.
@item 'left', 'output'
LTI model of the left/output frequency weighting @var{V}.
Default value is an identity matrix.
@item 'right', 'input'
LTI model of the right/input frequency weighting @var{W}.
Default value is an identity matrix.
@item 'method'
Approximation method for the L-infinity norm to be used as follows:
@table @var
@item 'sr', 's'
Use the square-root Singular Perturbation Approximation method.
@item 'bfsr', 'p'
Use the balancing-free square-root Singular Perturbation Approximation method. Default method.
@end table
@item 'alpha'
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix @var{G.A}. For a continuous-time
system, ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system, 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time systems and
1 for discrete-time systems.
@item 'tol1'
If @var{'order'} is not specified, @var{tol1} contains the tolerance for
determining the order of the reduced model.
For model reduction, the recommended value of @var{tol1} is
c*info.hsv(1), where c lies in the interval [0.00001, 0.001].
Default value is info.ns*eps*info.hsv(1).
If @var{'order'} is specified, the value of @var{tol1} is ignored.
@item 'tol2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given
model. TOL2 <= TOL1.
If not specified, ns*eps*info.hsv(1) is chosen.
@item 'gram-ctrb'
Specifies the choice of frequency-weighted controllability
Grammian as follows:
@table @var
@item 'standard'
Choice corresponding to a combination method [4]
of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method.
@item 'enhanced'
Choice corresponding to the stability enhanced
modified combination method of [4].
@end table
@item 'gram-obsv'
Specifies the choice of frequency-weighted observability
Grammian as follows:
@table @var
@item 'standard'
Choice corresponding to a combination method [4]
of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method.
@item 'enhanced'
Choice corresponding to the stability enhanced
modified combination method of [4].
@end table
@item 'alpha-ctrb'
Combination method parameter for defining the
frequency-weighted controllability Grammian.
abs(alphac) <= 1.
If alphac = 0, the choice of
Grammian corresponds to the method of Enns [1], while if
alphac = 1, the choice of Grammian corresponds
to the method of Lin and Chiu [2,3].
Default value is 0.
@item 'alpha-obsv'
Combination method parameter for defining the
frequency-weighted observability Grammian.
abs(alphao) <= 1.
If alphao = 0, the choice of
Grammian corresponds to the method of Enns [1], while if
alphao = 1, the choice of Grammian corresponds
to the method of Lin and Chiu [2,3].
Default value is 0.
@item 'equil', 'scale'
Boolean indicating whether equilibration (scaling) should be
performed on system @var{G} prior to order reduction.
Default value is true if @code{G.scaled == false} and
false if @code{G.scaled == true}.
Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs
is of utmost importance. The input and output scaling can @strong{not}
be done by the equilibration option or the @command{prescale} command
because these functions perform state transformations only.
Furthermore, signals should not be scaled simply to a certain range.
For all inputs (or outputs), a certain change should be of the same
importance for the model.
@end table
@strong{References}@*
[1] Enns, D.
Model reduction with balanced realizations: An error bound
and a frequency weighted generalization.
Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984.
[2] Lin, C.-A. and Chiu, T.-Y.
Model reduction via frequency-weighted balanced realization.
Control Theory and Advanced Technology, vol. 8,
pp. 341-351, 1992.
[3] Sreeram, V., Anderson, B.D.O and Madievski, A.G.
New results on frequency weighted balanced reduction
technique.
Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995.
[4] Varga, A. and Anderson, B.D.O.
Square-root balancing-free methods for the frequency-weighted
balancing related model reduction.
(report in preparation)
@strong{Algorithm}@*
Uses SLICOT AB09ID by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@chapter Controller Reduction
@section btaconred
@deftypefn{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{ncr}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{opt}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{ncr}, @var{opt}, @dots{})
Controller reduction by frequency-weighted Balanced Truncation Approximation (BTA).
Given a plant @var{G} and a stabilizing controller @var{K}, determine a reduced
order controller @var{Kr} such that the closed-loop system is stable and closed-loop
performance is retained.
The algorithm tries to minimize the frequency-weighted error
@iftex
@tex
$$ || V \ (K - K_r) \ W ||_{\infty} = min $$
@end tex
@end iftex
@ifnottex
@example
||V (K-Kr) W|| = min
inf
@end example
@end ifnottex
where @var{V} and @var{W} denote output and input weightings.
@strong{Inputs}
@table @var
@item G
LTI model of the plant.
It has m inputs, p outputs and n states.
@item K
LTI model of the controller.
It has p inputs, m outputs and nc states.
@item ncr
The desired order of the resulting reduced order controller @var{Kr}.
If not specified, @var{ncr} is chosen automatically according
to the description of key @var{'order'}.
@item @dots{}
Optional pairs of keys and values. @code{"key1", value1, "key2", value2}.
@item opt
Optional struct with keys as field names.
Struct @var{opt} can be created directly or
by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
@end table
@strong{Outputs}
@table @var
@item Kr
State-space model of reduced order controller.
@item info
Struct containing additional information.
@table @var
@item info.ncr
The order of the obtained reduced order controller @var{Kr}.
@item info.ncs
The order of the alpha-stable part of original controller @var{K}.
@item info.hsvc
The Hankel singular values of the alpha-stable part of @var{K}.
The @var{ncs} Hankel singular values are ordered decreasingly.
@end table
@end table
@strong{Option Keys and Values}
@table @var
@item 'order', 'ncr'
The desired order of the resulting reduced order controller @var{Kr}.
If not specified, @var{ncr} is chosen automatically such that states with
Hankel singular values @var{info.hsvc} > @var{tol1} are retained.
@item 'method'
Order reduction approach to be used as follows:
@table @var
@item 'sr', 'b'
Use the square-root Balance & Truncate method.
@item 'bfsr', 'f'
Use the balancing-free square-root Balance & Truncate method. Default method.
@end table
@item 'weight'
Specifies the type of frequency-weighting as follows:
@table @var
@item 'none'
No weightings are used (V = I, W = I).
@item 'left', 'output'
Use stability enforcing left (output) weighting
@iftex
@tex
$$ V = (I - G K)^{-1} G, \qquad W = I $$
@end tex
@end iftex
@ifnottex
@example
-1
V = (I-G*K) *G , W = I
@end example
@end ifnottex
@item 'right', 'input'
Use stability enforcing right (input) weighting
@iftex
@tex
$$ V = I, \qquad W = (I - G K)^{-1} G $$
@end tex
@end iftex
@ifnottex
@example
-1
V = I , W = (I-G*K) *G
@end example
@end ifnottex
@item 'both', 'performance'
Use stability and performance enforcing weightings
@iftex
@tex
$$ V = (I - G K)^{-1} G, \qquad W = (I - G K)^{-1} $$
@end tex
@end iftex
@ifnottex
@example
-1 -1
V = (I-G*K) *G , W = (I-G*K)
@end example
@end ifnottex
Default value.
@end table
@item 'feedback'
Specifies whether @var{K} is a positive or negative feedback controller:
@table @var
@item '+'
Use positive feedback controller. Default value.
@item '-'
Use negative feedback controller.
@end table
@item 'alpha'
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix @var{K.A}. For a continuous-time
controller, ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
controller, 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time controllers and
1 for discrete-time controllers.
@item 'tol1'
If @var{'order'} is not specified, @var{tol1} contains the tolerance for
determining the order of the reduced controller.
For model reduction, the recommended value of @var{tol1} is
c*info.hsvc(1), where c lies in the interval [0.00001, 0.001].
Default value is info.ncs*eps*info.hsvc(1).
If @var{'order'} is specified, the value of @var{tol1} is ignored.
@item 'tol2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given
controller. TOL2 <= TOL1.
If not specified, ncs*eps*info.hsvc(1) is chosen.
@item 'gram-ctrb'
Specifies the choice of frequency-weighted controllability
Grammian as follows:
@table @var
@item 'standard'
Choice corresponding to standard Enns' method [1]. Default method.
@item 'enhanced'
Choice corresponding to the stability enhanced
modified Enns' method of [2].
@end table
@item 'gram-obsv'
Specifies the choice of frequency-weighted observability
Grammian as follows:
@table @var
@item 'standard'
Choice corresponding to standard Enns' method [1]. Default method.
@item 'enhanced'
Choice corresponding to the stability enhanced
modified Enns' method of [2].
@end table
@item 'equil', 'scale'
Boolean indicating whether equilibration (scaling) should be
performed on @var{G} and @var{K} prior to order reduction.
Default value is false if both @code{G.scaled == true, K.scaled == true}
and true otherwise.
Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs
is of utmost importance. The input and output scaling can @strong{not}
be done by the equilibration option or the @command{prescale} command
because these functions perform state transformations only.
Furthermore, signals should not be scaled simply to a certain range.
For all inputs (or outputs), a certain change should be of the same
importance for the model.
@end table
@strong{Algorithm}@*
Uses SLICOT SB16AD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section cfconred
@deftypefn{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{opt}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @var{opt}, @dots{})
Reduction of state-feedback-observer based controller by coprime factorization (CF).
Given a plant @var{G}, state feedback gain @var{F} and full observer gain @var{L},
determine a reduced order controller @var{Kr}.
@strong{Inputs}
@table @var
@item G
LTI model of the open-loop plant (A,B,C,D).
It has m inputs, p outputs and n states.
@item F
Stabilizing state feedback matrix (m-by-n).
@item L
Stabilizing observer gain matrix (n-by-p).
@item ncr
The desired order of the resulting reduced order controller @var{Kr}.
If not specified, @var{ncr} is chosen automatically according
to the description of key @var{'order'}.
@item @dots{}
Optional pairs of keys and values. @code{"key1", value1, "key2", value2}.
@item opt
Optional struct with keys as field names.
Struct @var{opt} can be created directly or
by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
@end table
@strong{Outputs}
@table @var
@item Kr
State-space model of reduced order controller.
@item info
Struct containing additional information.
@table @var
@item info.hsv
The Hankel singular values of the extended system?!?.
The @var{n} Hankel singular values are ordered decreasingly.
@item info.ncr
The order of the obtained reduced order controller @var{Kr}.
@end table
@end table
@strong{Option Keys and Values}
@table @var
@item 'order', 'ncr'
The desired order of the resulting reduced order controller @var{Kr}.
If not specified, @var{ncr} is chosen automatically such that states with
Hankel singular values @var{info.hsv} > @var{tol1} are retained.
@item 'method'
Order reduction approach to be used as follows:
@table @var
@item 'sr-bta', 'b'
Use the square-root Balance & Truncate method.
@item 'bfsr-bta', 'f'
Use the balancing-free square-root Balance & Truncate method. Default method.
@item 'sr-spa', 's'
Use the square-root Singular Perturbation Approximation method.
@item 'bfsr-spa', 'p'
Use the balancing-free square-root Singular Perturbation Approximation method.
@end table
@item 'cf'
Specifies whether left or right coprime factorization is
to be used as follows:
@table @var
@item 'left', 'l'
Use left coprime factorization. Default method.
@item 'right', 'r'
Use right coprime factorization.
@end table
@item 'feedback'
Specifies whether @var{F} and @var{L} are fed back positively or negatively:
@table @var
@item '+'
A+BK and A+LC are both Hurwitz matrices.
@item '-'
A-BK and A-LC are both Hurwitz matrices. Default value.
@end table
@item 'tol1'
If @var{'order'} is not specified, @var{tol1} contains the tolerance for
determining the order of the reduced system.
For model reduction, the recommended value of @var{tol1} is
c*info.hsv(1), where c lies in the interval [0.00001, 0.001].
Default value is n*eps*info.hsv(1).
If @var{'order'} is specified, the value of @var{tol1} is ignored.
@item 'tol2'
The tolerance for determining the order of a minimal
realization of the coprime factorization controller.
TOL2 <= TOL1.
If not specified, n*eps*info.hsv(1) is chosen.
@item 'equil', 'scale'
Boolean indicating whether equilibration (scaling) should be
performed on system @var{G} prior to order reduction.
Default value is true if @code{G.scaled == false} and
false if @code{G.scaled == true}.
Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs
is of utmost importance. The input and output scaling can @strong{not}
be done by the equilibration option or the @command{prescale} command
because these functions perform state transformations only.
Furthermore, signals should not be scaled simply to a certain range.
For all inputs (or outputs), a certain change should be of the same
importance for the model.
@end table
@strong{Algorithm}@*
Uses SLICOT SB16BD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section fwcfconred
@deftypefn{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @var{opt}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @var{opt}, @dots{})
Reduction of state-feedback-observer based controller by frequency-weighted coprime factorization (FW CF).
Given a plant @var{G}, state feedback gain @var{F} and full observer gain @var{L},
determine a reduced order controller @var{Kr} by using stability enforcing frequency weights.
@strong{Inputs}
@table @var
@item G
LTI model of the open-loop plant (A,B,C,D).
It has m inputs, p outputs and n states.
@item F
Stabilizing state feedback matrix (m-by-n).
@item L
Stabilizing observer gain matrix (n-by-p).
@item ncr
The desired order of the resulting reduced order controller @var{Kr}.
If not specified, @var{ncr} is chosen automatically according
to the description of key @var{'order'}.
@item @dots{}
Optional pairs of keys and values. @code{"key1", value1, "key2", value2}.
@item opt
Optional struct with keys as field names.
Struct @var{opt} can be created directly or
by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
@end table
@strong{Outputs}
@table @var
@item Kr
State-space model of reduced order controller.
@item info
Struct containing additional information.
@table @var
@item info.hsv
The Hankel singular values of the extended system?!?.
The @var{n} Hankel singular values are ordered decreasingly.
@item info.ncr
The order of the obtained reduced order controller @var{Kr}.
@end table
@end table
@strong{Option Keys and Values}
@table @var
@item 'order', 'ncr'
The desired order of the resulting reduced order controller @var{Kr}.
If not specified, @var{ncr} is chosen automatically such that states with
Hankel singular values @var{info.hsv} > @var{tol1} are retained.
@item 'method'
Order reduction approach to be used as follows:
@table @var
@item 'sr', 'b'
Use the square-root Balance & Truncate method.
@item 'bfsr', 'f'
Use the balancing-free square-root Balance & Truncate method. Default method.
@end table
@item 'cf'
Specifies whether left or right coprime factorization is
to be used as follows:
@table @var
@item 'left', 'l'
Use left coprime factorization.
@item 'right', 'r'
Use right coprime factorization. Default method.
@end table
@item 'feedback'
Specifies whether @var{F} and @var{L} are fed back positively or negatively:
@table @var
@item '+'
A+BK and A+LC are both Hurwitz matrices.
@item '-'
A-BK and A-LC are both Hurwitz matrices. Default value.
@end table
@item 'tol1'
If @var{'order'} is not specified, @var{tol1} contains the tolerance for
determining the order of the reduced system.
For model reduction, the recommended value of @var{tol1} is
c*info.hsv(1), where c lies in the interval [0.00001, 0.001].
Default value is n*eps*info.hsv(1).
If @var{'order'} is specified, the value of @var{tol1} is ignored.
@end table
@strong{Algorithm}@*
Uses SLICOT SB16CD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@section spaconred
@deftypefn{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @var{ncr}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @var{opt}, @dots{})
@deftypefnx{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @var{ncr}, @var{opt}, @dots{})
Controller reduction by frequency-weighted Singular Perturbation Approximation (SPA).
Given a plant @var{G} and a stabilizing controller @var{K}, determine a reduced
order controller @var{Kr} such that the closed-loop system is stable and closed-loop
performance is retained.
The algorithm tries to minimize the frequency-weighted error
@iftex
@tex
$$ || V \ (K - K_r) \ W ||_{\infty} = min $$
@end tex
@end iftex
@ifnottex
@example
||V (K-Kr) W|| = min
inf
@end example
@end ifnottex
where @var{V} and @var{W} denote output and input weightings.
@strong{Inputs}
@table @var
@item G
LTI model of the plant.
It has m inputs, p outputs and n states.
@item K
LTI model of the controller.
It has p inputs, m outputs and nc states.
@item ncr
The desired order of the resulting reduced order controller @var{Kr}.
If not specified, @var{ncr} is chosen automatically according
to the description of key @var{'order'}.
@item @dots{}
Optional pairs of keys and values. @code{"key1", value1, "key2", value2}.
@item opt
Optional struct with keys as field names.
Struct @var{opt} can be created directly or
by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
@end table
@strong{Outputs}
@table @var
@item Kr
State-space model of reduced order controller.
@item info
Struct containing additional information.
@table @var
@item info.ncr
The order of the obtained reduced order controller @var{Kr}.
@item info.ncs
The order of the alpha-stable part of original controller @var{K}.
@item info.hsvc
The Hankel singular values of the alpha-stable part of @var{K}.
The @var{ncs} Hankel singular values are ordered decreasingly.
@end table
@end table
@strong{Option Keys and Values}
@table @var
@item 'order', 'ncr'
The desired order of the resulting reduced order controller @var{Kr}.
If not specified, @var{ncr} is chosen automatically such that states with
Hankel singular values @var{info.hsvc} > @var{tol1} are retained.
@item 'method'
Order reduction approach to be used as follows:
@table @var
@item 'sr', 's'
Use the square-root Singular Perturbation Approximation method.
@item 'bfsr', 'p'
Use the balancing-free square-root Singular Perturbation Approximation method. Default method.
@end table
@item 'weight'
Specifies the type of frequency-weighting as follows:
@table @var
@item 'none'
No weightings are used (V = I, W = I).
@item 'left', 'output'
Use stability enforcing left (output) weighting
@iftex
@tex
$$ V = (I - G K)^{-1} G, \qquad W = I $$
@end tex
@end iftex
@ifnottex
@example
-1
V = (I-G*K) *G , W = I
@end example
@end ifnottex
@item 'right', 'input'
Use stability enforcing right (input) weighting
@iftex
@tex
$$ V = I, \qquad W = (I - G K)^{-1} G $$
@end tex
@end iftex
@ifnottex
@example
-1
V = I , W = (I-G*K) *G
@end example
@end ifnottex
@item 'both', 'performance'
Use stability and performance enforcing weightings
@iftex
@tex
$$ V = (I - G K)^{-1} G, \qquad W = (I - G K)^{-1} $$
@end tex
@end iftex
@ifnottex
@example
-1 -1
V = (I-G*K) *G , W = (I-G*K)
@end example
@end ifnottex
Default value.
@end table
@item 'feedback'
Specifies whether @var{K} is a positive or negative feedback controller:
@table @var
@item '+'
Use positive feedback controller. Default value.
@item '-'
Use negative feedback controller.
@end table
@item 'alpha'
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix @var{K.A}. For a continuous-time
controller, ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
controller, 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time controllers and
1 for discrete-time controllers.
@item 'tol1'
If @var{'order'} is not specified, @var{tol1} contains the tolerance for
determining the order of the reduced controller.
For model reduction, the recommended value of @var{tol1} is
c*info.hsvc(1), where c lies in the interval [0.00001, 0.001].
Default value is info.ncs*eps*info.hsvc(1).
If @var{'order'} is specified, the value of @var{tol1} is ignored.
@item 'tol2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given
controller. TOL2 <= TOL1.
If not specified, ncs*eps*info.hsvc(1) is chosen.
@item 'gram-ctrb'
Specifies the choice of frequency-weighted controllability
Grammian as follows:
@table @var
@item 'standard'
Choice corresponding to standard Enns' method [1]. Default method.
@item 'enhanced'
Choice corresponding to the stability enhanced
modified Enns' method of [2].
@end table
@item 'gram-obsv'
Specifies the choice of frequency-weighted observability
Grammian as follows:
@table @var
@item 'standard'
Choice corresponding to standard Enns' method [1]. Default method.
@item 'enhanced'
Choice corresponding to the stability enhanced
modified Enns' method of [2].
@end table
@item 'equil', 'scale'
Boolean indicating whether equilibration (scaling) should be
performed on @var{G} and @var{K} prior to order reduction.
Default value is false if both @code{G.scaled == true, K.scaled == true}
and true otherwise.
Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs
is of utmost importance. The input and output scaling can @strong{not}
be done by the equilibration option or the @command{prescale} command
because these functions perform state transformations only.
Furthermore, signals should not be scaled simply to a certain range.
For all inputs (or outputs), a certain change should be of the same
importance for the model.
@end table
@strong{Algorithm}@*
Uses SLICOT SB16AD by courtesy of
@uref{http://www.slicot.org, NICONET e.V.}
@end deftypefn
@chapter Overloaded Operators
@section @@lti/ctranspose
Conjugate transpose or pertransposition of LTI objects.
Used by Octave for "sys'".
For a transfer-function matrix G, G' denotes the conjugate
of G given by G.'(-s) for a continuous-time system or G.'(1/z)
for a discrete-time system.
The frequency response of the pertransposition of G is the
Hermitian (conjugate) transpose of G(jw), i.e.
freqresp (G', w) = freqresp (G, w)'.
@strong{WARNING:} Do @strong{NOT} use this for dual problems,
use the transpose "sys.'" (note the dot) instead.
@section @@lti/horzcat
Horizontal concatenation of LTI objects. If necessary, object conversion
is done by sys_group. Used by Octave for "[sys1, sys2]".
@section @@lti/inv
Inversion of LTI objects.
@section @@lti/minus
Binary subtraction of LTI objects. If necessary, object conversion
is done by sys_group. Used by Octave for "sys1 - sys2".
@section @@lti/mldivide
Matrix left division of LTI objects. If necessary, object conversion
is done by sys_group in mtimes. Used by Octave for "sys1 \ sys2".
@section @@lti/mpower
Matrix power of LTI objects. The exponent must be an integer.
Used by Octave for "sys^int".
@section @@lti/mrdivide
Matrix right division of LTI objects. If necessary, object conversion
is done by sys_group in mtimes. Used by Octave for "sys1 / sys2".
@section @@lti/mtimes
Matrix multiplication of LTI objects. If necessary, object conversion
is done by sys_group. Used by Octave for "sys1 * sys2".
@section @@lti/plus
Binary addition of LTI objects. If necessary, object conversion
is done by sys_group. Used by Octave for "sys1 + sys2".
Operation is also known as "parallel connection".
@section @@lti/subsasgn
Subscripted assignment for LTI objects.
Used by Octave for "sys.property = value".
@section @@lti/subsref
Subscripted reference for LTI objects.
Used by Octave for "sys = sys(2:4, :)" or "val = sys.prop".
@section @@lti/transpose
Transpose of LTI objects. Used by Octave for "sys.'".
Useful for dual problems, i.e. controllability and observability
or designing estimator gains with @command{lqr} and @command{place}.
@section @@lti/uminus
Unary minus of LTI object. Used by Octave for "-sys".
@section @@lti/uplus
Unary plus of LTI object. Used by Octave for "+sys".
@section @@lti/vertcat
Vertical concatenation of LTI objects. If necessary, object conversion
is done by sys_group. Used by Octave for "[sys1; sys2]".
@chapter Miscellaneous
@section options
@deftypefn{Function File} {@var{opt} =} options (@var{"key1"}, @var{value1}, @var{"key2"}, @var{value2}, @dots{})
Create options struct @var{opt} from a number of key and value pairs.
For use with order reduction commands.
@strong{Inputs}
@table @var
@item key, property
The name of the property.
@item value
The value of the property.
@end table
@strong{Outputs}
@table @var
@item opt
Struct with fields for each key.
@end table
@strong{Example}
@example
@group
octave:1> opt = options ("method", "spa", "tol", 1e-6)
opt =
scalar structure containing the fields:
method = spa
tol = 1.0000e-06
@end group
@end example
@example
@group
octave:2> save filename opt
octave:3> # save the struct 'opt' to file 'filename' for later use
octave:4> load filename
octave:5> # load struct 'opt' from file 'filename'
@end group
@end example
@end deftypefn
@section strseq
@deftypefn {Function File} {@var{strvec} =} strseq (@var{str}, @var{idx})
Return a cell vector of indexed strings by appending the indices @var{idx}
to the string @var{str}.
@example
strseq ("x", 1:3) = @{"x1"; "x2"; "x3"@}
strseq ("u", [1, 2, 5]) = @{"u1"; "u2"; "u5"@}
@end example
@end deftypefn
@section test_control
@deftypefn {Script File} {} test_control
Execute all available tests at once.
The Octave control package is based on the @uref{http://www.slicot.org, SLICOT} library.
SLICOT needs a LAPACK library which is also a prerequisite for Octave itself.
In case of failing test, it is highly recommended to use
@uref{http://www.netlib.org/lapack/, Netlib's reference LAPACK}
for building Octave. Using ATLAS may lead to sign changes
in some entries in the state-space matrices.
In general, these sign changes are not 'wrong' and can be regarded as
the result of state transformations. Such state transformations
(but not input/output transformations) have no influence on the
input-output behaviour of the system. For better numerics,
the control package uses such transformations by default when
calculating the frequency responses and a few other things.
However, arguments like the Hankel singular Values (HSV) must not change.
Differing HSVs and failing algorithms are known for using Framework Accelerate
from Mac OS X 10.7.
@end deftypefn
@section BMWengine
@deftypefn{Function File} {@var{sys} =} BMWengine ()
@deftypefnx{Function File} {@var{sys} =} BMWengine (@var{"scaled"})
@deftypefnx{Function File} {@var{sys} =} BMWengine (@var{"unscaled"})
Model of the BMW 4-cylinder engine at ETH Zurich's control laboratory.
@example
@group
OPERATING POINT
Drosselklappenstellung alpha_DK = 10.3 Grad
Saugrohrdruck p_s = 0.48 bar
Motordrehzahl n = 860 U/min
Lambda-Messwert lambda = 1.000
Relativer Wandfilminhalt nu = 1
@end group
@end example
@example
@group
INPUTS
U_1 Sollsignal Drosselklappenstellung [Grad]
U_2 Relative Einspritzmenge [-]
U_3 Zuendzeitpunkt [Grad KW]
M_L Lastdrehmoment [Nm]
@end group
@end example
@example
@group
STATES
X_1 Drosselklappenstellung [Grad]
X_2 Saugrohrdruck [bar]
X_3 Motordrehzahl [U/min]
X_4 Messwert Lamba-Sonde [-]
X_5 Relativer Wandfilminhalt [-]
@end group
@end example
@example
@group
OUTPUTS
Y_1 Motordrehzahl [U/min]
Y_2 Messwert Lambda-Sonde [-]
@end group
@end example
@example
@group
SCALING
U_1N, X_1N 1 Grad
U_2N, X_4N, X_5N, Y_2N 0.05
U_3N 1.6 Grad KW
X_2N 0.05 bar
X_3N, Y_1N 200 U/min
@end group
@end example
@end deftypefn
@section Boeing707
@deftypefn {Function File} {@var{sys} =} Boeing707 ()
Creates a linearized state-space model of a Boeing 707-321 aircraft
at @var{v}=80 m/s
@iftex
@tex
($M = 0.26$, $G_{a0} = -3^{\circ}$, ${\alpha}_0 = 4^{\circ}$, ${\kappa}= 50^{\circ}$).
@end tex
@end iftex
@ifnottex
(@var{M} = 0.26, @var{Ga0} = -3 deg, @var{alpha0} = 4 deg, @var{kappa} = 50 deg).
@end ifnottex
System inputs: (1) thrust and (2) elevator angle.
System outputs: (1) airspeed and (2) pitch angle.
@strong{Reference}: R. Brockhaus: @cite{Flugregelung} (Flight
Control), Springer, 1994.
@end deftypefn
@section WestlandLynx
@deftypefn{Function File} {@var{sys} =} WestlandLynx ()
Model of the Westland Lynx Helicopter about hover.
@example
@group
INPUTS
main rotor collective
longitudinal cyclic
lateral cyclic
tail rotor collective
@end group
@end example
@example
@group
STATES
pitch attitude theta [rad]
roll attitude phi [rad]
roll rate (body-axis) p [rad/s]
pitch rate (body-axis) q [rad/s]
yaw rate xi [rad/s]
forward velocity v_x [ft/s]
lateral velocity v_y [ft/s]
vertical velocity v_z [ft/s]
@end group
@end example
@example
@group
OUTPUTS
heave velocity H_dot [ft/s]
pitch attitude theta [rad]
roll attitude phi [rad]
heading rate psi_dot [rad/s]
roll rate p [rad/s]
pitch rate q [rad/s]
@end group
@end example
@example
@group
Reference:
Skogestad, S. and Postlethwaite I.
Multivariable Feedback Control: Analysis and Design
Second Edition
Wiley 2005
http://www.nt.ntnu.no/users/skoge/book/2nd_edition/matlab_m/matfiles.html
@end group
@end example
@end deftypefn
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