ss
Create state-space model, convert to state-space model
Syntax
sys = ss(a,b,c,d)
sys = ss(a,b,c,d,Ts)
sys = ss(d)
sys = ss(a,b,c,d,ltisys)
sys_ss = ss(sys)
sys_ss = ss(sys,'minimal')
sys_ss = ss(sys,'explicit')
sys_ss = ss(sys,
'measured')
sys_ss = ss(sys, 'noise')
sys_ss = ss(sys, 'augmented')
Description
Use ss to create state-space models (ss model
objects) with real- or complex-valued matrices or to convert dynamic
system models to state-space model form. You can also use ss to
create Generalized state-space (genss)
models.
Creation of State-Space Models
sys = ss(a,b,c,d) creates
a state-space model object representing the continuous-time state-space
model
For a model with Nx states, Ny outputs,
and Nu inputs:
a is an Nx-by-Nx real-
or complex-valued matrix.
b is an Nx-by-Nu real-
or complex-valued matrix.
c is an Ny-by-Nx real-
or complex-valued matrix.
d is an Ny-by-Nu real-
or complex-valued matrix.
To set D = 0 , set d to
the scalar 0 (zero), regardless of the dimension.
sys = ss(a,b,c,d,Ts)
creates the discrete-time model
with sample time Ts (in seconds). Set Ts
= -1 or Ts = [] to leave the sample time
unspecified.
sys = ss(d)
specifies a static gain matrix D and is equivalent
to
sys = ss([],[],[],d)
sys = ss(a,b,c,d,ltisys)
creates a state-space model with properties inherited from the model ltisys (including
the sample time).
Any of the previous syntaxes can be followed by property name/property
value pairs.
'PropertyName',PropertyValue
Each pair specifies a particular property of the model, for
example, the input names or some notes on the model history. See Properties for more information about
available ss model object properties.
The following expression:
sys = ss(a,b,c,d,'Property1',Value1,...,'PropertyN',ValueN)
is equivalent to the sequence of commands:
sys = ss(a,b,c,d)
set(sys,'Property1',Value1,...,'PropertyN',ValueN)
Conversion to State Space
sys_ss = ss(sys) converts
a dynamic system model sys to state-space form.
The output sys_ss is an equivalent state-space
model (ss model object). This operation is known as state-space realization.
sys_ss = ss(sys,'minimal') produces
a state-space realization with no uncontrollable or unobservable states.
This state-space realization is equivalent to sys_ss = minreal(ss(sys)).
sys_ss = ss(sys,'explicit') computes
an explicit realization (E = I) of the dynamic
system model sys. If sys is
improper, ss returns an error.
Note:
Conversions to state space are not uniquely defined in the SISO
case. They are also not guaranteed to produce a minimal realization
in the MIMO case. For more information, see Recommended Working Representation. |
Conversion of Identified Models
An identified model is represented by an input-output equation
of the form
,
where u(t) is the set of measured input channels
and e(t) represents the noise channels. If
Λ = LL' represents the covariance of noise e(t),
this equation can also be written as
,
where
.
sys_ss = ss(sys) or sys_ss = ss(sys,
'measured') converts the measured component of an identified
linear model into the state-space form. sys is
a model of type idss, idproc, idtf, idpoly,
or idgrey. sys_ss represents
the relationship between u and y.
sys_ss = ss(sys, 'noise') converts the
noise component of an identified linear model into the state space
form. It represents the relationship between the noise input v(t) and
output y_noise = HL v(t). The noise input channels
belong to the InputGroup 'Noise'. The names of
the noise input channels are v@yname, where yname is
the name of the corresponding output channel. sys_ss has
as many inputs as outputs.
sys_ss = ss(sys, 'augmented') converts
both the measured and noise dynamics into a state-space model. sys_ss has ny+nu inputs
such that the first nu inputs represent the channels u(t) while
the remaining by channels represent the noise channels v(t). sys_ss.InputGroup contains
2 input groups- 'measured' and 'noise'.
sys_ss.InputGroup.Measured is set to 1:nu while sys_ss.InputGroup.Noise is
set to nu+1:nu+ny. sys_ss represents
the equation
Tip
An identified nonlinear model cannot be converted into a state-space
form. Use linear approximation functions such as linearize and linapp. |
Creation of Generalized State-Space Models
You can use the syntax:
gensys = ss(A,B,C,D)
to create a Generalized state-space (genss)
model when one or more of the matrices A, B, C, D is
a tunable realp or genmat model. For more information about
Generalized state-space models, see Models with Tunable Coefficients.
Properties
ss objects have the following properties:
<argumentlist>
State-space matrices.
a — State matrix A.
Square real- or complex-valued matrix with as many rows as states.
b — Input-to-state matrix B.
Real- or complex-valued matrix with as many rows as states and as
many columns as inputs.
c — State-to-output matrix C.
Real- or complex-valued matrix with as many rows as outputs and as
many columns as states.
d — Feedthrough matrix D.
Real- or complex-valued matrix with as many rows as outputs and as
many columns as inputs.
e — E matrix
for implicit (descriptor) state-space models. By default e
= [], meaning that the state equation is explicit. To specify
an implicit state equation E dx/dt = Ax + Bu,
set this property to a square matrix of the same size as a.
See dss for more information
about creating descriptor state-space models.
Logical value indicating whether scaling is enabled or disabled.
When Scaled = 0 (false), most numerical algorithms
acting on the state-space model automatically rescale the state vector
to improve numerical accuracy. You can disable such auto-scaling
by setting Scaled = 1 (true). For more information
about scaling, see prescale.
State names. For first-order models, set StateName to
a string. For models with two or more states, set StateName to
a cell array of strings . Use an empty string '' for
unnamed states.
State units. Use StateUnit to keep track
of the units each state is expressed in. For first-order models, set StateUnit to
a string. For models with two or more states, set StateUnit to
a cell array of strings. StateUnit has no effect
on system behavior.
Vector storing internal delays.
Internal delays arise, for example, when closing feedback loops
on systems with delays, or when connecting delayed systems in series
or parallel. For more information about internal delays, see Closing Feedback Loops with Time Delays in the Control System Toolbox™ User's
Guide.
For continuous-time models, internal delays are expressed in
the time unit specified by the TimeUnit property
of the model. For discrete-time models, internal delays are expressed
as integer multiples of the sampling period Ts.
For example, InternalDelay = 3 means a delay of
three sampling periods.
You can modify the values of internal delays. However, the
number of entries in sys.InternalDelay cannot change,
because it is a structural property of the model.
Input delay for each input channel, specified as a numeric vector.
For continuous-time systems, specify input delays in the time unit
stored in the TimeUnit property. For discrete-time
systems, specify input delays in integer multiples of the sampling
period Ts. For example, InputDelay = 3 means
a delay of three sampling periods.
For a system with Nu inputs, set InputDelay to
an Nu-by-1 vector. Each entry of this vector is
a numerical value that represents the input delay for the corresponding
input channel.
You can also set InputDelay to a scalar value
to apply the same delay to all channels.
Output delays. OutputDelay is a numeric vector
specifying a time delay for each output channel. For continuous-time
systems, specify output delays in the time unit stored in the TimeUnit property.
For discrete-time systems, specify output delays in integer multiples
of the sampling period Ts. For example, OutputDelay
= 3 means a delay of three sampling periods.
For a system with Ny outputs, set OutputDelay to
an Ny-by-1 vector, where each entry is a numerical
value representing the output delay for the corresponding output channel.
You can also set OutputDelay to a scalar value
to apply the same delay to all channels.
Sampling time. For continuous-time models, Ts = 0.
For discrete-time models, Ts is a positive scalar
representing the sampling period. This value is expressed in the unit
specified by the TimeUnit property of the model.
To denote a discrete-time model with unspecified sampling time, set Ts
= -1.
Changing this property does not discretize or resample the model.
Use c2d and d2c to convert between
continuous- and discrete-time representations. Use d2d to change the
sampling time of a discrete-time system.
String representing the unit of the time variable. For continuous-time
models, this property represents any time delays in the model. For
discrete-time models, it represents the sampling time Ts.
Use any of the following values:
'nanoseconds'
'microseconds'
'milliseconds'
'seconds'
'minutes'
'hours'
'days'
'weeks'
'months'
'years'
Changing this property changes the overall system behavior.
Use chgTimeUnit to convert
between time units without modifying system behavior.
Input channel names. Set InputName to a string
for single-input model. For a multi-input model, set InputName to
a cell array of strings.
Alternatively, use automatic vector expansion to assign input
names for multi-input models. For example, if sys is
a two-input model, enter:
sys.InputName = 'controls';
The input names automatically expand to {'controls(1)';'controls(2)'}.
You can use the shorthand notation u to refer
to the InputName property. For example, sys.u is
equivalent to sys.InputName.
Input channel names have several uses, including:
Identifying channels on model display and plots
Extracting subsystems of MIMO systems
Specifying connection points when interconnecting
models
Input channel units. Use InputUnit to keep
track of input signal units. For a single-input model, set InputUnit to
a string. For a multi-input model, set InputUnit to
a cell array of strings. InputUnit has no effect
on system behavior.
Input channel groups. The InputGroup property
lets you assign the input channels of MIMO systems into groups and
refer to each group by name. Specify input groups as a structure.
In this structure, field names are the group names, and field values
are the input channels belonging to each group. For example:
sys.InputGroup.controls = [1 2];
sys.InputGroup.noise = [3 5];
creates input groups named controls and noise that
include input channels 1, 2 and 3, 5, respectively. You can then extract
the subsystem from the controls inputs to all outputs
using:
sys(:,'controls')
Output channel names. Set OutputName to a
string for single-output model. For a multi-output model, set OutputName to
a cell array of strings.
Alternatively, use automatic vector expansion to assign output
names for multi-output models. For example, if sys is
a two-output model, enter:
sys.OutputName = 'measurements';
The output names to automatically expand to {'measurements(1)';'measurements(2)'}.
You can use the shorthand notation y to refer
to the OutputName property. For example, sys.y is
equivalent to sys.OutputName.
Output channel names have several uses, including:
Identifying channels on model display and plots
Extracting subsystems of MIMO systems
Specifying connection points when interconnecting
models
Output channel units. Use OutputUnit to keep
track of output signal units. For a single-output model, set OutputUnit to
a string. For a multi-output model, set OutputUnit to
a cell array of strings. OutputUnit has no effect
on system behavior.
Output channel groups. The OutputGroup property
lets you assign the output channels of MIMO systems into groups and
refer to each group by name. Specify output groups as a structure.
In this structure, field names are the group names, and field values
are the output channels belonging to each group. For example:
sys.OutputGroup.temperature = [1];
sys.InputGroup.measurement = [3 5];
creates output groups named temperature and measurement that
include output channels 1, and 3, 5, respectively. You can then extract
the subsystem from all inputs to the measurement outputs
using:
sys('measurement',:)
System name. Set Name to a string to label
the system.
Any text that you want to associate with the system. Set Notes to
a string or a cell array of strings.
Any type of data you wish to associate with system. Set UserData to
any MATLAB^{®} data type.
Sampling grid for model arrays, specified as a data structure.
For model arrays that are derived
by sampling one or more independent variables, this property tracks
the variable values associated with each model in the array. This
information appears when you display or plot the model array. Use
this information to trace results back to the independent variables.
Set the field names of the data structure to the names of the
sampling variables. Set the field values to the sampled variable values
associated with each model in the array. All sampling variables should
be numeric and scalar valued, and all arrays of sampled values should
match the dimensions of the model array.
For example, suppose you create a 11-by-1
array of linear models, sysarr, by taking snapshots
of a linear time-varying system at times t = 0:10.
The following code stores the time samples with the linear models.
sysarr.SamplingGrid = struct('time',0:10)
Similarly, suppose you create a 6-by-9
model array, M, by independently sampling two variables, zeta and w.
The following code attaches the (zeta,w) values
to M.
[zeta,w] = ndgrid(<6 values of zeta>,<9 values of w>)
M.SamplingGrid = struct('zeta',zeta,'w',w)
When you display M,
each entry in the array includes the corresponding zeta and w values.
M
M(:,:,1,1) [zeta=0.3, w=5] =
25
--------------
s^2 + 3 s + 25
M(:,:,2,1) [zeta=0.35, w=5] =
25
----------------
s^2 + 3.5 s + 25
...
</argumentlist>Examples
Example 1
Discrete-Time State-Space Model
Create a state-space model with a sampling time of 0.25 s and
the following state-space matrices:
To do this, enter the following commands:
A = [0 1;-5 -2];
B = [0;3];
C = [0 1];
D = 0;
sys = ss(A,B,C,D,0.25);
The last argument sets the sampling time.
Example 2
Discrete-Time State-Space Model with Custom
State and Input Names
Create a discrete-time model with matrices A,B,C,D and
sample time 0.05 second.
sys = ss(A,B,C,D,0.05,'statename',{'position' 'velocity'},...
'inputname','force',...
'notes','Created 01/16/11');
This model has two states labeled position and velocity,
and one input labeled force (the dimensions of A,B,C,D should
be consistent with these numbers of states and inputs). Finally, a
note is attached with the date of creation of the model.
Example 3
Convert Transfer Function Model to State-Space
Model
Convert a transfer function model to a state-space model.
by typing
H = [tf([1 1],[1 3 3 2]) ; tf([1 0 3],[1 1 1])];
sys = ss(H);
size(sys)
State-space model with 2 outputs, 1 input, and 5 states.
The number of states is equal to the cumulative order of the
SISO entries of H(s).
To obtain a minimal realization of H(s),
type
sys = ss(H,'min');
size(sys)
State-space model with 2 outputs, 1 input, and 3 states.
The resulting state-space model has order of three, which is
the minimum number of states needed to represent H(s).
You can see this number of states by factoring H(s)
as the product of a first-order system with a second-order system.
Example 4
Descriptor State-Space Model
Create a descriptor state-space model.
a = [2 -4; 4 2];
b = [-1; 0.5];
c = [-0.5, -2];
d = [-1];
e = [1 0; -3 0.5];
% Create a descriptor state-space model.
sys1 = dss(a,b,c,d,e);
% Compute an explicit realization.
sys2 = ss(sys1,'explicit')
These commands produce the result:
a =
x1 x2
x1 2 -4
x2 20 -20
b =
u1
x1 -1
x2 -5
c =
x1 x2
y1 -0.5 -2
d =
u1
y1 -1
Continuous-time model.
The result is an explicit state-space model (E =
I). A Bode plot shows that sys1 and sys2 are
equivalent.
bode(sys1,sys2)
Example 5
Generalized State-Space Model
This example shows how to create a state-space (genss) model having both fixed and tunable
parameters.
Create a state-space model having the following state-space
matrices:
where a and b are tunable
parameters, whose initial values are –1 and 3, respectively.
Create the tunable parameters using realp.
a = realp('a',-1);
b = realp('b',3);
Define a generalized matrix using algebraic expressions
of a and b.
A = [1 a+b;0 a*b]
A is a generalized matrix whose Blocks property
contains a and b. The initial
value of A is M = [1 2;0 -3],
from the initial values of a and b.
Create the fixed-value state-space matrices.
B = [-3.0;1.5];
C = [0.3 0];
D = 0;
Use ss to
create the state-space model.
sys = ss(A,B,C,D)
sys is a generalized LTI model (genss)
with tunable parameters a and b.
Example 6
Extract the measured and noise components of an identified polynomial
model into two separate state-space models. The former (measured component)
can serve as a plant model while the latter can serve as a disturbance
model for control system design.
load icEngine
z = iddata(y,u,0.04);
sys = ssest(z, 3);
sysMeas = ss(sys, 'measured')
sysNoise = ss(sys, 'noise')
Alternatively, use can simply use ss(sys) to
extract the measured component.
More About
expand all
For TF to SS model conversion, ss(sys_tf) returns
a modified version of the controllable canonical form. It uses an
algorithm similar to tf2ss, but further rescales
the state vector to compress the numerical range in state matrix A and
to improve numerics in subsequent computations.
For ZPK to SS conversion, ss(sys_zpk) uses
direct form II structures, as defined in signal processing texts.
See Discrete-Time Signal Processing by Oppenheim
and Schafer for details.
For example, in the following code, A and sys.a differ
by a diagonal state transformation:
n=[1 1];
d=[1 1 10];
[A,B,C,D]=tf2ss(n,d);
sys=ss(tf(n,d));
A
A =
-1 -10
1 0
sys.a
ans =
-1 -5
2 0
For details, see balance.
See Also
dss | frd | get | set | ssdata | tf | zpk
Tutorials