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Create statespace model, convert to statespace model
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')
Use ss to create statespace models (ss model objects) with real or complexvalued matrices or to convert dynamic system models to statespace model form. You can also use ss to create Generalized statespace (genss) models.
sys = ss(a,b,c,d) creates a statespace model object representing the continuoustime statespace model
For a model with Nx states, Ny outputs, and Nu inputs:
a is an NxbyNx real or complexvalued matrix.
b is an NxbyNu real or complexvalued matrix.
c is an NybyNx real or complexvalued matrix.
d is an NybyNu real or complexvalued 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 discretetime 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 statespace 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)
sys_ss = ss(sys) converts a dynamic system model sys to statespace form. The output sys_ss is an equivalent statespace model (ss model object). This operation is known as statespace realization.
sys_ss = ss(sys,'minimal') produces a statespace realization with no uncontrollable or unobservable states. This statespace 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. 
An identified model is represented by an inputoutput 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 statespace 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 statespace 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 statespace form. Use linear approximation functions such as linearize and linapp. 
You can use the syntax:
gensys = ss(A,B,C,D)
to create a Generalized statespace (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 statespace models, see Models with Tunable Coefficients.
ss objects have the following properties:
a,b,c,d,e 
Statespace matrices.

Scaled 
Logical value indicating whether scaling is enabled or disabled. When Scaled = 0 (false), most numerical algorithms acting on the statespace model automatically rescale the state vector to improve numerical accuracy. You can disable such autoscaling by setting Scaled = 1 (true). For more information about scaling, see prescale. Default: 0 (false) 
StateName 
State names. For firstorder 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. Default: Empty string '' for all states 
StateUnit 
State units. Use StateUnit to keep track of the units each state is expressed in. For firstorder 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. Default: Empty string '' for all states 
InternalDelay 
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 continuoustime models, internal delays are expressed in the time unit specified by the TimeUnit property of the model. For discretetime 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. 
InputDelay 
Input delay for each input channel, specified as a scalar value or numeric vector. For continuoustime systems, specify input delays in the time unit stored in the TimeUnit property. For discretetime 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 Nuby1 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. Default: 0 
OutputDelay 
Output delays. OutputDelay is a numeric vector specifying a time delay for each output channel. For continuoustime systems, specify output delays in the time unit stored in the TimeUnit property. For discretetime 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 Nyby1 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. Default: 0 for all output channels 
Ts 
Sampling time. For continuoustime models, Ts = 0. For discretetime 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 discretetime 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 discretetime representations. Use d2d to change the sampling time of a discretetime system. Default: 0 (continuous time) 
TimeUnit 
String representing the unit of the time variable. For continuoustime models, this property represents any time delays in the model. For discretetime models, it represents the sampling time Ts. Use any of the following values:
Changing this property changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior. Default: 'seconds' 
InputName 
Input channel names. Set InputName to a string for singleinput model. For a multiinput model, set InputName to a cell array of strings. Alternatively, use automatic vector expansion to assign input names for multiinput models. For example, if sys is a twoinput 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:
Default: Empty string '' for all input channels 
InputUnit 
Input channel units. Use InputUnit to keep track of input signal units. For a singleinput model, set InputUnit to a string. For a multiinput model, set InputUnit to a cell array of strings. InputUnit has no effect on system behavior. Default: Empty string '' for all input channels 
InputGroup 
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') Default: Struct with no fields 
OutputName 
Output channel names. Set OutputName to a string for singleoutput model. For a multioutput model, set OutputName to a cell array of strings. Alternatively, use automatic vector expansion to assign output names for multioutput models. For example, if sys is a twooutput 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:
Default: Empty string '' for all input channels 
OutputUnit 
Output channel units. Use OutputUnit to keep track of output signal units. For a singleoutput model, set OutputUnit to a string. For a multioutput model, set OutputUnit to a cell array of strings. OutputUnit has no effect on system behavior. Default: Empty string '' for all input channels 
OutputGroup 
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',:) Default: Struct with no fields 
Name 
System name. Set Name to a string to label the system. Default: '' 
Notes 
Any text that you want to associate with the system. Set Notes to a string or a cell array of strings. Default: {} 
UserData 
Any type of data you wish to associate with system. Set UserData to any MATLAB^{®} data type. Default: [] 
SamplingGrid 
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 11by1 array of linear models, sysarr, by taking snapshots of a linear timevarying 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 6by9 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 ... Default: [] 
DiscreteTime StateSpace Model
Create a statespace model with a sampling time of 0.25 s and the following statespace 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.
DiscreteTime StateSpace Model with Custom State and Input Names
Create a discretetime 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.
Convert Transfer Function Model to StateSpace Model
Convert a transfer function model to a statespace model.
by typing
H = [tf([1 1],[1 3 3 2]) ; tf([1 0 3],[1 1 1])]; sys = ss(H); size(sys) Statespace 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) Statespace model with 2 outputs, 1 input, and 3 states.
The resulting statespace 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 firstorder system with a secondorder system.
Descriptor StateSpace Model
Create a descriptor statespace 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 statespace 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 Continuoustime model.
The result is an explicit statespace model (E = I). A Bode plot shows that sys1 and sys2 are equivalent.
bode(sys1,sys2)
Generalized StateSpace Model
This example shows how to create a statespace (genss) model having both fixed and tunable parameters.
Create a statespace model having the following statespace 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 fixedvalue statespace matrices.
B = [3.0;1.5]; C = [0.3 0]; D = 0;
Use ss to create the statespace model.
sys = ss(A,B,C,D)
sys is a generalized LTI model (genss) with tunable parameters a and b.
Extract the measured and noise components of an identified polynomial model into two separate statespace 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.
dss  frd  get  set  ssdata  tf  zpk