Documentation

# 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

`$\begin{array}{l}\stackrel{˙}{x}=Ax+Bu\\ y=Cx+Du\end{array}$`

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

`$\begin{array}{l}x\left[n+1\right]=Ax\left[n\right]+Bu\left[n\right]\\ y\left[n\right]=Cx\left[n\right]+Du\left[n\right]\end{array}$`

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:

 `A,B,C,D,E` 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. `Scaled` 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`. Default: 0 (false) `StateName` State names, specified as one of the following: Character vector — For first-order models, for example, `'velocity'`.Cell array of character vectors — For models with two or more states`''` — For unnamed states. Default: `''` for all states `StateUnit` State units, specified as one of the following: Character vector — For first-order models, for example, `'velocity'`Cell array of character vectors — For models with two or more states`''` — For states without specified units Use `StateUnit` to keep track of the units each state is expressed in. `StateUnit` has no effect on system behavior. Default: `''` 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. 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 sample time `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 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 sample time `Ts`. For example, ```InputDelay = 3``` means a delay of three sample times. 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. Default: 0 `OutputDelay` 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 sample time `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. Default: 0 for all output channels `Ts` Sample 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 sample 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 sample time of a discrete-time system. Default: `0` (continuous time) `TimeUnit` Units for the time variable, the sample time `Ts`, and any time delays in the model, specified as one of the following values:`'nanoseconds'``'microseconds'``'milliseconds'``'seconds'` `'minutes'``'hours'``'days'``'weeks'``'months'``'years'` Changing this property has no effect on other properties, and therefore changes the overall system behavior. Use `chgTimeUnit` to convert between time units without modifying system behavior. Default: `'seconds'` `InputName` Input channel names, specified as one of the following: Character vector — For single-input models, for example, `'controls'`.Cell array of character vectors — For multi-input models. 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 plotsExtracting subsystems of MIMO systemsSpecifying connection points when interconnecting models Default: `''` for all input channels `InputUnit` Input channel units, specified as one of the following: Character vector — For single-input models, for example, `'seconds'`.Cell array of character vectors — For multi-input models. Use `InputUnit` to keep track of input signal units. `InputUnit` has no effect on system behavior. Default: `''` 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, specified as one of the following: Character vector — For single-output models. For example, `'measurements'`.Cell array of character vectors — For multi-output models. 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 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 plotsExtracting subsystems of MIMO systemsSpecifying connection points when interconnecting models Default: `''` for all output channels `OutputUnit` Output channel units, specified as one of the following: Character vector — For single-output models. For example, `'seconds'`.Cell array of character vectors — For multi-output models. Use `OutputUnit` to keep track of output signal units. `OutputUnit` has no effect on system behavior. Default: `''` for all output 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, specified as a character vector. For example, `'system_1'`. Default: `''` `Notes` Any text that you want to associate with the system, stored as a string or a cell array of character vectors. The property stores whichever data type you provide. For instance, if `sys1` and `sys2` are dynamic system models, you can set their `Notes` properties as follows: ```sys1.Notes = "sys1 has a string."; sys2.Notes = 'sys2 has a character vector.'; sys1.Notes sys2.Notes``` ```ans = "sys1 has a string." ans = 'sys2 has a character vector.' ``` Default: `[0×1 string]` `UserData` Any type of data you want to associate with system, specified as 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 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 ...``` For model arrays generated by linearizing a Simulink® model at multiple parameter values or operating points, the software populates `SamplingGrid` automatically with the variable values that correspond to each entry in the array. For example, the Simulink Control Design™ commands `linearize` and `slLinearizer` populate `SamplingGrid` in this way. Default: `[]`

## Examples

### Create Discrete-Time State-Space Model

Create a state-space model with a sample time of 0.25 seconds and the following state-space matrices:

`$A=\left[\begin{array}{cc}0& 1\\ -5& -2\end{array}\right]\phantom{\rule{1em}{0ex}}B=\left[\begin{array}{c}0\\ 3\end{array}\right]\phantom{\rule{1em}{0ex}}C=\left[\phantom{\rule{0.1em}{0ex}}\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{1em}{0ex}}D=\left[\phantom{\rule{0.1em}{0ex}}0\phantom{\rule{0.1em}{0ex}}\right]$`

Specify the state-space matrices.

```A = [0 1;-5 -2]; B = [0;3]; C = [0 1]; D = 0;```

Specify the sample time.

`Ts = 0.25;`

Create the state-space model.

`sys = ss(A,B,C,D,Ts);`

### Specify State and Input Names for Discrete-Time State-Space Model

Create state-space matrices and specify sample time.

```A = [0 1;-5 -2]; B = [0;3]; C = [0 1]; D = 0; Ts = 0.05;```

Create state-space model, specifying the state and input names.

```sys = ss(A,B,C,D,Ts,'StateName',{'Position' 'Velocity'},... 'InputName','Force');```

The number of state and input names must be consistent with the dimensions of `A`, `B`, `C`, and `D`.

### Convert Transfer Function to State-Space Model

Compute the state-space model of the following transfer function:

`$H\left(s\right)=\left[\begin{array}{c}\frac{s+1}{{s}^{3}+3{s}^{2}+3s+2}\\ \frac{{s}^{2}+3}{{s}^{2}+s+1}\end{array}\right]$`

Create the transfer function model.

`H = [tf([1 1],[1 3 3 2]) ; tf([1 0 3],[1 1 1])];`

Convert this model to a state-space model.

`sys = ss(H);`

Examine the size of the state-space model.

`size(sys)`
```State-space model with 2 outputs, 1 inputs, and 5 states. ```

The number of states is equal to the cumulative order of the SISO entries in H(s).

To obtain a minimal realization of H(s), enter

```sys = ss(H,'minimal'); size(sys)```
```State-space model with 2 outputs, 1 inputs, and 3 states. ```

The resulting model has an order of three, which is the minimum number of states needed to represent H(s). To see this number of states, refactor H(s) as the product of a first-order system and a second-order system.

`$H\left(s\right)=\left[\begin{array}{cc}\frac{1}{s+2}& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}\frac{s+1}{{s}^{2}+s+1}\\ \frac{{s}^{2}+3}{{s}^{2}+s+1}\end{array}\right]$`

### Explicit Realization of Descriptor State-Space Model

Create a descriptor state-space model (EI).

```a = [2 -4; 4 2]; b = [-1; 0.5]; c = [-0.5, -2]; d = [-1]; e = [1 0; -3 0.5]; sysd = dss(a,b,c,d,e);```

Compute an explicit realization of the system (E = I).

`syse = ss(sysd,'explicit')`
```syse = 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 state-space model. ```

Confirm that the descriptor and explicit realizations have equivalent dynamics.

`bodeplot(sysd,syse,'g--')`

### Create State-Space Model with Both Fixed and Tunable Parameters

This example shows how to create a state-space `genss` model having both fixed and tunable parameters.

`$A=\left[\begin{array}{cc}1& a+b\\ 0& ab\end{array}\right],\phantom{\rule{1em}{0ex}}B=\left[\begin{array}{c}-3.0\\ 1.5\end{array}\right],\phantom{\rule{1em}{0ex}}C=\left[\begin{array}{cc}0.3& 0\end{array}\right],\phantom{\rule{1em}{0ex}}D=0,$`

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 `[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 = Generalized continuous-time state-space model with 1 outputs, 1 inputs, 2 states, and the following blocks: a: Scalar parameter, 2 occurrences. b: Scalar parameter, 2 occurrences. Type "ss(sys)" to see the current value, "get(sys)" to see all properties, and "sys.Blocks" to interact with the blocks. ```

`sys` is a generalized LTI model (`genss`) with tunable parameters `a` and `b`. Confirm that the `A` property of `sys` is stored as a generalized matrix.

`sys.A`
```ans = Generalized matrix with 2 rows, 2 columns, and the following blocks: a: Scalar parameter, 2 occurrences. b: Scalar parameter, 2 occurrences. Type "double(ans)" to see the current value, "get(ans)" to see all properties, and "ans.Blocks" to interact with the blocks. ```

### Extract Components from Identified State-Space Model

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 `ss(sys)` to extract the measured component.

## Algorithms

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

### Topics

#### Learn how to automatically tune PID controller gains

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