# genss

Generalized state-space model

## Description

Generalized state-space (`genss`

) models are state-space models
that include tunable parameters or components. `genss`

models arise when
you combine numeric LTI models with models containing tunable components (Control Design
Blocks). For more information about numeric LTI models and Control Design Blocks, see
Models with Tunable Coefficients.

You can use generalized state-space models to represent control systems having a
mixture of fixed and tunable components. Use generalized state-space models for control
design tasks such as parameter studies and parameter tuning with commands such as
`systune`

and `looptune`

.

## Creation

To construct a `genss`

model:

Use

`series`

,`parallel`

,`lft`

, or`connect`

, or the arithmetic operators`+`

,`-`

,`*`

,`/`

,`\`

, and`^`

, to combine numeric LTI models with Control Design Blocks.Use

`tf`

or`ss`

with one or more input arguments that is a tunable parameter (`realp`

) or generalized matrix (`genmat`

) instead of a numeric value or array.Use the

`genss`

command to convert any numeric LTI model or control design block. For example, the following code converts`sys`

to a`genss`

model`gensys`

.gensys = genss(sys)

Converting

`frd`

and`genfrd`

models using`genss`

is not supported.Use commands like

`getIOTransfer`

(Simulink Control Design) or`getLoopTransfer`

(Simulink Control Design) to extract a`genss`

model from an`slTuner`

(Simulink Control Design) interface. The extracted`genss`

model contains all the tunable blocks and analysis points specified in the interface.

## Properties

`Blocks`

— Control design blocks

structure

Control design blocks included in the generalized LTI model or generalized matrix,
specified as a structure. The field names of `Blocks`

are the
`Name`

property of each control design block.

You can change some attributes of these control design blocks using dot notation. For
example, if the generalized LTI model or generalized matrix `M`

contains a `realp`

tunable parameter `a`

, change the
current value of
`a`

.

M.Blocks.a.Value = -1;

`A,B,C,D`

— Dependency of state-space matrices on tunable and uncertain parameters

`genmat`

object | `umat`

object | double array

`genmat`

Dependency of state-space matrices on tunable and uncertain parameters,
stored as a generalized matrix (`genmat`

), uncertain
matrix (`umat`

), or double array.

These properties model the dependency of the state-space matrices on
static Control Design Blocks, `realp`

,
`ureal`

, `ucomplex`

, or
`ucomplexm`

. Dynamic Control Design Blocks such as
`tunableGain`

or `tunableSS`

set
to their current values, and internal delays are set to zero.

When the corresponding state-space matrix does not depend on any static Control Design Blocks, these properties evaluate to double matrices.

For an example, see Dependence of State-Space Matrices on Parameters.

`E`

— E matrix

double matrix

E matrix, stored as a double matrix when the generalized state-space
equations are implicit. The value `E = []`

means that the
generalized state-space equations are explicit. For more information about
implicit state-space models, see State-Space Models.

`StateName`

— State names

`''`

for each state (default) | character vector | cell array of character vectors

State names, stored 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 example,

`{'position';'velocity'}`

.`''`

— For unnamed states.

You can assign state names to a `genss`

model only when all
its Control Design Blocks are static. Otherwise, specify the state names for
the component models before interconnecting them to create the
`genss`

model. When you do so, the `genss`

model tracks the assigned state names. For an example, see Track State Names in Generalized State-Space Model.

`StateUnit`

— State unit labels

character vector | cell array of character vectors | `''`

State unit labels, stored as one of the following:

Character vector — For first-order models, for example,

`'m/s'`

.Cell array of character vectors — For models with two or more states, for example,

`{'m';'m/s'}`

.`''`

— For unnamed states.

`StateUnit`

labels the units of each state for
convenience, and has no effect on system behavior.

You can assign state units to a `genss`

model only when all
its Control Design Blocks are static. Otherwise, specify the state units for
the component models before interconnecting them to create the
`genss`

model. When you do so, the `genss`

model tracks the assigned state units. For an example, see Track State Names in Generalized State-Space Model.

`InternalDelay`

— Internal delays

scalar | vector

Internal delays, specified as a scalar or vector. 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.

`InputDelay`

— Input delays

`0`

for all output channels | scalar | vector

Input delays, specified as a scalar or vector with length equal to the number of inputs.

For a system with *N _{u}* inputs, set

`InputDelay`

to an
*N*-by-1 vector, where each entry is a numerical value representing the input delay for the corresponding input channel. Specify

_{u}`InputDelay`

as a scalar value to apply the same delay to all
input channels.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 sampling periods.

`OutputDelay`

— Output delays

`0`

for all output channels | scalar | vector

Output delays, specified as a scalar or vector with length equal to the number of outputs.

For a system with *N _{y}* outputs, set

`OutputDelay`

to an
*N*-by-1 vector, where each entry is a numerical value representing the output delay for the corresponding output channel. Specify

_{y}`OutputDelay`

as a scalar value to apply the same delay to
all output channels.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.

`Ts`

— Sample time

`0`

(continuous time) (default) | `-1`

(discrete time) | scalar

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.

`TimeUnit`

— Time variable units

`'seconds'`

(default)

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.

`InputName`

— Input channel names

character vector | cell array of character vectors

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 plots

Extracting subsystems of MIMO systems

Specifying connection points when interconnecting models

`InputUnit`

— Input channel units

character vector | cell array of character vectors

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.

`InputGroup`

— Input channel groups

structure

Input channel groups for assigning the input channels of MIMO systems, specified 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')`

`OutputName`

— Output channel names

character vector | cell array of character vectors

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 plots

Extracting subsystems of MIMO systems

Specifying connection points when interconnecting models

`OutputUnit`

— Output channel units

character vector | cell array of character vectors

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.

`OutputGroup`

— Output channel groups

`struct`

(default)

Output channel groups for assigning the output channels of MIMO systems, specified 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.OutputGroup.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',:)`

`Name`

— System name

character vector

System name, specified as a character vector. For example,
`'system_1'`

.

`Notes`

— Any text that you want to associate with the system

string | cell array of character vectors

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.'

`UserData`

— Any type of data you want to associate with system

any MATLAB^{®} data

Any type of data you want to associate with system, specified as any MATLAB data type.

`SamplingGrid`

— Sampling grid for model arrays

`[]`

(default) | structure

Sampling grid for model arrays, specified as a 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`

(Simulink Control Design) and `slLinearizer`

(Simulink Control Design) populate `SamplingGrid`

in this way.

## Object Functions

The following lists contain a representative subset of the functions you can use with
`genss`

models. In general, many functions applicable to numeric LTI
models are also applicable to `genss`

models.

### Extract Responses

`getIOTransfer` | Closed-loop transfer function from generalized model of control system |

`getLoopTransfer` | Open-loop transfer function of control system represented by
`genss` model |

`getSensitivity` | Sensitivity function from generalized model of control system |

`getCompSensitivity` | Complementary sensitivity function from generalized model of control system |

### Access Blocks and Values

`getValue` | Current value of generalized model |

`getBlockValue` | Get current value of Control Design Block in Generalized Model |

`setBlockValue` | Modify value of Control Design Block in Generalized Model |

### Linear Analysis

`bode` | Bode frequency response of dynamic system |

`sigma` | Singular values of frequency response of dynamic system |

`nyquist` | Nyquist response of dynamic system |

`step` | Step response of dynamic system |

`lsim` | Compute time response simulation data of dynamic system to arbitrary inputs |

`margin` | Gain margin, phase margin, and crossover frequencies |

### Model Interconnection

## Examples

### Tunable Low-Pass Filter

In this example, you will create a low-pass filter with one tunable parameter *a*:

$$F=\frac{a}{s+a}$$

Since the numerator and denominator coefficients of a `tunableTF`

block are independent, you cannot use `tunableTF`

to represent `F`

. Instead, construct `F`

using the tunable real parameter object `realp`

.

Create a real tunable parameter with an initial value of `10`

.

`a = realp('a',10)`

a = Name: 'a' Value: 10 Minimum: -Inf Maximum: Inf Free: 1 Real scalar parameter.

Use `tf`

to create the tunable low-pass filter `F`

.

numerator = a; denominator = [1,a]; F = tf(numerator,denominator)

Generalized continuous-time state-space model with 1 outputs, 1 inputs, 1 states, and the following blocks: a: Scalar parameter, 2 occurrences. Type "ss(F)" to see the current value and "F.Blocks" to interact with the blocks.

`F`

is a `genss`

object which has the tunable parameter `a`

in its `Blocks`

property. You can connect `F`

with other tunable or numeric models to create more complex control system models. For an example, see Control System with Tunable Components.

### 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)

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 and "sys.Blocks" to interact with the blocks.

`sys`

is a generalized LTI model (`genss`

) with tunable parameters `a`

and `b`

.

### Control System Model with Both Numeric and Tunable Components

This example shows how to create a tunable model of a control system that has both fixed plant and sensor dynamics and tunable control components.

Consider the control system of the following illustration.

Suppose that the plant response is $$G(s)=1/(s+1{)}^{2}$$, and that the model of the sensor dynamics is $$S(s)=5/(s+4)$$. The controller $$C$$ is a tunable PID controller, and the prefilter $$F=a/(s+a)$$ is a low-pass filter with one tunable parameter, *a*.

Create models representing the plant and sensor dynamics. Because the plant and sensor dynamics are fixed, represent them using numeric LTI models.

G = zpk([],[-1,-1],1); S = tf(5,[1 4]);

To model the tunable components, use Control Design Blocks. Create a tunable representation of the controller *C*.

C = tunablePID('C','PID');

`C`

is a `tunablePID`

object, which is a Control Design Block with a predefined proportional-integral-derivative (PID) structure.

Create a model of the filter $$F=a/(s+a)$$ with one tunable parameter.

```
a = realp('a',10);
F = tf(a,[1 a]);
```

`a`

is a `realp`

(real tunable parameter) object with initial value 10. Using `a`

as a coefficient in `tf`

creates the tunable `genss`

model object `F`

.

Interconnect the models to construct a model of the complete closed-loop response from *r* to *y*.

T = feedback(G*C,S)*F

Generalized continuous-time state-space model with 1 outputs, 1 inputs, 5 states, and the following blocks: C: Tunable PID controller, 1 occurrences. a: Scalar parameter, 2 occurrences. Type "ss(T)" to see the current value and "T.Blocks" to interact with the blocks.

`T`

is a `genss`

model object. In contrast to an aggregate model formed by connecting only numeric LTI models, `T`

keeps track of the tunable elements of the control system. The tunable elements are stored in the `Blocks`

property of the `genss`

model object. Examine the tunable elements of `T`

.

T.Blocks

`ans = `*struct with fields:*
C: [1x1 tunablePID]
a: [1x1 realp]

When you create a `genss`

model of a control system that has tunable components, you can use tuning commands such as `systune`

to tune the free parameters to meet design requirements you specify.

### Track State Names in Generalized State-Space Model

Create a `genss`

model with labeled state names. To do so, label the states of the component LTI models before connecting them. For instance, connect a two-state fixed-coefficient plant model and a one-state tunable controller.

A = [-1 -1; 1 0]; B = [1; 0]; C = [0 1]; D = 0; G = ss(A,B,C,D); G.StateName = {'Pstate1','Pstate2'}; C = tunableSS('C',1,1,1); L = G*C;

The `genss`

model `L`

preserves the state names of the components that created it. Because you did not assign state names to the tunable component `C`

, the software automatically does so. Examine the state names of `L`

to confirm them.

L.StateName

`ans = `*3x1 cell*
{'Pstate1'}
{'Pstate2'}
{'C.x1' }

The automatic assignment of state names to control design blocks allows you to trace which states in the generalized model are contributed by tunable components.

State names are also preserved when you convert a `genss`

model to a fixed-coefficient state-space model. To confirm, convert `L`

to `ss`

form.

Lfixed = ss(L); Lfixed.StateName

`ans = `*3x1 cell*
{'Pstate1'}
{'Pstate2'}
{'C.x1' }

State unit labels, stored in the `StateUnit`

property of the `genss`

model, behave similarly.

### Dependence of State-Space Matrices on Parameters

Create a generalized model with a tunable parameter, and examine the dependence of the `A`

matrix on that parameter. To do so, examine the `A`

property of the generalized model.

```
G = tf(1,[1 10]);
k = realp('k',1);
F = tf(k,[1 k]);
L1 = G*F;
L1.A
```

Generalized matrix with 2 rows, 2 columns, and the following blocks: k: Scalar parameter, 2 occurrences. Type "double(ans)" to see the current value and "ans.Blocks" to interact with the blocks.

The `A`

property is a generalized matrix that preserves the dependence on the real tunable parameter `k`

. The state-space matrix properties `A`

, `B`

, `C`

, and `D`

only preserve dependencies on static parameters. When the `genss`

model has dynamic control design blocks, these are set to their current value for evaluating the state-space matrix properties. For example, examine the `A`

matrix property of a `genss`

model with a tunable PI block.

C = tunablePID('C','PI'); L2 = G*C; L2.A

`ans = `*2×2*
-10.0000 0.0010
0 0

Here, the `A`

matrix is stored as a double matrix, whose value is the `A`

matrix of the current value of `L2`

.

L2cur = ss(L2); L2cur.A

`ans = `*2×2*
-10.0000 0.0010
0 0

Additionally, extracting state-space matrices using `ssdata`

sets all control design blocks to their current or nominal values, including static blocks. Thus, the following operations all return the current value of the `A`

matrix of `L1`

.

[A,B,C,D] = ssdata(L1); A

`A = `*2×2*
-10 1
0 -1

double(L1.A)

`ans = `*2×2*
-10 1
0 -1

L1cur = ss(L1); L1cur.A

`ans = `*2×2*
-10 1
0 -1

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