State-space model

Use `ss`

to create real-valued or complex-valued state-space
models, or to convert dynamic system models to
state-space model form. You can also use `ss`

to create generalized
state-space (`genss`

) models or uncertain state-space
(`uss`

) models.

A state-space model is a mathematical representation of a physical system as a set of
input, output, and state variables related by first-order differential equations. The state
variables define the values of the output variables. The `ss`

model object
can represent SISO or MIMO state-space models in continuous time or discrete time.

In continuous-time, a state-space model is of the following form:

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

Here, `x`

, `u`

and `y`

represent the states inputs and outputs respectively, while `A`

,
`B`

, `C`

and `D`

are the state-space
matrices. The `ss`

object represents a state-space model in MATLAB^{®} storing `A`

, `B`

, `C`

and
`D`

along with other information such as sample time, names and delays
specific to the inputs and outputs.

You can create a state-space model object by either specifying the state, input and output
matrices directly, or by converting a model of another type (such as a transfer function model
`tf`

) to state-space form. For more information, see State-Space Models. You
can use an `ss`

model object to:

Perform linear analysis

Represent a linear time-invariant (LTI) model to perform control design

Combine with other LTI models to represent a more complex system

creates a continuous-time state-space model object of the following form:`sys`

= ss(`A`

,`B`

,`C`

,`D`

)

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

For instance, consider a plant with `Nx`

states,
`Ny`

outputs, and `Nu`

inputs. The state-space
matrices are:

`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.

returns the minimal state-space realization with no uncontrollable or unobservable
states. This realization is equivalent to `sys`

= ss(`ssSys`

,'minimal')`minreal(ss(sys))`

where
matrix `A`

has the smallest possible dimension.

Conversion to state-space form is not uniquely defined in the SISO case. It is also not guaranteed to produce a minimal realization in the MIMO case. For more information, see Recommended Working Representation.

returns an explicit state-space realization `sys`

= ss(`ssSys`

,'explicit')*(E = I)* of the dynamic
system state-space model `SSsys`

. `ss`

returns an
error if `ssSys`

is improper. For more information on explicit
state-space realization, see State-Space Models.

`A`

— State matrix`Nx`

-by-`Nx`

matrixState matrix, specified as an `Nx`

-by-`Nx`

matrix where, `Nx`

is the number of states. This input sets the value
of property A.

`B`

— Input-to-state matrix`Nx`

-by-`Nu`

matrixInput-to-state matrix, specified as an
`Nx`

-by-`Nu`

matrix where, `Nx`

is the number of states and `Nu`

is the number of inputs. This input
sets the value of property B.

`C`

— State-to-output matrix`Ny`

-by-`Nx`

matrixState-to-output matrix, specified as an
`Ny`

-by-`Nx`

matrix where, `Nx`

is the number of states and `Ny`

is the number of outputs. This input
sets the value of property C.

`D`

— Feedthrough matrix`Ny`

-by-`Nu`

matrixFeedthrough matrix, specified as an `Ny`

-by-`Nu`

matrix where, `Ny`

is the number of outputs and `Nu`

is the number of inputs. This input sets the value of property D.

`ts`

— Sample timescalar

Sample time, specified as a scalar. For more information, see Ts property.

`ltiSys`

— Dynamic system to convert to state-space formdynamic system model | model array

Dynamic system to convert to state-space form, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems that you can convert include:

Continuous-time or discrete-time numeric LTI models, such as

`tf`

,`zpk`

,`ss`

, or`pid`

models.Generalized or uncertain LTI models such as

`genss`

or`uss`

models. (Using uncertain models requires Robust Control Toolbox™ software.)The resulting state-space model assumes

current values of the tunable components for tunable control design blocks.

nominal model values for uncertain control design blocks.

Identified LTI models, such as

`idtf`

,`idss`

,`idproc`

,`idpoly`

, and`idgrey`

models. To select the component of the identified model to convert, specify`component`

. If you do not specify`component`

,`ss`

converts the measured component of the identified model by default. (Using identified models requires System Identification Toolbox™ software.)

`component`

— Component of identified model`'measured'`

(default) | `'noise'`

| `'augmented'`

Component of identified model to convert, specified as one of the following:

`'measured'`

— Convert the measured component of`sys`

.`'noise'`

— Convert the noise component of`sys`

`'augmented'`

— Convert both the measured and noise components of`sys`

.

`component`

only applies when `sys`

is an
identified LTI model.

For more information on identified LTI models and their measured and noise components, see Identified LTI Models.

`ssSys`

— Dynamic system model to convert to minimal realization or explicit form`ss`

model objectDynamic system model to convert to minimal realization or explicit form, specified
as an `ss`

model object.

`sys`

— Output system model`ss`

model object | `genss`

model object | `uss`

model objectOutput system model, returned as:

A state-space (

`ss`

) model object, when the inputs`A`

,`B`

,`C`

and`D`

are numeric matrices or when converting from another model object type.A generalized state-space model (

`genss`

) object, when one or more of the matrices`A`

,`B`

,`C`

and`D`

includes tunable parameters, such as`realp`

parameters or generalized matrices (`genmat`

). For an example, see Create State-Space Model with Both Fixed and Tunable Parameters.An uncertain state-space model (

`uss`

) object, when one or more of the inputs`A`

,`B`

,`C`

and`D`

includes uncertain matrices. Using uncertain models requires Robust Control Toolbox software.

`A`

— State matrix`Nx`

-by-`Nx`

matrixState matrix, specified as an `Nx`

-by-`Nx`

matrix
where `Nx`

is the number of states. The state-matrix can be represented
in many ways depending on the desired state-space model realization such as:

Model Canonical Form

Companion Canonical Form

Observable Canonical Form

Controllable Canonical Form

For more information, see Canonical State-Space Realizations.

`B`

— Input-to-state matrix`Nx`

-by-`Nu`

matrixInput-to-state matrix, specified as an
`Nx`

-by-`Nu`

matrix where `Nx`

is
the number of states and `Nu`

is the number of inputs.

`C`

— State-to-output matrix`Ny`

-by-`Nx`

matrixState-to-output matrix, specified as an
`Ny`

-by-`Nx`

matrix where `Nx`

is
the number of states and `Ny`

is the number of outputs.

`D`

— Feedthrough matrix`Ny`

-by-`Nu`

matrixFeedthrough matrix, specified as an `Ny`

-by-`Nu`

matrix where `Ny`

is the number of outputs and `Nu`

is
the number of inputs. `D`

is also called as the static gain matrix
which represents the ratio of the output to the input under steady state
condition.

`E`

— Matrix for implicit state-space models[] (default) |

`Nx`

-by-`Nx`

matrixMatrix for implicit or descriptor state-space models, specified as a
`Nx`

-by-`Nx`

matrix. `E`

is empty
by default, 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`0`

(default) | `1`

Logical value indicating whether scaling is enabled or disabled, specified as either
`0`

or `1`

.

When `Scaled`

is set to `0`

(disabled), then most
numerical algorithms acting on the state-space model `sys`

automatically rescale the state vector to improve numerical accuracy. You can prevent
such auto-scaling by setting `Scaled`

to `1`

(enabled).

For more information about scaling, see `prescale`

.

`StateName`

— State names`' '`

(default) | character vector | cell array of character vectorsState 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

`StateName`

is empty `' '`

for all states by
default.

`StateUnit`

— State units`' '`

(default) | character vector | cell array of character vectorsState units, specified 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

Use `StateUnit`

to keep track of the units of each state.
`StateUnit`

has no effect on system behavior.
`StateUnit`

is empty `' '`

for all states by
default.

`InternalDelay`

— Internal delays in the modelvector

Internal delays in the model, specified as a 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.

You can modify the values of internal delays using the property
`InternalDelay`

. However, the number of entries in
`sys.InternalDelay`

cannot change, because it is a structural
property of the model.

`InputDelay`

— Input delay`0`

(default) | scalar | `Nu`

-by-1 vectorInput delay for each input channel, specified as one of the following:

Scalar — Specify the input delay for a SISO system or the same delay for all inputs of a multi-input system.

`Nu`

-by-1 vector — Specify separate input delays for input of a multi-input system, where`Nu`

is the number of inputs.

For continuous-time systems, specify input delays in the time unit specified by the `TimeUnit`

property. For discrete-time systems, specify input delays in integer multiples of the sample time, `Ts`

.

For more information, see Time Delays in Linear Systems.

`OutputDelay`

— Output delay`0`

(default) | scalar | `Ny`

-by-1 vectorOutput delay for each output channel, specified as one of the following:

Scalar — Specify the output delay for a SISO system or the same delay for all outputs of a multi-output system.

`Ny`

-by-1 vector — Specify separate output delays for output of a multi-output system, where`Ny`

is the number of outputs.

For continuous-time systems, specify output delays in the time unit specified by the `TimeUnit`

property. For discrete-time systems, specify output delays in integer multiples of the sample time, `Ts`

.

For more information, see Time Delays in Linear Systems.

`Ts`

— Sample time`0`

(default) | positive scalar | `-1`

Sample time, specified as:

`0`

for continuous-time systems.A positive scalar representing the sampling period of a discrete-time system. Specify

`Ts`

in the time unit specified by the`TimeUnit`

property.`-1`

for a discrete-time system with an unspecified sample time.

Changing `Ts`

does not discretize or resample the model. To convert between continuous-time and discrete-time representations, use `c2d`

and `d2c`

. To change the sample time of a discrete-time system, use `d2d`

.

`TimeUnit`

— Time variable units`'seconds'`

(default) | `'nanoseconds'`

| `'microseconds'`

| `'milliseconds'`

| `'minutes'`

| `'hours'`

| `'days'`

| `'weeks'`

| `'months'`

| `'years'`

| ...Time variable units, specified as one of the following:

`'nanoseconds'`

`'microseconds'`

`'milliseconds'`

`'seconds'`

`'minutes'`

`'hours'`

`'days'`

`'weeks'`

`'months'`

`'years'`

Changing `TimeUnit`

has no effect on other properties, but changes the overall system behavior. Use `chgTimeUnit`

to convert between time units without modifying system behavior.

`InputName`

— Input channel names`''`

(default) | character vector | cell array of character vectorsInput channel names, specified as one of the following:

A character vector, for single-input models.

A cell array of character vectors, for multi-input models.

`''`

, no names specified for any input channels.

Alternatively, you can assign input names for multi-input models using automatic vector expansion. 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`

.

Use `InputName`

to:

Identify channels on model display and plots.

Extract subsystems of MIMO systems.

Specify connection points when interconnecting models.

`InputUnit`

— Input channel units`''`

(default) | character vector | cell array of character vectorsInput channel units, specified as one of the following:

A character vector, for single-input models.

A cell array of character vectors, for multi-input models.

`''`

, no units specified for any input channels.

Use `InputUnit`

to specify input signal units. `InputUnit`

has no effect on system behavior.

`InputGroup`

— Input channel groupsstructure

Input channel groups, specified as a structure. Use `InputGroup`

to assign the input channels of MIMO systems into groups and refer to each group by name. The field names of `InputGroup`

are the group names and the field values are the input channels of 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`

and `2`

, and `3`

and `5`

, respectively. You can then extract the subsystem from the `controls`

inputs to all outputs using:

`sys(:,'controls')`

By default, `InputGroup`

is a structure with no fields.

`OutputName`

— Output channel names`''`

(default) | character vector | cell array of character vectorsOutput channel names, specified as one of the following:

A character vector, for single-output models.

A cell array of character vectors, for multi-output models.

`''`

, no names specified for any output channels.

Alternatively, you can assign output names for multi-output models using automatic vector expansion. 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 also use the shorthand notation `y`

to refer to the `OutputName`

property. For example, `sys.y`

is equivalent to `sys.OutputName`

.

Use `OutputName`

to:

Identify channels on model display and plots.

Extract subsystems of MIMO systems.

Specify connection points when interconnecting models.

`OutputUnit`

— Output channel units`''`

(default) | character vector | cell array of character vectorsOutput channel units, specified as one of the following:

A character vector, for single-output models.

A cell array of character vectors, for multi-output models.

`''`

, no units specified for any output channels.

Use `OutputUnit`

to specify output signal units. `OutputUnit`

has no effect on system behavior.

`OutputGroup`

— Output channel groupsstructure

Output channel groups, specified as a structure. Use `OutputGroup`

to assign the output channels of MIMO systems into groups and refer to each group by name. The field names of `OutputGroup`

are the group names and the field values are the output channels of 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`

and `5`

, respectively. You can then extract the subsystem from all inputs to the `measurement`

outputs using:

`sys('measurement',:)`

By default, `OutputGroup`

is a structure with no fields.

`Name`

— System name`''`

(default) | character vectorSystem name, specified as a character vector. For example, `'system_1'`

.

`Notes`

— User-specified text`{}`

(default) | character vector | cell array of character vectorsUser-specified text that you want to associate with the system, specified as a character vector or cell array of character vectors. For example, `'System is MIMO'`

.

`UserData`

— User-specified data`[]`

(default) | any MATLAB data typeUser-specified data that you want to associate with the system, specified as any MATLAB data type.

`SamplingGrid`

— Sampling grid for model arraysstructure array

Sampling grid for model arrays, specified as a structure array.

Use `SamplingGrid`

to track the variable values associated with each model in a model array, including identified linear time-invariant (IDLTI) model arrays.

Set the field names of the 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 must be numeric scalars, and all arrays of sampled values must match the dimensions of the model array.

For example, you can create an 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, you can create a 6-by-9 model array, `M`

, by independently sampling two variables, `zeta`

and `w`

. The following code maps 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 instance, the Simulink
Control Design™ commands `linearize`

and `slLinearizer`

populate `SamplingGrid`

automatically.

By default, `SamplingGrid`

is a structure with no fields.

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

model objects. In general, any function applicable to Dynamic System Models is
applicable to an `ss`

object.

`step` | Step response plot of dynamic system; step response data |

`impulse` | Impulse response plot of dynamic system; impulse response data |

`lsim` | Simulate time response of dynamic system to arbitrary inputs |

`bode` | Bode plot of frequency response, or magnitude and phase data |

`nyquist` | Nyquist plot of frequency response |

`nichols` | Nichols chart of frequency response |

`bandwidth` | Frequency response bandwidth |

Create the SISO state-space model defined by the following state-space matrices:

$$A=\left[\begin{array}{cc}-1.5& -2\\ 1& 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}B=\left[\begin{array}{c}0.5\\ 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}C=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}D=0$$

Specify the A, B, C and D matrices, and create the state-space model.

A = [-1.5,-2;1,0]; B = [0.5;0]; C = [0,1]; D = 0; sys = ss(A,B,C,D)

sys = A = x1 x2 x1 -1.5 -2 x2 1 0 B = u1 x1 0.5 x2 0 C = x1 x2 y1 0 1 D = u1 y1 0 Continuous-time state-space model.

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

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

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);

For this example, consider a cube rotating about its corner with inertia tensor `J`

and a damping force `F`

of 0.2 magnitude. The input to the system is the driving torque while the angular velocities are the outputs. The state-space matrices for the cube are:

$$\begin{array}{l}A=-{J}^{-1}F,\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}B={J}^{-1},\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}C=I,\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}D=0,\\ where,\phantom{\rule{0.2777777777777778em}{0ex}}J=\left[\begin{array}{ccc}8& -3& -3\\ -3& 8& -3\\ -3& -3& 8\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}and\phantom{\rule{0.2777777777777778em}{0ex}}F=\left[\begin{array}{ccc}0.2& 0& 0\\ 0& 0.2& 0\\ 0& 0& 0.2\end{array}\right]\end{array}$$

Specify the `A`

, `B`

, `C`

and `D`

matrices, and create the continuous-time state-space model.

J = [8 -3 -3; -3 8 -3; -3 -3 8]; F = 0.2*eye(3); A = -J\F; B = inv(J); C = eye(3); D = 0; sys = ss(A,B,C,D)

sys = A = x1 x2 x3 x1 -0.04545 -0.02727 -0.02727 x2 -0.02727 -0.04545 -0.02727 x3 -0.02727 -0.02727 -0.04545 B = u1 u2 u3 x1 0.2273 0.1364 0.1364 x2 0.1364 0.2273 0.1364 x3 0.1364 0.1364 0.2273 C = x1 x2 x3 y1 1 0 0 y2 0 1 0 y3 0 0 1 D = u1 u2 u3 y1 0 0 0 y2 0 0 0 y3 0 0 0 Continuous-time state-space model.

`sys`

is MIMO since the system contains 3 inputs and 3 outputs observed from matrices `C`

and `D`

. For more information on MIMO state-space models, see MIMO State-Space Models.

Create a state-space model using the following discrete-time, multi-input, multi-output state matrices with sample time `ts = 0.2`

seconds:

$$A=\left[\begin{array}{cc}-7& 0\\ 0& -10\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}B=\left[\begin{array}{cc}5& 0\\ 0& 2\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}C=\left[\begin{array}{cc}1& -4\\ -4& 0.5\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}D=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]$$

Specify the state-space matrices and create the discrete-time MIMO state-space model.

A = [-7,0;0,-10]; B = [5,0;0,2]; C = [1,-4;-4,0.5]; D = [0,-2;2,0]; ts = 0.2; sys = ss(A,B,C,D,ts)

sys = A = x1 x2 x1 -7 0 x2 0 -10 B = u1 u2 x1 5 0 x2 0 2 C = x1 x2 y1 1 -4 y2 -4 0.5 D = u1 u2 y1 0 -2 y2 2 0 Sample time: 0.2 seconds 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 the state-space model, specifying the state and input names using name-value pairs.

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`

.

Naming the inputs and outputs can be useful when dealing with response plots for MIMO systems.

step(sys)

Notice the input name `Force`

in the title of the step response plot.

For this example, create a state-space model with the same time and input unit properties inherited from another state-space model. Consider the following state-space models:

$$\begin{array}{l}{A}_{1}=\left[\begin{array}{cc}-1.5& -2\\ 1& 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{B}_{1}=\left[\begin{array}{c}0.5\\ 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{C}_{1}=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{D}_{1}=5\\ {A}_{2}=\left[\begin{array}{cc}7& -1\\ 0& 2\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{B}_{2}=\left[\begin{array}{c}0.85\\ 2\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{C}_{2}=\left[\begin{array}{cc}10& 14\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{D}_{2}=2\end{array}$$

First, create a state-space model `sys1`

with the `TimeUnit`

and `InputUnit`

property set to '`minutes`

'.

A1 = [-1.5,-2;1,0]; B1 = [0.5;0]; C1 = [0,1]; D1 = 5; sys1 = ss(A1,B1,C1,D1,'TimeUnit','minutes','InputUnit','minutes');

Verify that the time and input unit properties of `sys1`

are set to '`minutes`

'.

propValues1 = [sys1.TimeUnit,sys1.InputUnit]

`propValues1 = `*1x2 cell*
{'minutes'} {'minutes'}

Create the second state-space model with properties inherited from `sys1`

.

A2 = [7,-1;0,2]; B2 = [0.85;2]; C2 = [10,14]; D2 = 2; sys2 = ss(A2,B2,C2,D2,sys1);

Verify that the time and input units of `sys2`

have been inherited from `sys1`

.

propValues2 = [sys2.TimeUnit,sys2.InputUnit]

`propValues2 = `*1x2 cell*
{'minutes'} {'minutes'}

In this example, you will create a static gain MIMO state-space model.

Consider the following two-input, two-output static gain matrix `m`

:

$$D=\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]$$

Specify the gain matrix and create the static gain state-space model.

D = [2,4;3,5]; sys1 = ss(D)

sys1 = D = u1 u2 y1 2 4 y2 3 5 Static gain.

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.

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

For this example, extract the measured and noise components of an identified polynomial model into two separate state-space models.

Load the Box-Jenkins polynomial model `ltiSys`

in `identifiedModel.mat`

.

load('identifiedModel.mat','ltiSys');

`ltiSys`

is an identified discrete-time model of the form: $$y(t)=\frac{B}{F}u(t)+\frac{C}{D}e(t)$$, where $$\frac{B}{F}$$ represents the measured component and $$\frac{C}{D}$$ the noise component.

Extract the measured and noise components as state-space models.

`sysMeas = ss(ltiSys,'measured') `

sysMeas = A = x1 x2 x1 1.575 -0.6115 x2 1 0 B = u1 x1 0.5 x2 0 C = x1 x2 y1 -0.2851 0.3916 D = u1 y1 0 Input delays (sampling periods): 2 Sample time: 0.04 seconds Discrete-time state-space model.

`sysNoise = ss(ltiSys,'noise')`

sysNoise = A = x1 x2 x3 x1 1.026 -0.26 0.3899 x2 1 0 0 x3 0 0.5 0 B = v@y1 x1 0.25 x2 0 x3 0 C = x1 x2 x3 y1 0.319 -0.04738 0.07106 D = v@y1 y1 0.04556 Input groups: Name Channels Noise 1 Sample time: 0.04 seconds Discrete-time state-space model.

The measured component can serve as a plant model, while the noise component can be used as a disturbance model for control system design.

Create a descriptor state-space model (*E* ≠ *I*).

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--')`

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`

.

For this example, consider a SISO state-space model defined by the following state-space matrices:

$$A=\left[\begin{array}{cc}-1.5& -2\\ 1& 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}B=\left[\begin{array}{c}0.5\\ 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}C=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}D=0$$

Considering an input delay of 0.5 seconds and an output delay of 2.5 seconds, create a state-space model object to represent the A, B, C and D matrices.

A = [-1.5,-2;1,0]; B = [0.5;0]; C = [0,1]; D = 0; sys = ss(A,B,C,D,'InputDelay',0.5,'OutputDelay',2.5)

sys = A = x1 x2 x1 -1.5 -2 x2 1 0 B = u1 x1 0.5 x2 0 C = x1 x2 y1 0 1 D = u1 y1 0 Input delays (seconds): 0.5 Output delays (seconds): 2.5 Continuous-time state-space model.

You can also use the `get`

command to display all the properties of a MATLAB object.

get(sys)

A: [2x2 double] B: [2x1 double] C: [0 1] D: 0 E: [] Scaled: 0 StateName: {2x1 cell} StateUnit: {2x1 cell} InternalDelay: [0x1 double] InputDelay: 0.5000 OutputDelay: 2.5000 Ts: 0 TimeUnit: 'seconds' InputName: {''} InputUnit: {''} InputGroup: [1x1 struct] OutputName: {''} OutputUnit: {''} OutputGroup: [1x1 struct] Notes: [0x1 string] UserData: [] Name: '' SamplingGrid: [1x1 struct]

For more information on specifying time delay for an LTI model, see Specifying Time Delays.

For this example, consider a state-space system object that represents the following state matrices:

$$A=\left[\begin{array}{ccc}-1.2& -1.6& 0\\ 1& 0& 0\\ 0& 1& 0\end{array}\right],\phantom{\rule{1em}{0ex}}B=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\phantom{\rule{1em}{0ex}}C=\left[\begin{array}{ccc}0& 0.5& 1.3\end{array}\right],\phantom{\rule{1em}{0ex}}D=0,$$

Create a state-space object `sys`

using the `ss`

command.

A = [-1.2,-1.6,0;1,0,0;0,1,0]; B = [1;0;0]; C = [0,0.5,1.3]; D = 0; sys = ss(A,B,C,D);

Next, compute the closed-loop state-space model for a unit negative gain and find the poles of the closed-loop state-space system object `sysFeedback`

.

sysFeedback = feedback(sys,1); P = pole(sysFeedback)

`P = `*3×1 complex*
-0.2305 + 1.3062i
-0.2305 - 1.3062i
-0.7389 + 0.0000i

The feedback loop for unit gain is stable since all poles have negative real parts. Checking the closed-loop poles provides a binary assessment of stability. In practice, it is more useful to know how robust (or fragile) stability is. One indication of robustness is how much the loop gain can change before stability is lost. You can use the root locus plot to estimate the range of `k`

values for which the loop is stable.

rlocus(sys)

Changes in the loop gain are only one aspect of robust stability. In general, imperfect plant modeling means that both gain and phase are not known exactly. Since modeling errors have the most detrimental effect near the gain crossover frequency (frequency where open-loop gain is 0dB), it also matters how much phase variation can be tolerated at this frequency.

You can display the gain and phase margins on a Bode plot as follows.

bode(sys) grid

For a more detailed example, see Assessing Gain and Phase Margins.

For this example, design a 2-DOF PID controller with a target bandwidth of 0.75 rad/s for a system represented by the following matrices:

$$A=\left[\begin{array}{cc}-0.5& -0.1\\ 1& 0\end{array}\right],\phantom{\rule{1em}{0ex}}B=\left[\begin{array}{c}1\\ 0\end{array}\right],\phantom{\rule{1em}{0ex}}C=\left[\begin{array}{cc}0& 1\end{array}\right],\phantom{\rule{1em}{0ex}}D=0.$$

Create a state-space object `sys`

using the `ss`

command.

A = [-0.5,-0.1;1,0]; B = [1;0]; C = [0,1]; D = 0; sys = ss(A,B,C,D)

sys = A = x1 x2 x1 -0.5 -0.1 x2 1 0 B = u1 x1 1 x2 0 C = x1 x2 y1 0 1 D = u1 y1 0 Continuous-time state-space model.

Using the target bandwidth, use `pidtune`

to generate a 2-DOF controller.

```
wc = 0.75;
C2 = pidtune(sys,'PID2',wc)
```

C2 = 1 u = Kp (b*r-y) + Ki --- (r-y) + Kd*s (c*r-y) s with Kp = 0.513, Ki = 0.0975, Kd = 0.577, b = 0.344, c = 0 Continuous-time 2-DOF PID controller in parallel form.

Using the type `'PID2'`

causes `pidtune`

to generate a 2-DOF controller, represented as a `pid2`

object. The display confirms this result. The display also shows that `pidtune`

tunes all controller coefficients, including the setpoint weights `b`

and `c`

, to balance performance and robustness.

For interactive PID tuning in the Live Editor, see the Tune PID Controller Live Editor task. This task lets you interactively design a PID controller and automatically generates MATLAB code for your live script.

For interactive PID tuning in a standalone app, use PID Tuner. See PID Controller Design for Fast Reference Tracking for an example of designing a controller using the app.

Consider a state-space plant `G`

with five inputs and four outputs and a state-space feedback controller `K`

with three inputs and two outputs. The outputs 1, 3, and 4 of the plant `G`

must be connected the controller `K`

inputs, and the controller outputs to inputs 4 and 2 of the plant.

For this example, consider two continuous-time state-space models for both `G`

and `K`

represented by the following set of matrices:

$${A}_{G}=\left[\begin{array}{ccc}-3& 0.4& 0.3\\ -0.5& -2.8& -0.8\\ 0.2& 0.8& -3\end{array}\right],\phantom{\rule{1em}{0ex}}{B}_{G}=\left[\begin{array}{ccccc}0.4& 0& 0.3& 0.2& 0\\ -0.2& -1& 0.1& -0.9& -0.5\\ 0.6& 0.9& 0.5& 0.2& 0\end{array}\right],\phantom{\rule{1em}{0ex}}{C}_{G}=\left[\begin{array}{ccc}0& -0.1& -1\\ 0& -0.2& 1.6\\ -0.7& 1.5& 1.2\\ -1.4& -0.2& 0\end{array}\right],\phantom{\rule{1em}{0ex}}{D}_{G}=\left[\begin{array}{ccccc}0& 0& 0& 0& -1\\ 0& 0.4& -0.7& 0& 0.9\\ 0& 0.3& 0& 0& 0\\ 0.2& 0& 0& 0& 0\end{array}\right]$$

$${A}_{K}=\left[\begin{array}{ccc}-0.2& 2.1& 0.7\\ -2.2& -0.1& -2.2\\ -0.4& 2.3& -0.2\end{array}\right],\phantom{\rule{1em}{0ex}}{B}_{K}=\left[\begin{array}{ccc}-0.1& -2.1& -0.3\\ -0.1& 0& 0.6\\ 1& 0& 0.8\end{array}\right],\phantom{\rule{1em}{0ex}}{C}_{K}=\left[\begin{array}{ccc}-1& 0& 0\\ -0.4& -0.2& 0.3\end{array}\right],\phantom{\rule{1em}{0ex}}{D}_{K}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -1.2\end{array}\right]$$

AG = [-3,0.4,0.3;-0.5,-2.8,-0.8;0.2,0.8,-3]; BG = [0.4,0,0.3,0.2,0;-0.2,-1,0.1,-0.9,-0.5;0.6,0.9,0.5,0.2,0]; CG = [0,-0.1,-1;0,-0.2,1.6;-0.7,1.5,1.2;-1.4,-0.2,0]; DG = [0,0,0,0,-1;0,0.4,-0.7,0,0.9;0,0.3,0,0,0;0.2,0,0,0,0]; sysG = ss(AG,BG,CG,DG)

sysG = A = x1 x2 x3 x1 -3 0.4 0.3 x2 -0.5 -2.8 -0.8 x3 0.2 0.8 -3 B = u1 u2 u3 u4 u5 x1 0.4 0 0.3 0.2 0 x2 -0.2 -1 0.1 -0.9 -0.5 x3 0.6 0.9 0.5 0.2 0 C = x1 x2 x3 y1 0 -0.1 -1 y2 0 -0.2 1.6 y3 -0.7 1.5 1.2 y4 -1.4 -0.2 0 D = u1 u2 u3 u4 u5 y1 0 0 0 0 -1 y2 0 0.4 -0.7 0 0.9 y3 0 0.3 0 0 0 y4 0.2 0 0 0 0 Continuous-time state-space model.

AK = [-0.2,2.1,0.7;-2.2,-0.1,-2.2;-0.4,2.3,-0.2]; BK = [-0.1,-2.1,-0.3;-0.1,0,0.6;1,0,0.8]; CK = [-1,0,0;-0.4,-0.2,0.3]; DK = [0,0,0;0,0,-1.2]; sysK = ss(AK,BK,CK,DK)

sysK = A = x1 x2 x3 x1 -0.2 2.1 0.7 x2 -2.2 -0.1 -2.2 x3 -0.4 2.3 -0.2 B = u1 u2 u3 x1 -0.1 -2.1 -0.3 x2 -0.1 0 0.6 x3 1 0 0.8 C = x1 x2 x3 y1 -1 0 0 y2 -0.4 -0.2 0.3 D = u1 u2 u3 y1 0 0 0 y2 0 0 -1.2 Continuous-time state-space model.

Define the `feedout`

and `feedin`

vectors based on the inputs and outputs to be connected in a feedback loop.

feedin = [4 2]; feedout = [1 3 4]; sys = feedback(sysG,sysK,feedin,feedout,-1)

sys = A = x1 x2 x3 x4 x5 x6 x1 -3 0.4 0.3 0.2 0 0 x2 1.18 -2.56 -0.8 -1.3 -0.2 0.3 x3 -1.312 0.584 -3 0.56 0.18 -0.27 x4 2.948 -2.929 -2.42 -0.452 1.974 0.889 x5 -0.84 -0.11 0.1 -2.2 -0.1 -2.2 x6 -1.12 -0.26 -1 -0.4 2.3 -0.2 B = u1 u2 u3 u4 u5 x1 0.4 0 0.3 0.2 0 x2 -0.44 -1 0.1 -0.9 -0.5 x3 0.816 0.9 0.5 0.2 0 x4 -0.2112 -0.63 0 0 0.1 x5 0.12 0 0 0 0.1 x6 0.16 0 0 0 -1 C = x1 x2 x3 x4 x5 x6 y1 0 -0.1 -1 0 0 0 y2 -0.672 -0.296 1.6 0.16 0.08 -0.12 y3 -1.204 1.428 1.2 0.12 0.06 -0.09 y4 -1.4 -0.2 0 0 0 0 D = u1 u2 u3 u4 u5 y1 0 0 0 0 -1 y2 0.096 0.4 -0.7 0 0.9 y3 0.072 0.3 0 0 0 y4 0.2 0 0 0 0 Continuous-time state-space model.

size(sys)

State-space model with 4 outputs, 5 inputs, and 6 states.

`sys`

is the resultant closed loop state-space model obtained by connecting the specified inputs and outputs of `G`

and `K`

.

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