# StateSpaceModel

StateSpaceModel[{a,b,c,d}]

represents the standard state-space model with state matrix a, input matrix b, output matrix c, and transmission matrix d.

StateSpaceModel[{a,b,c,d,e}]

represents a descriptor state-space model with descriptor matrix e.

StateSpaceModel[sys]

gives a state-space model corresponding to the systems model sys.

StateSpaceModel[eqns,{{x1,x10},},{{u1,u10},},{g1,},τ]

gives the state-space model obtained by Taylor linearization about the point (xi0,ui0) of the differential or difference equations eqns with outputs gi and independent variable τ.

# Details and Options  • StateSpaceModel can represent scalar and multivariate systems in continuous or discrete time.
• Time delays can be represented in any state-space model, by using SystemsModelDelay in any of the matrices.
• A continuous-time system modeled by the equations with states , control inputs , and outputs can be specified as StateSpaceModel[{a,b,c,d}].
• A discrete-time system modeled by the equations with states , control inputs , outputs , and sampling period τ can be specified as StateSpaceModel[{a,b,c,d},SamplingPeriod->τ].
• Continuous-time and discrete-time descriptor state-space systems can be specified as follows:
•  StateSpaceModel[{a,b,c,d,e}] StateSpaceModel[{a,b,c,d,e},SamplingPeriod->τ] • For a system with n states, p inputs, and q outputs, the matrices a, b, c, d and e should have dimensions {n,n}, {n,p}, {q,n}, {q,p}, and {n,n}.
• The following short inputs can be used:
•  StateSpaceModel[{a,b,c}] StateSpaceModel[{a,b}] StateSpaceModel[{a,b,c,Automatic,e}] StateSpaceModel[{a,b,Automatic,Automatic,e}] • In StateSpaceModel[sys] the following systems can be converted:
•  AffineStateSpaceModel approximate Taylor conversion NonlinearStateSpaceModel approximate Taylor conversion TransferFunctionModel exact conversion
• When converting from transfer-function model sys, the controllable realization is used.
• For equational input, default linearization points xi0 and uj0 are taken to be zero.
• The following options can be given:
•  SamplingPeriod Automatic the sampling period StateSpaceRealization Automatic the canonical realization DescriptorStateSpace Automatic standard or descriptor realization SystemsModelLabels Automatic the labels for the input, output, and state variables

# Examples

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## Basic Examples(5)

A state-space model of an integrator:

A secondorder single-input, single-output system:

The state-space model of a transfer-function object:

The state-space model of a system with sampling period τ:

The state-space model of a set of ODEs:

## Scope(31)

### Basic Uses(17)

A second-order system:

A fourth-order system:

A system with two inputs:

A system with two outputs:

Direct feedthrough is assumed to be zero:

Specify feedthrough:

The feedthrough is the sum of the inputs:

A discrete-time model:

A symbolic model:

The state-space model of a transfer function:

Perform symbolic conversions:

Taylor linearize an AffineStateSpaceModel:

The linearization of an AffineStateSpaceModel with nonzero equilibrium values:

Taylor linearize a NonlinearStateSpaceModel:

Linearize a nonlinear state-space model:

The linear state-space model of an ODE:

An ODE with a derivative control term:

Use Normal to obtain the matrices:

### Descriptor Systems(8)

A descriptor system:

A singular descriptor system:

Use Automatic to create a descriptor system with default outputs:

Systems can include both differential and algebraic equations:

The resulting model:

The model with the equations intact:

They are identical after pole-zero cancellations.

A discrete-time descriptor system from difference equations:

A zero descriptor matrix gives an algebraic system:

For descriptor systems, Normal returns all five matrices:

Invert the descriptor matrix to create a standard state-space model:

### Time-Delay Systems(5)

An output-delay system:

A system with two input delays:

A discrete-time system with a delay in the state matrix:

Create a time delay system directly from delay-differential equations:

Delays in the differential terms create neutral time-delay systems:

## Generalizations & Extensions(2)

If the transmission matrix is not specified, the model is assumed to have zero feedthrough:

If the outputs are not specified, they are assumed to be the states:

## Options(8)

### SamplingPeriod(4)

A continuous-time model:

A discrete-time model with sampling period 2:

SamplingPeriod is None for continuous-time systems:

A symbolic sampling period:

Specify a numerical value:

### StateSpaceRealization(3)

The controllable companion form:

The observable companion form:

The Jordan form:

The realizations of a discrete-time model:

### SystemsModelLabels(1)

Label the inputs, outputs, and states:

## Applications(7)

### Chemical Systems(1)

Equations governing the concentration in a two-stage chemical reactor with constant flow rate:

A state-space model for the reactor:

Control the inputs with unity feedback:

Substitute numeric values and simulate the response to a disturbance:

### Electrical Systems(2)

A series resistance-inductance-capacitance (RLC) circuit:

The same RLC circuit modeled as a descriptor state space:

Both models are equivalent:

A circuit with two loops modeled with Kirchhoff's laws: The state-space model with the algebraic constraints:

A standard state-space representation:

### Mechanical Systems(4)

Linearize an inverted pendulum model:

State-space model of a typical mechanical mass-spring-damper system:

A two-stage mass-spring-damper system with delayed feedback:

The cutting force required in a lathe depends on the cutting depth from the previous rotation:

## Properties & Relations(14)

The state-space representation of a system is not unique:

Similar state-space models have identical transfer functions:

The controllable and observable companion forms are duals of each other:

Compute their dual representations:

The eigenvalues of the state matrix are invariant:

The state matrix satisfies its characteristic equation (CayleyHamilton theorem):

A controllable system:

An uncontrollable system:

A controllable system with non-distinct eigenvalues:

An uncontrollable system with non-distinct eigenvalues:

An observable system:

An unobservable system:

An observable system with non-distinct eigenvalues:

An unobservable system with non-distinct eigenvalues:

Obtain the transfer function representation:

The state-space model of an improper transfer function is singular: