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
open allclose allBasic Examples (5)
Scope (31)
Basic Uses (17)
Direct feedthrough is assumed to be zero:
The feedthrough is the sum of the inputs:
The state-space model of a transfer function:
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)
Use Automatic to create a descriptor system with default outputs:
Systems can include both differential and algebraic equations:
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:
Generalizations & Extensions (2)
Options (8)
SamplingPeriod (4)
A discrete-time model with sampling period 2:
SamplingPeriod is None for continuous-time systems:
StateSpaceRealization (3)
Applications (7)
Chemical Systems (1)
Electrical Systems (2)
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 (Cayley–Hamilton theorem):
A controllable system with non-distinct eigenvalues:
An uncontrollable system with non-distinct eigenvalues:
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:
Text
Wolfram Research (2010), StateSpaceModel, Wolfram Language function, https://reference.wolfram.com/language/ref/StateSpaceModel.html (updated 2014).
CMS
Wolfram Language. 2010. "StateSpaceModel." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/StateSpaceModel.html.
APA
Wolfram Language. (2010). StateSpaceModel. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StateSpaceModel.html