This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# StateSpaceModel

 StateSpaceModel represents the state-space model with state matrix a, input matrix b, output matrix c, and transmission matrix d. StateSpaceModel[tf] gives a state-space realization of a proper TransferFunctionModel object tf. StateSpaceModelgives the state-space model obtained by Taylor linearization of , or , about the point (, ). StateSpaceModelgives the state-space model obtained by Taylor linearization about the point (, ) of the ordinary differential or difference equations eqns with output y and independent variable .
• A continuous-time system modeled by the equations , with states , control inputs , and outputs , can be specified as StateSpaceModel.
• The block diagram of the continuous-time system is given by:
• A discrete-time system modeled by the equations , with states , control inputs , outputs , and sampling period , can be specified as StateSpaceModel[{a, b, c, d}, SamplingPeriod->].
• The block diagram of the discrete-time system is given by:
• The matrices should have the following dimensions:
 b c d
• StateSpaceModel assumes c to be the identity matrix and d to be a zero matrix, or .
• The following options can be given:
 SamplingPeriod None the sampling period StateSpaceRealization Automatic the canonical realization SystemsModelLabels None the labels for the input, output, and state variables
 "Controllable" the controllable form "ControllableCompanion" the controllable companion form "Observable" the observable form "ObservableCompanion" the observable companion form
• By default, the controllable form of tf is computed.
• The controllable form is obtained by computing the controllable companion form of the subsystems corresponding to each input and combining the results.
• For the proper transfer-function matrix , the controllable companion form is
• The identity matrix has dimension .
• The observable form is obtained by computing the observable companion form of the subsystems corresponding to each output, and combining the results.
• For the proper transfer-function matrix , the observable companion form is
• Analogous results are obtained for discrete-time systems.
A state-space model of an integrator:
A second-order, single-input, single-output system:
The state-space model of a transfer-function object:
The state-space model of a system with sampling period :
The state-space model of a set of ODEs:
A state-space model of an integrator:
 Out[1]=

A second-order, single-input, single-output system:
 Out[1]=

The state-space model of a transfer-function object:
 Out[1]=

The state-space model of a system with sampling period :
 Out[1]=

The state-space model of a set of ODEs:
 Out[1]=
 Scope   (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:
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:
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   (7)
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:
The controllable companion form:
The observable companion form:
The Jordan form:
The realizations of a discrete-time model:
Label the inputs, outputs, and states:
 Applications   (3)
Linearize an inverted pendulum model:
State-space model of a typical mechanical mass-spring-damper system:
A typical resistance-inductance-capacitance (RLC) circuit:
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:
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:
New in 8