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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.
StateSpaceModel
gives the state-space model obtained by Taylor linearization of , or , about the point ().
StateSpaceModel
gives 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:
a
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:
SamplingPeriodNonethe sampling period
StateSpaceRealizationAutomaticthe canonical realization
SystemsModelLabelsNonethe 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:
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A second-order, single-input, single-output system:
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The state-space model of a transfer-function object:
In[1]:=
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The state-space model of a system with sampling period :
In[1]:=
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The state-space model of a set of ODEs:
In[1]:=
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Out[1]=
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:
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:
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:
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