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

Examples

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Basic Examples  (4)

Construct a state-space model from state, input, output and transmission matrices:

Its order:

Construct a state-space model from a transfer function model:

Test its controllability and observability:

Construct a state-space model from a set of ordinary differential equations (ODEs):

Its response to nonzero initial conditions:

Construct a discrete-time state-space model by specifying its sampling period:

Its step response:

Scope  (41)

Basic Uses  (12)

A state-space model with a state, input, output and transmission matrices:

Its response to a unit-step input:

A state-space model specified using only state, input and output matrices:

The transmission matrix is assumed to be a zero matrix:

A state-space model specified using only state and input matrices:

The states are assumed to be the outputs:

Both models have the same output response:

A discrete-time model with a sampling period of 0.1:

Test its controllability and observability:

A model with 3 states, 2 inputs and 1 output:

Count the number of inputs and outputs:

A model with 3 states, 1 input and 2 outputs:

The model dimensions:

A multiple-input multiple-output (MIMO) model:

The model dimensions:

A symbolic state-space model:

Compute the analytical unit-step response:

Specify the state variables:

They are visible in the AffineStateSpaceModel representation:

If no state variables are specified, they are chosen automatically:

Specify the state, input and output variables:

The NonlinearStateSpaceModel representation:

Use a state-space model to design a pole-placement controller:

The controller design that places the poles at :

The open- and closed-loop poles:

Plot the response of the closed-loop system to a set of initial conditions:

Descriptor Models  (6)

Descriptor models are used to model algebraic equations:

It modeled the equation , where is the input signal and is the output:

By default, the descriptor matrix e is assumed to be the identity matrix:

It is equivalent to the following model with an identity descriptor matrix:

Both models have the same state response:

Use Automatic to create a descriptor system with all the states as outputs:

Its state and output responses are the same:

It may be possible to convert a descriptor model to a standard model:

A model with a singular descriptor matrix:

It is still possible to convert it to a standard system:

It may not be possible to convert a descriptor model to a standard one in all cases:

Time-Delay Models  (3)

Use SystemsModelDelay to model a system with delays:

It has a delayed output response:

A system with two input delays:

Its output response:

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

Its transfer function model:

Model Conversions (13)

A state-space model of a transfer-function model:

The state-space representation is not unique:

Both yield same original transfer-function model:

A state-space model of a discrete-time transfer-function model:

A state-space model of an improper transfer-function model yields a descriptor system:

A state-space model of an affine state-space model discards all nonlinearities:

The original model cannot be fully recovered from the linearized model:

If the affine state-space model is linear, no information is lost:

And the original model can be fully recovered:

A state-space model of a nonlinear state-space model discards all nonlinearities:

The original model cannot be fully recovered from the linearized model:

Their output responses are not the same:

If the nonlinear state-space model is linear, no information is lost:

And the original model can be fully recovered:

A state-space model of a system of ODEs:

If the ODEs have nonlinear terms they are approximated:

The nonlinear representation does not approximate the nonlinear terms:

A state-space model of a set of difference equations:

A state-space model of a set of differential-algebraic equations:

Prevent elimination of the algebraic equation:

A state-space model of a set of difference-algebraic equations:

Prevent elimination of the algebraic equation:

A state-space model of a delay-differential equation:

Random Processes (6)

The state-space representation of a MAProcess:

An ARProcess:

An ARMAProcess:

An ARIMAProcess:

A SARIMAProcess:

A SARMAProcess:

The list of available properties:

Options  (15)

By default, the appearance is selected to fit the display in the notebook:

DescriptorStateSpace  (3)

Convert a standard state-space model to a descriptor model:

It may sometimes be possible to convert a descriptor state-space model to a standard one:

Obtain the state-space representation of a difference equation as a descriptor model:

SamplingPeriod  (3)

By default, a continuous-time model is constructed:

Explicitly construct a continuous-time model:

A discrete-time model with sampling period τ:

Assign a numerical value to τ:

Its output response:

StateSpaceRealization  (4)

By default, the controllable companion realization is computed:

Explicitly compute the controllable companion realization:

The observable companion realization:

It is the dual of the controllable companion realization:

The controllable and controllable companion realizations for a MIMO transfer-function model:

The observable and observable companion realizations for a MIMO transfer-function model:

They have a dual relationship with the controllable and controllable companion realizations:

SystemsModelLabels  (4)

Label the inputs, outputs and states:

Label only the inputs and outputs:

Label only the inputs:

Label only the outputs:

Applications  (27)

Mechanical Systems  (11)

Compute a state-space model of a mass-spring damper system using Newton's second law:

Assemble the equation of the system using :

It is linear and can be put into a linear state-space form without any approximations:

The response of the model to a unit-step input:

Compute a state-space model of an inverted pendulum using the Lagrangian:

The position of :

Its velocity:

The kinetic energy of the cart and pendulum:

The potential energy of the pendulum:

The Lagrangian:

The generalized forces:

The equations of motion:

A state-space model:

The nonpositive eigenvalues make it an unstable system:

Compute the state-space model of a vibration absorber using the Lagrangian and the Rayleigh dissipation function:

The kinetic energy of the system:

The potential energy:

The Lagrangian:

The Rayleigh dissipation function:

The system's equations of motion:

Its equilibrium position:

A state-space model of the system:

A numerical model:

Its response to an oscillatory disturbance:

Compute the state-space model of a multidomain system consisting of a motor with a load on a flexible shaft:

The kinetic energy:

The potential energy:

The Lagrangian:

The generalized torques and :

The equations of motion:

A state-space model:

The state response of a numerical model to an input torque from the motor:

Compute an approximate discrete-time state-space model of a ball and beam system starting from continuous-time equations:

The kinetic energy of the rolling ball, assuming no slipping:

Its potential energy, assuming small tilt-angles of the beam:

Its Lagrangian:

Its equation of motion:

The Kirchhoff voltage equation of the tilt-motor, assuming no armature inductance:

The tilt-motor's equation of motion:

The ball and beam's equations:

Construct a numerical state-space model of the ball and beam:

Discretize the model with a sampling period of 50 :

The displacement of the ball due to a nudge while keeping the beam level:

The displacement computed using the discrete-time model:

In both models, the ball will stop rolling due to friction:

An input voltage that tilts the beam back and forth once:

The displacement of the ball due to the tilts:

The displacement computed using the discrete-time model:

In both models, the ball ends up balanced on a level beam:

Compute state-space models of a pendulum about various equilibrium positions and compare them:

The sinusoidal term makes the model nonlinear:

The equilibrium positions of the pendulum are vertically downward and upward:

A state-space model of the pendulum linearized around a generic equilibrium value :

The state-space models around the two equilibrium positions for a set of parameter values:

The response of the model linearized around 0 is stable, while the one around 180° is unstable:

This is because the two equilibrium points have stable and unstable eigenvalues:

Compute the state-space model of a Wilberforce pendulum that has coupled dynamics and compare the efficacy of different inputs on the pendulum:

A model of the system:

A set of numerical values for the parameters:

Construct a state-space model of the system:

Its output response reveals the coupled dynamics between and :

The system is independently controllable with either the force or torque :

A system specification where is the sole feedback input:

And one where the torque is the sole feedback input:

A pole placement controller to damp the oscillations for each system:

Obtain the closed-loop systems:

The longitudinal oscillations are damped more effectively by the torque :

The rotational oscillations are damped more effectively by the force :

Obtain the feedback gains of each model:

It takes less effort to dampen the oscillations using :

This can also be seen by quantifying the control effort:

Compute the state-space model of a lathe's cutting process. The delays in the model are necessary to capture the chattering behavior of the system:

A model of the lathe's cutting process:

A state-space model:

The chattering can be seen in the delay model:

Obtain a delay-free approximation:

But the delay-free model does not capture the same chattering:

State-space models allow for state-feedback control techniques like pole placement. Place the poles of the inverted pendulum on the left-hand side of the -plane to balance the pendulum: »

A controller that balances the pendulum using a set of closed-loop poles:

The closed-loop system balances the pendulum when the cart is disturbed:

Verify the closed-loop poles:

The control effort:

State-space models are the basis for computing optimal state feedback gains that minimize a cost function. Dampen the oscillations of a flexible shaft using optimal control: »

Set the input current as the sole feedback input:

A set of state and control weight matrices and :

Compute the optimal controller that minimizes the cost function :

The oscillations are damped by the controller:

The optimal feedback gains:

The control effort:

State-space models are also used to solve tracking problems. Design a controller that tracks the position of a ball on a beam: »

Set the system specification to track the ball's position:

A set of state and control weight matrices and :

Compute the tracking controller:

Obtain the closed-loop system:

The closed-loop system places the ball in the middle of the beam:

The controller model:

The control effort:

Aerospace Systems  (6)

State-space model are useful in modeling aerospace systems. Construct a state-space model of a satellite's attitude dynamics starting from Euler's equations of motion:

Euler's equations with principal moments of inertia , , :

An operating point:

The operating point is an equilibrium point:

Construct a state-space model:

The satellite's attitude is unregulated if disturbed:

Verify the controllability of the model:

State-space models are used in model analysis. Construct a state-space model of a Harrier VTOL jet and assess its controllability:

The kinetic energy:

The potential energy:

The Lagrangian:

The generalized forces:

The equations of motion:

Its equilibrium position:

A state-space model:

A numerical model:

Both inputs are needed for the aircraft to be fully controllable:

State-space models are useful for the analysis of MIMO systems where multiple inputs affect a given output. Use the state-space model representation to compare the effectiveness of the aileron and the rudder on the yaw dynamics of a Boeing 747 using state feedback control:

A state-space model of the aircraft's lateral motion from a set of state, input and output matrices:

Obtain a state-space model with the rudder and another with aileron as the sole input:

The closed-loop systems of both systems with an LQR controller:

Using the rudder results in a faster yaw response:

As well as a smaller roll angle:

State-space models are useful for the design of discrete-time state feedback controllers. Obtain a state-space model of the 747's longitudinal dynamics and improve its handling qualities using discrete-time state-feedback control:

A nonlinear model of the aircraft's longitudinal dynamics:

A linearized state-space model:

Its response to an initial perturbation in the pitch angle is sluggish and oscillatory:

This is because the eigenvalues of the linear system are close to the imaginary axis:

A set control weights and sampling period:

Compute the optimal controller that minimizes an approximated discrete-time cost function:

The discrete-time closed-loop system:

The handling qualities are improved:

The control effort:

Discrete-time controllers can also be designed by approximating the continuous-time models. Design a discrete-time controller to stabilize a Harrier VTOL jet by approximating its continuous-time state-space model: »

A state-space model of the Harrier's horizontal dynamics:

Discretize the model with a sampling period of :

Without a controller, the jet's horizontal position is unregulated if its pitch is disturbed:

Design a discrete-time state feedback controller to stabilize the jet:

The discrete-time controller model:

The closed-loop system:

The jet is stabilized with respect to an initial disturbance by the controller:

State-space models can also be used for the design of output feedback controllers such as the estimator regulator. Design an estimator regulator that tracks the angular rate required for a satellite to maintain its nadir-pointing orientation using a discrete-time model: »

A model of the satellite:

Discretize the model with a sampling period of 0.3:

The radius and gravitational constant mass product of the Earth:

The semimajor axis of the satellite's circular orbit for an altitude of 470 km:

The orbital period is around 94 minutes:

The angular rate is the number of degrees over the orbital period:

Set the system specification to track the satellite's angular rate in the direction:

Compute a set of estimator gains:

And a state-feedback controller:

Assemble the estimator regulator:

The closed-loop system:

The satellite now tracks the required angular rate to keep its nadir-pointing orientation:

The controller model:

The control effort:

Biological Systems  (2)

Symbolic state-space models can be used to simulate models with parameters. Construct a state-space model of the metabolism of a drug and simulate it:

A model the drug's concentrations and in the GI tract and bloodstream:

A state-space model:

The analytical expression of its output response to a constant ingestion rate:

Plot the response for different values for the constants and :

State-space models can be used in the design of observers. Starting from an HIV infection model, design an estimator to estimate the free virus population:

A model of the infection:

Its equilibrium points and parameter values:

A state-space model:

A nonlinear model:

A state-output estimator:

Compare the nonlinear model's free virus population to the estimated virus population:

Chemical Systems  (3)

State-space models are useful for modeling chemical reaction processes. Construct a state-space model of a fermentation process and simulate its response to an exponential decay in the dilution rate:

A model of the fermentation process:

A state-space model:

A numerical model:

An exponential decay in the dilution rate leads to the halting of the fermentation process:

State-space models are ideal for modeling the chemical dynamics of a continuously stirred-tank reactor (CSTR). Construct a state-space model for the polymerization of methyl-methacrylate (MMA):

The reactor model:

Its equilibrium points:

A state-space model:

Its poles are on the left side of the plane, indicating the model is stable:

The MMA concentration decreases due to an increase in the initiator volumetric flow, indicating PMMA synthesis:

State-space models can be used to model systems with delays. Obtain a state-space model for a distillation column from its WoodBerry transfer function model and compare its response to a delay-free approximation of the same model:

The WoodBerry transfer function model of the distillation column:

A state-space model:

Its response to a step-input disturbance to the feed is delayed:

Obtain a delay-free approximation of the model:

The delay and delay-free system's responses to a disturbance in the feed input:

Electrical Systems  (4)

Construct a state-space model of a DC motor with armature and field voltage inputs and analyze its controllability:

The model's equations:

Its equilibrium point:

A state-space model:

The field voltage is necessary to control the motor:

Symbolic state-space models can be used to simulate models with parameters. Construct a symbolic state-space model of an operational amplifier (op amp) circuit from its governing equations and analyze its output phase and amplitude with different parameter values:

The governing equations using Kirchhoff's current law (KCL):

The transfer function model:

A state-space model:

Obtain two state-space models with different values for and :

The first set of capacitance values attenuates the input:

The second set inverts the input:

State-space models can represent systems with a mix of differential and algebraic equations. Construct a descriptor state-space model of an RLC circuit from its differential equations and a standard state-space model from its differential equations:

The equations of the individual components:

Kirchhoff's voltage law:

A descriptor state-space model is obtained because the Kirchhoff equation is algebraic:

The circuit's response to an AC voltage:

Using purely differential equations, a standard state-space model is obtained:

Both models are equivalent:

State-space models are used for solving tracking problems. Design an estimator-based tracking controller for a DC motor in the presence of varying torque loads and sensor noise: »

A model of the system:

The model specification for the controller design:

Compute a set of estimator gains:

Compute a state-feedback controller:

The closed-loop system:

A noisy signal to simulate the sensor noise:

A signal that simulates a varying torque load:

A reference signal to track:

Simulate the system's response:

Its output tracks the reference signal:

Its control signal:

Information Systems  (1)

State-space models can be used to model systems based on difference equations. Compute a state-space model of a webserver dynamics and simulate its response to the maximum number of requests and keep-alive times:

A difference equation model of the system:

A state-space model:

A set of values for inputs and :

Simulate the system using the input signal:

Properties & Relations  (20)

The eigenvalues of the state matrix are invariant under a similarity transformation:

The eigenvalues are equal:

The transfer function model of state-space models related through a similarity transformation are the same:

The transfer function models are the same:

The controllability property is generally not invariant under a similarity transformation:

The observability property is generally not invariant under a similarity transformation:

The controllable or controllable canonical realization is controllable:

But it is not necessarily observable:

The observable or observable canonical realization is observable:

But it is not necessarily controllable:

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

The dual of the controllable companion is the observable companion:

And vice versa:

A state-space model's state matrix satisfies its own characteristic polynomial:

The characteristic polynomial:

It satisfies its characteristic polynomial as per the CayleyHamilton theorem:

The characteristic polynomial of the state matrix is the denominator of a transfer function model:

The eigenvalues of the state matrix determine the speed of the system's response:

Its output response:

The system response is determined by the exponents:

They are the eigenvalues of the state matrix:

Thus if the state matrix's eigenvalues are negative, the response decays exponentially to zero:

Its output response:

It decays to zero:

This is because the eigenvalues are all negative:

If the eigenvalues are complex and on the left-hand plane, the response is oscillatory and decays to 0:

Its output response contains sinusoids:

And it decays to 0:

This is because its eigenvalues are a complex pair in the left-hand plane:

The closer the eigenvalue pair is to the negative real axis, the more damped the oscillations:

The second model's eigenvalues are closer to the negative real axis than the first's:

The second model's response is more damped:

The further the complex eigenvalue pair is from the origin, the faster the response:

The second model's eigenvalues are further away from the origin:

The second model's response is faster:

The response of the first model to a unit-step input:

If an eigenvalue pair is on the imaginary axis and the rest are negative, the response has undamped oscillations:

Its output response has undamped sinusoidal expressions:

This is because its eigenvalues include a negative value and a pair on the imaginary axis:

If one of the eigenvalues is zero and the rest are negative, the response will have a nonzero offset:

Its output response:

It converges on a nonzero value:

This is because one of its eigenvalues is zero:

If multiple eigenvalues are zero, the response is unstable and diverges to :

Its output response:

It diverges to :

This is because its eigenvalues include more than one zero:

If an eigenvalue is positive, the response is unstable and diverges to :

Its output response:

It diverges to :

This is because there is a positive eigenvalue:

If a discrete-time model's eigenvalues are within the unit circle, its response decays to zero:

Its output response decays to zero:

This is because its eigenvalues are within the unit circle:

If any eigenvalue is outside the unit circle, the response is unstable:

Its response is unstable and diverges to :

This is because not all its eigenvalues are inside the unit circle:

Possible Issues  (4)

Nonlinearities in a model are approximated:

Use a nonlinear model to prevent the approximation:

Compare the linear and nonlinear step responses:

The state matrix and input matrix must have the same number of rows:

Otherwise, the state-space model cannot be constructed:

The state matrix and output matrix must have the same number of columns:

Otherwise, the state-space model cannot be constructed:

The transmission matrix must have the same number of rows as the output matrix and the same number of columns as the input matrix:

Otherwise, the state-space model cannot be constructed

Wolfram Research (2010), StateSpaceModel, Wolfram Language function, https://reference.wolfram.com/language/ref/StateSpaceModel.html (updated 2014).

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

BibTeX

@misc{reference.wolfram_2024_statespacemodel, author="Wolfram Research", title="{StateSpaceModel}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/StateSpaceModel.html}", note=[Accessed: 05-October-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_statespacemodel, organization={Wolfram Research}, title={StateSpaceModel}, year={2014}, url={https://reference.wolfram.com/language/ref/StateSpaceModel.html}, note=[Accessed: 05-October-2024 ]}