StateSpaceModel
✖
StateSpaceModel
represents the standard state-space model with state matrix a, input matrix b, output matrix c, and transmission matrix d.
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 is also known as an LTI system (linear time-invariant).
- StateSpaceModel is typically used as a linearized model of a system for controller design.
- 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].
- Descriptor systems can be specified as follows:
-
StateSpaceModel[{a,b,c,d,e}] StateSpaceModel[{a,b,c,d,e},SamplingPeriod->τ] - Time delay systems can be represented by using SystemsModelDelay in any of the matrices.
- 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:
-
DescriptorStateSpace Automatic standard or descriptor realization ExternalTypeSignature Automatic variable types for embedded code SamplingPeriod Automatic the sampling period StateSpaceRealization Automatic the canonical realization SystemsModelLabels Automatic the labels for the input, output, and state variables
https://wolfram.com/xid/0v5g4hmcblbdp-rq3arg
https://wolfram.com/xid/0v5g4hmcblbdp-lw9rxy
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
Construct a state-space model from state, input, output and transmission matrices:
https://wolfram.com/xid/0v5g4hmcblbdp-g0wufm
https://wolfram.com/xid/0v5g4hmcblbdp-bh53xt
https://wolfram.com/xid/0v5g4hmcblbdp-cnvdyh
Construct a state-space model from a transfer function model:
https://wolfram.com/xid/0v5g4hmcblbdp-erjg84
Test its controllability and observability:
https://wolfram.com/xid/0v5g4hmcblbdp-cblc54
Construct a state-space model from a set of ordinary differential equations (ODEs):
https://wolfram.com/xid/0v5g4hmcblbdp-jg1nmb
https://wolfram.com/xid/0v5g4hmcblbdp-ftm6jl
Its response to nonzero initial conditions:
https://wolfram.com/xid/0v5g4hmcblbdp-mg8smz
Construct a discrete-time state-space model by specifying its sampling period:
https://wolfram.com/xid/0v5g4hmcblbdp-ioktwb
https://wolfram.com/xid/0v5g4hmcblbdp-jtszev
Scope (41)Survey of the scope of standard use cases
Basic Uses (12)
A state-space model with a state, input, output and transmission matrices:
https://wolfram.com/xid/0v5g4hmcblbdp-d6lmne
https://wolfram.com/xid/0v5g4hmcblbdp-rf83ji
Its response to a unit-step input:
https://wolfram.com/xid/0v5g4hmcblbdp-0jvyd4
A state-space model specified using only state, input and output matrices:
https://wolfram.com/xid/0v5g4hmcblbdp-0og1ug
https://wolfram.com/xid/0v5g4hmcblbdp-r2gn5i
The transmission matrix is assumed to be a zero matrix:
https://wolfram.com/xid/0v5g4hmcblbdp-e1b3yh
A state-space model specified using only state and input matrices:
https://wolfram.com/xid/0v5g4hmcblbdp-dm8njr
https://wolfram.com/xid/0v5g4hmcblbdp-z762h3
The states are assumed to be the outputs:
https://wolfram.com/xid/0v5g4hmcblbdp-tn2q9k
Both models have the same output response:
https://wolfram.com/xid/0v5g4hmcblbdp-pxrbm1
A discrete-time model with a sampling period of 0.1:
https://wolfram.com/xid/0v5g4hmcblbdp-fewq4i
Test its controllability and observability:
https://wolfram.com/xid/0v5g4hmcblbdp-h7vxi
A model with 3 states, 2 inputs and 1 output:
https://wolfram.com/xid/0v5g4hmcblbdp-lj6dqo
Count the number of inputs and outputs:
https://wolfram.com/xid/0v5g4hmcblbdp-bqvy8t
A model with 3 states, 1 input and 2 outputs:
https://wolfram.com/xid/0v5g4hmcblbdp-l6f2f8
https://wolfram.com/xid/0v5g4hmcblbdp-nxj90n
A multiple-input multiple-output (MIMO) model:
https://wolfram.com/xid/0v5g4hmcblbdp-flmokp
https://wolfram.com/xid/0v5g4hmcblbdp-hsv6c
https://wolfram.com/xid/0v5g4hmcblbdp-s1w2nb
Compute the analytical unit-step response:
https://wolfram.com/xid/0v5g4hmcblbdp-yrcae1
https://wolfram.com/xid/0v5g4hmcblbdp-e6sz0w
They are visible in the AffineStateSpaceModel representation:
https://wolfram.com/xid/0v5g4hmcblbdp-zvbytx
If no state variables are specified, they are chosen automatically:
https://wolfram.com/xid/0v5g4hmcblbdp-752o0d
https://wolfram.com/xid/0v5g4hmcblbdp-kp3j8r
Specify the state, input and output variables:
https://wolfram.com/xid/0v5g4hmcblbdp-8pebnf
The NonlinearStateSpaceModel representation:
https://wolfram.com/xid/0v5g4hmcblbdp-i7i7wo
Use a state-space model to design a pole-placement controller:
https://wolfram.com/xid/0v5g4hmcblbdp-zqewg8
The controller design that places the poles at :
https://wolfram.com/xid/0v5g4hmcblbdp-ii254m
The open- and closed-loop poles:
https://wolfram.com/xid/0v5g4hmcblbdp-h6ogkk
Plot the response of the closed-loop system to a set of initial conditions:
https://wolfram.com/xid/0v5g4hmcblbdp-nm8qp8
Descriptor Models (6)
Descriptor models are used to model algebraic equations:
https://wolfram.com/xid/0v5g4hmcblbdp-waeem5
It modeled the equation , where is the input signal and is the output:
https://wolfram.com/xid/0v5g4hmcblbdp-wjx5pv
By default, the descriptor matrix e is assumed to be the identity matrix:
https://wolfram.com/xid/0v5g4hmcblbdp-c84byg
https://wolfram.com/xid/0v5g4hmcblbdp-sa1n98
It is equivalent to the following model with an identity descriptor matrix:
https://wolfram.com/xid/0v5g4hmcblbdp-qt1f9o
Both models have the same state response:
https://wolfram.com/xid/0v5g4hmcblbdp-qoy0it
Use Automatic to create a descriptor system with all the states as outputs:
https://wolfram.com/xid/0v5g4hmcblbdp-bpptcq
Its state and output responses are the same:
https://wolfram.com/xid/0v5g4hmcblbdp-j6qulp
It may be possible to convert a descriptor model to a standard model:
https://wolfram.com/xid/0v5g4hmcblbdp-ogrg6
A model with a singular descriptor matrix:
https://wolfram.com/xid/0v5g4hmcblbdp-wjt44l
It is still possible to convert it to a standard system:
https://wolfram.com/xid/0v5g4hmcblbdp-l4oh82
It may not be possible to convert a descriptor model to a standard one in all cases:
https://wolfram.com/xid/0v5g4hmcblbdp-huhvmx
https://wolfram.com/xid/0v5g4hmcblbdp-1a3a24
Time-Delay Models (3)
Use SystemsModelDelay to model a system with delays:
https://wolfram.com/xid/0v5g4hmcblbdp-8u8y0f
It has a delayed output response:
https://wolfram.com/xid/0v5g4hmcblbdp-10plr6
A system with two input delays:
https://wolfram.com/xid/0v5g4hmcblbdp-k3aop8
https://wolfram.com/xid/0v5g4hmcblbdp-ikai3z
A discrete-time system with a delay in the state matrix:
https://wolfram.com/xid/0v5g4hmcblbdp-6f8q3l
https://wolfram.com/xid/0v5g4hmcblbdp-g4oe4x
Model Conversions (13)
A state-space model of a transfer-function model:
https://wolfram.com/xid/0v5g4hmcblbdp-gtgquz
https://wolfram.com/xid/0v5g4hmcblbdp-0bc5fz
The state-space representation is not unique:
https://wolfram.com/xid/0v5g4hmcblbdp-n4me6q
Both yield same original transfer-function model:
https://wolfram.com/xid/0v5g4hmcblbdp-0rnjom
A state-space model of a discrete-time transfer-function model:
https://wolfram.com/xid/0v5g4hmcblbdp-t75gdy
A state-space model of an improper transfer-function model yields a descriptor system:
https://wolfram.com/xid/0v5g4hmcblbdp-mm831l
A state-space model of an affine state-space model discards all nonlinearities:
https://wolfram.com/xid/0v5g4hmcblbdp-l8sbgo
The original model cannot be fully recovered from the linearized model:
https://wolfram.com/xid/0v5g4hmcblbdp-mz5edt
If the affine state-space model is linear, no information is lost:
https://wolfram.com/xid/0v5g4hmcblbdp-2dqdcp
And the original model can be fully recovered:
https://wolfram.com/xid/0v5g4hmcblbdp-qct1wr
A state-space model of a nonlinear state-space model discards all nonlinearities:
https://wolfram.com/xid/0v5g4hmcblbdp-6zlduc
https://wolfram.com/xid/0v5g4hmcblbdp-5r3gqi
The original model cannot be fully recovered from the linearized model:
https://wolfram.com/xid/0v5g4hmcblbdp-74cacl
Their output responses are not the same:
https://wolfram.com/xid/0v5g4hmcblbdp-vsgqc3
If the nonlinear state-space model is linear, no information is lost:
https://wolfram.com/xid/0v5g4hmcblbdp-r1lcej
And the original model can be fully recovered:
https://wolfram.com/xid/0v5g4hmcblbdp-4rggh2
A state-space model of a system of ODEs:
https://wolfram.com/xid/0v5g4hmcblbdp-n1ev2e
https://wolfram.com/xid/0v5g4hmcblbdp-kamlam
If the ODEs have nonlinear terms they are approximated:
https://wolfram.com/xid/0v5g4hmcblbdp-32zxwz
https://wolfram.com/xid/0v5g4hmcblbdp-esoq9v
The nonlinear representation does not approximate the nonlinear terms:
https://wolfram.com/xid/0v5g4hmcblbdp-qgsdyy
A state-space model of a set of difference equations:
https://wolfram.com/xid/0v5g4hmcblbdp-hte2r0
https://wolfram.com/xid/0v5g4hmcblbdp-brlxe6
A state-space model of a set of differential-algebraic equations:
https://wolfram.com/xid/0v5g4hmcblbdp-nisyud
https://wolfram.com/xid/0v5g4hmcblbdp-xyobxj
Prevent elimination of the algebraic equation:
https://wolfram.com/xid/0v5g4hmcblbdp-851pke
A state-space model of a set of difference-algebraic equations:
https://wolfram.com/xid/0v5g4hmcblbdp-d2uula
https://wolfram.com/xid/0v5g4hmcblbdp-qcykqp
Prevent elimination of the algebraic equation:
https://wolfram.com/xid/0v5g4hmcblbdp-vf2iu9
A state-space model of a delay-differential equation:
https://wolfram.com/xid/0v5g4hmcblbdp-hjrh6e
https://wolfram.com/xid/0v5g4hmcblbdp-orgy0d
Random Processes (6)
The state-space representation of a MAProcess:
https://wolfram.com/xid/0v5g4hmcblbdp-cr0q04
An ARProcess:
https://wolfram.com/xid/0v5g4hmcblbdp-31th93
An ARMAProcess:
https://wolfram.com/xid/0v5g4hmcblbdp-7irygx
An ARIMAProcess:
https://wolfram.com/xid/0v5g4hmcblbdp-zy54sz
https://wolfram.com/xid/0v5g4hmcblbdp-g7aonq
A SARMAProcess:
https://wolfram.com/xid/0v5g4hmcblbdp-35goyq
The list of available properties:
https://wolfram.com/xid/0v5g4hmcblbdp-sdn80s
Options (15)Common values & functionality for each option
By default, the appearance is selected to fit the display in the notebook:
https://wolfram.com/xid/0v5g4hmcblbdp-yz4upi
https://wolfram.com/xid/0v5g4hmcblbdp-b4ty5d
DescriptorStateSpace (3)
Convert a standard state-space model to a descriptor model:
https://wolfram.com/xid/0v5g4hmcblbdp-4qekec
It may sometimes be possible to convert a descriptor state-space model to a standard one:
https://wolfram.com/xid/0v5g4hmcblbdp-wf1k6x
Obtain the state-space representation of a difference equation as a descriptor model:
https://wolfram.com/xid/0v5g4hmcblbdp-5yw02l
SamplingPeriod (3)
By default, a continuous-time model is constructed:
https://wolfram.com/xid/0v5g4hmcblbdp-yj5hrw
https://wolfram.com/xid/0v5g4hmcblbdp-tphdif
Explicitly construct a continuous-time model:
https://wolfram.com/xid/0v5g4hmcblbdp-f44r6o
https://wolfram.com/xid/0v5g4hmcblbdp-it3syj
A discrete-time model with sampling period τ:
https://wolfram.com/xid/0v5g4hmcblbdp-gz8awl
Assign a numerical value to τ:
https://wolfram.com/xid/0v5g4hmcblbdp-f9to1m
https://wolfram.com/xid/0v5g4hmcblbdp-iqtiek
StateSpaceRealization (4)
By default, the controllable companion realization is computed:
https://wolfram.com/xid/0v5g4hmcblbdp-2ppw3s
Explicitly compute the controllable companion realization:
https://wolfram.com/xid/0v5g4hmcblbdp-en59ku
The observable companion realization:
https://wolfram.com/xid/0v5g4hmcblbdp-g1j07t
It is the dual of the controllable companion realization:
https://wolfram.com/xid/0v5g4hmcblbdp-q4c70x
The controllable and controllable companion realizations for a MIMO transfer-function model:
https://wolfram.com/xid/0v5g4hmcblbdp-y3sj26
The observable and observable companion realizations for a MIMO transfer-function model:
https://wolfram.com/xid/0v5g4hmcblbdp-5n4do3
https://wolfram.com/xid/0v5g4hmcblbdp-svz56
They have a dual relationship with the controllable and controllable companion realizations:
https://wolfram.com/xid/0v5g4hmcblbdp-zcucjd
https://wolfram.com/xid/0v5g4hmcblbdp-z0vrk2
SystemsModelLabels (4)
Label the inputs, outputs and states:
https://wolfram.com/xid/0v5g4hmcblbdp-bmfe0x
https://wolfram.com/xid/0v5g4hmcblbdp-vo49kw
Label only the inputs and outputs:
https://wolfram.com/xid/0v5g4hmcblbdp-fkf1jg
https://wolfram.com/xid/0v5g4hmcblbdp-gvek9d
https://wolfram.com/xid/0v5g4hmcblbdp-zc61mm
https://wolfram.com/xid/0v5g4hmcblbdp-44u4gh
Applications (27)Sample problems that can be solved with this function
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 :
https://wolfram.com/xid/0v5g4hmcblbdp-0famh2
It is linear and can be put into a linear state-space form without any approximations:
https://wolfram.com/xid/0v5g4hmcblbdp-vc811
https://wolfram.com/xid/0v5g4hmcblbdp-cj5syt
The response of the model to a unit-step input:
https://wolfram.com/xid/0v5g4hmcblbdp-zagvd3
Compute a state-space model of an inverted pendulum using the Lagrangian:
https://wolfram.com/xid/0v5g4hmcblbdp-e3irwj
https://wolfram.com/xid/0v5g4hmcblbdp-pvpkrx
The kinetic energy of the cart and pendulum:
https://wolfram.com/xid/0v5g4hmcblbdp-czh3yv
The potential energy of the pendulum:
https://wolfram.com/xid/0v5g4hmcblbdp-d5tdi7
https://wolfram.com/xid/0v5g4hmcblbdp-hmmgq2
https://wolfram.com/xid/0v5g4hmcblbdp-lalimf
https://wolfram.com/xid/0v5g4hmcblbdp-k0kedw
https://wolfram.com/xid/0v5g4hmcblbdp-6hvxz
The nonpositive eigenvalues make it an unstable system:
https://wolfram.com/xid/0v5g4hmcblbdp-sw60xe
Compute the state-space model of a vibration absorber using the Lagrangian and the Rayleigh dissipation function:
The kinetic energy of the system:
https://wolfram.com/xid/0v5g4hmcblbdp-c20lyo
https://wolfram.com/xid/0v5g4hmcblbdp-lx27vo
https://wolfram.com/xid/0v5g4hmcblbdp-h2ni1k
The Rayleigh dissipation function:
https://wolfram.com/xid/0v5g4hmcblbdp-b0nucl
The system's equations of motion:
https://wolfram.com/xid/0v5g4hmcblbdp-26dmq
https://wolfram.com/xid/0v5g4hmcblbdp-gmctfo
A state-space model of the system:
https://wolfram.com/xid/0v5g4hmcblbdp-jebdj1
https://wolfram.com/xid/0v5g4hmcblbdp-eu6kmt
Its response to an oscillatory disturbance:
https://wolfram.com/xid/0v5g4hmcblbdp-dxu15u
Compute the state-space model of a multidomain system consisting of a motor with a load on a flexible shaft:
https://wolfram.com/xid/0v5g4hmcblbdp-b00zug
https://wolfram.com/xid/0v5g4hmcblbdp-j3z5y8
https://wolfram.com/xid/0v5g4hmcblbdp-kg6j1z
https://wolfram.com/xid/0v5g4hmcblbdp-eoxlef
https://wolfram.com/xid/0v5g4hmcblbdp-i615wg
https://wolfram.com/xid/0v5g4hmcblbdp-b9qe89
The state response of a numerical model to an input torque from the motor:
https://wolfram.com/xid/0v5g4hmcblbdp-c1j274
https://wolfram.com/xid/0v5g4hmcblbdp-jj7l5g
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:
https://wolfram.com/xid/0v5g4hmcblbdp-g3k8a8
Its potential energy, assuming small tilt-angles of the beam:
https://wolfram.com/xid/0v5g4hmcblbdp-f1cg
https://wolfram.com/xid/0v5g4hmcblbdp-jl0c4
https://wolfram.com/xid/0v5g4hmcblbdp-ktrixh
The Kirchhoff voltage equation of the tilt-motor, assuming no armature inductance:
https://wolfram.com/xid/0v5g4hmcblbdp-l8d2k
The tilt-motor's equation of motion:
https://wolfram.com/xid/0v5g4hmcblbdp-jx83r8
The ball and beam's equations:
https://wolfram.com/xid/0v5g4hmcblbdp-hqte2a
Construct a numerical state-space model of the ball and beam:
https://wolfram.com/xid/0v5g4hmcblbdp-ib1glv
Discretize the model with a sampling period of 50 :
https://wolfram.com/xid/0v5g4hmcblbdp-m7rkrl
The displacement of the ball due to a nudge while keeping the beam level:
https://wolfram.com/xid/0v5g4hmcblbdp-fih6r
The displacement computed using the discrete-time model:
https://wolfram.com/xid/0v5g4hmcblbdp-nc6p9
In both models, the ball will stop rolling due to friction:
https://wolfram.com/xid/0v5g4hmcblbdp-ey10kg
An input voltage that tilts the beam back and forth once:
https://wolfram.com/xid/0v5g4hmcblbdp-d8rxjl
The displacement of the ball due to the tilts:
https://wolfram.com/xid/0v5g4hmcblbdp-c9o9xb
The displacement computed using the discrete-time model:
https://wolfram.com/xid/0v5g4hmcblbdp-fx57f8
In both models, the ball ends up balanced on a level beam:
https://wolfram.com/xid/0v5g4hmcblbdp-juomsi
Compute state-space models of a pendulum about various equilibrium positions and compare them:
The sinusoidal term makes the model nonlinear:
https://wolfram.com/xid/0v5g4hmcblbdp-h8ushn
The equilibrium positions of the pendulum are vertically downward and upward:
https://wolfram.com/xid/0v5g4hmcblbdp-lo27c8
A state-space model of the pendulum linearized around a generic equilibrium value :
https://wolfram.com/xid/0v5g4hmcblbdp-gsihcn
The state-space models around the two equilibrium positions for a set of parameter values:
https://wolfram.com/xid/0v5g4hmcblbdp-bts53y
The response of the model linearized around 0 is stable, while the one around 180° is unstable:
https://wolfram.com/xid/0v5g4hmcblbdp-cntm5h
This is because the two equilibrium points have stable and unstable eigenvalues:
https://wolfram.com/xid/0v5g4hmcblbdp-md9aaw
Compute the state-space model of a Wilberforce pendulum that has coupled dynamics and compare the efficacy of different inputs on the pendulum:
https://wolfram.com/xid/0v5g4hmcblbdp-fb1ayf
A set of numerical values for the parameters:
https://wolfram.com/xid/0v5g4hmcblbdp-jxgbpr
Construct a state-space model of the system:
https://wolfram.com/xid/0v5g4hmcblbdp-cr4zu9
Its output response reveals the coupled dynamics between and :
https://wolfram.com/xid/0v5g4hmcblbdp-b64x9e
The system is independently controllable with either the force or torque :
https://wolfram.com/xid/0v5g4hmcblbdp-oe51s
A system specification where is the sole feedback input:
https://wolfram.com/xid/0v5g4hmcblbdp-e0grv2
And one where the torque is the sole feedback input:
https://wolfram.com/xid/0v5g4hmcblbdp-e0rc0h
A pole placement controller to damp the oscillations for each system:
https://wolfram.com/xid/0v5g4hmcblbdp-dsfbx
Obtain the closed-loop systems:
https://wolfram.com/xid/0v5g4hmcblbdp-bl26j6
The longitudinal oscillations are damped more effectively by the torque :
https://wolfram.com/xid/0v5g4hmcblbdp-30vnu
https://wolfram.com/xid/0v5g4hmcblbdp-bf45pp
The rotational oscillations are damped more effectively by the force :
https://wolfram.com/xid/0v5g4hmcblbdp-fqp2g0
Obtain the feedback gains of each model:
https://wolfram.com/xid/0v5g4hmcblbdp-k2sjsz
It takes less effort to dampen the oscillations using :
https://wolfram.com/xid/0v5g4hmcblbdp-dyypy0
This can also be seen by quantifying the control effort:
https://wolfram.com/xid/0v5g4hmcblbdp-ckinwo
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:
https://wolfram.com/xid/0v5g4hmcblbdp-lcps15
https://wolfram.com/xid/0v5g4hmcblbdp-e4szo4
The chattering can be seen in the delay model:
https://wolfram.com/xid/0v5g4hmcblbdp-bl84db
Obtain a delay-free approximation:
https://wolfram.com/xid/0v5g4hmcblbdp-gxwmeo
But the delay-free model does not capture the same chattering:
https://wolfram.com/xid/0v5g4hmcblbdp-bkqm82
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: »
https://wolfram.com/xid/0v5g4hmcblbdp-egiagz
A controller that balances the pendulum using a set of closed-loop poles:
https://wolfram.com/xid/0v5g4hmcblbdp-k1g7jc
The closed-loop system balances the pendulum when the cart is disturbed:
https://wolfram.com/xid/0v5g4hmcblbdp-dnwtc2
https://wolfram.com/xid/0v5g4hmcblbdp-fg7e8e
https://wolfram.com/xid/0v5g4hmcblbdp-bi2qg3
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: »
https://wolfram.com/xid/0v5g4hmcblbdp-ywz40
Set the input current as the sole feedback input:
https://wolfram.com/xid/0v5g4hmcblbdp-bjc8q8
A set of state and control weight matrices and :
https://wolfram.com/xid/0v5g4hmcblbdp-d6y3ub
Compute the optimal controller that minimizes the cost function :
https://wolfram.com/xid/0v5g4hmcblbdp-d7uhru
The oscillations are damped by the controller:
https://wolfram.com/xid/0v5g4hmcblbdp-vappt
https://wolfram.com/xid/0v5g4hmcblbdp-dooukl
https://wolfram.com/xid/0v5g4hmcblbdp-fzp6v1
State-space models are also used to solve tracking problems. Design a controller that tracks the position of a ball on a beam: »
https://wolfram.com/xid/0v5g4hmcblbdp-fml9l8
Set the system specification to track the ball's position:
https://wolfram.com/xid/0v5g4hmcblbdp-la02il
A set of state and control weight matrices and :
https://wolfram.com/xid/0v5g4hmcblbdp-bxagiz
Compute the tracking controller:
https://wolfram.com/xid/0v5g4hmcblbdp-cm9syz
Obtain the closed-loop system:
https://wolfram.com/xid/0v5g4hmcblbdp-sx0dw
The closed-loop system places the ball in the middle of the beam:
https://wolfram.com/xid/0v5g4hmcblbdp-fnyutt
https://wolfram.com/xid/0v5g4hmcblbdp-hf6gk
https://wolfram.com/xid/0v5g4hmcblbdp-nuulzt
https://wolfram.com/xid/0v5g4hmcblbdp-cdppma
https://wolfram.com/xid/0v5g4hmcblbdp-ca4rfj
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 , , :
https://wolfram.com/xid/0v5g4hmcblbdp-pwwjo7
https://wolfram.com/xid/0v5g4hmcblbdp-krwwgn
The operating point is an equilibrium point:
https://wolfram.com/xid/0v5g4hmcblbdp-lfb3p
Construct a state-space model:
https://wolfram.com/xid/0v5g4hmcblbdp-byq27f
The satellite's attitude is unregulated if disturbed:
https://wolfram.com/xid/0v5g4hmcblbdp-fpf3en
Verify the controllability of the model:
https://wolfram.com/xid/0v5g4hmcblbdp-c9axp1
State-space models are used in model analysis. Construct a state-space model of a Harrier VTOL jet and assess its controllability:
https://wolfram.com/xid/0v5g4hmcblbdp-frkx66
https://wolfram.com/xid/0v5g4hmcblbdp-hh8vm5
https://wolfram.com/xid/0v5g4hmcblbdp-bg5nwh
https://wolfram.com/xid/0v5g4hmcblbdp-dqfrcs
https://wolfram.com/xid/0v5g4hmcblbdp-f2j9l
https://wolfram.com/xid/0v5g4hmcblbdp-mub4lz
https://wolfram.com/xid/0v5g4hmcblbdp-fbpjju
https://wolfram.com/xid/0v5g4hmcblbdp-ui7mh
Both inputs are needed for the aircraft to be fully controllable:
https://wolfram.com/xid/0v5g4hmcblbdp-ngsuwm
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:
https://wolfram.com/xid/0v5g4hmcblbdp-lvsk8
https://wolfram.com/xid/0v5g4hmcblbdp-dfrbq8
Obtain a state-space model with the rudder and another with aileron as the sole input:
https://wolfram.com/xid/0v5g4hmcblbdp-cd9hot
The closed-loop systems of both systems with an LQR controller:
https://wolfram.com/xid/0v5g4hmcblbdp-gkfmqv
https://wolfram.com/xid/0v5g4hmcblbdp-bnxrgs
Using the rudder results in a faster yaw response:
https://wolfram.com/xid/0v5g4hmcblbdp-eki9uy
As well as a smaller roll angle:
https://wolfram.com/xid/0v5g4hmcblbdp-elbvfg
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:
https://wolfram.com/xid/0v5g4hmcblbdp-lc42rc
A linearized state-space model:
https://wolfram.com/xid/0v5g4hmcblbdp-bfzd53
Its response to an initial perturbation in the pitch angle is sluggish and oscillatory:
https://wolfram.com/xid/0v5g4hmcblbdp-cg3wus
This is because the eigenvalues of the linear system are close to the imaginary axis:
https://wolfram.com/xid/0v5g4hmcblbdp-3jb9
A set control weights and sampling period:
https://wolfram.com/xid/0v5g4hmcblbdp-2dtxm
https://wolfram.com/xid/0v5g4hmcblbdp-cwnvq8
Compute the optimal controller that minimizes an approximated discrete-time cost function:
https://wolfram.com/xid/0v5g4hmcblbdp-wzj6d
The discrete-time closed-loop system:
https://wolfram.com/xid/0v5g4hmcblbdp-ex2cph
The handling qualities are improved:
https://wolfram.com/xid/0v5g4hmcblbdp-euuecx
https://wolfram.com/xid/0v5g4hmcblbdp-c166a1
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:
https://wolfram.com/xid/0v5g4hmcblbdp-nf19
Discretize the model with a sampling period of :
https://wolfram.com/xid/0v5g4hmcblbdp-ikbxpy
Without a controller, the jet's horizontal position is unregulated if its pitch is disturbed:
https://wolfram.com/xid/0v5g4hmcblbdp-cmzs80
Design a discrete-time state feedback controller to stabilize the jet:
https://wolfram.com/xid/0v5g4hmcblbdp-1gwpe
https://wolfram.com/xid/0v5g4hmcblbdp-gk5p17
The discrete-time controller model:
https://wolfram.com/xid/0v5g4hmcblbdp-brf45z
https://wolfram.com/xid/0v5g4hmcblbdp-itmo1q
The jet is stabilized with respect to an initial disturbance by the controller:
https://wolfram.com/xid/0v5g4hmcblbdp-gpmf31
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: »
https://wolfram.com/xid/0v5g4hmcblbdp-bb1hzf
Discretize the model with a sampling period of 0.3:
https://wolfram.com/xid/0v5g4hmcblbdp-ct4060
The radius and gravitational constant mass product of the Earth:
https://wolfram.com/xid/0v5g4hmcblbdp-chii4j
The semimajor axis of the satellite's circular orbit for an altitude of 470 km:
https://wolfram.com/xid/0v5g4hmcblbdp-dtwigb
The orbital period is around 94 minutes:
https://wolfram.com/xid/0v5g4hmcblbdp-f02qej
The angular rate is the number of degrees over the orbital period:
https://wolfram.com/xid/0v5g4hmcblbdp-bgh66h
Set the system specification to track the satellite's angular rate in the direction:
https://wolfram.com/xid/0v5g4hmcblbdp-du2ev9
Compute a set of estimator gains:
https://wolfram.com/xid/0v5g4hmcblbdp-dkawyw
And a state-feedback controller:
https://wolfram.com/xid/0v5g4hmcblbdp-ls1c0x
Assemble the estimator regulator:
https://wolfram.com/xid/0v5g4hmcblbdp-elfaov
https://wolfram.com/xid/0v5g4hmcblbdp-d3ls2t
The satellite now tracks the required angular rate to keep its nadir-pointing orientation:
https://wolfram.com/xid/0v5g4hmcblbdp-bin8mm
https://wolfram.com/xid/0v5g4hmcblbdp-i3mxuc
https://wolfram.com/xid/0v5g4hmcblbdp-g57jhl
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:
https://wolfram.com/xid/0v5g4hmcblbdp-c08scx
https://wolfram.com/xid/0v5g4hmcblbdp-frulv2
The analytical expression of its output response to a constant ingestion rate:
https://wolfram.com/xid/0v5g4hmcblbdp-9gdqh
Plot the response for different values for the constants and :
https://wolfram.com/xid/0v5g4hmcblbdp-wx1zra
https://wolfram.com/xid/0v5g4hmcblbdp-7ffkzh
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:
https://wolfram.com/xid/0v5g4hmcblbdp-gnrfl
Its equilibrium points and parameter values:
https://wolfram.com/xid/0v5g4hmcblbdp-goy304
https://wolfram.com/xid/0v5g4hmcblbdp-e0d1x7
https://wolfram.com/xid/0v5g4hmcblbdp-iib8qe
https://wolfram.com/xid/0v5g4hmcblbdp-w49a7
https://wolfram.com/xid/0v5g4hmcblbdp-j0jdp
Compare the nonlinear model's free virus population to the estimated virus population:
https://wolfram.com/xid/0v5g4hmcblbdp-ownbpp
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:
https://wolfram.com/xid/0v5g4hmcblbdp-if1w5d
https://wolfram.com/xid/0v5g4hmcblbdp-fa94ry
https://wolfram.com/xid/0v5g4hmcblbdp-ehyosn
An exponential decay in the dilution rate leads to the halting of the fermentation process:
https://wolfram.com/xid/0v5g4hmcblbdp-bxf7mi
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):
https://wolfram.com/xid/0v5g4hmcblbdp-g6vmtq
https://wolfram.com/xid/0v5g4hmcblbdp-ho6ju4
https://wolfram.com/xid/0v5g4hmcblbdp-ifvvii
Its poles are on the left side of the plane, indicating the model is stable:
https://wolfram.com/xid/0v5g4hmcblbdp-b39sjp
The MMA concentration decreases due to an increase in the initiator volumetric flow, indicating PMMA synthesis:
https://wolfram.com/xid/0v5g4hmcblbdp-x8n6j
State-space models can be used to model systems with delays. Obtain a state-space model for a distillation column from its Wood–Berry transfer function model and compare its response to a delay-free approximation of the same model:
The Wood–Berry transfer function model of the distillation column:
https://wolfram.com/xid/0v5g4hmcblbdp-fgo5sj
https://wolfram.com/xid/0v5g4hmcblbdp-gcjw3h
Its response to a step-input disturbance to the feed is delayed:
https://wolfram.com/xid/0v5g4hmcblbdp-ih528h
Obtain a delay-free approximation of the model:
https://wolfram.com/xid/0v5g4hmcblbdp-dl6gxy
The delay and delay-free system's responses to a disturbance in the feed input:
https://wolfram.com/xid/0v5g4hmcblbdp-e8m1l2
https://wolfram.com/xid/0v5g4hmcblbdp-ml117a
Electrical Systems (4)
Construct a state-space model of a DC motor with armature and field voltage inputs and analyze its controllability:
https://wolfram.com/xid/0v5g4hmcblbdp-ma75ma
https://wolfram.com/xid/0v5g4hmcblbdp-c860xb
https://wolfram.com/xid/0v5g4hmcblbdp-38u397
https://wolfram.com/xid/0v5g4hmcblbdp-v4wwwi
https://wolfram.com/xid/0v5g4hmcblbdp-bl85xe
The field voltage is necessary to control the motor:
https://wolfram.com/xid/0v5g4hmcblbdp-c2112a
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):
https://wolfram.com/xid/0v5g4hmcblbdp-qvbajw
https://wolfram.com/xid/0v5g4hmcblbdp-04wlvo
https://wolfram.com/xid/0v5g4hmcblbdp-z9yhjw
Obtain two state-space models with different values for and :
https://wolfram.com/xid/0v5g4hmcblbdp-h7ymgw
The first set of capacitance values attenuates the input:
https://wolfram.com/xid/0v5g4hmcblbdp-4rwbq9
The second set inverts the input:
https://wolfram.com/xid/0v5g4hmcblbdp-yv0wcc
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:
https://wolfram.com/xid/0v5g4hmcblbdp-bbljtg
https://wolfram.com/xid/0v5g4hmcblbdp-oxl961
A descriptor state-space model is obtained because the Kirchhoff equation is algebraic:
https://wolfram.com/xid/0v5g4hmcblbdp-4h7huq
The circuit's response to an AC voltage:
https://wolfram.com/xid/0v5g4hmcblbdp-d7wx
Using purely differential equations, a standard state-space model is obtained:
https://wolfram.com/xid/0v5g4hmcblbdp-gxadg1
https://wolfram.com/xid/0v5g4hmcblbdp-3q83pb
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: »
https://wolfram.com/xid/0v5g4hmcblbdp-bkzh03
The model specification for the controller design:
https://wolfram.com/xid/0v5g4hmcblbdp-d234ls
Compute a set of estimator gains:
https://wolfram.com/xid/0v5g4hmcblbdp-c5k3vk
Compute a state-feedback controller:
https://wolfram.com/xid/0v5g4hmcblbdp-iwwdeb
https://wolfram.com/xid/0v5g4hmcblbdp-cd98ib
A noisy signal to simulate the sensor noise:
https://wolfram.com/xid/0v5g4hmcblbdp-iug21b
https://wolfram.com/xid/0v5g4hmcblbdp-sf6obs
A signal that simulates a varying torque load:
https://wolfram.com/xid/0v5g4hmcblbdp-mftraj
https://wolfram.com/xid/0v5g4hmcblbdp-enacz4
https://wolfram.com/xid/0v5g4hmcblbdp-bubwf
Simulate the system's response:
https://wolfram.com/xid/0v5g4hmcblbdp-qpt1w
Its output tracks the reference signal:
https://wolfram.com/xid/0v5g4hmcblbdp-x1k50
https://wolfram.com/xid/0v5g4hmcblbdp-p17ij
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:
https://wolfram.com/xid/0v5g4hmcblbdp-67ty2
https://wolfram.com/xid/0v5g4hmcblbdp-kyrgse
A set of values for inputs and :
https://wolfram.com/xid/0v5g4hmcblbdp-qanjgd
https://wolfram.com/xid/0v5g4hmcblbdp-2e45d
Simulate the system using the input signal:
https://wolfram.com/xid/0v5g4hmcblbdp-e3gn42
Properties & Relations (20)Properties of the function, and connections to other functions
The eigenvalues of the state matrix are invariant under a similarity transformation:
https://wolfram.com/xid/0v5g4hmcblbdp-f4ef1n
https://wolfram.com/xid/0v5g4hmcblbdp-bdvpcr
https://wolfram.com/xid/0v5g4hmcblbdp-z06h00
The transfer function model of state-space models related through a similarity transformation are the same:
https://wolfram.com/xid/0v5g4hmcblbdp-rool0x
https://wolfram.com/xid/0v5g4hmcblbdp-gvvhyd
The transfer function models are the same:
https://wolfram.com/xid/0v5g4hmcblbdp-ig0g19
The controllability property is generally not invariant under a similarity transformation:
https://wolfram.com/xid/0v5g4hmcblbdp-emie0p
https://wolfram.com/xid/0v5g4hmcblbdp-8dciay
https://wolfram.com/xid/0v5g4hmcblbdp-fhmlqv
The observability property is generally not invariant under a similarity transformation:
https://wolfram.com/xid/0v5g4hmcblbdp-nisptv
https://wolfram.com/xid/0v5g4hmcblbdp-y9dg80
https://wolfram.com/xid/0v5g4hmcblbdp-192b3y
The controllable or controllable canonical realization is controllable:
https://wolfram.com/xid/0v5g4hmcblbdp-eo4qda
https://wolfram.com/xid/0v5g4hmcblbdp-cc9r2j
But it is not necessarily observable:
https://wolfram.com/xid/0v5g4hmcblbdp-31znib
The observable or observable canonical realization is observable:
https://wolfram.com/xid/0v5g4hmcblbdp-4iz7yw
https://wolfram.com/xid/0v5g4hmcblbdp-s4lwbs
But it is not necessarily controllable:
https://wolfram.com/xid/0v5g4hmcblbdp-pr7jco
The controllable companion and observable companion realizations are duals of each other:
https://wolfram.com/xid/0v5g4hmcblbdp-g6t0kx
The dual of the controllable companion is the observable companion:
https://wolfram.com/xid/0v5g4hmcblbdp-8ibizj
https://wolfram.com/xid/0v5g4hmcblbdp-3aj972
A state-space model's state matrix satisfies its own characteristic polynomial:
https://wolfram.com/xid/0v5g4hmcblbdp-ihzcdo
The characteristic polynomial:
https://wolfram.com/xid/0v5g4hmcblbdp-cak2yn
It satisfies its characteristic polynomial as per the Cayley–Hamilton theorem:
https://wolfram.com/xid/0v5g4hmcblbdp-gv02nx
The characteristic polynomial of the state matrix is the denominator of a transfer function model:
https://wolfram.com/xid/0v5g4hmcblbdp-20iyjw
https://wolfram.com/xid/0v5g4hmcblbdp-v3mjwe
The eigenvalues of the state matrix determine the speed of the system's response:
https://wolfram.com/xid/0v5g4hmcblbdp-jomiq3
https://wolfram.com/xid/0v5g4hmcblbdp-39rhz5
https://wolfram.com/xid/0v5g4hmcblbdp-d2tx4j
The system response is determined by the exponents:
https://wolfram.com/xid/0v5g4hmcblbdp-toq0ie
They are the eigenvalues of the state matrix:
https://wolfram.com/xid/0v5g4hmcblbdp-tux8nr
Thus if the state matrix's eigenvalues are negative, the response decays exponentially to zero:
https://wolfram.com/xid/0v5g4hmcblbdp-jurp6w
https://wolfram.com/xid/0v5g4hmcblbdp-jywk59
https://wolfram.com/xid/0v5g4hmcblbdp-ee4u2
This is because the eigenvalues are all negative:
https://wolfram.com/xid/0v5g4hmcblbdp-gle0rx
If the eigenvalues are complex and on the left-hand plane, the response is oscillatory and decays to 0:
https://wolfram.com/xid/0v5g4hmcblbdp-bas642
Its output response contains sinusoids:
https://wolfram.com/xid/0v5g4hmcblbdp-davuje
https://wolfram.com/xid/0v5g4hmcblbdp-f8cnph
This is because its eigenvalues are a complex pair in the left-hand plane:
https://wolfram.com/xid/0v5g4hmcblbdp-lf1uc
The closer the eigenvalue pair is to the negative real axis, the more damped the oscillations:
https://wolfram.com/xid/0v5g4hmcblbdp-4a6ybi
The second model's eigenvalues are closer to the negative real axis than the first's:
https://wolfram.com/xid/0v5g4hmcblbdp-y0mjj5
The second model's response is more damped:
https://wolfram.com/xid/0v5g4hmcblbdp-4l5cho
The further the complex eigenvalue pair is from the origin, the faster the response:
https://wolfram.com/xid/0v5g4hmcblbdp-pwf9wa
The second model's eigenvalues are further away from the origin:
https://wolfram.com/xid/0v5g4hmcblbdp-tx8rus
The second model's response is faster:
https://wolfram.com/xid/0v5g4hmcblbdp-jczu9w
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:
https://wolfram.com/xid/0v5g4hmcblbdp-futncn
Its output response has undamped sinusoidal expressions:
https://wolfram.com/xid/0v5g4hmcblbdp-djcucs
https://wolfram.com/xid/0v5g4hmcblbdp-r9oyb
This is because its eigenvalues include a negative value and a pair on the imaginary axis:
https://wolfram.com/xid/0v5g4hmcblbdp-b5zwzg
If one of the eigenvalues is zero and the rest are negative, the response will have a nonzero offset:
https://wolfram.com/xid/0v5g4hmcblbdp-kfs1zm
https://wolfram.com/xid/0v5g4hmcblbdp-cm5seu
It converges on a nonzero value:
https://wolfram.com/xid/0v5g4hmcblbdp-d6kys0
This is because one of its eigenvalues is zero:
https://wolfram.com/xid/0v5g4hmcblbdp-4upz8
If multiple eigenvalues are zero, the response is unstable and diverges to ∞:
https://wolfram.com/xid/0v5g4hmcblbdp-85e532
https://wolfram.com/xid/0v5g4hmcblbdp-qo1w7k
https://wolfram.com/xid/0v5g4hmcblbdp-zu899k
This is because its eigenvalues include more than one zero:
https://wolfram.com/xid/0v5g4hmcblbdp-dv4a11
If an eigenvalue is positive, the response is unstable and diverges to ∞:
https://wolfram.com/xid/0v5g4hmcblbdp-qf00sx
https://wolfram.com/xid/0v5g4hmcblbdp-kwrpw
https://wolfram.com/xid/0v5g4hmcblbdp-c7li3x
This is because there is a positive eigenvalue:
https://wolfram.com/xid/0v5g4hmcblbdp-lm6j5
If a discrete-time model's eigenvalues are within the unit circle, its response decays to zero:
https://wolfram.com/xid/0v5g4hmcblbdp-pp6w3q
Its output response decays to zero:
https://wolfram.com/xid/0v5g4hmcblbdp-b60y3k
This is because its eigenvalues are within the unit circle:
https://wolfram.com/xid/0v5g4hmcblbdp-h4ktip
If any eigenvalue is outside the unit circle, the response is unstable:
https://wolfram.com/xid/0v5g4hmcblbdp-kiyh33
Its response is unstable and diverges to ∞:
https://wolfram.com/xid/0v5g4hmcblbdp-htmxhg
This is because not all its eigenvalues are inside the unit circle:
https://wolfram.com/xid/0v5g4hmcblbdp-n68y5w
Possible Issues (4)Common pitfalls and unexpected behavior
Nonlinearities in a model are approximated:
https://wolfram.com/xid/0v5g4hmcblbdp-d68tuk
https://wolfram.com/xid/0v5g4hmcblbdp-bew9lj
Use a nonlinear model to prevent the approximation:
https://wolfram.com/xid/0v5g4hmcblbdp-hkdt8b
Compare the linear and nonlinear step responses:
https://wolfram.com/xid/0v5g4hmcblbdp-dxx15p
The state matrix and input matrix must have the same number of rows:
https://wolfram.com/xid/0v5g4hmcblbdp-ca99df
Otherwise, the state-space model cannot be constructed:
https://wolfram.com/xid/0v5g4hmcblbdp-bku9e9
The state matrix and output matrix must have the same number of columns:
https://wolfram.com/xid/0v5g4hmcblbdp-lpx2ai
Otherwise, the state-space model cannot be constructed:
https://wolfram.com/xid/0v5g4hmcblbdp-05fxf
The transmission matrix must have the same number of rows as the output matrix and the same number of columns as the input matrix:
https://wolfram.com/xid/0v5g4hmcblbdp-bkz2rh
Otherwise, the state-space model cannot be constructed
https://wolfram.com/xid/0v5g4hmcblbdp-by5wpg
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).
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.
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
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: 10-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_statespacemodel, organization={Wolfram Research}, title={StateSpaceModel}, year={2014}, url={https://reference.wolfram.com/language/ref/StateSpaceModel.html}, note=[Accessed: 10-January-2025
]}