idpoly

Polynomial model with identifiable parameters

Description

An idpoly model represents a system as a continuous-time or discrete-time polynomial model with identifiable (estimable) coefficients. Use idpoly to create a polynomial model or to convert Dynamic System Models to polynomial form.

A polynomial model of a system with input vector u, output vector y, and disturbance e takes the following form in discrete time:

$A\left(q\right)y\left(t\right)=\frac{B\left(q\right)}{F\left(q\right)}u\left(t\right)+\frac{C\left(q\right)}{D\left(q\right)}e\left(t\right)$

The variables A, B, C, D, and F are polynomials expressed with the time-shift operator q-1. For instance, the A polynomial takes this form:

$A\left(q\right)=1+{a}_{1}{q}^{-1}+{a}_{2}{q}^{-2}+\cdots +{a}_{na}{q}^{-na}$

Here, na is the order of the A polynomial. q-1y(t) is equivalent to y(t-1).

For example, if A(q) = 1 + a1q-1 + a2q-2, then A(y(t)) = 1 + a1(t-1) + a2(t-2).

The C, D, and F polynomials take the same form as the A polynomial, starting with 1. The B polynomial does not start with 1.

In continuous time, a polynomial model takes the following form:

$A\left(s\right)Y\left(s\right)=\frac{B\left(s\right)}{F\left(s\right)}U\left(s\right)+\frac{C\left(s\right)}{D\left(s\right)}E\left(s\right)$

U(s) contains the Laplace transformed inputs to sys. Y(s) contains the Laplace transformed outputs. E(s) contains the Laplace transform of the disturbances for each output.

For idpoly models, the coefficients of the polynomials A, B, C, D, and F can be estimable parameters. The idpoly model stores the values of these matrix elements in the A, B, C, D, and F properties of the model.

Time-series models are special cases of polynomial models for systems without measured inputs. For AR models, B and F are empty, and C and D are 1 for all outputs. For ARMA models, B and F are empty, while D is 1.

Although idpoly supports continuous-time models, idtf and idproc enable more choices for estimation of continuous-time models. Therefore, for most continuous-time applications, these other model types are preferable.

Creation

You can obtain an idpoly model in one of three ways.

• Estimate the idpoly model based on output or input-output measurements of a system by using commands such as polyest, arx, armax, oe, bj, iv4, or ivar. These commands estimate the values of the free polynomial coefficients. The estimated values are stored in the A, B, C, D, and F properties of the resulting idpoly model. The Report property of the resulting model stores information about the estimation, such as information on the handling of initial conditions and options used in estimation.

When you obtain an idpoly model by estimation, you can extract estimated coefficients and their uncertainties from the model using commands such as polydata, getpar, or getcov.

• Create an idpoly model using the idpoly command. You can create an idpoly model to configure an initial parameterization for estimation of a polynomial model to fit measured response data. When you do so, you can specify constraints on the polynomial coefficients. For example, you can fix the values of some coefficients, or specify minimum or maximum values for the free coefficients. You can then use the configured model as an input argument to polyest to estimate parameter values with those constraints.

• Convert an existing dynamic system model to an idpoly model using the idpoly command.

Description

Create Input-Output Polynomial Model

example

sys = idpoly(A,B,C,D,F,NoiseVariance,Ts) creates a polynomial model with identifiable coefficients. A, B, C, D, and F specify the initial values of the coefficients. NoiseVariance specifies the initial value of the variance of the white noise source. Ts is the model sample time.

example

sys = idpoly(A,B,C,D,F,NoiseVariance,Ts,Name,Value) creates a polynomial model using additional options specified by one or more name-value pair arguments.

Create Time-Series Model

sys = idpoly(A) creates a time-series model with only an autoregressive term. In this case, sys represents the AR model given by A(q) y(t) = e(t). The noise e(t) has variance 1. A specifies the initial values of the estimable coefficients.

example

sys = idpoly(A,[],C,D,[],NoiseVariance,Ts) creates a time-series model with an autoregressive and a moving average term. The inputs A, C, and D, specify the initial values of the estimable coefficients. NoiseVariance specifies the initial value of the noise e(t). Ts is the model sample time. (Omit NoiseVariance and Ts to use their default values.)

If D is set to [], then sys represents the ARMA model given by

$A\left(q\right)y\left(t\right)=C\left(q\right)e\left(t\right)$

example

sys = idpoly(A,[],C,D,[],NoiseVariance,Ts,Name,Value) creates a time-series model using additional options specified by one or more name-value pair arguments.

Convert Dynamic System Model to Polynomial Model

example

sys = idpoly(sys0) converts the dynamic system model sys0 to idpoly model form. sys0 can be any dynamic system model.

sys = idpoly(sys0,'split') converts sys0 to idpoly model form, and treats the last Ny input channels of sys0 as noise channels in the returned model. sys0 must be a numeric tf (Control System Toolbox), zpk (Control System Toolbox), or ss (Control System Toolbox) model object. Also, sys0 must have at least as many inputs as outputs.

Input Arguments

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Dynamic system, specified as a dynamic system model to convert to an idpoly model.

When sys0 is an identified model, its estimated parameter covariance is lost during conversion. If you want to translate the estimated parameter covariance during the conversion, use translatecov.

For the syntax sys = idpoly(sys0,'split'), sys0 must meet the following requirements.

• sys0 is a numeric tf (Control System Toolbox), zpk (Control System Toolbox), or ss (Control System Toolbox) model object.

• sys0 has at least as many inputs as outputs.

• The subsystem sys0(:,Ny+1:Nu) must be biproper.

Properties

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Values of the polynomial coefficients, specified as row vectors for SISO models or, for MIMO models, cell arrays of row vectors that correspond to each of the A, B, C, D, and F polynomials. For each polynomial, the coefficients are stored in the following order:

• Ascending powers of z–1 or q–1 (for discrete-time polynomial models).

• Descending powers of s or p (for continuous-time polynomial models).

The leading coefficients of A, C, D, and F are always 1.

For MIMO models with Ny outputs and Nu inputs, A, B, C, D, and F are cell arrays of row vectors. Each entry in the cell array contains the coefficients of a particular polynomial that relates input, output, and noise values.

PolynomialDimension and ConstraintsRelation Described
A

Ny-by-Ny cell array of row vectors

• Diagonal entries — Fixed to 1

• Off-diagonal entries — Fixed to 0

A{i,j} contains coefficients that relate the output yi to the output yj.

For example, for a two-output system, A is a 2-by-2 cell array, such as:

A{1,1} = [1 .1]
A{1,2} = [0.4 -0.6]
A{2,1} = 0
A{2,2} = [1 0.2 0.3]

B,F

Ny-by-Nu array of row vectors

• B — Not constrained

• F — Fixed to 1

B{i,j} and F{i,j} contain coefficients that relate the output yi to the input uj.

For example, for a two-output system, B and F are 2-by-1 cell arrays, such as:

B{1,1} = [0.1 0.2]
B{2,1} = [0.5 0.3]
F{1,1} = [1 0.8]
F{2,1} = [1 0.4]

C,D

Ny-by-1 array of row vectors

• C — Fixed to 1

• D — Fixed to 1

C{i} and D{i} contain coefficients that relate the output yi to the noise ei.

For example, for a two-output system, C and D are 2-by-1 cell arrays, such as:

C{1,1} = [1 0.3]
C{2,1} = [1 0.5 0.3]
D{1,1} = [1 0.7]
D{2,1} = [1 0.1 0.2]

For a time series model (a model with no measured inputs), B = [] and F = [].

If you obtain an idpoly model sys by identification using a function such as polyest or arx, then sys.A, sys.B, sys.C, sys.D, and sys.F contain the estimated values of the polynomial coefficients.

If you create an idpoly model sys using the idpoly command, sys.A, sys.B, sys.C, sys.D, and sys.F contain the initial coefficient values that you specify with the A,B,C,D,F input arguments. Use NaN for any coefficient whose initial value is not known. Use [] for any polynomial that is not present in the model structure that you want to create. For example, to create an ARX model, use [] for C, D, and F. For an ARMA time series model, use [] for B and F. Default initial values when you create an idpoly model are:

• B — []

• C — 1 for all outputs

• D — 1 for all outputs

• F — []

For an idpoly model sys, each property sys.A, sys.B, sys.C, sys.D, and sys.F is an alias of the corresponding Value entry in the Structure property of sys. For example, sys.A is an alias of the value of the property sys.Structure.A.Value.

Polynomial model display variable, specified as one of the following values:

• 'z^-1' — Default for discrete-time models

• 'q^-1' — Equivalent to 'z^-1'

• 's' — Default for continuous-time models

• 'p' — Equivalent to 's'

The value of Variable is reflected in the display, and also affects the interpretation of the A, B, C, D, and F coefficient vectors for discrete-time models. When Variable is set to 'z^-1' or 'q^-1', the coefficient vectors are ordered as ascending powers of the variable.

Transport delays, specified as a numeric array containing a separate transport delay for each input-output pair or as a scalar that applies the same delay to each input-output pair.

For continuous-time systems, transport delays are expressed in the time unit stored in the TimeUnit property. For discrete-time systems, transport delays are expressed as integers denoting a delay of a multiple of the sample time Ts.

For a MIMO system with Ny outputs and Nu inputs, IODelay is an Ny-by-Nu array. Each entry of this array is a numerical value representing the transport delay for the corresponding input-output pair. You can set IODelay to a scalar value to apply the same delay to all input-output pairs.

If you create an idpoly model sys using the idpoly command, sys.IODelay contains the initial values of the transport delay that you specify with a name-value pair argument.

If you obtain an idpoly model sys by identification using a function such as polyest or arx, then sys.IODelay contains the estimated values of the transport delay.

For an idpoly model sys, the property sys.IODelay is an alias for the value of the property sys.Structure.IODelay.Value.

Logical vector denoting the presence or absence of integration on noise channels, specified as a logical vector with length equal to the number of outputs.

IntegrateNoise(i) = true indicates that the noise channel for the ith output contains an integrator. In this case, the corresponding D polynomial contains an additional term that is not represented in the property sys.D. This integrator term is equal to 1/s for continuous-time systems and 1/(1-z-1) for discrete-time systems.

Property-specific information about the estimable parameters of the idpoly model, specified as a structure.

For a system with Ny outputs and Nu inputs, the dimensions of the Structure elements are as follows:

• sys.Structure.ANy-by-Ny

• sys.Structure.BNy-by-Nu

• sys.Structure.CNy-by-1

• sys.Structure.DNy-by-1

• sys.Structure.FNy-by-Nu

sys.Structure.A, sys.Structure.B, sys.Structure.C, sys.Structure.D, and sys.Structure.F contain information about the polynomial coefficients. sys.Structure.IODelay contains information about the transport delay. sys.Structure.IntegrateNoise contains information about the integration terms on the noise. Each parameter in Structure contains the following fields.

FieldDescriptionExamples
ValueParameter values. Each property is an alias of the corresponding Value entry in the Structure property of sys. NaN represents unknown parameter values.sys.Structure.A.Value contains the initial or estimated values of the SISO A polynomial. sys.A is an alias of the value of this property. sys.A{i,j} is the alias of the MIMO property sys.Structure.A(i,j).Value.
MinimumMinimum value that the parameter can assume during estimation sys.Structure.IODelay.Minimum = 0.1 constrains the transport delay to values greater than or equal to 0.1. sys.Structure.IODelay.Minimum must be greater than or equal to zero.
MaximumMaximum value that the parameter can assume during estimation
FreeBoolean specifying whether the parameter is a free estimation variable. If you want to fix the value of a parameter during estimation, set the corresponding Free to false. For fixed values, such as the leading coefficients of the values of A polynomial, which are always equal to 1, the corresponding value of Free is always false.If B is a 3-by-3 matrix,sys.Structure.B.Free = eye(3) fixes all of the off-diagonal entries in B to the values specified in sys.Structure.B.Value. In this case, only the diagonal entries in B are estimable.
ScaleScale of the value of the parameter. The estimation algorithm does not use Scale.
InfoStructure array that contains the fields Label and Unit for storing parameter labels and units. Specify parameter labels and units as character vectors.Example: 'Time'

An inactive polynomial, such as the B polynomial in a time series model, is not available as a parameter in the Structure property. For example, sys = idpoly([1 -0.2 0.5]) creates an AR model. sys.Structure contains the fields sys.Structure.A and sys.Structure.IntegrateNoise. However, there is no field in Structure corresponding to B, C, D, F, or IODelay.

Variance (covariance matrix) of the model innovations e, specified as a scalar or a positive semidefinite matrix.

• SISO model — Scalar

• MIMO model with Ny outputs — Ny-by-Ny positive semidefinite matrix

An identified model includes a white Gaussian noise component e(t). NoiseVariance is the variance of this noise component. Typically, the model estimation function (such as polyest) determines this variance.

Summary report that contains information about the estimation options and results for a state-space model obtained using estimation commands, such as polyest, armax, oe, and bj. Use Report to find estimation information for the identified model, including:

• Estimation method

• Estimation options

• Search termination conditions

• Estimation data fit and other quality metrics

If you create the model by construction, the contents of Report are irrelevant.

m = idpoly({[1 0.5]},{[1 5]},{[1 0.01]});
m.Report.OptionsUsed
ans =

[]

If you obtain the model using estimation commands, the fields of Report contain information on the estimation data, options, and results.

m = polyest(z2,[2 2 3 3 2 1]);
m.Report.OptionsUsed
Option set for the polyest command:

InitialCondition: 'auto'
Focus: 'prediction'
EstimateCovariance: 1
Display: 'off'
InputOffset: []
OutputOffset: []
Regularization: [1x1 struct]
SearchMethod: 'auto'
SearchOptions: [1x1 idoptions.search.identsolver]

For more information on this property and how to use it, see the Output Arguments section of the corresponding estimation command reference page and Estimation Report.

Input delay for each input channel, specified as a scalar value or numeric vector. For continuous-time systems, specify input delays in the time unit stored in the TimeUnit property. For discrete-time systems, specify input delays in integer multiples of the sample time Ts. For example, setting InputDelay to 3 specifies a delay of three sample times.

For a system with Nu inputs, set InputDelay to an Nu-by-1 vector. Each entry of this vector is a numerical value that represents the input delay for the corresponding input channel.

You can also set InputDelay to a scalar value to apply the same delay to all channels.

In estimation, InputDelay is a fixed constant of the model. The software uses the IODelay property for estimating time delays. To specify initial values and constraints for estimation of time delays, use sys.Structure.IODelay.

Output delay for each output channel, specified as 0. This value is fixed for identified systems such as idpoly.

Sample time, specified as one of the following.

• Discrete-time model with an unspecified sample time — -1

• Continuous-time model — 0

• Discrete-time model with a specified sampling time — Positive scalar representing the sampling period expressed in the unit specified by the TimeUnit property of the model

Changing this property does not discretize or resample the model. Use c2d and d2c to convert between continuous- and discrete-time representations. Use d2d to change the sample time of a discrete-time system.

Units for the time variable, the sample time Ts, and any time delays in the model, specified as a scalar.

Changing this property does not resample or convert the data. Modifying the property changes only the interpretation of the existing data. Use chgTimeUnit to convert data to different time units

Input channel names, specified as a character vector or cell array.

• Single-input model — Character vector, for example, 'controls'

• Multi-input model — Cell array of character vectors

Alternatively, use automatic vector expansion to assign input names for multi-input models. For example, if sys is a two-input model, enter the following:

sys.InputName = 'controls';

The input names automatically expand to {'controls(1)';'controls(2)'}.

When you estimate a model using an iddata object data, the software automatically sets InputName to data.InputName.

You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName.

You can use input channel names in several ways, including:

• To identify channels on model display and plots.

• To extract subsystems of MIMO systems.

• To specify connection points when interconnecting models.

Input channel units, specified as a character vector or cell array:

• Single-input model — Character vector

• Multi-input Model — Cell array of character vectors

Use InputUnit to keep track of input signal units. InputUnit has no effect on system behavior.

Input channel groups, specified as a structure. The InputGroup property lets you divide the input channels of MIMO systems into groups so that you can refer to each group by name. In the InputGroup structure, set field names to the group names, and field values to the input channels belonging to each group.

For example, create input groups named controls and noise that include input channels 1, 2 and 3, 5, respectively.

sys.InputGroup.controls = [1 2];
sys.InputGroup.noise = [3 5];

You can then extract the subsystem from the controls inputs to all outputs using the following syntax:

sys(:,'controls')

Output channel names, specified as a character vector or cell array.

• Single-input model — Character vector, for example, 'measurements'

• Multi-input model — Cell array of character vectors

Alternatively, use automatic vector expansion to assign output names for multi-output models. For example, if sys is a two-output model, enter the following:

sys.OutputName = 'measurements';

The output names automatically expand to {'measurements(1)';'measurements(2)'}.

When you estimate a model using an iddata object data, the software automatically sets OutputName to data.OutputName.

You can use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName.

You can use output channel names in several ways, including:

• To identify channels on model display and plots.

• To extract subsystems of MIMO systems.

• To specify connection points when interconnecting models.

Output channel units, specified as a character vector or cell array.

• Single-input model — Character vector, for example, 'seconds'

• Multi-input Model — Cell array of character vectors

Use OutputUnit to keep track of output signal units. OutputUnit has no effect on system behavior.

Output channel groups, specified as a structure. The OutputGroup property lets you divide the output channels of MIMO systems into groups and refer to each group by name. In the OutputGroup structure, set field names to the group names, and field values to the output channels belonging to each group.

For example, create output groups named temperature and measurement that include output channels 1, and 3, 5, respectively.

sys.OutputGroup.temperature = ;
sys.OutputGroup.measurement = [3 5];

You can then extract the subsystem from all inputs to the measurement outputs using the following syntax:

sys('measurement',:)

System name, specified as a character vector, for example, 'system_1'.

Any text that you want to associate with the system, specified as a string.

• For a single note, specify Notes as a string or a character vector

• For multiple notes, specify Notes as a string array.

The property preserves the string or character data type that you specify. When you specify a character vector, the software packages the character vector in a 1-by-1 cell array.

For example, if sys1, sys2, and sys3 are dynamic system models, you can set their Notes properties as follows.

sys1.Notes = "sys1 has a string.";
sys2.Notes = ["sys2 has a first string";"sys3 has a second string"];
sys3.Notes = 'sys3 has a character vector.';
sys1.Notes
sys2.Notes
sys3.Notes
ans =

"sys1 has a string."

ans =

2×1 string array

"sys2 has a first string"
"sys2 has a second string"

ans =

1×1 cell array

{'sys3 has a character vector'}

Data to associate with the system, specified as any MATLAB data type.

Sampling grid for model arrays, specified as a structure.

For arrays of identified linear (IDLTI) models that you derive by sampling one or more independent variables, this property tracks the variable values associated with each model. This information appears when you display or plot the model array. Use this information to trace results back to the independent variables.

Set the field names of the data structure to the names of the sampling variables. Set the field values to the sampled variable values associated with each model in the array. All sampling variables must be numeric and scalar valued, and all arrays of sampled values must match the dimensions of the model array.

For example, suppose that you collect data at various operating points of a system. You can identify a model for each operating point separately and then stack the results together into a single system array. You can tag the individual models in the array with information regarding the operating point.

nominal_engine_rpm = [1000 5000 10000];
sys.SamplingGrid = struct('rpm', nominal_engine_rpm)

Here, sys is an array containing three identified models obtained at 1000, 5000, and 10000 rpm, respectively.

For model arrays that you generate by linearizing a Simulink® model at multiple parameter values or operating points, the software populates SamplingGrid automatically with the variable values that correspond to each entry in the array.

Object Functions

In general, any function applicable to Dynamic System Models is applicable to an idpoly model object. These functions are of four general types.

• Functions that operate and return idpoly model objects enable you to transform and manipulate idpoly models. For instance:

• Use merge to merge estimated idpoly models.

• Use c2d to convert an idpoly model from continuous to discrete time. Use d2c to convert an idpoly model from discrete to continuous time.

• Functions that perform analytical and simulation functions on idpoly models, such as bode and sim

• Functions that retrieve or interpret model information, such as advice and getpar

• Functions that convert idpoly models into a different model type, such as idtf for time domain or idfrd for frequency domain

The following lists contain a representative subset of the functions that you can use with idpoly models.

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 translatecov Translate parameter covariance across model transformation operations setpar Set attributes such as values and bounds of linear model parameters chgTimeUnit Change time units of dynamic system d2d Resample discrete-time model d2c Convert model from discrete to continuous time c2d Convert model from continuous to discrete time merge Merge estimated models

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 sim Simulate response of identified model predict Predict state and state estimation error covariance at next time step using extended or unscented Kalman filter, or particle filter compare Compare identified model output and measured output impulse Impulse response plot of dynamic system; impulse response data step Step response plot of dynamic system; step response data bode Bode plot of frequency response, or magnitude and phase data

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 tfdata Access transfer function data get Access model property values getpar Obtain attributes such as values and bounds of linear model parameters getcov Parameter covariance of identified model advice Analysis and recommendations for data or estimated linear models

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 idtf Transfer function model with identifiable parameters idss State-space model with identifiable parameters idfrd Frequency response data or model

Examples

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Create an idpoly model representing the single-input, single-output ARMAX model described by the following equation:

$\mathit{y}\left(\mathit{t}\right)+0.5\mathit{y}=\mathit{u}\left(\mathit{t}\right)+5\mathit{u}\left(\mathit{t}-1\right)+2\mathit{u}\left(\mathit{t}-2\right)+\mathit{e}\left(\mathit{t}\right)+0.01\mathit{e}\left(\mathit{t}-1\right)$

$\mathit{y}$ is the output, $\mathit{u}$ is the input, and $\mathit{e}$ is the white-noise disturbance on $\mathit{y}$.

To create the idpoly model, define the A, B, and C polynomials that describe the relationships between the output, input, and noise values, respectively. Because there are no denominator terms in the system equation, D and F are 1.

A = [1 0.5];
B = [1 5 2];
C = [1 0.01];

Create an idpoly model with the specified coefficients.

sys = idpoly(A,B,C)
sys =
Discrete-time ARMAX model: A(z)y(t) = B(z)u(t) + C(z)e(t)
A(z) = 1 + 0.5 z^-1

B(z) = 1 + 5 z^-1 + 2 z^-2

C(z) = 1 + 0.01 z^-1

Sample time: unspecified

Parameterization:
Polynomial orders:   na=1   nb=3   nc=1   nk=0
Number of free coefficients: 5
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Created by direct construction or transformation. Not estimated.

The display shows all the polynomials and allows you to verify them. The display also states that there are five free coefficients.

Create an idpoly model with specified noise variance nv and sample time Ts. To do so, you must also include values of 1 for D and F.

Ts = 0.1;
nv = 0.01;
sys = idpoly(A,B,C,1,1,nv,Ts)
sys =
Discrete-time ARMAX model: A(z)y(t) = B(z)u(t) + C(z)e(t)
A(z) = 1 + 0.5 z^-1

B(z) = 1 + 5 z^-1 + 2 z^-2

C(z) = 1 + 0.01 z^-1

Sample time: 0.1 seconds

Parameterization:
Polynomial orders:   na=1   nb=3   nc=1   nk=0
Number of free coefficients: 5
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Created by direct construction or transformation. Not estimated.

The display shows a sample time of 0.1 seconds.

Specify an input-output delay iod of one sample when you create an idpoly model.

iod = 1;
sys = idpoly(A,B,C,1,1,nv,Ts,'IODelay',1)
sys =
Discrete-time ARMAX model: A(z)y(t) = B(z)u(t) + C(z)e(t)
A(z) = 1 + 0.5 z^-1

B(z) = 1 + 5 z^-1 + 2 z^-2

C(z) = 1 + 0.01 z^-1

Input delays (listed by channel): 1
Sample time: 0.1 seconds

Parameterization:
Polynomial orders:   na=1   nb=3   nc=1   nk=0
Number of free coefficients: 5
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Created by direct construction or transformation. Not estimated.

The display shows an input delay of one sample.

You can use sys to specify an initial parameterization for estimation with commands such as polyest or armax.

Create an idpoly model representing the single-output ARMA model described by the following equation:

$\mathit{y}\left(\mathit{t}\right)+0.5\mathit{y}=\mathit{e}\left(\mathit{t}\right)+0.01\mathit{e}\left(\mathit{t}-1\right)$

Because a time series has no measured inputs, this model contains only A and C polynomials.

A = [1 0.5];
C = [1 0.01];

Create a discrete-time time-series model without specifying a sample time.

sys = idpoly(A,[],C)
sys =
Discrete-time ARMA model: A(z)y(t) = C(z)e(t)
A(z) = 1 + 0.5 z^-1

C(z) = 1 + 0.01 z^-1

Sample time: unspecified

Parameterization:
Polynomial orders:   na=1   nc=1
Number of free coefficients: 2
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Created by direct construction or transformation. Not estimated.

Create a continuous-time time-series by specifying a sample time of 0 for the name-value pair argument 'Ts'.

sys = idpoly(A,[],C,'Ts',0)
sys =
Continuous-time ARMA model: A(s)y(t) = C(s)e(t)
A(s) = s + 0.5

C(s) = s + 0.01

Parameterization:
Polynomial orders:   na=1   nc=1
Number of free coefficients: 2
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Created by direct construction or transformation. Not estimated.

You can also set the sample time using the Ts input argument rather than the name-value pair argument, but the syntax is more complex. You must specify the D value as 1 or empty, and set both the F position and the noise variance position (if you are not specifying noise variance) to empty.

Ts = 0;
sys = idpoly(A,[],C,1,[],[],Ts)
sys =
Continuous-time ARMA model: A(s)y(t) = C(s)e(t)
A(s) = s + 0.5

C(s) = s + 0.01

Parameterization:
Polynomial orders:   na=1   nc=1
Number of free coefficients: 2
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Created by direct construction or transformation. Not estimated.

Create an idpoly model representing the one-input, two-output ARMAX model described by the following equations:

$\begin{array}{l}{y}_{1}\left(t\right)+0.5{y}_{1}\left(t-1\right)+0.9{y}_{2}\left(t-1\right)+0.1{y}_{2}\left(t-2\right)=\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}u\left(t\right)+5u\left(t-1\right)+2u\left(t-2\right)+{e}_{1}\left(t\right)+0.01{e}_{1}\left(t-1\right)\\ {y}_{2}\left(t\right)+0.05{y}_{2}\left(t-1\right)+0.3{y}_{2}\left(t-2\right)=\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}10u\left(t-2\right)+{e}_{2}\left(t\right)+0.1{e}_{2}\left(t-1\right)+0.02{e}_{2}\left(t-2\right).\end{array}$

${y}_{1}$ and ${y}_{2}$ are the two outputs, and $u$ is the input. ${e}_{1}$ and ${e}_{2}$ are the white noise disturbances on the outputs ${y}_{1}$ and ${y}_{2}$, respectively.

To create the idpoly model, define the A, B, and C polynomials that describe the relationships between the outputs, inputs, and noise values. (Because there are no denominator terms in the system equations, D and F are 1.)

Define the cell array containing the coefficients of the A polynomials.

A = cell(2,2);
A{1,1} = [1 0.5];
A{1,2} = [0 0.9 0.1];
A{2,1} = ;
A{2,2} = [1 0.05 0.3];

You can read the values of each entry in the A cell array from the left side of the equations describing the system. For example, A{1,1} describes the polynomial that gives the dependence of ${y}_{1}$ on itself. This polynomial is ${A}_{11}=1+0.5{q}^{-1}$, because each factor of ${q}^{-1}$ corresponds to a unit time decrement. Therefore, A{1,1} = [1 0.5], giving the coefficients of ${A}_{11}$ in increasing exponents of ${q}^{-1}$.

Similarly, A{1,2} describes the polynomial that gives the dependence of ${y}_{1}$ on ${y}_{2}$. From the equations, ${A}_{12}=0+0.9{q}^{-1}+0.1{q}^{-2}$. Thus, A{1,2} = [0 0.9 0.1].

The remaining entries in A are similarly constructed.

Define the cell array containing the coefficients of the B polynomials.

B = cell(2,1);
B{1,1} = [1 5 2];
B{2,1} = [0 0 10];

B describes the polynomials that give the dependence of the outputs ${y}_{1}$ and ${y}_{2}$ on the input $u$. From the equations, ${B}_{11}=1+5{q}^{-1}+2{q}^{-2}$. Therefore, B{1,1} = [1 5 2].

Similarly, from the equations, ${B}_{21}=0+0{q}^{-1}+10{q}^{-2}$. Therefore, B{2,1} = [0 0 10].

Define the cell array containing the coefficients of the C polynomials.

C = cell(2,1);
C{1,1} = [1 0.01];
C{2,1} = [1 0.1 0.02];

C describes the polynomials that give the dependence of the outputs ${y}_{1}$ and ${y}_{2}$ on the noise terms ${e}_{1}$ and ${e}_{2}$. The entries of C can be read from the equations similarly to those of A and B.

Create an idpoly model with the specified coefficients.

sys = idpoly(A,B,C)
sys =
Discrete-time ARMAX model:
Model for output number 1: A(z)y_1(t) = - A_i(z)y_i(t) + B(z)u(t) + C(z)e_1(t)
A(z) = 1 + 0.5 z^-1

A_2(z) = 0.9 z^-1 + 0.1 z^-2

B(z) = 1 + 5 z^-1 + 2 z^-2

C(z) = 1 + 0.01 z^-1

Model for output number 2: A(z)y_2(t) = B(z)u(t) + C(z)e_2(t)
A(z) = 1 + 0.05 z^-1 + 0.3 z^-2

B(z) = 10 z^-2

C(z) = 1 + 0.1 z^-1 + 0.02 z^-2

Sample time: unspecified

Parameterization:
Polynomial orders:   na=[1 2;0 2]   nb=[3;1]   nc=[1;2]
nk=[0;2]
Number of free coefficients: 12
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Created by direct construction or transformation. Not estimated.

The display shows all the polynomials and allows you to verify them. The display also states that there are 12 free coefficients. Leading terms of diagonal entries in A are always fixed to 1. Leading terms of all other entries in A are always fixed to 0.

You can use sys to specify an initial parameterization for estimation with commands such as polyest or armax.

Model a dynamic system using a transfer function. Then use idpoly to convert the transfer-function model into polynomial form.

Using idtf, construct the continuous-time, single-input, single-output (SISO) transfer function model described by the following equation:

$\mathit{G}\left(\mathit{s}\right)=\frac{\mathit{s}+4}{{\mathit{s}}^{2}+20\mathit{s}+5}$

num = [1 4];
den = [1 20 5];
G = idtf(num,den)
G =

s + 4
--------------
s^2 + 20 s + 5

Continuous-time identified transfer function.

Parameterization:
Number of poles: 2   Number of zeros: 1
Number of free coefficients: 4
Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Created by direct construction or transformation. Not estimated.

Convert the transfer function into polynomial form.

sys = idpoly(G)
sys =
Continuous-time OE model: y(t) = [B(s)/F(s)]u(t) + e(t)
B(s) = s + 4

F(s) = s^2 + 20 s + 5

Parameterization:
Polynomial orders:   nb=2   nf=2   nk=0
Number of free coefficients: 4
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Created by direct construction or transformation. Not estimated.

The display shows the polynomial form and the polynomial coefficients. 