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idpoly

Polynomial model with identifiable parameters

Syntax

sys = idpoly(A,B,C,D,F,NoiseVariance,Ts)
sys = idpoly(A,B,C,D,F,NoiseVariance,Ts,Name,Value)
sys = idpoly(A)
sys = idpoly(A,[],C,D,[],NoiseVariance,Ts)
sys = idpoly(A,[],C,D,[],NoiseVariance,Ts,Name,Value)
sys = idpoly(sys0)
sys = idpoly(sys0,'split')

Description

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. 
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.
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–1) y(t) = e(t). The noise e(t) has variance 1. A specifies the initial values of the estimable coefficients. 
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 = [], then sys represents the ARMA model given by:$A\left({q}^{-1}\right)y\left(t\right)=C\left({q}^{-1}\right)e\left(t\right).$
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.
sys = idpoly(sys0) converts any dynamic system model, sys0, to idpoly model form. 
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 (nonidentified) tf, zpk, or ss model object. Also, sys0 must have at least as many inputs as outputs.

Object Description

An idpoly model represents a system as a continuous-time or discrete-time polynomial model with identifiable (estimable) coefficients.

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

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

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) are the Laplace transformed inputs to sys. Y(s) are the Laplace transformed outputs. E(s) is the Laplace transform of the disturbance.

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.

There are three ways to obtain an idpoly model:

• Estimate the idpoly model based on output or input-output measurements of a system, 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 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.

Examples

collapse all

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

 

and are the two outputs, and is the input. and are the white noise disturbances on the outputs and 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, B 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} = [0]; 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 on itself. This polynomial is , because each factor of corresponds to a unit time decrement. Therefore, A{1,1} = [1 0.5], giving the coefficients of in increasing exponents of .

Similarly, A{1,2} describes the polynomial that gives the dependence of on . From the equations, . 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 and on the input . From the equations, . Therefore, B{1,1} = [1 5 2].

Similarly, from the equations, . 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 and on the noise terms and . 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 parametrization for estimation with such commands as polyest or armax.

Input Arguments

A,B,C,D,F

Initial values of polynomial coefficients.

For SISO models, specify the initial values of the polynomial coefficients as row vectors. Specify the coefficients in order of:

• 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 must be 1. Use NaN for any coefficient whose initial value is not known.

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.

PolynomialDimensionRelation Described
ANy-by-Ny array of row vectorsA{i,j} contains coefficients of relation between output yi and output yj
B,FNy-by-Nu array of row vectorsB{i,j} and F{i,j}contain coefficients of relations between output yi and input uj
C,DNy-by-1 array of row vectorsC{i} and D{i}contain coefficients of relations between output yi and noise ei

The leading coefficients of the diagonal entries of A (A{i,i},i=1:Ny) must be 1. The leading coefficients of the off-diagonal entries of A must be zero, for causality. The leading coefficients of all entries of C, D, and F , must be 1.

Use [] for any polynomial that is not present in the wanted model structure. For example, to create an ARX model, use [] for C, D, and F. For an ARMA time series, use [] for B and F.

Default: B = []; C = 1 for all outputs; D = 1 for all outputs; F = []

Ts

Sample time. For continuous-time models, Ts = 0. For discrete-time models, Ts is a positive scalar representing the sample time expressed in the unit specified by the TimeUnit property of the model. To denote a discrete-time model with unspecified sample time, set Ts = -1.

Default: –1 (discrete-time model with unspecified sample time)

NoiseVariance

The variance (covariance matrix) of the model innovations e.

An identified model includes a white, Gaussian noise component e(t). NoiseVariance is the variance of this noise component. Typically, a model estimation function (such as polyest) determines this variance. Use this input to specify an initial value for the noise variance when you create an idpoly model.

For SISO models, NoiseVariance is a scalar. For MIMO models, NoiseVariance is a Ny-by-Ny matrix, where Ny is the number of outputs in the system.

Default: Ny-by-Ny identity matrix

sys0

Dynamic system.

Any dynamic system to be converted into an idpoly object.

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 be a numeric (non-identified) tf, zpk, or ss model object. Also, sys0 must have at least as many inputs as outputs. Finally, the subsystem sys0(:,Ny+1:Nu) must be biproper.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Use Name,Value arguments to specify additional properties of idpoly models during model creation. For example, idpoly(A,B,C,D,F,1,0,'InputName','Voltage') creates an idpoly model with the InputName property set to Voltage.

Properties

idpoly object properties include:

A,B,C,D,F

Values of 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, and F input arguments, respectively.

If you obtain an idpoly model by identification, then sys.A, sys.B, sys.C, sys.D, and sys.F contain the estimated values of the coefficients.

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

For SISO polynomial models, the values of the numerator coefficients are stored as a row vector in order of:

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

• Descending powers of s or p (for continuous-time transfer functions).

The leading coefficients of A, C, and D are fixed to 1. Any coefficient whose initial value is not known is stored as NaN.

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.

PolynomialDimensionRelation Described
ANy-by-Ny array of row vectorsA{i,j} contains coefficients of relation between output yi and output yj
B,FNy-by-Nu array of row vectorsB{i,j} and F{i,j}contain coefficients of relations between output yi and input uj
C,DNy-by-1 array of row vectorsC{i} and D{i}contain coefficients of relations between output yi and noise ei

The leading coefficients of the diagonal entries of A (A{i,i}, i=1:Ny) are fixed to 1. The leading coefficients of the off-diagonal entries of A are fixed to zero. The leading coefficients of all entries of C, D, and F , are fixed to 1.

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

Default: B = []; C = 1 for all outputs; D = 1 for all outputs; F = []

Variable

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. For Variable = 'z^-1' or 'q^-1', the coefficient vectors are ordered as ascending powers of the variable.

IODelay

Transport delays. IODelay is a numeric array specifying a separate transport delay for each input/output pair.

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 argument pair.

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

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 delay of a multiple of the sample time Ts.

For a MIMO system with Ny outputs and Nu inputs, set IODelay is a Ny-by-Nu array, where each entry 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.

Default: 0 for all input/output pairs

IntegrateNoise

Logical vector, denoting presence or absence of integration on noise channels.

Specify IntegrateNoise as a logical vector of 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 which is not represented in the property sys.D. This integrator term is equal to [1 0] for continuous-time systems, and equal to [1 -1] for discrete-time systems.

Default: 0 for all output channels

Structure

Information about the estimable parameters of the idpoly model. 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 contain information about the integration terms on the noise. Each contains the following fields:

• Value — Parameter values. For example, sys.Structure.A.Value contains the initial or estimated values of the A coefficients.

NaN represents unknown parameter values.

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

For MIMO models, sys.A{i,j} is an alias to sys.Structure.A(i,j).Value, and similarly for the other identifiable coefficient values.

• Minimum — Minimum value that the parameter can assume during estimation. For example, 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.

• Maximum — Maximum value that the parameter can assume during estimation.

• Free — Logical value 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 = false. For example, if B is a 3-by-3 matrix, sys.Structure.B.Free = eyes(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.

For fixed values, such as the leading coefficients in sys.Structure.B.Value, the corresponding value of Free is always false.

• Scale — Scale of the parameter's value. Scale is not used in estimation.

• Info — Structure array for storing parameter units and labels. The structure has Label and Unit fields.

Specify parameter units and labels as character vectors. For example, 'Time'.

For a MIMO model 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

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, sys.Structure.IODelay, and sys.Structure.IntegrateNoise. However, there is no field in sys.Structure corresponding to B, C, D, or F.

NoiseVariance

The variance (covariance matrix) of the model innovations e.

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 arx) determines this variance.

For SISO models, NoiseVariance is a scalar. For MIMO models, NoiseVariance is a Ny-by-Ny matrix, where Ny is the number of outputs in the system.

Report

Summary report that contains information about the estimation options and results when the polynomial model is obtained using estimation commands, such as polyest, armax, oe, and bj. Use Report to query a model for how it was estimated, including its:

• Estimation method

• Estimation options

• Search termination conditions

• Estimation data fit and other quality metrics

The contents of Report are irrelevant if the model was created by construction.

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

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

load iddata2 z2; m = polyest(z2,[2 2 3 3 2 1]); m.Report.OptionsUsed
Option set for the polyest command: InitialCondition: 'auto' Focus: 'prediction' EstCovar: 1 Display: 'off' InputOffset: [] OutputOffset: [] Regularization: [1x1 struct] SearchMethod: 'auto' SearchOption: [1x1 idoptions.search.identsolver] Advanced: [1x1 struct]

Report is a read-only property.

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.

InputDelay

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, InputDelay = 3 means 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.

Default: 0

OutputDelay

Output delays.

For identified systems, such as idpoly, OutputDelay is fixed to zero.

Ts

Sample time. For continuous-time models, Ts = 0. For discrete-time models, Ts is a positive scalar representing the sample time expressed in the unit specified by the TimeUnit property of the model. To denote a discrete-time model with unspecified sample time, set Ts = -1.

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.

Default: –1 (discrete-time model with unspecified sample time)

TimeUnit

Units for the time variable, the sample time Ts, and any time delays in the model, specified as one of the following values:

• 'nanoseconds'

• 'microseconds'

• 'milliseconds'

• 'seconds'

• 'minutes'

• 'hours'

• 'days'

• 'weeks'

• 'months'

• 'years'

Changing this property has no effect on other properties, and therefore changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior.

Default: 'seconds'

InputName

Input channel names, specified as one of the following:

• Character vector — For single-input models, for example, 'controls'.

• Cell array of character vectors — For multi-input models.

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

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.

Input channel names have several uses, including:

• Identifying channels on model display and plots

• Extracting subsystems of MIMO systems

• Specifying connection points when interconnecting models

Default: '' for all input channels

InputUnit

Input channel units, specified as one of the following:

• Character vector — For single-input models, for example, 'seconds'.

• Cell array of character vectors — For multi-input models.

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

Default: '' for all input channels

InputGroup

Input channel groups. The InputGroup property lets you assign the input channels of MIMO systems into groups and refer to each group by name. Specify input groups as a structure. In this structure, field names are the group names, and field values are the input channels belonging to each group. For example:

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

creates input groups named controls and noise that include input channels 1, 2 and 3, 5, respectively. You can then extract the subsystem from the controls inputs to all outputs using:

sys(:,'controls')

Default: Struct with no fields

OutputName

Output channel names, specified as one of the following:

• Character vector — For single-output models. For example, 'measurements'.

• Cell array of character vectors — For multi-output models.

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

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.

Output channel names have several uses, including:

• Identifying channels on model display and plots

• Extracting subsystems of MIMO systems

• Specifying connection points when interconnecting models

Default: '' for all output channels

OutputUnit

Output channel units, specified as one of the following:

• Character vector — For single-output models. For example, 'seconds'.

• Cell array of character vectors — For multi-output models.

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

Default: '' for all output channels

OutputGroup

Output channel groups. The OutputGroup property lets you assign the output channels of MIMO systems into groups and refer to each group by name. Specify output groups as a structure. In this structure, field names are the group names, and field values are the output channels belonging to each group. For example:

sys.OutputGroup.temperature = [1]; sys.InputGroup.measurement = [3 5];

creates output groups named temperature and measurement that include output channels 1, and 3, 5, respectively. You can then extract the subsystem from all inputs to the measurement outputs using:

sys('measurement',:)

Default: Struct with no fields

Name

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

Default: ''

Notes

Any text that you want to associate with the system, specified as a character vector or cell array of character vectors. For example, 'System is MIMO'.

Default: {}

UserData

Any type of data you want to associate with system, specified as any MATLAB® data type.

Default: []

SamplingGrid

Sampling grid for model arrays, specified as a data structure.

For arrays of identified linear (IDLTI) models that are derived 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 should be numeric and scalar valued, and all arrays of sampled values should match the dimensions of the model array.

For example, if 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)

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

For model arrays generated 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. For example, the Simulink Control Design™ commands linearize and slLinearizer populate SamplingGrid in this way.

Default: []

Tips

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