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Polynomial model with identifiable parameters
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 sampling 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 sampling time. (Omit NoiseVariance and Ts to use their default values.)
If D = [], then sys represents the ARMA model given by:
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.
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:
In continuous time, a polynomial model takes the following form:
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 such commands as polyest, arx, armax, oe, bj, iv4, or ivar. These estimation 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.
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:
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 N_{y} outputs and N_{u} 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.
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 desired 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 |
Sampling time. For continuous-time models, Ts = 0. For discrete-time models, Ts is a positive scalar representing the sampling period expressed in the unit specified by the TimeUnit property of the model. To denote a discrete-time model with unspecified sampling time, set Ts = -1. Default: –1 (discrete-time model with unspecified sampling 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 N_{y}-by-N_{y} matrix, where N_{y} is the number of outputs in the system. Default: N_{y}-by-N_{y} 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. |
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.
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:
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 N_{y} outputs and N_{u} 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.
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 |
String specifying the polynomial model display variable. Variable requires one of the following values:
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, specify transport are expressed as integers denoting delay of a multiple of the sampling period 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:
For a MIMO model with Ny outputs and Nu inputs, the dimensions of the Structure elements are as follows:
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 N_{y}-by-N_{y} matrix, where N_{y} is the number of outputs in the system. | ||||||||||||
Report |
Information about the estimation process. Report contains the following fields:
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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 sampling period Ts. For example, InputDelay = 3 means a delay of three sampling periods. 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, like idpoly, OutputDelay is fixed to zero. | ||||||||||||
Ts |
Sampling time. For continuous-time models, Ts = 0. For discrete-time models, Ts is a positive scalar representing the sampling period expressed in the unit specified by the TimeUnit property of the model. To denote a discrete-time model with unspecified sampling 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 sampling time of a discrete-time system. Default: –1 (discrete-time model with unspecified sampling time) | ||||||||||||
TimeUnit |
String representing the unit of the time variable. For continuous-time models, this property represents any time delays in the model. For discrete-time models, it represents the sampling time Ts. Use any of the following values:
Changing this property changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior. Default: 'seconds' | ||||||||||||
InputName |
Input channel names. Set InputName to a string for single-input model. For a multi-input model, set InputName to a cell array of strings. 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:
Default: Empty string '' for all input channels | ||||||||||||
InputUnit |
Input channel units. Use InputUnit to keep track of input signal units. For a single-input model, set InputUnit to a string. For a multi-input model, set InputUnit to a cell array of strings. InputUnit has no effect on system behavior. Default: Empty string '' 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. Set OutputName to a string for single-output model. For a multi-output model, set OutputName to a cell array of strings. 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 to 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:
Default: Empty string '' for all input channels | ||||||||||||
OutputUnit |
Output channel units. Use OutputUnit to keep track of output signal units. For a single-output model, set OutputUnit to a string. For a multi-output model, set OutputUnit to a cell array of strings. OutputUnit has no effect on system behavior. Default: Empty string '' for all input 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. Set Name to a string to label the system. Default: '' | ||||||||||||
Notes |
Any text that you want to associate with the system. Set Notes to a string or a cell array of strings. Default: {} | ||||||||||||
UserData |
Any type of data you wish to associate with system. Set UserData to 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. Default: [] |
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