idnlhw/findop

Compute operating point for Hammerstein-Wiener model

Syntax

``````[X,U] = findop(sys,'steady',InputLevel,OutputLevel)``````
``````[X,U] = findop(sys,spec)``````
``````[X,U] = findop(___,Options)``````
``````[X,U,Report] = findop(___)``````
``````[X,U] = findop(sys,'snapshot',T,Uin)``````
``````[X,U] = findop(sys,'snapshot',T,Uin,X0)``````

Description

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``````[X,U] = findop(sys,'steady',InputLevel,OutputLevel)``` returns the operating-point state values, `X`, and input values, `U`, for the `idnlhw` model, `sys`, using steady-state input and output specifications.```

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``````[X,U] = findop(sys,spec)``` returns the steady-state operating point for `sys` using the operating point specification in `spec`.```

example

``````[X,U] = findop(___,Options)``` specifies optimization search options for all of the previous syntaxes.```

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``````[X,U,Report] = findop(___)``` returns a summary report on the optimization search results for all of the previous syntaxes.```

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``````[X,U] = findop(sys,'snapshot',T,Uin)``` returns the operating point for `sys` at a simulation snapshot at time, `T`, using the specified input, `Uin`. The initial states of `sys` are assumed to be zero.```

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``````[X,U] = findop(sys,'snapshot',T,Uin,X0)``` specifies the initial states of the simulation.```

Examples

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Load the estimation data and estimate a Hammerstein-Wiener model.

```load twotankdata; z = iddata(y,u,1); M = nlhw(z,[5 1 3]);```

Find the steady-state operating point where the input level is set to `1` and the output is unknown.

`[X,U] = findop(M,'steady',1,NaN);`

Estimate a Hammerstein-Wiener model.

```load iddata7; orders = [4*ones(1,2) 2*ones(1,2) 3*ones(1,2)]; M = nlhw(z7,orders,[],idPiecewiseLinear);```

Create a default operating point specification object.

`spec = operspec(M);`

Set the values for the input signals.

```spec.Input.Value(1) = -1; spec.Input.Value(2) = 1;```

Set the maximum and minimum values for the output signal.

```spec.Output.Max = 10; spec.Output.Min = -10;```

Find the steady-state operating point using the given specifications.

`[X,U] = findop(M,spec);`

Load the estimation data and estimate a Hammerstein-Wiener model.

```load twotankdata; z = iddata(y,u,1); M = nlhw(z,[5 1 3]);```

Create a default `findopOptions` option set.

`opt = findopOptions(M);`

Modify the option set to specify a steepest descent gradient search method with a maximum of 50 iterations.

```opt.SearchMethod = 'grad'; opt.SearchOptions.MaxIterations = 50;```

Find the steady-state operating point using the specified options.

`[X,U] = findop(M,'steady',1,NaN,opt);`

Load the estimation data and estimate a Hammerstein-Wiener model.

```load iddata7; orders = [4*ones(1,2) 2*ones(1,2) 3*ones(1,2)]; M = nlhw(z7,orders,[],idPiecewiseLinear);```

Find the steady-state operating point where input 1 is set to `1` and input 2 is unrestricted. The initial guess for the output value is `2`.

`[X,U,R] = findop(M,'steady',[1 NaN],2);`

Display the summary report.

`disp(R);`
``` SearchMethod: 'auto' WhyStop: 'Near (local) minimum, (norm(g) < tol).' Iterations: 5 FinalCost: 1.9722e-31 FirstOrderOptimality: 1.6481e-16 SignalLevels: [1x1 struct] ```

Load the estimation data estimate a Hammerstein-Wiener model.

```load twotankdata; z = iddata(y,u,1); M = nlhw(z,[5 1 3]);```

Find the simulation snapshot after 10 seconds, assuming initial states of zero.

`[X,U] = findop(M,'snapshot',10,z);`

Load the estimation data and estimate a Hammerstein-Wiener model.

```load twotankdata z = iddata(y,u,1); M = nlhw(z,[5 1 3]);```

Create an initial state vector.

`X0 = [10;10;5;5;1;1;0];`

Find the simulation snapshot after 10 seconds using the specified initial states.

`[X,U] = findop(M,'snapshot',10,z,X0);`

Input Arguments

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Hammerstein-Wiener model, specified as an `idnlhw` object.

Steady-state input level for computing the operating point, specified as a vector. The length of `InputLevel` must equal the number of inputs specified in `sys`.

The optimization algorithm assumes that finite values in `InputLevel` are fixed input values. Use `NaN` to specify unknown input signals with initial guesses of `0`. The minimum and maximum bounds for all inputs have default values of `-Inf` and `+Inf` respectively.

Steady-state output level for computing the operating point, specified as a vector. The length of `OutputLevel` must equal the number of outputs specified in `sys`.

The values in `OutputLevel` indicate initial guesses for the optimization algorithm. Use `NaN` to specify unknown output signals with initial guesses of `0`. The minimum and maximum bounds for all outputs have default values of `-Inf` and `+Inf` respectively.

Operating-point specifications, such as minimum and maximum input/output constraints and known inputs, specified as an `idnlhw/operspec` object.

Operating point snapshot time, specified as a positive scalar. The value of `T` must be in the range [T0, N*Ts], where N is the number of input samples, Ts is the sample time and T0 is the input start time (`Uin.Tstart`).

Snapshot simulation input, specified as one of the following:

• Time-domain `iddata` object with a sample time and input size that matches `sys`.

• Matrix with as many columns as there are input channels. If the matrix has N rows, the input data is assumed to correspond to the time vector `(1:N)*sys.Ts`.

Initial states of the simulation, specified as a column vector with length equal to the number of states in `sys`. `X0` provides the initial conditions at the time corresponding to the first input sample (`Uin.Start`, if `Uin` is an `iddata` object, or `sys.Ts` if `Uin` is a double matrix).

For more information about the states of an `idnlhw` model, see Definition of idnlhw States.

Operating point search options, specified as a `findopOptions` option set.

Output Arguments

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Operating point state values, returned as a column vector of length equal to the number of model states.

Operating point input values, returned as a column vector of length equal to the number of inputs.

Search result summary report, returned as a structure with the following fields:

FieldDescription
`SearchMethod`Search method used for iterative parameter estimation. See `SearchMethod` in `findopOptions` for more information.
`WhyStop`Search algorithm termination condition.
`Iterations`Number of estimation iterations performed.
`FinalCost`Final value of the minimization objective function (sum of the squared errors).
`FirstOrderOptimality`$\infty$-norm of the search gradient vector when the search algorithm terminates.
`SignalLevels`Structure containing the fields `Input` and `Output`, which are the operating point input and output signal levels respectively.

Algorithms

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`findop` computes the operating point from steady-state operating point specifications or at a simulation snapshot.

Computing the Operating Point from Steady-State Specifications

To compute the steady-state operating point, call `findop` using either of the following syntaxes:

```[X,U] = findop(sys,'steady',InputLevel,OutputLevel) [X,U] = findop(sys,spec)```

`findop` uses a different approach to compute the steady-state operating point depending on how much information you provide for this computation:

• When you specify values for all input levels (no `NaN` values). For a given input level, U, the equilibrium state values are X = inv(I-A)*B*f(U), where `[A,B,C,D] = ssdata(model.LinearModel)`, and f() is the input nonlinearity.

• When you specify known and unknown input levels. `findop` uses numerical optimization to minimize the norm of the error and compute the operating point. The total error is the union of contributions from e1 and e2 , e(t) = (e1(t)e2(t)), such that:

• e1 applies for known outputs and the algorithm minimizes e1 = y- g(L(x,f(u))), where f is the input nonlinearity, L(x,u) is the linear model with states x, and g is the output nonlinearity.

• e2 applies for unknown outputs and the error is a measure of whether these outputs are within the specified minimum and maximum bounds. If a variable is within its specified bounds, the corresponding error is zero. Otherwise, the error is equal to the distance from the nearest bound. For example, if a free output variable has a value z and its minimum and maximum bounds are L and U, respectively, then the error is e2= max[z-U, L-z, 0].

The independent variables for the minimization problem are the unknown inputs. In the error definition e, both the input u and the states x are free variables. To get an error expression that contains only unknown inputs as free variables, the algorithm `findop` specifies the states as a function of inputs by imposing steady-state conditions: x = inv(I-A)*B*f(U), where A and B are state-space parameters corresponding to the linear model L(x,u). Thus, substituting x = inv(I-A)*B*f(U) into the error function results in an error expression that contains only unknown inputs as free variables computed by the optimization algorithm.

Computing the Operating Point at a Simulation Snapshot

When you use the syntax `[X,U] = findop(sys,'snapshot',T,UIN,X0)`, the algorithm simulates the model output until the snapshot time, `T`. At the snapshot time, the algorithm computes the inputs for the linear model block of the Hammerstein-Wiener model (`LinearModel` property of the`idnlhw` object) by transforming the given inputs using the input nonlinearity: w = f(u). `findop` uses the resulting w to compute `x` until the snapshot time using the following equation: x(t+1) = Ax(t) + Bw(t), where `[A,B,C,D] = ssdata(model.LinearModel)`.

Note

For snapshot-based computations, `findop` does not perform numerical optimization.

Version History

Introduced in R2008a

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