idnlhw/findop
Compute operating point for Hammerstein-Wiener model
Syntax
Description
[
returns
the operating-point state values, X
,U
] =
findop(sys
,'steady',InputLevel
,OutputLevel
)X
, and input
values, U
, for the idnlhw
model, sys
,
using steady-state input and output specifications.
Examples
Find Steady-State Hammerstein-Wiener Operating Point Using Default Specifications
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);
Find Hammerstein-Wiener Operating Point Using Additional Specifications
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);
Find Hammerstein-Wiener Operating Point Using Custom Options
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);
Retrieve Hammerstein-Wiener Operating Point Search Report
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]
Find Hammerstein-Wiener Simulation Snapshot Using Default Initial States
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);
Find Hammerstein-Wiener Simulation Snapshot Using Initial State Specifications
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
sys
— Hammerstein-Wiener model
idnlhw
object
Hammerstein-Wiener model, specified as an idnlhw
object.
InputLevel
— Steady-state input level
vector
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.
OutputLevel
— Steady-state output level
vector
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.
spec
— Operating-point specifications
operspec
object
Operating-point specifications, such as minimum and maximum input/output constraints and known
inputs, specified as an idnlhw/operspec
object.
T
— Operating point snapshot time
positive scalar
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
).
Uin
— Snapshot simulation input
iddata
object | matrix
Snapshot simulation input, specified as one of the following:
Time-domain
iddata
object with a sample time and input size that matchessys
.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
.
X0
— Initial states
column vector
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.
Options
— Operating point search options
findopOptions
option set
Operating point search options, specified as a findopOptions
option
set.
Output Arguments
X
— Operating point state values
column vector
Operating point state values, returned as a column vector of length equal to the number of model states.
U
— Operating point input values
column vector
Operating point input values, returned as a column vector of length equal to the number of inputs.
Report
— Search result summary
structure
Search result summary report, returned as a structure with the following fields:
Field | Description |
---|---|
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 | -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
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 theidnlhw
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 R2008aR2018a: Advanced Options are deprecated for SearchOptions
when SearchMethod
is 'lsqnonlin'
Specification of lsqnonlin
- related advanced options are deprecated,
including the option to invoke parallel processing when estimating using the
lsqnonlin
search method, or solver, in Optimization Toolbox™.
See Also
idnlhw
| findopOptions
| idnlhw/operspec
| sim
| idnlarx/findop
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