ucover :: Functions (Robust Control Toolbox™)

ucover

Purpose

Fit an uncertain model to set of LTI responses

Syntax

usys = (Parray,Pnom,ord)
usys = (Parray,Pnom,ord1,ord2,utype)
[usys,info] = (Parray,...)
[usys_new,info_new] = (Pnom,info,ord1_new,ord2_new)

Description

usys = (Parray,Pnom,ord) returns an uncertain model usys with nominal value Pnom and whose range of behaviors includes all responses in the LTI array Parray. The uncertain model structure is of the form , where

is an ultidyn object that represents uncertain dynamics with unit peak gain.
W is a stable, minimum-phase shaping filter that adjusts the amount of uncertainty at each frequency.

ord is the number of states (order) of W. Pnom and Parray can be ss, tf, zpk, or zpk models. usys is of class ufrd when Pnom is an frd model and is an uss model otherwise.

usys = (Parray,Pnom,ord1,ord2,utype) specifies the order ord1 and ord2 of each diagonal entry of W1 and W2, where W1 and W2 are diagonal, stable, minimum-phase shaping filters. utype specifies the uncertain model structure, as described in Uncertain Model Structures, and can be 'InputMult' (default), 'OutputMult' or 'Additive'. ord1 and ord2 can be:

[], which implies that the corresponding filter is 1.
Scalar, which implies that the corresponding filter is scalar-valued.
Vectors with as many entries as diagonal entries in W1 and W2.

[usys,info] = (Parray,...) returns a structure info that contains optimization information. info.W1opt and Info.W2opt contain the values of W1 and W2 computed on a frequency grid and info.W1 and info.W2 contain the fitted shaping filters W1 and W2.

[usys_new,info_new] = (Pnom,info,ord1_new,ord2_new) improves the fit using initial filter values in info and new orders ord1_new and ord2_new of W1 and W2. This syntax speeds up command execution by reusing previously computed information. Use this syntax when you are changing filter orders in an iterative workflow.

Definitions

Uncertain Model Structures

When fitting the responses of LTI models in Parray, the gaps between Parray and the nominal response Pnom of the uncertain model are modeled as uncertainty on the system dynamics. To model the frequency distribution of these unmodeled dynamics, ucover measures the gap between Pnom and Parray at each frequency and selects a shaping filter W whose magnitude approximates the maximum gap between Pnom and Parray. The following figure shows the relative gap between the nominal response and six LTI responses, enveloped using a second-order shaping filter.

The software then sets the uncertainty to , where is an ultidyn object that represents unit-gain uncertain dynamics. This ensures that the amount of uncertainty at each frequency is specified by the magnitude of W and therefore closely tracks the gap between Pnom and Parray.

There are three possible uncertainty model structures:

Input Multiplicative of the form .
Output Multiplicative of the form .
Additive of the form .

Use additive uncertainty to model the absolute gaps between Pnom and Parray, and multiplicative uncertainty to model relative gaps.

Note For SISO models, input and output multiplicative uncertainty are equivalent. For MIMO systems with more outputs than inputs, the input multiplicative structure may be too restrictive and not adequately cover the range of models.

The model structure

that you obtain using usys = ucover(Parray,Pnom,ord), corresponds to and .

Examples

Fit an uncertain model to multiple LTI responses:

Create the nominal plant.

```
Pnom = tf(2,[1 -2]);
```

Create an LTI array whose responses the uncertain model should fit.

p1 = Pnom*tf(1,[.06 1]); 
p2 = Pnom*tf([-.02 1],[.02 1]); 
p3 = Pnom*tf(50^2,[1 2*.1*50 50^2]); 
array = stack(1,p1,p2,p3);
Parray = frd(array,logspace(-1,3,60));

Plot relative errors between the nominal plant response and the three models in the LTI array.

```
bodemag((Pnom-Parray)/Pnom)
```

The set of relative errors is shown in the following figure. If you use a multiplicative uncertainty model structure, the magnitude of the shaping filter should fit the maximum relative errors at each frequency.

Use a 1st-order shaping filter to fit the maximum relative errors.

[P,Info] = ucover(Parray,Pnom,1);

Plot a Bode magnitude plot to see how well the shaping filter fits the relative errors.

bodemag((Pnom-Parray)/Pnom,'b--',Info.W1,'r'); grid

The plot, as shown in the following figure, shows that the filter W1 is too conservative and exceeds the maximum relative error at most frequencies.

To obtain a tighter fit, rerun the function using a 4th-order filter.

```
[P,Info] = ucover(Parray,Pnom,4);
```

Evaluate the fit by plotting the Bode magnitude plot.

bodemag((Pnom-Parray)/Pnom,'b--',Info.W1,'r'); grid

The plot, as shown in the following figure, shows that magnitude of W1 closely matches the minimum uncertainty amount.

Algorithm

The ucover command designs the minimum-phase shaping filters W1 and W2 in two steps:

Computes the optimal values of W1 and W2 on a frequency grid.
Fits W1 and W2 values with the dynamic filters of the specified orders using the fitmagfrd command.