This example shows how to use the Robust Control Toolbox™ command `ucover`

to model a family of LTI responses as an uncertain system. This command is useful to fit an uncertain model to a set of frequency responses representative of the system variability, or to reduce the complexity of an existing uncertain model to facilitate the synthesis of robust controllers with `musyn`

.

In this first example, we have a family of models describing the plant behavior under various operating conditions. The nominal plant model is a first-order unstable system.

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

Pnom = 2 ----- s - 2 Continuous-time transfer function.

The other models are variations of `Pnom`

. They all have a single unstable pole but the location of this pole may vary with the operating condition.

p1 = Pnom*tf(1,[.06 1]); % extra lag p2 = Pnom*tf([-.02 1],[.02 1]); % time delay p3 = Pnom*tf(50^2,[1 2*.1*50 50^2]); % high frequency resonance p4 = Pnom*tf(70^2,[1 2*.2*70 70^2]); % high frequency resonance p5 = tf(2.4,[1 -2.2]); % pole/gain migration p6 = tf(1.6,[1 -1.8]); % pole/gain migration

To apply robust control tools, we can replace this set of models with a single uncertain plant model whose range of behaviors includes `p1`

through `p6`

. This is one use of the command `ucover`

. This command takes an array of LTI models `Parray`

and a nominal model `Pnom`

and models the difference `Parray-Pnom`

as multiplicative uncertainty in the system dynamics.

Because `ucover`

expects an array of models, use the `stack`

command to gather the plant models `p1`

through `p6`

into one array.

Parray = stack(1,p1,p2,p3,p4,p5,p6);

Next, use `ucover`

to "cover" the range of behaviors `Parray`

with an uncertain model of the form

P = Pnom * (1 + Wt * Delta)

where all uncertainty is concentrated in the "unmodeled dynamics" `Delta`

(a `ultidyn`

object). Because the gain of `Delta`

is uniformly bounded by 1 at all frequencies, a "shaping" filter `Wt`

is used to capture how the relative amount of uncertainty varies with frequency. This filter is also referred to as the uncertainty weighting function.

Try a 4th-order filter `Wt`

for this example:

```
orderWt = 4;
Parrayg = frd(Parray,logspace(-1,3,60));
[P,Info] = ucover(Parrayg,Pnom,orderWt,'InputMult');
```

The resulting model `P`

is a single-input, single-output uncertain state-space (USS) object with nominal value `Pnom`

.

P

P = Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 5 states. The model uncertainty consists of the following blocks: Parrayg_InputMultDelta: Uncertain 1x1 LTI, peak gain = 1, 1 occurrences Type "P.NominalValue" to see the nominal value, "get(P)" to see all properties, and "P.Uncertainty" to interact with the uncertain elements.

tf(P.NominalValue)

ans = 2 ----- s - 2 Continuous-time transfer function.

A Bode magnitude plot confirms that the shaping filter `Wt`

"covers" the relative variation in plant behavior. As a function of frequency, the uncertainty level is 30% at 5 rad/sec (-10dB = 0.3) , 50% at 10 rad/sec, and 100% beyond 29 rad/sec.

Wt = Info.W1; bodemag((Pnom-Parray)/Pnom,'b--',Wt,'r'); grid title('Relative Gaps vs. Magnitude of Wt')

You can now use the uncertain model `P`

to design a robust controller for the original family of plant models, see Simultaneous Stabilization Using Robust Control for details.

In this second example, we start with a detailed uncertain model of the plant. This model consists of first-order dynamics with uncertain gain and time constant, in series with a mildly underdamped resonance and significant unmodeled dynamics. This model is created using the `ureal`

and `ultidyn`

commands for specifying uncertain variables:

gamma = ureal('gamma',2,'Perc',30); % uncertain gain tau = ureal('tau',1,'Perc',30); % uncertain time-constant wn = 50; xi = 0.25; P = tf(gamma,[tau 1]) * tf(wn^2,[1 2*xi*wn wn^2]); % Add unmodeled dynamics and set SampleStateDim to 5 to get representative % sample values of the uncertain model P delta = ultidyn('delta',[1 1],'SampleStateDim',5,'Bound',1); W = makeweight(0.1,20,10); P = P * (1+W*delta)

P = Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 4 states. The model uncertainty consists of the following blocks: delta: Uncertain 1x1 LTI, peak gain = 1, 1 occurrences gamma: Uncertain real, nominal = 2, variability = [-30,30]%, 1 occurrences tau: Uncertain real, nominal = 1, variability = [-30,30]%, 1 occurrences Type "P.NominalValue" to see the nominal value, "get(P)" to see all properties, and "P.Uncertainty" to interact with the uncertain elements.

A collection of step responses illustrates the plant variability.

```
step(P,4)
title('Sampled Step Responses of Uncertain System')
```

The uncertain plant model `P`

contains 3 uncertain elements. For control design purposes, it is often desirable to simplify this uncertainty model while approximately retaining its overall variability. This is another use of the command `ucover`

.

To use `ucover`

in this context, first map the uncertain model `P`

into an array of LTI models using `usample`

. This command samples the uncertain elements in an uncertain system and returns the corresponding LTI models, each model representing one possible behavior of the uncertain system. In this example, sample `P`

at 60 points (the random number generator is seeded for repeatability):

```
rng(0,'twister');
Parray = usample(P,60);
```

Next, use `ucover`

to cover all behaviors in `Parray`

by a simple uncertainty model `Usys`

. Choose the nominal value of `P`

as center of the cover, and use a 2nd-order filter to model the frequency distribution of the unmodeled dynamics.

```
orderWt = 2;
Parrayg = frd(Parray,logspace(-3,3,60));
[Usys,Info] = ucover(Parrayg,P.NominalValue,orderWt,'InputMult');
```

A Bode magnitude plot shows how the filter magnitude (in red) "covers" the relative variability of the plant frequency response (in blue).

Wt = Info.W1; bodemag((P.NominalValue-Parray)/P.NominalValue,'b--',Wt,'r') title('Relative Gaps (blue) vs. Shaping Filter Magnitude (red)')

You can now use the simplified uncertainty model `Usys`

to design a robust controller for the original plant, see First-Cut Robust Design for details.

In this third example, we start with 40 frequency responses of a 2-input, 2-output system. This data has been collected with a frequency analyzer under various operating conditions. A two-state nominal model is fitted to the most typical response:

A = [-5 10;-10 -5]; B = [1 0;0 1]; C = [1 10;-10 1]; D = 0; Pnom = ss(A,B,C,D);

The frequency response data is loaded into a 40-by-1 array of FRD models:

```
load ucover_demo
size(Pdata)
```

40x1 array of FRD models. Each model has 2 outputs, 2 inputs, and 120 frequency points.

Plot this data and superimpose the nominal model.

bode(Pdata,'b--',Pnom,'r',{.1,1e3}), grid legend('Frequency response data','Nominal model','Location','NorthEast')

Because the response variability is modest, try modeling this family of frequency responses using an additive uncertainty model of the form

P = Pnom + w * Delta

where `Delta`

is a 2-by-2 `ultidyn`

object representing the unmodeled dynamics and `w`

is a scalar weighting function reflecting the frequency distribution of the uncertainty (variability in Pdata).

Start with a first-order filter `w`

and compare the magnitude of `w`

with the minimum amount of uncertainty needed at each frequency:

[P1,InfoS1] = ucover(Pdata,Pnom,1,'Additive'); w = InfoS1.W1; bodemag(w,'r',InfoS1.W1opt,'g',{1e-1 1e3}) title('Scalar Additive Uncertainty Model') legend('First-order w','Min. uncertainty amount','Location','SouthWest')

The magnitude of `w`

should closely match the minimum uncertainty amount. It is clear that the first-order fit is too conservative and exceeds this minimum amount at most frequencies. Try again with a third-order filter `w`

. For speed, reuse the data in `InfoS1`

to avoid recomputing the optimal uncertainty scaling at each frequency.

[P3,InfoS3] = ucover(Pnom,InfoS1,3,'Additive'); w = InfoS3.W1; bodemag(w,'r',InfoS3.W1opt,'g',{1e-1 1e3}) title('Scalar Additive Uncertainty Model') legend('Third-order w','Min. uncertainty amount','Location','SouthWest')

The magnitude of `w`

now closely matches the minimum uncertainty amount. Among additive uncertainty models, `P3`

provides a tight cover of the behaviors in `Pdata`

. Note that `P3`

has a total of 8 states (2 from the nominal part and 6 from `w`

).

P3

P3 = Uncertain continuous-time state-space model with 2 outputs, 2 inputs, 8 states. The model uncertainty consists of the following blocks: Pdata_AddDelta: Uncertain 2x2 LTI, peak gain = 1, 1 occurrences Type "P3.NominalValue" to see the nominal value, "get(P3)" to see all properties, and "P3.Uncertainty" to interact with the uncertain elements.

You can refine this additive uncertainty model by using non-scalar uncertainty weighting functions, for example

P = Pnom + W1*Delta*W2

where `W1`

and `W2`

are 2-by-2 diagonal filters. In this example, restrict use `W2=1`

and allow both diagonal entries of W1 to be third order.

```
[PM,InfoM] = ucover(Pdata,Pnom,[3;3],[],'Additive');
```

Compare the two entries of `W1`

with the minimum uncertainty amount computed earlier. Note that at all frequencies, one of the diagonal entries of `W1`

has magnitude much smaller than the scalar filter `w`

. This suggests that the diagonally-weighted uncertainty model yields a less conservative cover of the frequency response family.

bodemag(InfoS1.W1opt,'g*',... InfoM.W1opt(1,1),'r--',InfoM.W1(1,1),'r',... InfoM.W1opt(2,2),'b--',InfoM.W1(2,2),'b',{1e-1 1e3}); title('Diagonal Additive Uncertainty Model') legend('Scalar Optimal Weight',... 'W1(1,1), pointwise optimal',... 'W1(1,1), 3rd-order fit',... 'W1(2,2), pointwise optimal',... 'W1(2,2), 3rd-order fit',... 'Location','SouthWest')

The degree of conservativeness of one cover over another can be partially quantified by considering the two frequency-dependent quantities:

Fd2s = norm(inv(W1)*w) , Fs2d = norm(W1/w)

These quantities measure by how much one uncertainty model needs to be scaled to cover the other. For example, the uncertainty model `Pnom + W1*Delta`

needs to be enlarged by a factor `Fd2s`

to include all of the models represented by the uncertain model `Pnom + w*Delta`

.

Plot `Fd2s`

and `Fs2d`

as a function of frequency.

Fd2s = fnorm(InfoS1.W1opt*inv(InfoM.W1opt)); Fs2d = fnorm(InfoM.W1opt*inv(InfoS1.W1opt)); semilogx(fnorm(Fd2s),'b',fnorm(Fs2d),'r'), grid axis([0.1 1000 0.5 2.6]) xlabel('Frequency (rad/s)'), ylabel('Magnitude') title('Scale factors relating different covers') legend('Diagonal to Scalar factor',... 'Scalar to Diagonal factor','Location','SouthWest');

This plot shows that:

`Fs2d = 1`

in a large frequency range so`Pnom+w*Delta`

includes all the behaviors modeled by`Pnom+W1*Delta`

In that same frequency range,

`Pnom+W1*Delta`

does not include all of the behaviors modeled by`Pnom+w*Delta`

and, in fact, would need to be enlarged by a factor between 1.2 and 2.6 in order to do so.

In the frequency range [1 20], neither uncertainty model contains the other, but at all frequencies, making

`Pnom+W1*Delta`

cover`Pnom+w*Delta`

requires a much smaller scaling factor than the converse.

This indicates that the `Pnom+W1*Delta`

model provides a less conservative cover of the frequency response data in `Pdata`

.