Robustness analysis with LTI parametric uncertainties

The file Demo_001.m is found in IQClab’s folder demos. This demo performs a $\mu$ – and IQC-robustness analysis for an uncertain plant that is affected by LTI parametric uncertainties. Here it is possible to vary several inputs:

The uncertainty block:
1. One parametric uncertainty that is repeated twice, or
2. Two different parametric uncertainties that are repeated once
Relaxation type:
1. DG-scalings
2. Convex hull relaxation
3. Partial convexity
4. Zeroth order Poyla relaxation
Performance metric:
1. Induced $L_2$ -gain
2. $H_2$ -norm
3. Robust stability test

The uncertain system is given by $\Delta\star M$ with the open-loop LTI plant $M=ss(A,B,C,D)$ , where

$A=\left(\begin{array}{cc}-2&-3\\1&1\end{array}\right)$ , $B=\left(\begin{array}{ccc}1&0&1\\0&0&0\end{array}\right)$ , $C=\left(\begin{array}{cc}1&0\\0&0\\1&0\end{array}\right)$ , $D=\left(\begin{array}{ccc}1&-2&0\\1&-1&d_{23}\\0&1&0\end{array}\right)$ ,

while:

$d_{23}=1$ for Option 3.1 and Option 3.3,
$d_{23}=0$ for Option 3.2.

On the other hand, the uncertainty block is defined by:

$\Delta=\Delta_1=\delta I_2$ with $\delta\in[-\alpha,\alpha]$ , for Option 1.1, or
$\Delta=\Delta_2=\mathrm{diag}(\delta_1,\delta_2)$ with $\delta_1\in[-\alpha_1,\alpha_1]$ , $\delta_2\in[-\alpha_2,\alpha_2]$ for Option 1.2.

The demo file Demo_001.m allows to run an IQC-analysis for various values of $\alpha\in[0,1]$ and within the file one can change the inputs mentioned above. For illustration purposes, the following 5 lines of code specify an IQC-analysis for the uncertain plant $\Delta_1(\delta)\star M$ , $\delta\in[-\alpha,\alpha]$ for $\alpha=0.5$ and the induced $L_2$ -gain as performance metric. In addition, the following parameters are considered:

Relaxation type: ‘CH’ (Option 2.2)
Length of the basis function: 2
Solution check: ‘on’
Enforce strictness of the LMIs: $\epsilon=1e-8$

% Define uncertain plant
M = ss([-2,-3;1,1],[1,0,1;0,0,0],[1,0;0,0;1,0],[1,-2,0;1,-1,1;0,1,0]);

% Define uncertainty block
de = iqcdelta('de','InputChannel',1:2,'OutputChannel',1:2,'Bounds',[-0.5,0.5]);

% Assign IQC-multiplier to uncertainty block
de = iqcassign(de,'ultis','Length',3,'RelaxationType','CH');

% Define performance block
pe = iqcdelta('pe','ChannelClass','P','InputChannel',3, 'OutputChannel',3,'PerfMetric','L2');

% Perform IQC-analysis
prob = iqcanalysis(M,{de,pe},'SolChk','on','eps',1e-8);

To continue, if running the IQC analysis in Demo_001.m for

$\Delta_1$ (Option 1.1)
Convex hull relaxation (Option 2.2)
Induced $L_2$ -gain performance (Option 3.1)

you obtain as output the worst-case induced $L_2$ -gain for increasing values of $\alpha\in[0,1]$ computed by the $\mu$ -tools (command: wcgain) and the IQC-tools for different lengths of the basis function. This yields the results shown in the following figure. As can be seen, the IQC-analysis produces worst-case induced $L_2$ -gains (i.e. $H_\infty$ -norms in this example), which are consisted with the $\mu$ -analysis. For static multipliers (case: $l=1$ , red dashed-line) the $\gamma$ -level increases faster than the others, at the benefit of a lower computational load. On the other hand, for basis-lengths larger than $1$ , the results coincide the $\mu$ -analysis up to values of $\alpha=0.85$ . Choosing higher basis-lengths would allow to converge even further.