LPV Analysis

Contents

Problem Statement

The following example illustrates how the LPV approach can help to analyze a subtle difference between two parameter dependent systems.

Consider a first order Linear Time-Invariant (LTI) system G:

and a time-varying parameter $\delta(t)$, subject to $-1 \leq \delta \leq 1$ and $-\beta \leq \dot{\delta} \leq \beta$, in a parallel interconnection as shown in Figure 1.

Figure 1: A parallel interconnection of two first order systems.

The parallel signal paths in Figure 1 describe two systems: $G\delta$ and $\delta G$. The systems differ only in the position of the $\delta$ parameter. In one system $\delta$ is applied to the input of $G$, while it is applied to the output of $G$ in the other. The output of the interconnection, $e$, is the difference between the outputs of the two systems.

Question: Is there any difference between placing the scalar $\delta$ before or after $G$ in the signal path?

Modeling

The following code buils up the system G, the time-varying parameter $\delta$, and the interconnection shown in Figure 1:

% Define the LTI System G
G = ss(-1,1,1,0);

% Define a scalar parameter 'delta' with values at 20 grid points between -1 and 1
Vals = linspace(-1,1,10);
delta = pgrid('delta',Vals);

% Define a parameter dependent system that describes the interconnection in Figure 1
H = delta*G-G*delta;

LTI Analysis

If $\delta(t)$ is constant, then the gain from $d$ to $e$ can be easily computed. The follwing code computes the induced $L_2$ norm from $d$ to $e$ (i.e. the infinity norm of H), and plots how it changes as a function of the parameter $\delta$.

% Compute the induced L2 norm of H
n = norm(H,inf);

% Plot how this norm varies with the value of the parameter delta
lpvplot(n)
title('Induced L_2 norm of H as a function of a constant \delta')
ylabel('Induced L_2 norm of H')

LTI Analysis Result

When $\delta$ is held constant, the induced $L_2$ norm of H is zero for all values of $\delta$. Judging from the LTI analysis, the position of the parameter in the signal path has no effect. The LTI analysis is not capable of discriminating between the two systems: $G\delta$ and $\delta G$.

LPV Analysis

Now compute the induced $L_2$ norm of H while taking into account the time-varying nature of $\delta$. The following code computes the induced $L_2$ norm of H for any trajectory of $\delta$ which satisfies: $-1 \leq \delta \leq 1$ and $-\infty \leq \dot{\delta} \leq \infty$.

syslpvnorm = lpvnorm(H)
syslpvnorm =
    1.0024

LPV Analysis Results

The LPV analysis yields a non-zero induced $L_2$ norm for H when $\delta$ is allowed to vary with time. This means that there exists some trejectory of $\delta$, subject to $-1 \leq \delta \leq 1$ and $-\infty \leq \dot{\delta} \leq \infty$. such that the two different signal paths through the interconnection in Figure 1 do not yield the same result.

The previous analysis assumed that $\delta$ could change arbitrarily fast with time, i.e. $-\infty \leq \dot{\delta} \leq \infty$. Lets repeat the previous analysis with different bounds on the rate of variation of $\delta$.

The following code computes the induced $L_2$ norm of H when the rate of variation of $\delta$ is constrained: $-\beta \leq \dot{\delta} \leq \beta$, with $\beta<\infty$.

% Define basis functions for the analysis algorithm.
bf = basis(delta,'delta',1);
Xb = [1;bf;bf^2;bf^3];
% Define a set of rate bounds (beta) to try: 15 values between 0.01 and 4.
rb = logspace(-2,log10(4),15);
for i=1:numel(rb)
  % Set the rate bounds of H to be +/- rb(i)
  H.Parameter.delta.RateBounds = [-rb(i) rb(i)];
  % Compute the induced L2 norm of H, subject to a time-varying delta
  % lying between -1 and 1, with d/dt(delta) between +/- rb(i)
  NormBounds(i) = lpvnorm(H,Xb);
end


plot(rb,NormBounds)
xlabel('Rate bound on \delta: -\beta \leq d/dt(\delta) \leq \beta')
ylabel('Induced L_2 norm of H')
title('Induced L_2 norm of H as a function of the rate bound on \delta')

LPV simulation

Lets compare the time-domain response of the two signal paths n $H$, i.e. the systems $G \delta$ and $\delta G$. We will use lpvstep to compare the parameter dependent step response:

Start by defining a time vector

t = 0:0.01:10;

Define a structure whose fields describe the trajectory of the parameter $\delta$:

ptraj.time = t;
ptraj.delta = sin(t);

Plot the step response of $G \delta$ and $\delta G$ when $\delta(t) = sin(t)$. The time domain response highlights the difference between the two signal paths when $\delta$ is treated as a time-varying parameter.

lpvstep(delta*G,ptraj)
hold on
lpvstep(G*delta,ptraj)
legend('\delta G','G\delta','location','best')

Summary

The preceding example demonstrates the power of the LPV approach, and its ability to augment traditional LTI analysis methods with results that take into account the time-varying nature of system components.

If the time varying nature of $\delta$ is ignored, an analysis in the LTI framework indicates that the two signal paths in Figure 1 are equivalent. However, if the the time-varying nature of $\delta$ is taken into account, an analysis in the LPV framework demonstrates that the position of $\delta$ in this interconnection can have a drastic effect on the results.

If $\delta$ varies slowly with time the difference between the two signal paths is small, e.g. its on the order of 1% when $|\dot{\delta}| \leq 0.01$. However, when $\delta$ changes faster, the difference becomes significant, e.g. the difference between the two singal paths is on the order of 20% when $|\dot{\delta}| = 0.2$, .

Reference

This example was published by Tamas Peni and Peter Seiler in [1]

  1. T. Peni, and P. Seiler, "Computation of a lower bound for the induced L2 norm of LPV systems," accepted to the American Control Conference, 2015.