Feedback Interconnection of Passive Systems

This example illustrates the properties of a feedback interconnection of passive systems.

Feedback Interconnection of Passive Systems

Consider an interconnection of two subsystems $$G_1$$ and $$G_2$$ in feedback. The interconnected system $$H$$ maps the input $$r$$ to the output $$y_1$$.

If both systems $$G_1$$ and $$G_2$$ are passive, then the interconnected system $$H$$ is guaranteed to be passive. Take for example

$$G_1(s)=\frac{s^2+s+1}{s^2+s+4},G_2(s)=\frac{s+2}{s+5}.$$

Both systems are passive as confirmed by

G1 = tf([1,1,1],[1,1,4]); 
isPassive(G1)

ans = logical
   1

G2 = tf([1,2],[1,5]);
isPassive(G2)

ans = logical
   1

The interconnected system is therefore passive.

H = feedback(G1,G2);
isPassive(H)

ans = logical
   1

This is confirmed by verifying that the Nyquist plot of $$H$$ is positive real.

nyquist(H)

Passivity Indices for Feedback Interconnection

There is a relationship between the passivity indices of $$G_1$$ and $$G_2$$ and the passivity indices of the interconnected system $$H$$. Let $${\nu }_1$$ and $${\nu }_2$$ denote the input passivity indices for $$G_1$$ and $$G_2$$, and let $${\rho }_1$$ and $${\rho }_2$$ denote the output passivity indices. If all these indices are positive, then the input passivity index $$\nu$$ and the output passivity index $$\rho$$ for the feedback interconnection $$H$$ satisfy

$$\nu \ge \frac{{\nu }_1{\rho }_2}{{\nu }_1+{\rho }_2},\rho \ge {\rho }_1+{\nu }_2.$$

In other words, we can infer some minimum level of input and output passivity for the closed-loop system $$H$$ from the input and output passivity indices of $$G_1$$ and $$G_2$$. For details, see the paper by Zhu, F. and Xia, M and Antsaklis, P.J., "Passivity analysis and passivation of feedback systems using passivity indices," American Control Conference , 2014, pp. 1833-1838. Verify the lower bound for the input passivity index $$\nu$$.

% Input passivity index for G1
nu1 = getPassiveIndex(G1,'input');
% Output passivity index for G2
rho2 = getPassiveIndex(G2,'output');
% Input passivity index for H
nu = getPassiveIndex(H,'input')

nu = 
0.1293

% Lower bound
nu1*rho2/(nu1+rho2)

ans = 
7.1402e-11

Similarly, verify the lower bound for the output passivity index of $$H$$.

% Output passivity index for G1
rho1 = getPassiveIndex(G1,'output');
% Input passivity index for G2
nu2 = getPassiveIndex(G2,'input');
% Output passivity index for H
rho = getPassiveIndex(H,'output')

rho = 
0.4441

% Lower bound
rho1+nu2

ans = 
0.4000