Series Interconnection of Passive Systems

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

Series Interconnection of Passive Systems

Consider an interconnection of two subsystems $$G_1$$ and $$G_2$$ in series. The interconnected system $$H$$ is given by the mapping from input $$u$$ to output $$y_2$$.

In contrast with parallel and feedback interconnections, passivity of the subsystems $$G_1$$ and $$G_2$$ does not guarantee passivity for the interconnected system $$H$$. Take for example

$$G_1(s)=\frac{5s^2+3s+1}{s^2+2s+1},G_2(s)=\frac{s^2+s+5s+0.1}{s^3+2s^2+3s+4}.$$

Both systems are passive as confirmed by

G1 = tf([5 3 1],[1,2,1]);
isPassive(G1)

ans = logical
   1

G2 = tf([1,1,5,.1],[1,2,3,4]);
isPassive(G2)

ans = logical
   1

However the series interconnection of $$G_1$$ and $$G_2$$ is not passive:

H = G2*G1;
isPassive(H)

ans = logical
   0

This is confirmed by verifying that the Nyquist plot of $$G_2{G}_1$$ is not positive real.

nyquist(H)

Passivity Indices for Series Interconnection

While the series interconnection of passive systems is not passive in general, there is a relationship between the passivity indices of $$G_1$$ and $$G_2$$ and the passivity indices of $$H=G_2{G}_1$$. 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 series interconnection $$H$$ satisfy

$$\nu \ge -\frac{0.125}{{\rho }_1{\rho }_2},\rho \ge -\frac{0.125}{{\nu }_1{\nu }_2}.$$

In other words, the shortage of passivity at the inputs or outputs of $$H$$ is no worse than the right-hand-side expressions. For details, see the paper by Arcak, M. and Sontag, E.D., "Diagonal stability of a class of cyclic systems and its connection with the secant criterion," Automatica, Vol 42, No. 9, 2006, pp. 1531-1537. Verify these lower bounds for the example above.

% Output passivity index for G1
rho1 = getPassiveIndex(G1,'output');
% Output passivity index for G2
rho2 = getPassiveIndex(G2,'output');
% Input passivity index for H=G2*G1
nu = getPassiveIndex(H,'input')

nu = 
-1.2886

% Lower bound
-0.125/(rho1*rho2)

ans = 
-2.4194

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

% Input passivity index for G1
nu1 = getPassiveIndex(G1,'input');
% Input passivity index for G2
nu2 = getPassiveIndex(G2,'input');
% Output passivity index for H=G2*G1
rho = getPassiveIndex(H,'output')

rho = 
-0.6966

% Lower bound
-0.125/(nu1*nu2)

ans = 
-6.0000