Nonlinear Control Sy.. - Free

9.4. EXAMPLES 229
5- Very strictly-passive Systems: The system V) is said to be very strictly passive
if it is dissipative with respect to Q = -eI, R = -bI, and S = I. In this case,
substituting these values in (9.10), we obtain
or
-E(y,Y)T - b(u, U)T + (y, U)T > -O(x1) - O(xo) > -O(x0)
rT
J U T y dt = (U, Y)T ? CU, U)T + E(y, Y)T + 13-
0
The following lemma states a useful results, which is a direct consequence of these
definitions.
Lemma 9.2 If -0 is strictly output passive, then it has a finite L2 gain.
Proof: The proof is left as an exercise (Exercise 9.2).
9.4 Examples
9.4.1 Mass-Spring System with Friction
Consider the mass-spring system moving on a horizontal surface, shown in Figure 9.2.
Assuming for simplicity that the friction between the mass and the surphase is negligible,
we obtain the following equation of the motion:
ml+0±+kx= f
where m represents the mass, k is the spring constant, p, the viscous friction force associated
with the spring, and f is an external force. Defining state variables x1 = x, and ±1 = x2,
and assuming that the desired output variable is the velocity vector, x2, we obtain the
following state space realization:
1 = X2
2 = -mx1 - P- x2 + 1
= X2
Y
To study the dissipative properties of this system, we proceed to find the total energy stored
in the system at any given time. We have
E = 1kx2 + 21
mx2
I 0