Nonlinear Control Sy.. - Free

9.12. EXERCISES 253
9.12 Exercises
(9.1) Prove Lemma 9.1.
(9.2) Prove Lemma 9.2.
(9.3) Consider the pendulum-on-a-cart system discussed in Chapter 1. Assuming for simplicity
that the moment of inertia of the pendulum is negligible, and assuming that
the output equation is
y=±=X2
(a) Find the kinetic energy K1 of the cart.
(b) Find the kinetic energy K2 of the pendulum.
(c) Find the potential energy P in the cart-pendulum system.
(d) Using the previous results, find the total energy E = K1 + K2 + P = K + P
stored in the cart-pendulum system.
(e) Defining variables
_ x M+m mlcoso u _ f
0
q= 9 M= L ml cos ml2B '
show that the energy stored in the pendulum can be expressed as
1
E = 24TM(q)4 +mgl(cos0 - 1)
(f) Computing the derivative of E along the trajectories of the system, show that the
pendulum-on-a-cart system with input u = f and output ± is a passive system.
(9.4) Consider the system
(a) Determine whether i is
-a21-22+u, a>0
X1-X X2
= xl
(i) passive.
(ii) strictly passive.
(iii) strictly output-passive.
(iv) has a finite G2 gain.
(b) If the answer to part (a)(iv) is affirmative, find an upper bound on the L2 gain.