SOLVED EE6415 Tutorial 6

1. Consider the discrete time system x[k + 1] = Ax[k] with A =   1.3 0.2 0.2 −1 0.4 −0.4 −0.4 −0.2 0.7   Given Q =   2 0 0 0 1 0 0 0 3   for the discrete time Lyapunov equation AT P A − P = −Q (a) Find P (b) Find eigenvalues of P (c) Comment on the stability of the system 2. Consider the system x˙(t) =  2.8 9.6 9.6 −2.8  x(t) Given Q =  20 0 0 20 , find P for the Lyapunov equation AT P + P A = −Q. (a) Write the matrix P, is it unique? (b) Find the expression (or value) for the eigenvalues of P. 1 (c) Comment on the stability of x = 0 of the system. 3. Consider a second order system given as ω¨ + g(ω) ˙ω + ω = 0 with equilibrium point ω = ˙ω = 0. Find the conditions on the g(0) such that convergence to the origin can be guaranteed ? 4. Comment on the stability of the origin for the following system: (a) x˙ 1 = x2(1 − x 2 1 ) x˙ 2 = −(x1 + x2)(1 − x 2 1 ) (b) x˙ 1 = x 3 1 + x 2 1x2 x˙ 2 = −x2 + x 2 2 + x1x2 − x 3 1 (c) x˙ 1 = x2 x˙ 2 = −x 3 1 − x 3 2 5. Consider the system x˙ 1 = x1 − x 3 1 + x2 x˙ 2 = 3x1 − x2 (a) Find all equilibrium point of the system. (b) Using linearization, study the stability of each equilibrium point. (c) Using quadratic Lyapunov functions, estimate the region of attraction of each asymptotically stable equilibrium point. Try to make your estimate as large as possible. (d) Construct the phase portrait of the system and show on it the exact regions as attraction as well as your estimates 6. Consider the system x˙ 1 = h(t)x2 − g(t)x 3 1 x˙ 2 = −h(t)x1 − g(t)x 3 2 where h(t) and g(t) are bounded, continuously differentiable functions and g(t) > 0, for all t ≥ 0. (a) Is the equilibrium point x = 0 uniformly asymptotically stable? (b) Is it exponentially stable? (c) Is it globally uniformly asymptotically stable? (d) Is it globally exponentially stable? 7. Suppose the set M in LaSalle’s theorem consists of a finite number of isolated points. Show that limx→∞ x(t) exists and equals one of these points. 2 8. Consider the function given below: V (x) = (x1 + x2) 2 1 + (x1 + x2) 2 + (x1 − x2) 2 Is this a valid Lyapunov candidate function to deduce the global asymptotic stability of an equilibrium point? Justify your answer by: (a) Proving/Disproving that it is radially unbounded. (b) Plotting the level sets, and inferring from the plot. 3