6) Consider the forced system below: x1 = sin(x2)cos(x1) +...

6) Consider the forced system below: $x_1 = \sin(x_2)\cos(x_1) + x_1^3 + u$ $x_2 = -x_1x_2\cos(x_1)$ Based on the candidate Lyapunov function $V(x) = \frac{1}{2}(x_1^2 + x_2^2)$, get a control law for feedback $u(x_1, x_2)$ such that the resulting system is stable ($x_1(t) \to 0$) Demonstrate the result by simulation

Consider the nonlinear controlled dynamical system dot{x}_1(t) = x_1(t)x_2(t), x_1(0) = x_{10}, t ge 0, dot{x}_2(t) = x_1^2(t) + u(t), x_2(0) = x_{20}. 1. Show that there is no linear feedback control of the form u = k_1x_1 + k_2x_2 such that the closed-loop system has an asymptotically stable equilibrium at the origin. You may have to study the Chetaev's instability theorem and using the function V(x_1, x_2) = x_1^2 - x_2^2. 2. Verify the above conclusion by numerical simulations. 3. Using the function V(x_1, x_2) = x_1^2 + x_2^2 as a candidate Lyapunov function and derive a nonlinear feedback control that renders global asymptotic stability of the origin. 4. If L(u) = int_0^infty (x_1^2(t) + x_2^2(t) + u^2(t))dt is the performance of a control law u then suggest a numerical method to optimize over some set of stabilizing feedback control laws.

Adi S.

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