Question

Consider the nonlinear controlled dynamical system dot{x}_1(t) = x_1(t)x_2(t), x_1(0) = x_{10}, t ge 0, (3.1) dot{x}_2(t) = x_1^2(t) + u(t), x_2(0) = x_{20}. (3.2) 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.

          Consider the nonlinear controlled dynamical system

dot{x}_1(t) = x_1(t)x_2(t), x_1(0) = x_{10}, t ge 0, (3.1)
dot{x}_2(t) = x_1^2(t) + u(t), x_2(0) = x_{20}. (3.2)

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.

Consider the nonlinear controlled dynamical system

dotx1(t) = x1(t)x2(t), x1(0) = x10, t ge 0, (3.1)
dotx2(t) = x1^2(t) + u(t), x2(0) = x20. (3.2)

1. Show that there is no linear feedback control of the form u = k1x1 + k2x2 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(x1, x2) = x1^2 - x2^2.
2. Verify the above conclusion by numerical simulations.
3. Using the function V(x1, x2) = x1^2 + x2^2 as a candidate Lyapunov function and derive a nonlinear feedback control that renders global asymptotic stability of the origin.
4. If L(u) = int0^infty (x1^2(t) + x2^2(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.

Added by Rafael M.

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