Optimal Control Problem with EulerLagrange A system is descr

Optimal Control Problem with Euler-Lagrange

A system is described by the following third-order linear difference equation:

A system is described by the following third-order linear difference equation: x(k + 1) = [0 1 0 0 0 1 -1/4 1/2 1/2] x(k) + [0 0 1] u(k) Recall the discrete linear regulator problem from Lecture 5. Assume that the state and control are unconstrained, and that the performance measure to be minimized is given by J = 1/2 x^T(N) [2 0 0 0 2 0 0 0 2] x (N) + 1/2 sigma_k = 0^N - 1 {X^T (k) [1 0 0 0 2 0 0 0 2] x (k) + u^2 (k)} = 1/2 x^T (N) Hx(N) + 1/2 sigma_k = 0^N - 1 {x^T (k) Q x (k) + u^2 (k)} Is the system state controllable? If the system is controllable, then write a MATLAB program (M-file) that computes the optimal control u * (k), i.e. compute the feedback gains F (k). Produce a single (combined) plot showing the resulting feedback gains vs. index k. Include a copy of your M-file in the homework submittal as well as your plot. Again using MATLAB, and assuming that the initial state is given by x(0) = [2 2 2]^T, write a routine (include in the previous M-file) to apply the optimal control u * (k) and plot the results (states and control on a combined plot) in order to demonstrate that the regulator problem has been solved. Submit a copy of your M-file and your plot.

Solution

again using matlab,and assuing that iitial state is given by

write a rouine

if the system is ccontrollable then write a matlab program that computer the optimal contrel

compute the produce a single plot showing the resultig feedback gais vs index k.include a copyofyour m-file