Consider the following optimal control problem: 1 ∙tf S x¹Zx+du² dt 2 x(t) = Cx(t) + Su(t), x(to) = xo, for some fixed i

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Consider The Following Optimal Control Problem 1 Tf S X Zx Du Dt 2 X T Cx T Su T X To Xo For Some Fixed I 1 (190.58 KiB) Viewed 21 times

Consider the following optimal control problem: 1 ∙tf S x¹Zx+du² dt 2 x(t) = Cx(t) + Su(t), x(to) = xo, for some fixed initial condition X₁ ER² as well as Z≥ 0 and d > 0. a) For tf <∞, the solution of this problem is a time-varying feedback. Which form does this feedback have? How do you determine it? Write all the neces- sary equations and briefly explain how to solve them. b) Now let tf →∞. How is the solution for the infinite horizon different from the finite horizon case from task a)? Write all the necessary equations to determine the solution of the optimal control problem with infinite horizon. c) Consider the algebraic Riccati equation (ARE) with ¹=[-2]. 0 d) V(xo) := min u(t)ER s.t. e) A B = C = [1 0], Q=C¹C, R = 1. Which one of the following is a solution (P) to the ARE? Is the solution unique? Justify your answers. a) c) √2 0 √2+1 -13 -√3 0 A Voo (xo) : min ∞ 1 b) d) u(t)ER 2 √2+1 0 0 √2 0 이 :] Is the resulting feedback u(t) = Kx(t) with K = R-¹BTP for system x = Ax+Bu and y = Cx (asymptotically) stabilizing? Justify your answer. Let the infinite cost V(xo) for the optimal control problem be (P) x¹Zx+du² dt to s.t. x(t)= Cx(t) + Su(t), x(to) = xo, (P) What form does Vo(xo) have? Write the formula. Consider C = A, S = B, Z=Q, d= R with A, B, Q, R as defined in task c). What is the infinite cost for Xo = = [24 - 1]¹? Consider now the infinite horizon optimal control from task b), but now with input constraints -1 ≤ u ≤ 3.3. What is the solution for the constrained opti- mal control problem? Hint: propose a control law that fulfills the constraints using the solution from task b). It is not necessary to solve any equations.