Q2. (a) Consider the following finite-horizon linear-quadratic optimal [10 marks] control problem N-1 PN (x): Minimise(x7Qx+u...

 

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Q2. (a) Consider the following finite-horizon linear-quadratic optimal [10 marks] control problem N-1 PN (x): Minimise(x7Qx+u Ru₂) + 1xPxN, (14) subject to: X+1 = Ax₂ + But € N[N-1) Xo = x, where R € R"X" is a symmetric positive definite matrix and Q,P, € R™ are symmetric positive semidefinite. We have a very efficient software that can solve convex quadratic optimisation problems of the form Minimisez ¹Hz + h¹z, ZERK where HE R*XK is a symmetric positive definite matrix. Write the above optimal control problem, PN(X), in a way that can be solved by this software. (b) Consider the following finite-horizon optimal control problem N-1 PN (X): Minimise(x²Qxx + u Ru₂) + q*x₂] + XP/XN +97XN subject to: X₁+1 = Axt + Bu, Vt € N,N-1}, Xo = x, where R € R"" is a symmetric positive definite matrix and Q.P; € R¹ are symmetric positive semidefinite. Use the dynamic programming procedure to solve this problem (determine the optimal sequence of control actions, u(x),..., u₁(x), and the value function of the problem, V*(x).) [15 marks] Q2. (a) Consider the following finite-horizon linear-quadratic optimal [10 marks] control problem N-1 PN (x): Minimise(x7Qx+u Ru₂) + 1xPxN, (14) subject to: X+1 = Ax₂ + But € N[N-1) Xo = x, where R € R"X" is a symmetric positive definite matrix and Q,P, € R™ are symmetric positive semidefinite. We have a very efficient software that can solve convex quadratic optimisation problems of the form Minimisez ¹Hz + h¹z, ZERK where HE R*XK is a symmetric positive definite matrix. Write the above optimal control problem, PN(X), in a way that can be solved by this software. (b) Consider the following finite-horizon optimal control problem N-1 PN (X): Minimise(x²Qxx + u Ru₂) + q*x₂] + XP/XN +97XN subject to: X₁+1 = Axt + Bu, Vt € N,N-1}, Xo = x, where R € R"" is a symmetric positive definite matrix and Q.P; € R¹ are symmetric positive semidefinite. Use the dynamic programming procedure to solve this problem (determine the optimal sequence of control actions, u(x),..., u₁(x), and the value function of the problem, V*(x).) [15 marks]

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a To write the given optimal control problem in a form that can be solved by the given software we can define the state vector z and the input vector ... View the full answer