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Unformatted text preview: EE364b Prof. S. Boyd EE364b Homework 7 1. MPC for output tracking. We consider the linear dynamical system x ( t + 1) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) , t = 0 ,...,T − 1 , with state x ( t ) ∈ R n , input u ( t ) ∈ R m , and output y ( t ) ∈ R p . The matrices A and B are known, and x (0) = 0. The goal is to choose the input sequence u (0) ,...,u ( T − 1) to minimize the output tracking cost J = T summationdisplay t =1 bardbl y ( t ) − y des ( t ) bardbl 2 2 , subject to bardbl u ( t ) bardbl ∞ ≤ U max , t = 0 ,...,T − 1. In the remainder of this problem, we will work with the specific problem instance with data A = 1 1 0 0 1 1 0 0 1 , B = . 5 1 , C = bracketleftBig − 1 0 1 bracketrightBig , T = 100, and U max = 0 . 1. The desired output trajectory is given by y des ( t ) = t < 30 , 10 30 ≤ t < 70 , t ≥ 70 . (a) Find the optimal input u ⋆ , and the associated optimal cost J ⋆ . (b) Rolling look-ahead. Now consider the input obtained using an MPC-like method: At time t , we find the values u ( t ) ,...,u ( t + N − 1) that minimize t + N summationdisplay τ = t +1 bardbl y ( τ ) − y des ( τ ) bardbl 2 2 , subject to bardbl u ( τ ) bardbl ∞ ≤ U max , τ = t,...,t + N − 1, and the state dynamics, with x ( t ) fixed at its current value. We then use u ( t ) as the input to the system. (This is an informal description, but you can figure out what we mean.) In a tracking context, we call N the amount of look-ahead , since it tells us how much of the future of the desired output signal we are allowed to access when we decide on the current input. Find the input signal for look-ahead values N = 8, N = 10, and N = 12. Compare the cost J obtained in these three cases to the optimal cost J ⋆ found in part (a)....
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- Fall '09
- Upper and lower bounds, partial binary tree, Solve new problems, fixedvals