Ee363 Homework 3 Solutions
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چکیده
1. Solution of a two-point boundary value problem. We consider a linear dynamical system ˙ x = Ax, with x(t) ∈ R n. There is an n-dimensional subspace of solutions of this equation, so to single out one of the trajectories we can impose, roughly speaking, n equations. In the most common situation, we specify x(0) = x 0 , in which case the unique solution is x(t) = e tA x 0. This is called an initial value problem since we specify the initial value of the state. In a final value problem, we specify the final state: x(T) = x f. In this case the unique solution is x(t) = e (t−T)A x f. In a two-point boundary value problem we impose conditions on the initial and final states. (a) Find the solution to the two-point boundary value problem ˙ x = Ax, F x(0) + Gx(T) = h. h. Your answer can contain a matrix exponential. What condition must hold to ensure that there is a unique solution to this equa-tion? (b) Express the two-point boundary value problem that arises in the continuous time LQR problem (i.e., with the Hamiltonian system) in the form given above, and then find the solution to this boundary value problem. (You may leave matrix exponentials in your solution.) How is the optimal input u obtained from this solution?
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Ee363 Homework 6 Solutions
0 = d dt ‖x(t)‖ = 2x(t) ẋ(t) = 2x(t)Ax(t) = x(t) (A+ A )x(t) for all x(t), which occurs if and only A+A = 0, which is the same as A = −A, i.e., A is skew-symmetric. There are many other ways to see this. For example, the norm of the state will be constant provided the velocity vector is always orthogonal to the position vector, i.e., ẋ(t)x(t) = 0. This also leads us to A + A = 0. Another approa...
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holds for all trajectories of the system, and for all t. Here we interpret u and y as power-conjugate quantities (i.e., quantities whose product gives power) such as voltage and current or force and velocity. The inequality above states that at all times, the total energy delivered to the system since t = 0 is nonnegative, i.e., it is impossible to extract any energy from a passive system. (a) ...
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تاریخ انتشار 2009