question archive Problem 1

Problem 1

Subject:Electrical EngineeringPrice: Bought3

Problem 1. (Continuation of Problem 4 of HW #3) Consider three cars moving on the same lane, whose initial locations at time t = 0 are x1 (0), x2(0), x3(0) E R, respectively. Suppose the dynamics of the cars are given by ic 1 (t) = 12(t) - 21(t) + u(t) (la) 2c 2(t) = 1(t)+ 3(t) - 202 (t) (1b) 2 2c3 ( t) = 202(t) - 23(t). (1c) (a) With u(t) = 1, t 2 0, find the expression of x(t) for t 2 0. As t -> co, describe the steady-state behaviors of xi(t), i = 1, 2, 3. (b) Let (Ad, Ba, Ca, Da) be the sampled discrete-time dynamics of the continuous-time system (1) with the sampling time T = 1 s. Find both the analytic expression of (Ad, Ba, Ca, Da) and the numerical values (using e.g. Matlab command c2d). (c) For the sampled system (Ad, Ba, Ca, Da), plot its response x[k] under the unit step input uk = 1 with initial state x[0] = [0 1 3]. One useful Matlab command is 1sim. (d) Suppose the u(t) term in (la) is moved to (1b), so that dynamics of the first two cars become 1 (t) = 12(t) - 1(t), 32(t) = 1(t) +3(t) 2 2 2 ( t ) + u (t) while the dynamics of the third cars remain unchanged. With u(t) = 1, t 2 0, find the expression of x(t) for t 2 0. As t -> co, describe the steady-state behaviors of xi (t), i = 1, 2, 3.

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