ESE 500: Homework 3 Due on October 16 (Monday) Part A: Matrix Exponential Problem 1. Without using Matlab, compute eAt , using two of your favourite methods, for: \u0014 \u0015 0 2 A= 1 2 It is recommended to read the additional course material about computing the matrix exponential. Problem 2 (Product of exponentials). Let A, B be two n n matrices. Prove that eAt eBt = e(A+B)t if and only if AB = BA. Hints: For the only \"only if\" part differentiate two times. For the \"if\" part differentiate once, and try to use the fact that e(A+B)t is the unique solution of X = (A + B)X, X(0) = I. Part B: Time-varying systems. Problem 3 (Transition matrix properties). Are the following valid transition matrices? If yes compute the matrix A(t). If not explain why. " # 2 e(tt0 ) 0 i) (t, t0 ) = 2 0 e(tt0 ) 2 2 t2 t2 t t0 0 e 2 cosh (t t0 ) e 2 sinh (t t0 ) ii) (t, t0 ) = t2 t2 , t2 t2 0 0 e 2 sinh (t t0 ) e 2 cosh (t t0 ) x x x x where cosh x = e +e and sinh x = e e . 2 2 Problem 4 (Computing the transition matrix). In the case of linear timevarying systems: x(t) = A(t)x(t) Rt A( )d in general we have (t, t0 ) 6= e t0 . However, there are special cases when this is true. For example in time-invariant linear systems, with A(t) = A we have: Rt A( )d (t, t0 ) = e t0 = eA(tt0 ) . 1 In fact there isRa general result that states: (t, t0 ) = e t t0 A( )d is true if A(t) commutes with \u0012Z t A(t) \u0013 A( )d \u0012Z t = t0 Rt t0 A( )d or \u0013 A( )d A(t), t0 see also book chapter 4.5. Answer the following: i) Suppose A(t) = A1 f (t) + A2 g(t), where A1 , A2 are n n constant matrices and f (t), g(t) R are functions. Suppose the matrices commute or A1 A2 = A2 A1 . Use the above result and Problem 2 to prove that: (t, t0 ) = e \u0011 \u0011 \u0010 R \u0010 R A2 tt g( )d A1 tt f ( )d e 0 0 . ii) Use i) to compute (t, t0 ), for: \u0014 A(t) = t 1 1 t \u0015 . Part C: Least Squares. Problem 5 (Minimum norm control). Read the notes about least squares. Consider the linear system: x = Ax + Bu, x(0) = 0, where A Rnn , B Rnp . Under zero initial conditions, the continuous time solution at time T is given by: Z T x(T ) = eA(T t) Bu(t)dt = L(u), 0 where L is a linear operator, which takes a continuous function u : [0, T ] Rp and returns x(T ) = L(u) Rn . Now, suppose we want to control our system to reach a given fixed target state xd at time T . In this case, the control objective is x(T ) = L(u) = xd . In general, under certain conditions1 , there are infinite ways we can choose u in order to achieve this. However, among all of those, we would like the control input u that minimizes the consumed energy. This can be posed as the following least squares problem: min kuk2C = hu, uiC s.t. L(u) = xd , (1) RT 0 Those conditions require W = 0 eA(T ) BB 0 eA (T ) d to be invertible. This is also equivalent to the system being controllable as we will see in following lectures. 1 where we denote by C the continuous functions from [0, T ] to Rp and the inner product in C is defined as: Z hu, viC = T u(t)0 v(t)dt. 0 In this problem, you are asked to find the optimal control input u of the above optimization problem, by solving the following: i) Find the adjoint operator L of L, i.e. L such that: hL(u), xiRn = hu, L (x)iC , where hy, xiRn = y 0 x is the standard inner product. The adjoint in this case takes an element x Rn and returns L (x), which is a function in C. ii) Show that LL = RT 0 0 eA(T ) BB 0 eA (T ) d . RT 0 iii) Assume that W = LL = 0 eA(T ) BB 0 eA (T ) d is invertible. Using the notes about least-squares find the solution u (t), t [0, T ] of (1) in terms of A, B, W, t. Part D: Discretization Problem 6. Suppose we have the system: \u0014 \u0015 \u0015 \u0014 \u0015 \u0014 0 0 0 1 u(t), x(0) = x(t) + x(t) = 0 1 3 4 \u0002 \u0003 y(t) = 1 0 x(t) Suppose we have sampling frequency Ts = 0.2 and u(t) = 1, for all t 0: i) Without using Matlab, compute the discretized system with the exact discretization method (second method in the book). ii) Compute the discretization using c2d function of Matlab. It should be the same with the one you computed before. iii) Simulate the continuous time system using ode45 of Matlab for t [0, 20]. Plot y(t). iv) Simulate the discrete time system, for k = 1, . . . , N , where N = 20/Ts . In the same plot as before, plot the discretized y(k) using function stem of Matlab. Observe that the discretization is exact and has no error. When you do the stem plot, make sure you regularize the discrete time axis to be the [0, 20] interval (multiply discrete time k by Ts ). Part E: Stability Problem 7 (Internal Stability). Suppose we have the continuous time systems: x = Ai x, i = 1, 2 where \u0014 A1 = 5 3 6 4 \u0015 \u0014 , A2 = 3 1 2 0 \u0015 and the discrete time systems: xk+1 = Ai xk , i = 3, 4 where \u0014 A3 = 0 1 1 0 \u0015 \u0014 , A4 = 0 2 2 0 \u0015 Which of the previous systems are i) asymptotically stable, ii) unstable, iii) marginally stable. Explain your answer. (We examine internal (or Lyapunov) stability, NOT BIBO stability.) Problem 8 (Lyapunov equation). Prove that the following system: \u0015 \u0014 2 1 x x = 2 4 is stable by solving the Lyapunov equation A0 P + P A =