Question

Question 7. (15 marks] Consider the discrete time system given by the state equation 07 x4 + 11-18 8/11 - 10/n VIK) = 10 11 **) 1. [3 marks) Determine if the system is (a) Lyapunov state, syptereally ) Bounded input Bounded Output (BIBO) stable. Provide brief explanations 2. (8 marks) Design a discrete-time state feedback control law of the form - Kxkl by finding the gain K to place the closed-loop eigenvalues at 0.5 3. [4 marks) Suppose the previous control law is modified to uk) - Nr(t) _Kx[k] using the previously designed state feedback gain K. Determine N so that the output yk of the closed-loop system achieves asymptotic tracking of arbitrary constant reference inputs, Would this design in general retain asymptotic constant reference tracking it the model of the plant is slightly altered? Question 8. (8 marks] Consider the following discrete-time system, 0.5 1 uk) y[k) = 10 11x[k]+v[k), 08 x[k+1) where E[v[k]?) - var/v[k]) = - 10 with an initial state x[0] - [1.0]'. For the Discrete-time Kalman filter, 1. (5 marks] Determine the solution (prediction error covariance) of the Hiccati amerence equation at k = 0 and k = 1, i.e. find P[0] and P[1]. Note: Choose the initial state estimate as *[0] = [0,-1] 2. [3 marks Calculate the raman gan ar k = 1, 1.e. MNO L},

Answer 1

Griven state Equation x [k+ 1) = [08]. x0+ [!] .UCK] Y CX) - COCK Real matrix matrix A-[io] Eigen values Zeroes (Both) Here are we take o as positive symmetric matei. Generally (!). checked. Then ATP & PA= As two poles lies in The system y Unstable. Alo BIBO unstable the origin 2 x[k+ 1). AxCkJ + B.UCK) 1: UCF]=-K.803 > *CK+1) = A xCk) -BK. xCk) = x (x+1) = (A-BR) 2[K]. Gives Closed Loop Eiger valies at s= -0.5 - SIA + BK | (5+ 0.52 s [ki K₂] sts to.25 +] + [!] Cri 5:] stki 2 11 st stor 2 st kis tk. 2 S + st 0.23 K = K2 = 0.25 [ki K₂] ('02 25 K

we (3) 95 modity UCK) = N.VC+) - K &[K] Previously assigned K [1 0.25) > "(k+1 = 4.4CK) + BUCK) = CK + 1) = A < [ ] - Bk *(K) +BN.V(+) = (A-BK] .*CK) + BN.V+). Feed Forward N → Scala - IN) -C.CA-BK) 1.6 No.8 Given x[k+1) = [°: Jack)+(8].00 Y[K] - [o ) 2 x ] + vt). SO A = [05:) B = 10] с [ 0 ] 10 Variance Initial state to] = Ciot - ELVCE)] For discrete Kalman Feiter Prediction - Error Co-variance ! ko and = 1 Pro) and P(1) call this sê (k+1) A. âcie) + B.UCK) + N(YCK) - C-Cr)) we Prediction - observer one-step ahead of YCK) measurones separation Principle... ů (k+ 1) = (A-BK - mc). à Ck) + mYCE 9+ y

Put K = 0 in â cx + 1) = (A-BK-mc).sê CK) + m.yck) > 2 (1) (A-BK-mc) n(o)t m.y(0) こ Ylo) = c.x (0) [o B[!] (0) Loo). à (0) А, В, , с we know K=1. put (о Simzlarly know & (2)