Consider the system 2 Ik+1 = Ik + Uk 0.5 y = [10] Tk A) The...

  

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Consider the system 2 Ik+1 = Ik + Uk 0.5 y = [10] Tk A) The point = 0 is an equilibrium of the system when u = 0. Determine the stability of (use the Matlab command eig if needed). (5/100 marks) B) Design a state-feedback controller uk =-[ko k] to stabilise the equilibrium point . Determine if it possible to compute the control gain such that the location of the closed-loop eigenvalues can be arbitrarily assigned. Compute the control gain such that the closed-loop eigenvalues are 21 = 0.2 and 22 = 0.5. Is it possible to select k = -2? Justify your answer. (20/100 marks) C) Determine if the system is completely observable. Write the estimation error dynamics of an observer in predictor and current forms. (7/100 marks) D) The estimation error dynamics of the observer in current form has eigenvalues at the origin if the gain is Le [1 0.25]. Compute the gain of the observer in predictor form Lp such that the estimation error dynamics also has eigenvalues at the origin. (8/100 marks) Consider the system 2 Ik+1 = Ik + Uk 0.5 y = [10] Tk A) The point = 0 is an equilibrium of the system when u = 0. Determine the stability of (use the Matlab command eig if needed). (5/100 marks) B) Design a state-feedback controller uk =-[ko k] to stabilise the equilibrium point . Determine if it possible to compute the control gain such that the location of the closed-loop eigenvalues can be arbitrarily assigned. Compute the control gain such that the closed-loop eigenvalues are 21 = 0.2 and 22 = 0.5. Is it possible to select k = -2? Justify your answer. (20/100 marks) C) Determine if the system is completely observable. Write the estimation error dynamics of an observer in predictor and current forms. (7/100 marks) D) The estimation error dynamics of the observer in current form has eigenvalues at the origin if the gain is Le [1 0.25]. Compute the gain of the observer in predictor form Lp such that the estimation error dynamics also has eigenvalues at the origin. (8/100 marks)