1. (20pts) Consider the following state-space system, x = [12] x + [1] u u, y...

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1. (20pts) Consider the following state-space system, x = [12] x + [1] u u, y = [0_1]X 3 a) (5pts) Derive the controllability and observability matrices and verify that the system is both observable and controllable. b) (10pts) Design a full-order observer, such that the observer poles are located at (-5 5j). Write the equation of the observer. c) (5pts) Noting that x2 is directly measured, design a reduced order observer for estimating the state x that places the observer pole at -5. Write the observer equation and show how numerical differentiation of x can be avoided in designing the reduced order observer. 1. (20pts) Consider the following state-space system, x = [12] x + [1] u u, y = [0_1]X 3 a) (5pts) Derive the controllability and observability matrices and verify that the system is both observable and controllable. b) (10pts) Design a full-order observer, such that the observer poles are located at (-5 5j). Write the equation of the observer. c) (5pts) Noting that x2 is directly measured, design a reduced order observer for estimating the state x that places the observer pole at -5. Write the observer equation and show how numerical differentiation of x can be avoided in designing the reduced order observer.