(a) Consider a single input system x = Ax+bu, where A and bare nxnand nx1 constant...

Transcribed Image Text:

(a) Consider a single input system x = Ax+bu, where A and bare nxnand nx1 constant matrices, respectively. Suppose that the system is controllable. Introduce the linear transformation =Px, where P = [b Ab A" b]. Find the transformed state-space equation and check its controllability. (b) Consider the linear system 1 x= 0 -1 0 1 y = [101] x 0 1x+1u -1 where x is the state, u is the control input and y is the output. Can one find a state feedback u=-Kx+r, where r is the reference input, so that the closed-loop poles can be at any desired locations? Design, if possible, a feedback gain K so that the closed-loop poles are located at -2, -1jl. (a) Consider a single input system x = Ax+bu, where A and bare nxnand nx1 constant matrices, respectively. Suppose that the system is controllable. Introduce the linear transformation =Px, where P = [b Ab A" b]. Find the transformed state-space equation and check its controllability. (b) Consider the linear system 1 x= 0 -1 0 1 y = [101] x 0 1x+1u -1 where x is the state, u is the control input and y is the output. Can one find a state feedback u=-Kx+r, where r is the reference input, so that the closed-loop poles can be at any desired locations? Design, if possible, a feedback gain K so that the closed-loop poles are located at -2, -1jl.