State Feedback

State Feedback

State Feedback Closed loop matrices R=0 the system is called regulator.

=> Faster/stable system State Feedback The CL state matrix is a function of K By appropriate changing K we can change eig(ACL) => Faster/stable system This method is called pole placement. WE MUST CHECK IF THE SYSTEM IS CONTROLLABLE

State Feedback Eigenvalues of CL system: 3-k CL eigenvalues at -10

State Feedback If the system is unstable create a controller that will stabilise the system. >> eig(A) >> rank(ctrb(A,B)) ans = -0.3723 5.3723 ans = 2

State Feedback -10 and -11 Matlab If the system is n x n then you need to solve an n x n system of equations! K=place(A,B, [-10 -11]) P=[-10 -20 -30]; C=[0 0 1], D=0. Matlab

State Feedback - LQR Matlab: K =[26 72]T State feedback Response of the system Reference signal Study the signal u=-Kx Place the poles at [-100 -110] K =[26 72]T P=[-10 -11]

State Feedback - LQR Compromise between speed and energy that we use. Similar problem/dilemma if we had an input. Solution: Linear Quadratic Regulator (optimum controller) Q and R are positive definite matrices Square symmetric matrices, positive eigenvalues for all nonzero X Q: Importance of the error, R: Importance of the energy that we use (assume that is stable)

State Feedback - LQR P is positive definite X=1 (assume that is stable) P is positive definite X=1 Reduced Riccati Equation

State Feedback - LQR Steps to design an LQR controller: To find the optimum P solve: 2.Find the optimum gain [K, P, E]=lqr(sys, Q, R) E are the eigenvalues of A-BK Find K, The eigenvalues of A-BK The response of the system For R=1 and Q=eye(2) and Q=2*eye(2) (X(0)=[1 1]). Matlab CAD Exercise

Estimating techniques Magic trick

Estimating techniques Example: A=[1 2;3 4]; B=[1 0]'; C=[1 0]; D=0;

Estimating techniques The error between the estimated and real state is A is unstable Homogeneous ODE A is slow A=[1 2;3 4]; B=[1 0]'; C=[1 0]; D=0; U=1,

Estimating techniques

Estimating techniques G=place(A’, C’, P) But the system must be observable Where to place these eigenvalues??? A=[1 2;3 4]; B=[1 0]'; C=[1 0]; D=0; U=1,