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State Feedback
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State Feedback
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State Feedback Closed loop matrices
R=0 the system is called regulator.
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=> 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
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State Feedback Eigenvalues of CL system: 3-k CL eigenvalues at -10
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State Feedback If the system is unstable create a controller that will stabilise the system. >> eig(A) >> rank(ctrb(A,B)) ans = 5.3723 ans = 2
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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, [ ]) P=[ ]; C=[0 0 1], D=0. Matlab
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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 [ ] K =[26 72]T P=[ ]
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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)
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State Feedback - LQR P is positive definite X=1
(assume that is stable) P is positive definite X=1 Reduced Riccati Equation
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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
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Estimating techniques
Magic trick
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Estimating techniques
Example: A=[1 2;3 4]; B=[1 0]'; C=[1 0]; D=0;
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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,
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Estimating techniques
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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,
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