State Feedback
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State Feedback. State Feedback. State Feedback. Closed loop matrices. R=0 the system is called regulator. State Feedback. The CL state matrix is a function of K. => Faster/stable system. By appropriate changing K we can change eig( A CL ). This method is called pole placement.

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State Feedback

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State feedback

State Feedback


State feedback

State Feedback


State feedback

State Feedback

Closed loop matrices

R=0 the system is called regulator.


State feedback

State Feedback

The CL state matrix is a function of K

=> Faster/stable system

By appropriate changing K we can change eig(ACL)

This method is called pole placement.

WE MUST CHECK IF THE SYSTEM IS CONTROLLABLE


State feedback

State Feedback

Eigenvalues of CL system:

3-k

CL eigenvalues at -10

k=13


State feedback

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

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

State Feedback - LQR

Matlab:

State feedback

Response of the system

Reference signal

Study the signal u=-Kx

Place the poles at [-100 -110]

Response of the system

Study the signal u=-Kx

K =[26 72]T

P=[-10 -11]


State feedback

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

for all nonzero X

Square symmetric matrices, positive eigenvalues

Q: Importance of the error, R: Importance of the energy that we use

(assume that

is stable)


State feedback

State Feedback - LQR

(assume that

is stable)

P is positive definite

X=1

Reduced Riccati Equation


State feedback

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


State feedback

Estimating techniques

Magic trick


State feedback

Estimating techniques

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


State feedback

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,


State feedback

Estimating techniques


State feedback

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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