1. State Feedback

2. State Feedback

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

4. => 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

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

6. 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

7. 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

8. 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=[ ]

9. 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)

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

11. 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

12. Estimating techniques
Magic trick

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

14. 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,

15. Estimating techniques

16. 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,