Chap 8. State Feedback and State Estimators

Outlines Introduction State feedback Regulation and tracking State estimator Feedback from estimated states State feedback-multivariable case State estimators-multivariable case Feedback from estimated states-multivariable case 2

1. Introduction Use controllability and observability Control design

2. State Feedback Single-variable system Input with negative state feedback Closed-loop

2. State Feedback Theorem 8.1

2. State Feedback Remark

2. State Feedback

2. State Feedback The power of state feedback

2. State Feedback Theorem 8.2

2. State Feedback

2. State Feedback Theorem 8.3

2. State Feedback

2. State Feedback Feedback transfer function Plant (A, b, c) With negative feedback bk Numerators are the same.

2. State Feedback

2. State Feedback Matlab command: place How to select the desired eigenvalues According to performance criteria Rise time Settling time Overshoot Desired regions

2. State Feedback: solving Lyap. Eq A different method to computing feedback gain However, new eigenvalues cannot contain any original eigenvalues of A Procedure

2. State Feedback: solving Lyap. Eq Justification Theorem 8.4

2. State Feedback: solving Lyap. Eq

3. Regulation and Tracking Regulation Reference r is 0, and the response is caused by some nonzero initial conditions Find a state feedback gain so that the response will die out at a desired rate Tracking Reference r is a constant a (more generally, a varying signal a(t)) Find a state feedback gain so that the response will approach a when t goes to infinity

3. Regulation and Tracking Plant (A, b, c) Regulation u = r-kx, (A-bk, b, c) Response: Tracking u = pr-kx, p is the feedforword gain TF: p:

3. Regulation and Tracking: stabilization Uncontrollable system State feedback Closed-loop

4. State Estimator State feedback requires all states are available, which may not be the case in practice State estimator / observer, will generate the estimate of the state

4. State Estimator Open-loop estimator System Duplication (if A and b are known) If tow systems have the same initial state Initial state can be computed, if original system is observable Thus, if original system is observable, open-loop estimator can be used 28

4. State Estimator Closed-loop estimator y is used in closed-loop estimator the output difference will be used to correct the estimated state or 29

4. State Estimator Define 30

4. State Estimator 31

4. State Estimator Theorem 8.O3 Justification Solving Lyapunov Equation: a different design 32

4. State Estimator Procedure 8.O1 33

4. State Estimator Justification 34

4. State Estimator: reduced dimension Basic idea Procedure 8.R1 35

4. State Estimator: reduced dimension Justification Theorem 8.6 36

4. State Estimator: reduced dimension 37

4. State Estimator: reduced dimension 38

4. State Estimator: reduced dimension 39

5. Feedback from Estimated States System Feedback from estimated states 40

5. Feedback from Estimated States 3 concerns Consider the system

5. Feedback from Estimated States Using equivalence transformation

5. Feedback from Estimated States Observation

6. State Feedback-Multivariable Case p-input State feedback K is p-by-n Theorem 8.M1

6. State Feedback-Multivariable Case Theorem 8.M3 Cyclic design Change the multi-input problem into a single-input problem Cyclic matrix: if its characteristic polynomial equals its minimal polynomial A is cyclic iff the Jordan form of A has one and only one Jordan block associated with each distinct eigenvalue

6. State Feedback-Multivariable Case Theorem 8.7 Justification

6. State Feedback-Multivariable Case

6. State Feedback-Multivariable Case Theorem 8.8 Justification

6. State Feedback-Multivariable Case

6. State Feedback-Multivariable Case An example

6. State Feedback-Multivariable Case State feedback cyclic design

6. State Feedback-Multivariable Case Lyapunov-Equation Method: extend the Lyapunov procedure to multivariable case Procedure 8.M1 Justification

6. State Feedback-Multivariable Case Theorem 8.M4

7. State Estimators-Multivariable Case n-dimensional p-input q-output system Full dimension estimator Error and error dynamics

7. State Estimators-Multivariable Case Reduced-dimensional estimators Procedure 8.MR1

7. State Estimators-Multivariable Case Justification

7. State Estimators-Multivariable Case Theorem 8.M6

8. Feedback from Estimated States- Multivariable case System Estimator Partition the inverse of P

8. Feedback from Estimated States- Multivariable case