Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08

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1 Fall 2007 線性系統 Linear Systems Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian NTU-EE Sep07 Jan08 Materials used in these lecture notes are adopted from Linear System Theory & Design, 3rd. Ed., by C.-T. Chen (1999) Outline NTUEE-LS8-DesignSISO-2 Introduction State Feedback (8.2) Regulation & Tracking (8.3) State Estimator (8.4) Feedback from Estimated States (8.5) State Feedback Multivariable Case (8.6) State Estimators Multivariable Case (8.7) Feedback from Estimated States Multivariable Case (8.8)

2 Outline NTUEE-LS8-DesignSISO-3 State Feedback (8.2) State Feedback in Controllable Canonical Form Eigenvalue Assignment by Solving the Lyapunov Eqn Regulation & Tracking (8.3) Robust Tracking and Disturbance Rejection Stabilization State Estimator (8.4) Full-Dimensional State Estimator Reduced-Dimensional State Estimator Feedback from Estimated States (8.5) Basic Concept of Feedback Control (8.1) NTUEE-LS8-DesignSISO-4 r u Plant y (system to be controlled) r: reference signal

3 Basic Concept of Feedback Control 2 r: reference signal NTUEE-LS8-DesignSISO-5 r + u Plant y - (system to be controlled) x state feedback control r + u Plant y - (system to be controlled) output feedback control State Feedback (SISO Systems) (8.2) NTUEE-LS8-DesignSISO-6 Closed-loop system :

4 State Feedback (SISO Systems) 2 NTUEE-LS8-DesignSISO-7 u + + y Theorem 8.1 (8.2) NTUEE-LS8-DesignSISO-8 Proof:

5 Example 8.1 (8.2) NTUEE-LS8-DesignSISO-9 Controllability: Observability: controllable and observable State feedback: Feedback system: Example NTUEE-LS8-DesignSISO-10 Controllability: Observability: Observability may change after state feedback

6 Example 8.2 (8.2) NTUEE-LS8-DesignSISO-11 Characteristic Polynomial: State feedback: Feedback system: Example NTUEE-LS8-DesignSISO-12 Characteristic Polynomial:

7 State Feedback in Controllable Canonical Form (8.2) NTUEE-LS8-DesignSISO-13 State Feedback in Controllable Canonical Form 2 NTUEE-LS8-DesignSISO-14

8 State Feedback in Controllable Canonical Form 3 NTUEE-LS8-DesignSISO-15 State Feedback in Controllable Canonical Form 4 NTUEE-LS8-DesignSISO-16

9 State Feedback in Controllable Canonical Form 5 NTUEE-LS8-DesignSISO-17 State Feedback in Controllable Canonical Form 6 NTUEE-LS8-DesignSISO-18

10 State Feedback in Controllable Canonical Form 7 NTUEE-LS8-DesignSISO-19 State Feedback in Controllable Canonical Form 8 NTUEE-LS8-DesignSISO-20

11 State Feedback in Controllable Canonical Form 9 NTUEE-LS8-DesignSISO-21 In Summary 1 NTUEE-LS8-DesignSISO-22

12 In Summary 2 NTUEE-LS8-DesignSISO-23 Theorem 8.2 (8.2) NTUEE-LS8-DesignSISO-24 (A,b,c) (A,b) (A,b,c)

13 Theorem Proof: If with the transformation Q, the state equation (A, b, c) _ is transformed into (A, b, c), then g(s) ^ is as shown. 2 3 Let Q = [ b Ab A b A b], Q 1 2 = 1 α1 α2 α3 0 1 α α 1 2, α NTUEE-LS8-DesignSISO-25 x = Q x = Q Q x = Q x α 4 1 α = = 1 0 Then A Q = = 1 A Q1, b Q1 b, α α c = c Q = [ cb cab ca b ca b] 1 Theorem And Q A α 4 α α = 1 = A Q 0 1 α NTUEE-LS8-DesignSISO-26 A = α1 α2 α3 α Q 2 b = b = [ ] b = [ ], c = c Q = [ β β β β ] Note: (1) β 1 = c b, β 2 = α 1 c b + c A b,, but not so important here _ (2) Q 1 = C, Q 2 = C 1.

14 Theorem 8.3 (8.2) NTUEE-LS8-DesignSISO-27 (A,b,c) Proof: Theorem Choose P such that NTUEE-LS8-DesignSISO-28 Use the state feedback gain Can obtain any desired characteristic polynomial after state feedback

15 Theorem And get NTUEE-LS8-DesignSISO-29 Thus Double check: Theorem Zeros are not affected by state feedback NTUEE-LS8-DesignSISO-30 Before state feedback: After state feedback: _ Observability may change with possible new pole-zero relation

16 Example 8.3 (8.2) NTUEE-LS8-DesignSISO-31 If the desired eigenvalues are 1.5±0.5j and 1±j, then Design Guidelines (8.2) Guidelines for selecting desired eigenvalues NTUEE-LS8-DesignSISO-32 More negative real parts of eigenvalues Faster response y(t), Larger system bandwidth (noise problem), and Larger input u(t) (actuator saturation problem) Clustered eigenvalues Response y(t) sensitive to parameter change, and Larger input u(t)

17 Design Guidelines 2 Suggested methods: NTUEE-LS8-DesignSISO-33 (1) Let the eigenvalues uniformly spread over the crescent region (2) Find k to minimize the performance index J Optimal Control weighting matrices Design Guidelines 3 NTUEE-LS8-DesignSISO-34 Feedback Control of Dynamic Systems, 4th Ed., By Franklin, Powell, Emami-Naeini, 2002

18 State Feedback of Discrete-Time Systems (SISO Systems) (8.2) NTUEE-LS8-DesignSISO-35 x[k+1] = u[k] = A x[k] + b u[k] r[k] k x[k] x[k+1] = (A bk) x[k] + b r[k] Thus everything is the same, except that the desired eigenvalues are different from the C.T. case. For example, the hatched area in the figure is suggested. Eigenvalue Assignment by Solving the Lyapunov Equation (8.2.1) NTUEE-LS8-DesignSISO-36 A different method of computing state feedback gain for eigenvalue assignment Restriction: Different sets of eigenvalues wrt eig(a)

19 Eigenvalue Assignment by Solving the Lyapunov Equation 2 NTUEE-LS8-DesignSISO-37 Note that: In the procedure if a solution T exists and is nonsingular, Then from steps 3 and 4, we have Thus A bk and F have the same (assigned) eigenvalues. A sufficient condition for T to exist is that A and F have no common eigenvalues (i.e., every eigenvalue must move). Theorem 8.4 (8.2.1) NTUEE-LS8-DesignSISO-38 Proof: (for n = 4 only) Let the characteristic polynomial of A be Then _ And for all eigenvalues λ i of F Thus has nonzero eigenvalues and is nonsingular

20 Theorem Also, observe the recursive relation NTUEE-LS8-DesignSISO-39 (+ T is Gain Selection (8.2.1) Given a set of desired eigenvalues, how to set F and k? NTUEE-LS8-DesignSISO-40 _ (1) Using the observable canonical form, set F to the companion form with the desired eigenvalues, and let k = [1 0 0]; _ (2) Using the modal form, set F to the diagonal form with the desired eigenvalues, and _ Let k have at least one nonzero element associated with each diagonal block of F, _ such as k = [ ], [ ], [ ].

21 Example 8.4 (8.2.1) (the same as Example 8.3) NTUEE-LS8-DesignSISO-41 The desired eigenvalues are 1.5±0.5j and 1±j, thus set No matter k is set to [ ] or [ ], the resulting k is the same as the one obtained in Example 8.3 (different k, different T, but the same kt 1 ). _ Theorem NTUEE-LS8-DesignSISO-42 For single-input systems, the state feedback gain k corresponding to a set of pre-assigned eigenvalues is unique.

22 State Feedback and State Estimation (MIMO) (8.6 & 8.7) NTUEE-LS8-DesignSISO-43 Regulation and Tracking (8.3) NTUEE-LS8-DesignSISO-44 Regulation: y(t) to follow r(t) = 0, t 0 Tracking: y(t) to follow r(t) = a, a constant, t 0 Servomechanism problem: y(t) to follow r(t), a non-constant reference signal, t 0

23 Regulation and Tracking 2 To solve the regulation problem is NTUEE-LS8-DesignSISO-45 to find a state feedback gain k such that A bk is a stable matrix, To solve the tracking problem, we need an extra feedforward gain p: r + u k x Regulation and Tracking 3 With the feedforward gain and state feedback, the transfer function is NTUEE-LS8-DesignSISO-46 Thus for y(t) to follow r(t) = a, a constant, choose k such that A bk is stable, and p such that Which requires β 4 0, i.e., the plant has no zeros at s = 0

24 Robust Tracking and Disturbance Rejection (8.3.1) NTUEE-LS8-DesignSISO-47 In practical applications, there will be disturbance, i.e., exogenous input affecting the system, and there will be uncertainty regarding the exact values of A, b, and c. Robust Tracking: Tracking with parameter uncertainty _ (β 4 and α 4 not known exactly) Disturbance Rejection: Elimination of the effect by disturbance Robust Tracking and Disturbance Rejection 2 Plant nominal equation: NTUEE-LS8-DesignSISO-48 constant disturbance An internal model control configuration with state feedback

25 Theorem 8.5 (8.3.1) NTUEE-LS8-DesignSISO-49 closed-loop system Proof: Theorem NTUEE-LS8-DesignSISO-50

26 Theorem NTUEE-LS8-DesignSISO-51 Theorem Block diagram manipulation: NTUEE-LS8-DesignSISO-52

27 Theorem NTUEE-LS8-DesignSISO-53 Taking determinants of the identity i.e., Theorem NTUEE-LS8-DesignSISO-54 For constant (step-type) disturbance Even with (small) parameter uncertainty, lim t y w (t) = 0, as long as roots of Δ f (s) all have negative real parts Robust constant-disturbance rejection

28 Theorem NTUEE-LS8-DesignSISO-55 Also, Thus even with parameter uncertainty robust tracking for constant reference Stabilization (8.3.2) NTUEE-LS8-DesignSISO-56 Stable systems Stabilizable systems

29 Stabilization 2 Without loss of generality, consider NTUEE-LS8-DesignSISO-57 Uncontrollable systems are stabilizable by state feedback if and only if the uncontrollable part of the system is stable State Estimator (Observer) for SISO Systems (8.4) NTUEE-LS8-DesignSISO-58 Given A, b, c, Can we estimate x(t) by measuring y(τ) and u(τ) for τ [0, t]?

30 State Estimator (Observer) for SISO Systems 2 Open-Loop Estimator: NTUEE-LS8-DesignSISO-59 and use y(τ) and u(τ) for τ [0, t] ^ to compute x(0) = x(0) system (model) estimator Disadvantage 1: Any inaccuracy in the determination of x(0) will make lim t x(t) ^ x(t) = for unstable A Disadvantage 2: Very sensitive to parameter uncertainty in A and b State Estimator (Observer) for SISO Systems 3 Closed-Loop Estimator: NTUEE-LS8-DesignSISO-60

31 State Estimator (Observer) for SISO Systems 4 Estimator Error: NTUEE-LS8-DesignSISO-61 Theorem 8.O3 (8.4) NTUEE-LS8-DesignSISO-62 Proof:

32 Procedure 8.O1 (8.4) A different method for designing state estimators: NTUEE-LS8-DesignSISO-63 Reduced-Dimensional State Estimator (8.4.1) NTUEE-LS8-DesignSISO-64 The equation y(t) = c x(t) already has one-dimensional information about x(t), so only an (n 1)-dimensional state estimator is needed to estimate the remaining information about x(t)

33 Theorem 8.6 (8.4.1) NTUEE-LS8-DesignSISO-65 Theorem Proof: (necessity for general n) If (F, l) not controllable NTUEE-LS8-DesignSISO-66 If (A, c) not observable

34 Theorem Proof: (sufficiency for n = 4) NTUEE-LS8-DesignSISO-67 As in Theorem 8.4, Let Then Theorem NTUEE-LS8-DesignSISO-68 If there is a nonzero vector r, such that Pr = 0, i.e., Then consider [l Fl F 2 l F 3 l] [l Fl F 2 l] (F, l) controllable (A, c) observable and Λ nonsingular

35 Theorem In the reduced-dimensional state estimator NTUEE-LS8-DesignSISO-69 Define the error signal, then Thus lim t e(t) = 0 if F is chosen to be stable, and lim t c yt () c cx() t c c xˆ( t) = limt = = () t = ( t) () t () t x x T z T Tx T T Feedback from Estimated States (8.5) NTUEE-LS8-DesignSISO-70 r + - u Plant y k Estimator

36 Feedback from Estimated States 2 NTUEE-LS8-DesignSISO-71 Consider the equivalence transformation P = P 1 Feedback from Estimated States 3 We see NTUEE-LS8-DesignSISO-72

37 State Feedback and State Estimation (MIMO) (8.6 & 8.7) 3NTUEE-LS8-DesignSISO-73 r + - u Plant y K Estimator State Feedback and State Estimation (MIMO) (8.6 & 8.7) NTUEE-LS8-DesignSISO-74

38 State Feedback and State Estimation (MIMO) (8.6 & 8.7) 2NTUEE-LS8-DesignSISO-75 State Feedback and State Estimation (MIMO) (8.6 & 8.7) 3NTUEE-LS8-DesignSISO-76

39 State Feedback and State Estimation (MIMO) (8.6 & 8.7) 3NTUEE-LS8-DesignSISO-77 r + - u Plant y K Estimator