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Modern Control Systems1 Lecture 08 State Feedback Controller Design 8.1 State Feedback and Stabilization 8.2 Full-Order Observer Design 8.3 Separation Principle 8.4 Reduced-Order Observer 8.5 State Feedback Control Design with Integrator

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Modern Control Systems2 Stabilization by State Feedback: Regulator Case Given Controllable There exists a state feedback matrix, F, such that Plant: State Feedback Law: Closed-Loop System: Theorem State Feedback and Stabilization

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Modern Control Systems3 State Feedback System (Regulator Case) C C A A B B D D F F

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Modern Control Systems4 State Feedback Design in Controllable Form (8.1)

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Modern Control Systems5 Comparing (8.1) and (8.2), we have Suppose the desired characteristic polynomial (8.2) (8.3)

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Modern Control Systems6 C C A A B B D D F F State Feedback: General Case (Non-Zero Input Case) State Feedback Control System

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Modern Control Systems7 State Feedback Design with Transformation to Controllable Form Controllable From:

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Modern Control Systems8 Transform to Controllable Form Coordinate Transform Matrix Controllable Form:

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Modern Control Systems9 Example (A, B) is in controllable from, we can derive the state feedback gain from eq. (8.3)

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Modern Control Systems10 Plant: State Feedback: Closed Loop System: Char. Equation: Suppose that the system is controllable, i.e. Obtain the State Feedback Matrix by Comparing Coefficients

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Modern Control Systems11 Then, for any desired pole locations: (8.4) We can obtain the desired char. polynomial By controllability, there exists a state feedback matrix K, such that From (8.4), we can solve for the state feedback gain K.

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Modern Control Systems12 Example Plant: Fig. State Feedback Design Example State Feedback:

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Modern Control Systems13 Desired pole locations: Percent Overshoot 5%, Settling Rise time 5 sec. By comparing coefficients on the both sides of 8.5), we obtain From (8.4), we get (8.5) Spec. for Step Response:

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Modern Control Systems14 Simulation Results Fig. Step response of above example

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Modern Control Systems15 The Matrix Polynomial Plant: State Feedback: Then the state feedback gain matrix is Ackermann Formula for SISO Systems

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Modern Control Systems16 Steady State Error From (3.6) By Final Value Theorem Lapalce Transform of the Error Variable Error Variable

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Modern Control Systems17 Full-Order Observer Supposeis the observer state Estimation error: Error Dynamics Equation: Plant: L: Observer gain Full-Order Observer Design

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Modern Control Systems18 Hence if all the eigenvalues of (A-LC) lie in LHP, then the error system is asy. stable and Fig. Full-Order Observer C A B C A B + L

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Modern Control Systems19 By duality between controllable from and obeservable form Given Observable There exists a observer matrix, L, such that Theorem we have the following theorem.

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Modern Control Systems20 The eigenvalues of can be assigned arbitrarily by proper choice of K. Since have same eigenvalues, if we choose then the eigenvalues of (A-LC) can be arbitrarily assigned.

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Modern Control Systems21 Separation Principle Plant: State Feedback Law using estimated state: Observer: Error Dynamics: State Equation: (8.6) (8.7)

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Modern Control Systems22 Equation (8.9) tells us that the eigenvalues of the observer-based state feedback system is consisted of eigenvalues of (A-BF) and (A-LC). Hence, the design of state feedback and observer gain can be done independently. From (8.6) and (8.7), we obtain the overall state equation Eigenvalues of the overall state equation (7.17) (8.8) (8.9) Separation Principle (Cont.)

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Modern Control Systems23 Observer-Based Control System Plant: Observer ： State Feedback Law:

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Modern Control Systems24 Fig. Observer-based control system C A B C A B L K

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Modern Control Systems25 C A B C A B L K Fig. Observer-based control system with compensating gain

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Modern Control Systems26 Consider the n-dimensional dynamical equation (8.10a) (8.10b) Here we assume that C has full rank, that is, rank C =q. Then, there exists a coordinate transformation which can be partitioned as Reduced-Order Observer Design (8.11)

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Modern Control Systems27 Since, we have which become Observer (8.12a) (8.13a) (8.12b) (8.13b) Observer: Plant: where

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Modern Control Systems28 Note that and w are function of known signals u and y. Now if the dynamical equation above is observable, an estimator of can be constructed. Theorem: The pair {A, C} in (8.10) or, equivalently, the pair in (8.12) is observable if and only if the pair in (8.13) is observable.

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Modern Control Systems29 Such that the eigenvalues of can be arbitrarily assigned by a proper choices of. The substitution of w and into (8.143) yields To eliminate the term of the derivative of y, by defining (8.14) (8.15) (8.16) Let the estimate of be

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Modern Control Systems30 Using (8.15), then the derivative of (8.16) becomes From (8.15), we see that is an estimate of. Define the following matrices

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Modern Control Systems31 Reduced-Order Observer: where

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Modern Control Systems32 Fig. Reduced-Order Observer C A B + +

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Modern Control Systems33 then we have Define Error Variable

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Modern Control Systems34 Since the eigenvalues of can be arbitrarily assigned, the rate of e(t) approaching zero or, equivalently, the rate of approaching can be determined by the designer. Now we combine with to form We get Then from

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Modern Control Systems35 Consider the n-dimensional dynamical equation Here we assume that C has full rank, that is, rank C =q. Define where R is any (n-q) n real constant matrix so that P is nonsingular. (8.17a) (8.17b) How to transform state equation to the form of (8.11)

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Modern Control Systems36 Compute the inverse of P as where Q 1 and Q 2 are n q and n (n-q) matrices. Hence, we have

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Modern Control Systems37 Now we transform (8.17) into (8.11), by the equivalence transformation which can be partitioned as

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Modern Control Systems38 SISO State Space System Integral Control: Augmented Plant:

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Modern Control Systems39 Closed-Loop System: State Feedback Control Design with Integrator

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Modern Control Systems40 C A B K Block diagram of the integral control system Fig.Block diagram of the integral control system

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Modern Control Systems41 Example Percent Overshoot 10%, Settling time 0.5 sec. Spec. for Step Response: State Feedback Design:

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Modern Control Systems42 From the steady state analysis in Sec. 3.4

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Modern Control Systems43 State Feedback Design with Error Integrator: (8.18) Closed-Loop System:

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Modern Control Systems44 By comparing coefficients on left hand sides of (8.19) and (8.20), we obtain From (8.18), we get the char. eq. of the closed-loop system is (8.19) (8.20) The desired char. eq. of the closed-loop system is

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Modern Control Systems45 Closed-Loop System: Final Value Theorem Steady State Error

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