1. State Feedback Controller Design
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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State Feedback and Stabilization
Stabilization by State Feedback: Regulator Case
Plant:
State Feedback Law:
Closed-Loop System:
Theorem
Given
Controllable
There exists a state feedback matrix, F, such that
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3. Modern Control SystemsA B D F
State Feedback System (Regulator Case)
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State Feedback Design in Controllable Form
(8.1)
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Suppose the desired characteristic polynomial
(8.2)
Comparing (8.1) and (8.2), we have
(8.3)
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State Feedback: General Case (Non-Zero Input Case) D B C A F
State Feedback Control System
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State Feedback Design with Transformation to Controllable Form
Controllable From:
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Transform to Controllable Form
Coordinate Transform Matrix
Controllable Form:
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Example
(A, B) is in controllable from, we can derive the state feedback gain from eq. (8.3)
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Obtain the State Feedback Matrix by Comparing Coefficients
Plant:
State Feedback:
Closed Loop System:
Char. Equation:
Suppose that the system is controllable, i.e.
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Then, for any desired pole locations:
We can obtain the desired char. polynomial
By controllability, there exists a state feedback matrix K, such that
(8.4)
From (8.4), we can solve for the state feedback gain K.
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Example
Plant:
State Feedback:
Fig. State Feedback Design Example
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Spec. for Step Response:
Percent Overshoot 5%, Settling Rise time 5 sec.
Desired pole locations:
From (8.4), we get
(8.5)
By comparing coefficients on the both sides of 8.5), we obtain
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Simulation Results
Fig. Step response of above example
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Ackermann Formula for SISO Systems
Plant:
State Feedback:
The Matrix Polynomial
Then the state feedback gain matrix is
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Steady State Error
Error Variable
Lapalce Transform of the Error Variable
From (3.6)
By Final Value Theorem
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Full-Order Observer Design
Full-Order Observer
Plant:
Suppose
is the observer state
L: Observer gain
Estimation error:
Error Dynamics Equation:
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Hence if all the eigenvalues of (A-LC) lie in LHP, then the error system
is asy. stable and C A B + L
Fig. Full-Order Observer
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By duality between controllable from and obeservable form
we have the following theorem.
Theorem
Given
Observable
There exists a observer matrix, L, such that
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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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Separation Principle
Plant:
State Feedback Law using estimated state:
State Equation:
(8.6)
Observer:
Error Dynamics:
(8.7)
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Separation Principle (Cont.)
From (8.6) and (8.7), we obtain the overall state equation
(8.8)
Eigenvalues of the overall state equation (7.17)
(8.9)
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.
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23. Observer-Based Control System
Plant:
Observer：
State Feedback Law:
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24. Modern Control SystemsA B L K
Fig. Observer-based control system
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25. Modern Control SystemsA B L K
Fig. Observer-based control system with compensating gain
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Reduced-Order Observer Design
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
(8.11)
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Since
, we have
(8.12a)
(8.12b)
which become
Plant:
(8.13a)
(8.13b)
where
Observer:
Observer
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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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Let the estimate of be
(8.14)
Such that the eigenvalues of can be arbitrarily assigned by a proper choices of The substitution of w and into (8.143) yields
(8.15)
To eliminate the term of the derivative of y, by defining
(8.16)
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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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Reduced-Order Observer:
where
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Fig. Reduced-Order Observer
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Define Error Variable
then we have
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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
Then from
We get
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How to transform state equation to the form of (8.11)
Consider the n-dimensional dynamical equation
(8.17a)
(8.17b)
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.
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Compute the inverse of P as
where Q1 and Q2 are nq and n(n-q) matrices. Hence, we have
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Now we transform (8.17) into (8.11),
by the equivalence transformation
which can be partitioned as
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SISO State Space System
Integral Control:
Augmented Plant:
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State Feedback Control Design with Integrator
Closed-Loop System:
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Block diagram of the integral control system B C A K
Fig.Block diagram of the integral control system
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Example
Spec. for Step Response:
Percent Overshoot 10%, Settling time 0.5 sec.
State Feedback Design:
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From the steady state analysis in Sec. 3.4
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State Feedback Design with Error Integrator:
Closed-Loop System:
(8.18)
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From (8.18), we get the char. eq. of the closed-loop system is
(8.19)
The desired char. eq. of the closed-loop system is
(8.20)
By comparing coefficients on left hand sides of (8.19) and (8.20), we obtain
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Closed-Loop System:
Final Value Theorem
Steady State Error
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