# Control Systems Design

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1 ELEC4410 Control Systems Design Lecture 17: State Feedback Julio H Braslavsky School of Electrical Engineering and Computer Science Lecture 17: State Feedback p1/23

2 Outline Overview of Control Design via State Space Methods State Feedback Design State Feedback Stabilisation Lecture 17: State Feedback p2/23

3 Overview of Control via State Space The linear systems theory that we ve been discussing is the basis for linear control design theory in state space, which we will discuss from this lecture on Linear state space control theory involves modifying the behaviour of an m-input, p-output, n-state system ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t), (OL) which we call the plant, or open loop state equation, by application of a control law of the form u(t) = Nr(t) Kx(t), (U) in which r(t) is the new (reference) input signal The matrix K is the state feedback gain and N the feedforward gain Lecture 17: State Feedback p3/23

4 Overview of Control via State Space Substitution of (U) into (OL) gives the closed-loop state equation ẋ(t) = (A BK)x(t) + BNr(t) y(t) = Cx(t) (CL) Obviously, the closed-loop system is also LTI Lecture 17: State Feedback p4/23

5 Overview of Control via State Space Substitution of (U) into (OL) gives the closed-loop state equation ẋ(t) = (A BK)x(t) + BNr(t) y(t) = Cx(t) (CL) Obviously, the closed-loop system is also LTI r u N B C ẋ A x y K State feedback with feedforward precompensation This type of control is said to be static, because u only depends on the present values of the state x and the reference r Note that it requires that all states of the system be measured Lecture 17: State Feedback p4/23

6 Overview of Control via State Space When not all the states of the system are measurable, we resource to their estimation by means of an observer, or state estimator, which reconstructs the state from measurements of the inputs u(t) and outputs y(t) r u ẋ x y N B C A K ^x Observer Output feedback by estimated state feedback The combination of state feedback and state estimation yields a dynamic output feedback controller Lecture 17: State Feedback p5/23

7 Overview of Control via State Space In this part of the course we will see techniques for the design of the state feedback gain K to achieve Lecture 17: State Feedback p6/23

8 Overview of Control via State Space In this part of the course we will see techniques for the design of the state feedback gain K to achieve pole placement (stabilisation) Lecture 17: State Feedback p6/23

9 Overview of Control via State Space In this part of the course we will see techniques for the design of the state feedback gain K to achieve pole placement (stabilisation) regulation and robust tracking Lecture 17: State Feedback p6/23

10 Overview of Control via State Space In this part of the course we will see techniques for the design of the state feedback gain K to achieve pole placement (stabilisation) regulation and robust tracking techniques for the design of observers Lecture 17: State Feedback p6/23

11 Overview of Control via State Space In this part of the course we will see techniques for the design of the state feedback gain K to achieve pole placement (stabilisation) regulation and robust tracking techniques for the design of observers Our basic aim in this part of the course is To learn how to design a linear control system by dynamic output feedback (state feedback + observer) to satisfy the desired closed-loop system specifications in stability and performance We start with SISO systems, and then move on to MIMO Over the end of the course, we will discuss some optimal designs Lecture 17: State Feedback p6/23

12 State Feedback Design We start by considering SISO systems, and the state feedback control scheme with feedforward precompensation For simplicity, let s assume for the moment that N = 1 r u ẋ B C A x y K Lecture 17: State Feedback p7/23

13 State Feedback Design We start by considering SISO systems, and the state feedback control scheme with feedforward precompensation For simplicity, let s assume for the moment that N = 1 r u ẋ B C A x y K An essential system property in state feedback is Controllability, and our first important observation is that Controllability is invariant with respect to state feedback Theorem (Invariance of Controllability with State Feedback) For any K R 1 n, the pair (A BK, B) is controllable if and only if the pair (A, B) is controllable Lecture 17: State Feedback p7/23

14 State Feedback Design It is interesting to note that, on the other hand, Observability is not invariant with respect to feedback Example (Loss of Observability after feedback) The system ẋ(t) = 1 2 x(t) + 0 u(t) [ ] y(t) = 1 2 x(t) is controllable and observable, since its controllability and observability matrices [ ] C = B AB = 0 2, and O = C = CA 7 4 are nonsingular Lecture 17: State Feedback p8/23

15 State Feedback Design Example (Continuation) The state feedback control u(t) = r(t) [ ] 3 1 x(t) yields the closed loop state equations ẋ(t) = 1 2 x(t) [ ] y(t) = 1 2 x(t) 0 u(t) 1 Lecture 17: State Feedback p9/23

16 State Feedback Design Example (Continuation) The state feedback control u(t) = r(t) [ ] 3 1 x(t) yields the closed loop state equations ẋ(t) = 1 2 x(t) [ ] y(t) = 1 2 x(t) 0 u(t) 1 The controllability matrix for the closed loop state equation is C K = [ ], which, as expected, is nonsingular, and verifies that the closed-loop system is controllable Lecture 17: State Feedback p9/23

17 State Feedback Design Example (Continuation) The state feedback control u(t) = r(t) [ ] 3 1 x(t) yields the closed loop state equations ẋ(t) = 1 2 x(t) [ ] y(t) = 1 2 x(t) 0 u(t) 1 The controllability matrix for the closed loop state equation is C K = [ ], which, as expected, is nonsingular, and verifies that the closed-loop system is controllable However, the observability matrix is O = [ ], which is singular, and thus the closed-loop system with this state feedback is not observable Lecture 17: State Feedback p9/23

18 State Feedback Design Example (Continuation) The state feedback control u(t) = r(t) [ ] 3 1 x(t) yields the closed loop state equations ẋ(t) = 1 2 x(t) [ ] y(t) = 1 2 x(t) 0 u(t) 1 The controllability matrix for the closed loop state equation is C K = [ ], which, as expected, is nonsingular, and verifies that the closed-loop system is controllable However, the observability matrix is O = [ ], which is singular, and thus the closed-loop system with this state feedback is not observable Observability is not invariant with respect to feedback Lecture 17: State Feedback p9/23

19 State Feedback Design The following example illustrates what can we achieve with state feedback Example (Eigenvalue assignment by state feedback) The plant ẋ(t) = [ ] x(t) + [ 1 0 ] u(t) has a matrix A with the characteristic polynomial (s) = (s 1) 2 9 = s 2 2s 8 = (s 4)(s + 2), ie, eigenvalues at s = 4 and s = 2 The system is unstable Lecture 17: State Feedback p10/23

20 State Feedback Design The following example illustrates what can we achieve with state feedback Example (Eigenvalue assignment by state feedback) The plant ẋ(t) = [ ] x(t) + [ 1 0 ] u(t) has a matrix A with the characteristic polynomial (s) = (s 1) 2 9 = s 2 2s 8 = (s 4)(s + 2), ie, eigenvalues at s = 4 and s = 2 The system is unstable The state feedback u = r [k 1 k 2 ]x gives the closed-loop system ẋ = (A BK)x + Br = ( [ ] [ ]) k 1 k x + [ 1 0 ] r = [ ] 1 k 1 3 k x + [ 1 0 ] r Lecture 17: State Feedback p10/23

21 State Feedback Design Example (Continuation) The new evolution matrix A K = A BK has characteristic polynomial K (s) = (s 1 + k 1 )(s 1) 3(3 k 2 ) = s 2 + (k 1 2)s + (3k 2 k 1 8) Clearly, the roots of K (s), or equivalently, the eigenvalues of the closed-loop system can be arbitrarily assigned by a suitable choice of k 1 and k 2 Lecture 17: State Feedback p11/23

22 State Feedback Design Example (Continuation) The new evolution matrix A K = A BK has characteristic polynomial K (s) = (s 1 + k 1 )(s 1) 3(3 k 2 ) = s 2 + (k 1 2)s + (3k 2 k 1 8) Clearly, the roots of K (s), or equivalently, the eigenvalues of the closed-loop system can be arbitrarily assigned by a suitable choice of k 1 and k 2 For instance, for both eigenvalues to be placed at 1 ± j2, the desired characteristic polynomial is (s + 1 j2)(s j2) = s 2 + 2s + 5 By equating k 1 1 = 2 and 3k 2 k 1 8 = 5 we get k 1 = 4 and k 2 = 17/3 Thus, the feedback gain K = [ 4 17/3 ] will shift the eigenvalues of the system from 4, 2 to 1 ± j2 Lecture 17: State Feedback p11/23

23 State Feedback Design The example shows that state feedback allows us to place the eigenvalues of the closedloop system at any position, and that the state feedback gain can be computed by direct substitution Lecture 17: State Feedback p12/23

24 State Feedback Design The example shows that state feedback allows us to place the eigenvalues of the closedloop system at any position, and that the state feedback gain can be computed by direct substitution However, The method of the example is not practical for systems of higher dimensions It s not clear what role did controllability play in this eigenvalue assignment Lecture 17: State Feedback p12/23

25 State Feedback Design To formulate a general result we need to use the Controller Canonical Form (CCF) discussed in Lecture 12 Recall that if C = [ B AB A n 1 B ] is full row rank, then we can always make a change of coordinates under which the state matrices have the form Ā = PAP 1 = α 1 α 2 α n 1 α n C = CP 1 = [ β 1 β 2 β n 1 β n ], B = PB = These matrices arise from the change of coordinates x = Px where P 1 = C C 1 with C = [ B AB A n 1 B ] C = [ B Ā B Ā n 1 B ] Lecture 17: State Feedback p13/23

26 State Feedback Design Theorem (Eigenvalue assignment by state feedback) if the state equation ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t), is controllable, then the state feedback control law u = r Kx, where K R 1 n, assigns the eigenvalues of the closed-loop state equation ẋ(t) = (A BK)x(t) + Br(t) y(t) = Cx(t), to any desired, arbitrary locations, provided that complex eigenvalues are assigned in conjugate pairs Lecture 17: State Feedback p14/23

27 State Feedback Design Proof: If the system is controllable, we can take it to its CCF by the change of coordinates x = Px, which yields Ā = P 1 AP and B = BP It is not difficult to verify that C [ B, Ā B,, Ā n 1 B] = P[B, AB,, A n 1 B] = PC, and thus P 1 = C C 1 Lecture 17: State Feedback p15/23

28 State Feedback Design Proof: If the system is controllable, we can take it to its CCF by the change of coordinates x = Px, which yields Ā = P 1 AP and B = BP It is not difficult to verify that C [ B, Ā B,, Ā n 1 B] = P[B, AB,, A n 1 B] = PC, and thus P 1 = C C 1 On substituting x = Px in the state feedback law u = r Kx = r KP 1 x r K x, Since Ā B K = P(A BK)P 1, we see that A BK and Ā B K are similar, and thus have the same eigenvalues Lecture 17: State Feedback p15/23

29 State Feedback Design Proof: If the system is controllable, we can take it to its CCF by the change of coordinates x = Px, which yields Ā = P 1 AP and B = BP It is not difficult to verify that C [ B, Ā B,, Ā n 1 B] = P[B, AB,, A n 1 B] = PC, and thus P 1 = C C 1 On substituting x = Px in the state feedback law u = r Kx = r KP 1 x r K x, Since Ā B K = P(A BK)P 1, we see that A BK and Ā B K are similar, and thus have the same eigenvalues Now, say that λ 1, λ 2,, λ n are the desired closed-loop eigenvalue locations We can then generate the desired characteristic polynomial K (s) = (s λ 1 )(s λ 2 ) (s λ n ) = s n + ᾱ 1 s n ᾱ n Lecture 17: State Feedback p15/23

30 State Feedback Design If we choose K = [(ᾱ 1 α 1 ), (ᾱ 2 α 2 ),, (ᾱ n α n )], the closed-loop state equation becomes (in the x coordinates) x(t) = (Ā B K) x(t) + Br(t) = = α 1 α 2 α n 1 α n ᾱ 1 ᾱ 2 ᾱ n 1 ᾱ n x(t) + (ᾱ 1 α 1 ) (ᾱ 2 α 2 ) (ᾱ n α n ) r(t) x(t) + Br( Lecture 17: State Feedback p16/23

31 State Feedback Design If we choose K = [(ᾱ 1 α 1 ), (ᾱ 2 α 2 ),, (ᾱ n α n )], the closed-loop state equation becomes (in the x coordinates) x(t) = (Ā B K) x(t) + Br(t) = = α 1 α 2 α n 1 α n ᾱ 1 ᾱ 2 ᾱ n 1 ᾱ n x(t) + (ᾱ 1 α 1 ) (ᾱ 2 α 2 ) (ᾱ n α n ) r(t) Because the closed-loop evolution matrix (Ā B K) is still in companion form, we see from the last expression that its characteristic polynomial is the desired one K (s) Finally, from K = KP 1, we get that K = KP x(t) + Br( Lecture 17: State Feedback p16/23

32 State Feedback Design The matrix C can be easily build from the coefficients α 1, α 2,, α n of the characteristic polynomial of A or Ā as C = 1 α 1 α 2 α n 2 α n α 1 α n 3 α n α n 4 α n α (note the inverse!) Lecture 17: State Feedback p17/23

33 State Feedback Design The matrix C can be easily build from the coefficients α 1, α 2,, α n of the characteristic polynomial of A or Ā as C = 1 α 1 α 2 α n 2 α n α 1 α n 3 α n α n 4 α n α (note the inverse!) In closed-loop, once the eigenvalue assignment is performed, the system transfer function from r to y is given by G(s) = β 1 s n 1 + β 2 s n β n s n + ᾱ 1 s n 1 + ᾱ 2 s n α n The transfer function has poles at the new, desired locations However, the zeros of the system are the same as in the open-loop system State feedback can arbitrarily assign the system poles (eigenvalues), but has no effect on the system zeros Lecture 17: State Feedback p17/23

34 State Feedback Design Procedure for pole placement by state feedback (Bass-Gura Formula) 1 Obtain the coefficients of the open loop characteristic polynomial (s) = s n + α 1 s n α n 2 Form the controllability matrices C = [ B AB A n 1 B ] and C = 1 α 1 α 2 α n 2 α n α 1 α n 3 α n α n 4 α n α (note the inverse!) 3 Select the coefficients of the desired closed-loop characteristic polynomial K (s) = s n + ᾱ 1 s n ᾱ n and build the statefeedback gain in x coordinates, K = [ (ᾱ 1 α 1 ) (ᾱ 2 α 2 ) (ᾱ n α n ) ] 4 Compute the state-feedback gain in the original x coordinates K = K CC 1 Lecture 17: State Feedback p18/23

35 State Feedback Stabilisation We have seen that if a state equation is controllable, then we can assign its eigenvalues arbitrarily by state feedback But, what happens when the state equation is not controllable? Lecture 17: State Feedback p19/23

36 State Feedback Stabilisation We have seen that if a state equation is controllable, then we can assign its eigenvalues arbitrarily by state feedback But, what happens when the state equation is not controllable? We know that we can take any state equation to the Controllable/Uncontrollable Canonical Form x C = x C Ā 12 ĀC x C + 0 Ā C x C }{{} Ā B C 0 u Because the evolution matrix Ā is block-triangular, its eigenvalues are the union of the eigenvalues of the diagonal blocks: Ā C and Ā C Lecture 17: State Feedback p19/23

37 State Feedback Stabilisation The state feedback law u = r Kx = r K x = r [ K C K C ] [ x C x C ] yields the closed-loop system x C B C K C Ā 12 B C = x ĀC K C x C + C 0 Ā C x C B C 0 r Lecture 17: State Feedback p20/23

38 State Feedback Stabilisation The state feedback law u = r Kx = r K x = r [ K C K C ] [ x C x C ] yields the closed-loop system x C B C K C Ā 12 B C = x ĀC K C x C + C 0 Ā C x C B C 0 r We see that the eigenvalues of Ā C are not affected by the state feedback, so they remain unchanged The value of K C is irrelevant the uncontrollable states cannot be affected Lecture 17: State Feedback p20/23

39 State Feedback Stabilisation r u x C Ā C x C Ā 12 B C x C Ā C x C K C K C State feedback in Controllable/Uncontrollable coordinates We conclude that the condition of Controllability is not only sufficient, but also necessary to place all eigenvalues of A BK in desired locations Lecture 17: State Feedback p21/23

40 State Feedback Stabilisation A notion of interest in control that is weaker than that of Controllability is that of Stabilisability Stabilisability The system ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t), is said to be stabilisable if all its uncontrollable states are asymptotically stable This condition is equivalent to asking that the matrix Ā C be Hurwitz Lecture 17: State Feedback p22/23

41 Summary We have presented an overview of the process of control design via state space methods It involves the design of Lecture 17: State Feedback p23/23

42 Summary We have presented an overview of the process of control design via state space methods It involves the design of A state feedback gain K Lecture 17: State Feedback p23/23

43 Summary We have presented an overview of the process of control design via state space methods It involves the design of A state feedback gain K A state estimator (observer) Lecture 17: State Feedback p23/23

44 Summary We have presented an overview of the process of control design via state space methods It involves the design of A state feedback gain K A state estimator (observer) We have shown that, if the system is controllable, it is possible to arbitrarily assign the eigenvalues of the closed-loop state equation by a suitable choice of K Lecture 17: State Feedback p23/23

45 Summary We have presented an overview of the process of control design via state space methods It involves the design of A state feedback gain K A state estimator (observer) We have shown that, if the system is controllable, it is possible to arbitrarily assign the eigenvalues of the closed-loop state equation by a suitable choice of K If the system is not controllable, then we can apply state feedback of the controllable states only uncontrollable states cannot be affected Lecture 17: State Feedback p23/23

46 Summary We have presented an overview of the process of control design via state space methods It involves the design of A state feedback gain K A state estimator (observer) We have shown that, if the system is controllable, it is possible to arbitrarily assign the eigenvalues of the closed-loop state equation by a suitable choice of K If the system is not controllable, then we can apply state feedback of the controllable states only uncontrollable states cannot be affected A system is thus stabilisable if those states that are not controllable are already stable Lecture 17: State Feedback p23/23

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