Design Methods for Control Systems

Transcription

1 Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term

2 Schedule November 25 MSt December 2 MSt Homework # 1 December 9 MSt December 16 MSt Homework # 2 January 6 GM January 13 GM Homework # 3 January 20 GM January 27 GM Homework # 4 Some time in February: Review session

3 Overview Ch Introduction to to feedback control theory Ch Classical control system design Ch LQ, LQG and H 2 control system design Ch Design of of multivariable control systems Ch Uncertainty models and robustness Ch H optimization and µ -synthesis

4 Scope and features Mature review of classical and modern control system design techniques Linear time-invariant systems 70% SISO- 30% MIMO Continuous-time MATLAB exercises Control toolbox Mu-Tools and Robust Control toolboxes

5 Overview Ch Introduction to to feedback control theory Introduction Introduction Types Types of of control control systems systems Design Design issues issues Configurations Configurations High-gain High-gain feedback feedback Stability Stability Closed-loop Closed-loop characteristic characteristic polynomial polynomial Nyquist Nyquist criterion criterion Stability Stability margins margins Performance Performance System System functions functions Low Low and and high high frequencies frequencies Robustness Robustness Robustness Robustness functions functions Loop Loop shaping shaping Limits Limits of of performance performance Two-degree-of-freedom control control systems systems

6 Types of control systems Regulator systems Servo or positioning systems Tracking systems

7 Design issues Targets Closed-loop stability Disturbance attenuation Good command response Robustness Limitations Plant capacity Measurement noise

8 Configurations Two degrees of freedom One degree of freedom

9 High-gain feedback-1 Loop gain γ = ψ oφ Signal balance e = r γ () e Feedback equation e+ γ () e = r

10 High-gain feedback-2 Feedback equation e+ γ () e = r High gain: γ ( e ) >> e Implies: γ () e r Hence: e << r r ψ ( y) So that y ψ 1 () r In case of unit feedback y r Good tracking

11 High-gain feedback-3 Loop gain δ = ( φ) o ( ψ) Signal balance: z = d δ ( z) High gain: δ ( z ) >> z Hence: z << d Good disturbance reduction

12 High-gain feedback-4 Need closed-loop stability Good tracking and disturbance attentuation are retained as long as the closed-loop system remains stable the gain remains high Under these conditions high-gain feedback implies robustness with respect to loop uncertainty

13 Pitfalls of high-gain feedback High-gain feedback has pitfalls: Naively making the gain large easily results in an unstable feedback system Even if the feedback system is stable overly large plant inputs may occur that exceed the plant capacity Measurement noise causes loss of performance

14 Stability-1 State space representation: xt &() = Axt () + Brt () et () ut () = Cxt () + Drt () zt ()

15 Stability-2 xt &() = Axt () + Brt () et () ut () = Cxt () + Drt () zt () The closed-loop system is stable if its state space representation is asymptotically stable Equivalent statements: xt () 0 as t for every solution of xt &() = Axt () All eigenvalues of A have strictly negative real parts All roots of det(si A) have strictly negative real parts

16 Stability-3 The control system is BIBO stable if every bounded input signal r results in bounded output signals e, u and z. BIBO = bounded input bounded output Asymptotic stability BIBO stability The converse is not true

17 Stability-4 Internal stability Inject internal signals into each exposed interconnection of the system, and define additional internal output signals after each injection point Then the system is internally stable if it is BIBO stable with respect to all inputs (external and internal) and all (external and internal) outputs

18 Stability-5 Example

19 Stability-6 If each component system is stabilizable and detectable ( has no hidden unstable modes ) then Stability Internal stability When input-output descriptions are used (such as transfer functions) internal stability is often easier to check than asymptotic stability

20 Closed-loop characteristic polynomial-1 State space representation of the open-loop system: Characteristic polynomial: State space representation of the closed-loop system: Characteristic polynomial: xt &() = Axt () + Bet () yt () = Cxt () + Det () χ () s = det( si A) xt &() = Axt (), cl 1 Acl = A B( I + D) C χ () s = det( si A ) cl cl

21 Closed-loop characteristic polynomial-2 P(s) C(s) L(s)=P(s)C(s) plant transfer matrix compensator transfer matrix loop gain transfer matrix Then χ cl det( I + L( s)) () s = χ() s det( I + L ( ))

22 Closed-loop characteristic polynomial-3 χ cl det( I + L( s)) () s = χ() s det( I + L ( )) SISO case: N() s Y() s Ls () = PsCs () () = Ds () X() s Then (within a nonzero constant factor) χ () s = D() s X() s χ cl () s = D() s X() s + N() s Y() s

23 Nyquist criterion Consider the SISO case The locus of L( jω), ω R in the complex plane is called the Nyquist plot of the loop gain The number of unstable closedloop poles = The number of times the Nyquist plot encircles the point 1 + The number of unstable open-loop poles

24 Generalized Nyquist criterion Consider the MIMO case The number of unstable closed-loop poles = The number of times the locus of det( I + L( jω)), ω R encircles the origin + The number of unstable open-loop poles (Principle of the argument)

25 Stability margins-1 In the SISO case, the point 1 is a critical point for the Nyquist plot of the closed-loop system. If the Nyquist plot is changed so that it crosses the point 1 then the system becomes unstable If the closed-loop system is stable but the Nyquist plot passes closely by 1 then the system is near-unstable, that is, has an oscillatory response the system may become unstable by small perturbations of the plant, that is, the system is not robust

26 Stability margins-2 There exist various stability margins. They measure how close the Nyquist plot gets to 1 Gain margin k m Phase margin φ m Modulus margin τ m (Landau)

27 System functions: L and S Loop gain L Sensitivity function S L = PC z 1 = v 1{ + L S

28 System functions: R Input disturbance ( proces ) sensitivity function R 1 z = Pv = SPr { 1+ L R

29 System functions: H and T Closed-loop transfer function H Complementary sensitivity function T z L L = Fr H = L {1 + L H T F

30 System functions: U Input ( control ) sensitivity function U C u = ( Fr m v) CP U

31 Measurement noise 1 PC PC z = v+ Fr m PC PC PC S T T

32 S, R, U and T 1 P z1 1+ PC 1+ PC v1 S R v1 z = = C PC v v U T 1+ PC 1+ PC 2 2 2

33 Design interrelations S T 1 = 1 + L L = 1 + L U= T / P R= SP H = T F S and T are suitable objects for manipulation

34 Low and high frequencies-1 Typical shapes for S and T

35 Low and high frequencies-2 Loop gain L large at low frequencies: small at high frequencies: L( jω ) >> 1, S 1/ L, T 1 L( jω ) << 1, S 1, T L Crossover region: L( jω) 1

36 Low and high frequencies-3 input sensitivity C 1/ P for low frequencies U = T / P = 1+ PC C for high frequencies input disturbance sensitivity R P 1/ C for low frequencies = SP = 1+ PC P for high frequencies closed-loop transfer function H = TF F corrects T

37 Robustness functions-1 Sufficient condition for stability under perturbation: L( jω) L ( jω) < 1 + L ( jω), o ω R o

38 Robustness functions-2 Equivalently, L( jω) Lo( jω) 1 + Lo( jω) <, ω L ( jω) L ( jω) o o R or L( jω) Lo ( jω) 1 <, ω L ( jω) T ( jω) o o R

39 Robustness functions-3 Bound on the relative size of perturbations: L( jω) L ( jω) L o o ( jω ) W 1 ( jω), ω R Sufficient and necessary condition for stability under all perturbations that satisfy the bound: W 1 1 ( jω) <, ω R To ( jω )

40 Robustness functions-4 Size of the smallest perturbation that may destabilize the system: W 1 1 ( jω) =, ω R T ( jω ) o

41 Robustness functions-5 The preceding discussion focuses on preventing the Nyquist plot of the loop gain L from crossing the point 1. Preventing the inverse Nyquist plot that is, the Nyquist plot of 1/L from crossing the point 1 also guarantees stability. Sufficient condition: < 1 +, ω R L( jω) L ( jω) L ( jω) o o

42 Robustness functions-6 Equivalently, 1 1 L( jω) Lo ( jω) 1 <, ω R 1 So ( jω ) L ( jω ) o

43 Robustness functions-7 Consider perturbations such that 1 1 L( jω) Lo ( jω) 1 L ( jω ) o W 2 ( jω), ω R Sufficient and necessary condition for robust stability: W 2 1 ( jω) <, So ( j ) ω R ω

44 Robustness functions-8 Size of the smallest perturbation that may destabilize the system: W 2 1 ( jω) =, ω R So ( jω )

45 Combined robustness test-1 Define δ L ( jω) = L( jω) L ( jω) L o o ( jω ) δ L 1 ( jω) = 1 1 L( jω) Lo ( jω) 1 L ( jω ) o

46 Combined robustness test-2 Then the perturbed closed-loop system is stable if ω R δ L ( j ) 1 ω < 1 S( jω ) or δ L ( jω) < 1 T( jω ) typically satisfied at low frequencies typically satisfied at high frequencies

47 Combined robustness test-3 Critical frequency region: crossover area

48 Loop shaping Low frequencies: large loop gain High frequencies: small loop gain In the crossover region the phase is constrained because of stability

49 Bode s gain-phase relationship-1 Between break frequencies the loop gain behaves as Hence L( jω) c( jω) n L( jω) cω n π arg L( jω ) n 2 Phase and magnitude do not behave independently Bode s gain-phase relationship describes the relation more accurately

50 Bode s gain-phase relationship-2 The limitations imposed by stability on the phase in the crossover region by Bode s gain-phase relationship limit the rate at which the loop gain decreases: If, say, π arg L( jω) in the crossover region 2 then 1 L( jω) cω in the crossover region

51 Limits of performance Bode s integral The Freudenberg-Looze equalities Limitations are imposed by causality the pole-zero configuration

52 Bode s integral-1 If L has at least two more poles than zeros then 0 log S( jω) dω = π Re pi 0 The p i are the right-half plane poles of the loop gain. i Proof: Use the Poisson integral from complex function theory

53 Bode s integral-1

54 Bode s integral-1

55 Bode s integral-2 Dual result: Suppose that the loop has integrating action of at least order 2. Then 0 1 log T(1/ jω) dω = π Re 0 z The z i are the right-half plane zeros of the loop gain. i i

56 Freudenberg-Looze equality-1 Let z be any right-half plane zero of the loop gain. Poisson s formula of complex function theory leads to the equality 0 log( S( jω) ) dw ( ω) = log B ( z) 0 z 1 poles Strengthens Bode s integral W z 1 ω Imz 1 ω + Im ( ω) = arctan + arctan π Re z π Re z z Increasing function. Rises most steeply at z. B () poles s = i p p i i + s s Blaschke product

57 Freudenberg-Looze equality-2 W z for different values of arg z (a) arg z = 0 (b) arg z is almost π/2 Frequencies where W z rises most steeply contribute most to the integral

58 Freudenberg-Looze equality-3 The bounds for S hold provided µ 1 ε Wz ( ω1 ) 1 W ( ω ) z 1 B ( 1 ) poles ( z) Wz ω The dependence of the right-hand side on the various parameters may be analyzed

59 Freudenberg-Looze equality-4 Effects of right-half plane zeros on S S may be made small up to the frequency min i z i. Attempting to make S small beyond this frequency makes S peak Right-half plane poles further impair the achievable reduction of S (in particular, nearlycancelling right-half plane pole-zero pairs)

60 Freudenberg-Looze equality-5 Rederivation of the Freudenberg-Looze equality while 1 1 L replacing L with 1/L, so that S = = = 1+ L L L interchanging the roles of the poles and the zeros T leads to 0 log( T( jω) ) dw ( ω) = log B ( p) 0 p 1 zeros

61 Freudenberg-Looze equality-6 0 log( T( jω) ) dw ( ω) = log B ( p) 0 p 1 zeros p is any right-half plane pole of the loop gain, and B () zeros s = i z z i i + s s In the application of the equality, interchange the roles of low and high frequencies

62 Freudenberg-Looze equality-7 Effects of right-half plane poles on T T may be made small above the frequency max i p i. Attempting to make T small below this frequency makes T peak Right-half plane zeros further impair the achievable reduction of T (in particular, nearly-cancelling right-half plane pole-zero pairs)

63 Freudenberg-Looze equality-8 Consequences for S and T

64 Two-degree-of-freedom systems-1 N Y Let P =, C = D X Then the closed-loop transfer function is PC NY H = F = F 1+ PC D cl Dcl = DX + NY with D cl the closed-loop characteristic polynomial

65 Two-degree-of-freedom systems-2 Other two-degree-of-freedom configuration: H = NX D cl F has zeros at the roots of N and X H NY = F has zeros at the Dcl roots of N and Y Can the zeros of H be made independent of the feedback compensator?

66 Two-degree-of-freedom systems-3 Further two-degree-of-freedom configuration N Y Y P =, C = C = D X X Need XX 1 2 = X, YY 1 2 = Y to achieve the same loop gain as in the two previous cases Have H PC = = 1+ PC NX Y D cl

67 Two-degree-of-freedom systems-4 H is independent of the compensator if we let so that H = NX Y D 2 1 cl Y = X = 1 Y = Y, X = X N C1 =, C2 = Y, H = X D cl

68 Two-degree-of-freedom systems-5 Resulting feedback system Equivalent configuration

69 Two-degree-of-freedom systems-6 Extension May choose F polynomial so that we obtain a 1½-degree-of-freedom system

70 Two-degree-of-freedom systems-7 Further extension: F polynomial, F o rational: 2½-degree-of-freedom system F =1, F o rational: H = NF D cl F o 2-degree-of-freedom system