Design Methods for Control Systems


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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 timeinvariant systems 70% SISO 30% MIMO Continuoustime MATLAB exercises Control toolbox MuTools 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 Highgain Highgain feedback feedback Stability Stability Closedloop Closedloop 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 Twodegreeoffreedom control control systems systems
6 Types of control systems Regulator systems Servo or positioning systems Tracking systems
7 Design issues Targets Closedloop stability Disturbance attenuation Good command response Robustness Limitations Plant capacity Measurement noise
8 Configurations Two degrees of freedom One degree of freedom
9 Highgain feedback1 Loop gain γ = ψ oφ Signal balance e = r γ () e Feedback equation e+ γ () e = r
10 Highgain feedback2 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 Highgain feedback3 Loop gain δ = ( φ) o ( ψ) Signal balance: z = d δ ( z) High gain: δ ( z ) >> z Hence: z << d Good disturbance reduction
12 Highgain feedback4 Need closedloop stability Good tracking and disturbance attentuation are retained as long as the closedloop system remains stable the gain remains high Under these conditions highgain feedback implies robustness with respect to loop uncertainty
13 Pitfalls of highgain feedback Highgain 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 Stability1 State space representation: xt &() = Axt () + Brt () et () ut () = Cxt () + Drt () zt ()
15 Stability2 xt &() = Axt () + Brt () et () ut () = Cxt () + Drt () zt () The closedloop 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 Stability3 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 Stability4 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 Stability5 Example
19 Stability6 If each component system is stabilizable and detectable ( has no hidden unstable modes ) then Stability Internal stability When inputoutput descriptions are used (such as transfer functions) internal stability is often easier to check than asymptotic stability
20 Closedloop characteristic polynomial1 State space representation of the openloop system: Characteristic polynomial: State space representation of the closedloop 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 Closedloop characteristic polynomial2 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 Closedloop characteristic polynomial3 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 openloop poles
24 Generalized Nyquist criterion Consider the MIMO case The number of unstable closedloop poles = The number of times the locus of det( I + L( jω)), ω R encircles the origin + The number of unstable openloop poles (Principle of the argument)
25 Stability margins1 In the SISO case, the point 1 is a critical point for the Nyquist plot of the closedloop system. If the Nyquist plot is changed so that it crosses the point 1 then the system becomes unstable If the closedloop system is stable but the Nyquist plot passes closely by 1 then the system is nearunstable, 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 margins2 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 Closedloop 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 frequencies1 Typical shapes for S and T
35 Low and high frequencies2 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 frequencies3 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 closedloop transfer function H = TF F corrects T
37 Robustness functions1 Sufficient condition for stability under perturbation: L( jω) L ( jω) < 1 + L ( jω), o ω R o
38 Robustness functions2 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 functions3 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 functions4 Size of the smallest perturbation that may destabilize the system: W 1 1 ( jω) =, ω R T ( jω ) o
41 Robustness functions5 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 functions6 Equivalently, 1 1 L( jω) Lo ( jω) 1 <, ω R 1 So ( jω ) L ( jω ) o
43 Robustness functions7 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 functions8 Size of the smallest perturbation that may destabilize the system: W 2 1 ( jω) =, ω R So ( jω )
45 Combined robustness test1 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 test2 Then the perturbed closedloop 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 test3 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 gainphase relationship1 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 gainphase relationship describes the relation more accurately
50 Bode s gainphase relationship2 The limitations imposed by stability on the phase in the crossover region by Bode s gainphase 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 FreudenbergLooze equalities Limitations are imposed by causality the polezero configuration
52 Bode s integral1 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 righthalf plane poles of the loop gain. i Proof: Use the Poisson integral from complex function theory
53 Bode s integral1
54 Bode s integral1
55 Bode s integral2 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 righthalf plane zeros of the loop gain. i i
56 FreudenbergLooze equality1 Let z be any righthalf 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 FreudenbergLooze equality2 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 FreudenbergLooze equality3 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 righthand side on the various parameters may be analyzed
59 FreudenbergLooze equality4 Effects of righthalf 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 Righthalf plane poles further impair the achievable reduction of S (in particular, nearlycancelling righthalf plane polezero pairs)
60 FreudenbergLooze equality5 Rederivation of the FreudenbergLooze 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 FreudenbergLooze equality6 0 log( T( jω) ) dw ( ω) = log B ( p) 0 p 1 zeros p is any righthalf 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 FreudenbergLooze equality7 Effects of righthalf 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 Righthalf plane zeros further impair the achievable reduction of T (in particular, nearlycancelling righthalf plane polezero pairs)
63 FreudenbergLooze equality8 Consequences for S and T
64 Twodegreeoffreedom systems1 N Y Let P =, C = D X Then the closedloop transfer function is PC NY H = F = F 1+ PC D cl Dcl = DX + NY with D cl the closedloop characteristic polynomial
65 Twodegreeoffreedom systems2 Other twodegreeoffreedom 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 Twodegreeoffreedom systems3 Further twodegreeoffreedom 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 Twodegreeoffreedom systems4 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 Twodegreeoffreedom systems5 Resulting feedback system Equivalent configuration
69 Twodegreeoffreedom systems6 Extension May choose F polynomial so that we obtain a 1½degreeoffreedom system
70 Twodegreeoffreedom systems7 Further extension: F polynomial, F o rational: 2½degreeoffreedom system F =1, F o rational: H = NF D cl F o 2degreeoffreedom system
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