Lecture 2 FRTN10 Multivariable Control
Course Outline L1 L5 Specifications, models and loop-shaping by hand 1 Introduction 2 Stability and robustness 3 Specifications and disturbance models 4 Control synthesis in frequency domain 5 Case study: DVD player L6 L8 Limitations on achievable performance L9 L11 Controller optimization: analytic approach L12 L14 Controller optimization: numerical approach L15 Course review
Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
Stability is crucial Examples: bicycle JAS 39 Gripen Mercedes A-class ABS brakes
Input output stability u S y= S(u) Ageneral system S is calledinput output stable (or L 2 stable or BIBOstable orjust stable )if its L 2 gain is finite: S = sup u S(u) u <
Input output stability of LTI systems ForanLTIsystem S withimpulse response g(t)andtransfer function G(s), the following stability conditions are equivalent: S is bounded g(t) decays exponentially All polesof G(s)arein the left half-plane (LHP),i.e.,allpoles have negative real part
Internal stability The LTI system dx dt = Ax+Bu y= Cx+Du is called internally stable if the following equivalent conditions hold: Thestate x decaysexponentially when u=0 All eigenvaluesof Aare in the LHP
Internal vs input output stability If x=ax+bu y= Cx+Du is internally stable then G(s)=C(sI A) 1 B+D is input output stable. Warning The opposite is not always true! There may be unstable pole-zero cancellations(that also render the system uncontrollable and/or unobservable), and these may not be seen in the transfer function!
Stability of feedback loops Assumescalaropen-loopsystem G 0 (s) Σ G 0 (s) 1 Theclosed-loopsystemis stable if and only if all solutionsto the characteristic equation 1+G 0 (s)=0 are in the left half-plane.
Simplified Nyquist criterion If G 0 (s)isstable,thentheclosed-loopsystem[1+g 0 (s)] 1 isstable if and only if the Nyquist curve of G 0 (s)doesnot encircle 1. Imaginary Axis 1 0 1 1 0.5 0 0.5 Real Axis (Note: Matlab gives a Nyquist plot for both positive and negative frequencies)
General Nyquist criterion Let P=numberof unstable(rhp) poles in G 0 (s) N=number ofclockwise encirclements of 1bythe Nyquist plot of G 0 (s) Thentheclosed-loopsystem[1+G 0 (s)] 1 has P+N unstablepoles
Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
Sensitivity and robustness How sensitive is the closed-loop system to model errors and disturbances? How do we measure the distance to instability? Is it possible to guarantee stability for all systems within some distance from the ideal model?
Amplitude and phase margins Amplitude margin A m : arg G 0 (iω 0 )= 180, G 0 (iω 0 ) = 1/A m Phasemarginϕ m : G 0 (iω c ) = 1, arg G 0 (iω c )=ϕ m 180 1 10 1 0.8 0.6 0.4 1/A m Gain 10 0 1 A m 10 1 ω c 0.2 0 0.2 10 G 0 (ω o ) 2 10 1 10 0 ϕ m 50 0.4 100 0.6 G 0 (ω c ) 0.8 1 Phase 150 200 ϕ m ω o 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 250 10 1 10 0
Mini-problem k(s+ 1) s 2 e + cs+ 1 sl 1 Nominally k= 1, c=1and L= 0. How muchmargin isthereineach parameter before the closed-loop system becomes unstable? Im 0 0.5 1 1.5 0 0.5 1 1.5 Re Magnitude (abs) Phase (deg) 10 0 10-1 0-30 -60 Gm = Inf, Pm = 109 deg (at 1.41 rad/s) -90 10-1 10 0 10 1 Frequency (rad/s)
Mini-problem
Sensitivity functions r C(s) P(s) y 1 S(s)= 1 1+ P(s)C(s) T(s)= P(s)C(s) 1+ P(s)C(s) sensitivity function complementary sensitivity function Notethat we alwayshave S(s)+ T(s)=1
Sensitivity towards changes in plant r C(s) P(s) y 1 Howsensitive is the closedloopto a(small) changein P? dt dp = C (1+ PC) 2= T P(1+ PC) Relativechange in T comparedto relative change in P: dt/t dp/p = 1 1+ PC = S
Sensitivity towards disturbances r d + e u C + P + v y 1 + n Open-loop response (C = 0) to process disturbances d, v: Closed-loop response: Y cl = Y ol = V+ PD 1 1+ PC V+ P 1+ PC D=S Y ol
Interpretation as stability margin The L 2 gain ofthe sensitivityfunctionmeasuresthe inverse ofthe distance between the Nyquist plot and the point 1: R 1 = sup ω 1 1+ P(iω)C(iω) = M s 1 Im R Re P(iω)C(iω)
Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
Robustness analysis How large plant uncertainty can be tolerated without risking instability? Example(multiplicative uncertainty): v (s) w P(s) C(s)
The Small Gain Theorem r 1 e 1 S 1 S 2 e 2 r 2 Assumethat S 1 and S 2 are stable. If S 1 S 2 < 1, thenthe closed-loopsystem(from (r 1, r 2 )to (e 1, e 2 ))is stable. Note 1: The theorem applies also to nonlinear, time-varying, and multivariable systems Note 2: The stability condition is sufficient but not necessary, so the results may be conservative
The Small Gain Theorem r 1 e 1 S 1 S 2 e 2 r 2 Assumethat S 1 and S 2 are stable. If S 1 S 2 < 1, thenthe closed-loopsystem(from (r 1, r 2 )to (e 1, e 2 ))is stable. Note 1: The theorem applies also to nonlinear, time-varying, and multivariable systems Note 2: The stability condition is sufficient but not necessary, so the results may be conservative
Proof e 1 = r 1 +S 2 (r 2 +S 1 (e 1 )) ( ) e 1 r 1 + S 2 r 2 + S 1 e 1 e 1 r 1 + S 2 r 2 1 S 1 S 2 This showsboundedgain from (r 1, r 2 ) to e 1. Thegain to e 2 is boundedin the sameway.
Application to robustness analysis v (s) w P(s) C(s) Thediagram canbe redrawnas v w PC 1+PC
Application to robustness analysis v w PC 1+PC Assumingthat T= PC 1+PC guarantees stability if is stable,the SmallGain Theorem T < 1
Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
Gain of multivariable systems Recall from Lecture 1 that forastable LTIsystem S. S = sup G(iω) = G ω How to calculate G(iω) for a multivariable system?
Vector norm and matrix gain Foravector x C n,we use the 2-norm x = x x= x 1 2 + + x n 2 (A denotestheconjugate transpose of A) Foramatrix A C n m,we use the L 2 -induced norm Ax x A := sup = sup A Ax x x x x = λ(a x A) λ(a A) denotesthelargest eigenvalue of A A. The ratio Ax / x is maximized when x is a corresponding eigenvector.
Example: Different gains in different directions: [ ] y1 = y 2 [ ] [ ] 2 4 u1 0 3 u 2 1.5 Input u= [0.309 0.951] T, u = 1 1 0.5 u 2 0 0.5 1 1.5 4 3 2 1 0 1 2 3 4 u 1 y=gu = [4.42 2.85] T, y = 5.26 5 y 2 0 5 15 10 5 0 5 10 15 (red):eigenvectors ; (blue): V ; (green): U A=U*S*V T
Singular Values Foramatrix A,its singular valuesσ i aredefined as σ i = λ i whereλ i arethe eigenvalues of A A. Let σ(a) denote the largest singular value and σ(a) the smallest singular value. Foralinear map y= Ax, it holds that σ(a) y x σ(a) The singular values are typically computed using singular value decomposition(svd): A=UΣV
Singular Values Foramatrix A,its singular valuesσ i aredefined as σ i = λ i whereλ i arethe eigenvalues of A A. Let σ(a) denote the largest singular value and σ(a) the smallest singular value. Foralinear map y= Ax, it holds that σ(a) y x σ(a) The singular values are typically computed using singular value decomposition(svd): A=UΣV
SVD example Matlab code for singular value decomposition of the matrix [ ] 2 4 A= 0 3 SVD: A=U S V whereboththematricesu andv areunitary(i.e. have orthonormal columns s.t. V V= I) and S is the diagonalmatrix with(sorteddecreasing) singular values σ i. Multiplying A with an input vector along the first column inv gives A V (:,1) =USV V (:,1) = [ 1 =US =U 0] (:,1) σ 1 That is, we get maximal gain σ 1 in the output direction U (:,1) if we use an input in directionv (:,1) (and minimal gain σ 2 if we use the second columnv (:,n) =V (:,2) ). >> A = [2 4; 0 3] A = 2 4 0 3 >> [U,S,V] = svd(a) U = 0.8416-0.5401 0.5401 0.8416 S = 5.2631 0 0 1.1400 V = 0.3198-0.9475 0.9475 0.3198 >> A*V(:,1) ans = 4.4296 2.8424 >> U(:,1)*S(1,1) ans = 4.4296 2.8424
Example: Gain of multivariable system Consider the transfer function matrix 2 G(s)= s+ 1 s s 2 + 0.1s+ 1 4 2s+ 1 3 s+ 1 >> s=zpk('s') >> G=[ 2/(s+1) 4/(2*s+1); s/(s^2+0.1*s+1) 3/(s+1)]; >> sigma(g) % plot sigma values of G wrt freq >> grid on >> norm(g,inf) % infinity norm = system gain ans = 10.3577
Singular Values 10 2 10 1 System: G Frequency (rad/sec): 0.0106 Singular Value (abs): 5.26 Singular Values (abs) 10 0 System: G Frequency (rad/sec): 0.0101 0.0106 Singular Value (abs): 1.14 10 1 10 2 10 2 10 1 10 0 10 1 10 2 Frequency (rad/sec) The singular values of the tranfer function matrix(prev slide). Note that G(0)= [2 4;03] whichcorrespondsto Ain thesvd-exampleabove. G = 10.3577.
Lecture 2 summary Input output stability: S < Sensitivity function: S := dt/t dp/p = 1 1+PC SmallGain Theorem: Thefeedbackinterconnectionof S 1 and S 2 is stable if S 1 S 2 < 1 Conservative compared to the Nyquist criterion Useful for robustness analysis Thegain ofamultivariable system G(s)is givenby sup ω σ(g(iω)),whereσis the largest singular value