Lecture 7 Lyapunov Stability Theory Theme: Generate parameter adjust ment rules which guara ntee stability The Big Picture 1.Int roduct ion Model-Reference Adaptive Systems The MiT rule pros and cons 1. The idea The stability problem 2. The mit rule Ordinary Differential Equat io ns ya punov's stability theory · Dynamical Systems 4. Desig n of mras using Lyapunov theory 2. Lyapunov's Stability Theory 5. Applications to adapt ive cont rol · The stability co nce 6. Bounded -input, bounded-o ut put st a bility Finding Lyapunov Funct io ns pplications to adaptive cont rol Applicatio n to adapt ive co nt rol 8. Rel at ions between mras and str · A simple pro blem 9. Co nclusions State feed back · Out put feed back Stability Concepts St able syst ems and st able solut ions Lyapunov Himself An example tability of mechanical syst ems A state space approach The key id C K.J. Astrom and BWittenmark
Lecture 7 Lyapunov Stability Theory Theme: Generate parameter adjustment rules which guarantee stability 1. Introduction The MIT rule pros and cons The stability problem Ordinary Di erential Equations Dynamical Systems 2. Lyapunov's Stability Theory The stability concept Lyapunov's idea Finding Lyapunov Functions 3. Application to adaptive control A simple problem State feedback Output feedback 4. Conclusions The Big Picture Model-Reference Adaptive Systems 1. The idea 2. The MIT Rule 3. Lyapunov's stability theory 4. Design of MRAS using Lyapunov theory 5. Applications to adaptive control 6. Bounded-input, bounded-output stability 7. Applications to adaptive control 8. Relations between MRAS and STR 9. Conclusions Lyapunov Himself Stability Concepts Linear systems are very special Stable systems and stable solutions An example Lyapunovs stability concepts { Stability of mechanical systems { A state space approach { The key ideas c K. J. Åström and B. Wittenmark 1
Stable solu Consider the mras desig ned with MIT rule Prelimi naries kaGs) Consider the solution a(t)=0to f(c)f(0) kG(s) ae=sinat w=l(a), 2(b)and 3(c) f(c)-f(y)川≤Lz-ylL>0 Sensit ivity to pert ur bat io ns LACAAVAAA Initial conditions Variat io ns in∫() (c) Stabil ity concepts Co ncl usio n Lyapunov Stability dEFINITION 1 Lya punov's Idea The solution a(t=0 is stable if for given e>0there exists a number &(e>0 such that nspiratio n from mechanics all solut ions wit h init ial co ndit io ns Stable and unst able equilibria -(O)川<6 ave the pro perty z(t川‖<εfor0≤t<∞ The solution is unsta ble if it is not st able. the solut io n is asymptotically stable if it is stable and s can be found such that all solutions with lz(0川‖< 8 have the property that‖(t)‖|→0 ast→o Notice Stability of a part icular solut ion ∫(c)<0 How to make it glo bal? C K.J. Astrom and B Wittenmark
Stable Solutions Consider the MRAS designed with MIT rule Model e y Process u Σ + – uc θ y m − γ s kG(s) k0G(s) Π Π uc = sin!t ! = 1(a), 2(b) and 3(c) 0 5 10 15 20 0 1 0 5 10 15 20 0 1 0 5 10 15 20 0 1 Time Time Time (a) ^ (b) ^ (c) ^ Conclusion? Preliminaries Consider the solution x(t)=0 to dx dt = f (x) f (0) = 0 Existence and uniqueness kf (x) f (y)k Lkx yk L > 0 Sensitivity to perturbations What perturbations? { Initial conditions { Variations in f (x) Stability concepts! Lyapunov Stability Definition 1 The solution x(t)=0 is stable if for given " > 0 there exists a number (") > 0 such that all solutions with initial conditions kx(0)k < have the property kx(t)k < " for 0 t < 1 The solution is unstable if it is not stable. The solution is asymptotically stable if it is stable and can be found such that all solutions with kx(0)k < have the property that kx(t)k ! 0 as t ! 1. Notice Stability of a particular solution Local concept How to make it global? Lyapunov's Idea Inspiration from mechanics Stable and unstable equilibria x1 x 2 x=0 V(x)=const dx dt Condition @V T @x f (x) < 0 c K. J. Åström and B. Wittenmark 2
a continuo usly differentiable func tion V in ding lyapunov funct io ns R-R is c alled positive definite in a reg ion rans fo rming o ne diffic ult pro blem to a U CRcont aining the origin if ot he 1.V(0)=0 Let the linear system (a)>0,c∈Uand≠ A func tion is c alled positive semidefinite if dt Condit io n 2 is replaced by V(a)20 be stable. Pie k Q positive defi nit e. T he THEORE quat ion If there exists a func tion v: rh r that ATP+PA=-Q positive defi nite suc h that has always a uniq ue solutio n wit h P positive a2 definite and the funt io n f(m)=-W(x) V(x) is neg at ive semidefinite, then the solut ion Lw t=0 is stable. If dv/ dt is negative defi yapunov tunc tion t hen the solut io n is also asym ptotic ally stable m p e rying Syst ems System mat rix a1 a2 亚=( A a ay definition 3 T he solution a(t)=0 is uniformly sta ble if fore>0 the 6(e) dependent of to, suc h that where g1>0 and 92>0 are posit ive. Assume t hat the mat rix p has the form (to0? inc reasing and a( 0)=0. It is said to belong Proofs are given in the mat hematics course on to class Koo if a=∞anda(r)→∞as mat ric es! C K.J. Astrom and BWittenmark
Formalities Definition 2 A continuously di erentiable function V : Rn ! R is called positive de nite in a region U Rn containing the origin if 1. V (0) = 0 2. V (x) > 0; x 2 U and x 6= 0 A function is called positive semide nite if Condition 2 is replaced by V (x) 0. Theorem 1 If there exists a function V : Rn ! R that is positive de nite such that dV dt = @V T @x dx dt = @V T @x f (x) = W(x) is negative semide nite, then the solution x(t)=0 is stable. If dV =dt is negative de nite, then the solution is also asymptotically stable. Finding Lyapunov Functions "Transforming one dicult problem to another" Let the linear system dx dt = Ax be stable. Pick Q positive de nite. The equation AT P + P A = Q has always a unique solution with P positive de nite and the funtion V (x) = xT P x is a Lyapunov function Example System matrix A = a1 a2 a3 a4 Q = q1 0 0 q2 where q1 > 0 and q2 > 0 are positive. Assume that the matrix P has the form P = p1 p2 p2 p3 The Lyapunov equation becomes 0 @ 2a1 2a3 0 a2 a1 + a4 a3 0 2a2 2a4 1 A 0 @ p1 p2 p3 1 A = 0 @ q1 0q2 1 A Conditions for existence. When is P > 0? Proofs are given in the mathematics course on matrices! Time-Varying Systems dx dt = f (x; t) Definition 3 The solution x(t)=0 is uniformly stable if for " > 0 there exists a number (") > 0, independent of t0, such that kx(t0)k 0, independent of t0, such that x(t) ! 0 as t ! 1, uniformly in t0, for all kx(t0)k < c. Definition 4 A continuous function : [0; a) ! [0;1) is said to belong to class K if it is strictly increasing and (0) = 0. It is said to belong to class K1 if a = 1 and (r) ! 1 as r ! 1. c K. J. Åström and B. Wittenmark 3
Lyapunov here m Desig n of Mras using T Heorem2 t here m Let a be an equilibrium point an D ∈R v b ● The idea co nt in uo usly differentiable funct io n such that Determine a cont roller st ruct ure a1(|)≤v(a,t)≤a2(|-) Derive the Error Equation fnd a l eter mine an adaptat ion law Onr+a2F(a,m≤-a3(|) a first order system for VT>0, where a1, a2, and as are class ● State feed back K functions. Then a= 0 is uniformly ● Out put feed back asym ptot ically st a ble adaptation of feed fo rw ard Gain Fin ding the adjustment Law Process model System model dy dt Desired res po nse dy ym+k。ue Desired equilibri d Co nt roller Int roduce the error E=y-ym and the error equat ion becomes Lyapunov funct ion de aE+(k8-ko v(e,()=re2+5(-B dr dv Candidate Lyapunov function +yuce V(e,6) Choosing the adjust ment rule d Time derivative d=e(-ae+6(0-k)u)+-k)a d C K.J. Astrom and B Wittenmark
Lyapunov Theorem Theorem 2 Let x = 0 be an equilibrium point and D = fx 2 Rn j kxk < rg. Let V be a continuously di erentiable function such that 1(kxk) V (x; t) 2(kxk) dV dt = @V @ t + @V @x f (x; t) 3(kxk) for 8t 0, where 1, 2, and 3 are class K functions. Then x = 0 is uniformly asymptotically stable. Design of MRAS using Lyapunov Theorem The idea { Determine a controller structure { Derive the Error Equation { Find a Lyapunov function { Determine an adaptation law A rst order system State feedback Output feedback Adaptation of Feedforward Gain Process model dy dt = ay + ku Desired response dym dt = ym + kouc Controller u = uc Introduce the error e = y ym and the error equation becomes de dt = ae + (k ko)uc Candidate Lyapunov function V (e; ) = 2 e 2 + k 2 ( k0 k )2 Time derivative dV dt = e(ae + k( k0 k )uc) + b( k0 k ) d dt = ae2 + (k k0) d dt + uce Finding the Adjustment Law System model dy dt = ay + ku d dt =?? Desired equilibrium e = 0 = 0 = b0 b Lyapunov function V (e; ) = 2 e 2 + k 2 ( 0)2 dV dt = ae2 + (b b0) d dt + uce Choosing the adjustment rule d dt = uce Gives dV dt = ae2 c K. J. Åström and B. Wittenmark 4
f feed f o n Lyapunov rule MT rule d e (a) d t M Rule () Process ∑)- Lv 叫G Model (c) Is there really mag ic in this world? A First order s Proc ess model a First Order System, co nt dy rivative of lyapunov func tion Desired res po nse dv de M - aMyM+ bMc Cont roller amne+-(b02 +a-am) 62 t yuce t he error Adapt at ion law de aMe-(692+a-am)y+(boo bM) Candidate for Lyapunov fun tion b C K.J. Astrom and BWittenmark
Adaptation of Feedforward Gain Lyapunov rule d dt = uce MIT Rule d dt = ye θ Σ – Model Process + Σ – Model Process + y y e e θ uc uc kG(s) kG(s) k0G(s) k0G(s) y m y m − γ s − γ s Π Π Π Π (a) (b) Is there really magic in this world? Simulation MIT rule 0 5 10 15 20 0 1 0 5 10 15 20 0 1 0 5 10 15 20 0 1 Time Time Time (a) ^ (b) ^ (c) ^ Lyapunov rule 0 5 10 15 20 0 1 0 5 10 15 20 0 1 0 5 10 15 20 0 1 Time Time Time (a) ^ (b) ^ (c) ^ A First Order System Process model dy dt = ay + bu Desired response dym dt = amym + bmuc Controller u = 1uc 2y The error e = y ym de dt = ame (b2 + a am)y + (b1 bm) uc Candidate for Lyapunov function V (e; 1; 2) = 1 2 e 2 + 1 b (b2 + a am)2 + 1 b (b1 bm)2 A First Order System, cont Derivative of Lyapunov function dV dt = e de dt + 1 (b2 + a am) d2 dt + 1 (b1 bm) d1 dt = ame 2 + 1 (b2 + a am) d2 dt ye + 1 (b1 bm) d1 dt + uce Adaptation law d1 dt = uce d2 dt = ye c K. J. Åström and B. Wittenmark 5
Com parison with mit rule A First Order System, cont abnor yapunav function V(e,61,6 2+(2+2-a2+(1-h)2 Its derivative MIT rle Negative semi-definite but nct negative definite de V=-2am edf 2ame-ame(b2+a-an by+(b1-bn) u) Error goes to zero but parameters do nat necessanly go to tHar correct values! simulation Process inputs and outputs State feed back ame idea Parameters Denve error eq uation 01 Find Lyapunov equation c K.J. Astrom and B Wit tenma
A First Order System, cont The Lyapunov function V (e; 1; 2) = 1 2 e 2 + 1 b (b2 + a am)2 + 1 b (b1 bm)2 Its derivative dV dt = ame 2 Negative semi-de nite but not negative de nite V = 2ame de dt = 2ame (ame (b2 + a am)y + (b1 bm) uc) Error goes to zero but parameters do not necessarily go to their correct values! Comparison with MIT rule Lyapunov − Σ Π + e u y Σ Π Π Π − + uc Gm (s) G(s) θ 1 θ 2 γ s − γ s MIT rule − Σ Π + e u y Σ Π Π Π − + uc Gm (s) G(s) θ 1 θ 2 γ s − γ s am s + am am s + am Simulation Process inputs and outputs 0 20 40 60 80 100 −1 1 0 20 40 60 80 100 −5 0 5 Time Time (a) ym y (b) u Parameters 0 20 40 60 80 100 0 2 4 0 20 40 60 80 100 −1 1 Time Time 1 2 = 5 = 1 = 0:2 = 5 = 1 = 0:2 State Feedback Same idea Find a controller structure Derive error equation Find Lyapunov equation c K. J. Åström and B. Wittenmark 6
state feed back t he Er ro r equat io n Process mode Process dt dx d Desired response to command sig nals Desired response d /Amam+B Cortrol Contrd T He clased-loop system Error /(A-BD)+BMu/A(6)+B(6) Az+bu-A B Parametric ation A(6)=A Hence Bc(8)=Bm df=Ame+(a-Am-BL)X+(BM-Bm) Compatibility conditions =Ame+(a(0)-Am )x+(Bc()-Bm)uc A-Am=BL =Ane+y(0-6) B=BM t he lyapunoy func tion THe eror eq aton de adaptat io n of Fee d fo rw ard gain ry Error v(e,6)/;ePe+(0-0)y(0-0°) e/(|G(p)-|oG(P)ua/‖G(p)(6-6°)u Hence Introduce a reaization of G(s T Qe+r(0-0)tpE+(0-8or do eQe+(0-8 e=cx WHere Q positive defi nite and Candidate for lyapunoy function AmP+PAm/-Q Adaptation law v/2aPa+(-9 dg WHere A P+PA/-Q dv DIscUSS O KJ. As TToM aNd B. WIITENMaEk
State Feedback Process model dx dt = Ax + Bu Desired response to command signals dxm dt = Amxm + Bmuc Control law u = M uc Lx The closed-loop system dx dt = (A BL)x + BMuc = Ac()x + Bc()uc Parametrization Ac( 0 ) = Am Bc( 0 ) = Bm Compatibility conditions A Am = BL Bm = BM The Error Equation Process dx dt = Ax + Bu Desired response dxm dt = Amxm + Bmuc Control law u = M uc Lx Error e = x xm de dt = dx dt dxm dt = Ax + Bu Amxm Bmuc Hence de dt = Ame + (A Am BL) x + (BM Bm) uc = Ame + (Ac() Am) x + (Bc () Bm) uc = Ame + 0 The Lyapunov Function The error equation de dt = Ame + 0 Try V (e; ) = 1 2 eT P e + ( 0)T ( 0) Hence dV dt = 2 eT Qe + ( 0 ) T P e + ( 0 )T d dt = 2 eT Qe + ( 0 )T d dt + T P e where Q positive de nite and ATmP + P Am = Q Adaptation law d dt = T P e gives dV dt = 2 e TQe Discuss! Adaptation of Feedforward Gain Error e = (kG(p) k0G(p))uc = kG(p)( 0)uc Introduce a realization of G(s) dx dt = Ax + B( 0 )uc e = Cx Candidate for Lyapunov function V = 1 2 x T P x + ( 0) 2 where AT P + P A = Q c K. J. Åström and B. Wittenmark 7
a d aptatio n of Fee d fo rw ard gain Dervative of v Process d V Px+xP. =A+B(-6) 2Ax+Bu(-)2 Adaptation law 2P(Ax+Bu(a)+(e-)正 B IP dt ub Pa Can wefi nd p such tHat Ada a BP=C d dt -yuc b Pa T He adaptation law tHen becomes Q DIScUSS Kalm an-Yakubovic hlem m a definition 5 m m ary A rationa transfer furction g witH real coefficierits is positive real(PR)if Lyapunov Stability T Hed Stability concep G(s)≥0KRs≥0 Lyapunov tHeorem a transfer function G is strictly positive reai How to use rt? (SPR)if G(s-E)is positive rea for some rea Adaptive laws witH Guaranteed stati Simpl e aeslgn b roceau re Find control law LEMMA 1 T He transfer fuction Denve Error Equation G(s=c(sI-A)-lB Find Lyapunov function CHoose adjustment law so tHat is strictly positive rea if and orly if tHere exist positive dfi nite matrices P and Q sucH tHat ●Rear AP+PA=-Q Strong similarities witH Mh rule No norma iz ation O K. AsTEoM aNd B. WitleNMalk
Adaptation of Feedforward Gain Derivative of V dV dt = 2 dxT dt P x + xT P dx dt + ( 0 ) d dt = 2 Ax + Buc( 0 )T P x + 2 xT P Ax + Buc ( 0 ) + ( 0 ) d dt = 2 xT Qx + ( 0 ) d dt + uc BT P x Adaptation law d dt = uc BT P x gives dV dt = 2 x TQx Discuss Output Feedback Process dx dt = Ax + B( 0 )uc e = Cx Adaptation law d dt = uc BT P x Can we nd P such that BT P = C The adaptation law then becomes d dt = uce Kalman-Yakubovich Lemma Definition 5 A rational transfer function G with real coecients is positive real (PR) if Re G(s) 0 for Re s 0 A transfer function G is strictly positive real (SPR) if G(s ") is positive real for some real " > 0. Lemma 1 The transfer function G(s) = C(sI A)1B is strictly positive real if and only if there exist positive de nite matrices P and Q such that AT P + P A = Q and BT P = C Summary Lyapunov Stability Theory { Stability concept { Lyapunovs theorem { How to use it? Adaptive laws with Guaranteed Stability Simple design procedure { Find control law { Derive Error Equation { Find Lyapunov Function { Choose adjustment law so that dV =dt 0 Remark { Strong similarities with MIT rule { Actually simpler { No normalization c K. J. Åström and B. Wittenmark 8