Lyapunov Stability Stability of Equilibrium Points

Transcription

1 Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x), x( t ) = x (L.1) r nnlinear time varying discrete time (D) systems f the frm x (k +1) = f (k, x (k)), x (k ) = x, (L.2) where x R n, t R +, k Z + and x is the initial state at t (k respectively). Definitin [Equilibrium State] An equilibrium state x e is such that f (t, x e ) = 0, fr all t, fr C systems. f (k, x e ) = x e, fr all k, fr D systems. Withut lss f generality, we will assume that 0 is an equilibrium state. Ntice that nn-linear systems (and sme linear systems) may have mre than ne equilibrium state. Definitin [Ref.1] [Stability and Unifrm Stability in the sense f Lyapunv] he equilibrium state 0 f (1) is (lcally) stable in the sense f Lyapunv if fr every ε > 0, there exists a δ( ε, t ) > 0 such that, if xt ( ) < δ then xt ()< ε fr all t > t (respectively k fr D). In additin, if δ can be chsen independent f t, i.e., δ( ε), then, the rigin is (lcally) unifrmly stable. Definitin [Ref.1] [Asympttic Stability and Unifrm Asympttic Stability] he equilibrium state 0 f (1) is (lcally) asympttically stable if 1. It is stable in the sense f Lyapunv and 2. here exists a δ ( t ) such that, if xt ( ) < δ, then, xt ( ) Æ0 as tæ. he equilibrium state 0 f (1) is (lcally) unifrmly asympttically stable if 1. It is unifrmly stable in the sense f Lyapunv and 3. here exists a δ, independent f t, such that, if xt ( ) < δ, then, x( t) 0 as t, unifrmly in t ; that is, fr each ε > 0, there exists =(ε)>0, independent f t, such that xt () < ε, t t + ( ε), xt ( ) < δ 0 0 Nte: he definitin f stability in the sense f Lyapunv is clsely related t that f cntinuity f slutins. An equilibrium is stable if all slutins starting at nearby pints stay nearby; therwise, it is unstable. It is asympttically stable if all slutins starting at nearby pints nt nly stay nearby, but als tend t the equilibrium pint as time appraches infinity. 1

2 Definitin [Glbal asympttic stability] he equilibrium state 0 f (1) is glbally asympttically stable, if it is asympttically stable fr any δ > 0. Definitin [Expnential stability] he equilibrium state 0 f (1) is expnentially stable, if it is stable in the sense f Lyapunv and there exists a δ > 0 and cnstants M < and α > 0 such that ( t t ) xt () e α Mx (L.3) fr all xt ( ) < δ. α is called the rate f expnential cnvergence. 2. Lyapunv Stability herems Fr Autnmus (r ime-invariant) Systems When f in (1) des nt depend n time t explicitly, i.e., x = f( x), x( t ) = x (L.4) fr cntinuus time r x (k +1) = f (x (k)), x (k ) = x, (L.5) fr discrete time, then, the system becmes an autnmus system. he behavir f an autnmus system is invariant t shifts in the time rigin. hus, the slutin x(t) depends n x 0 and t-t 0 nly, and is independent f t 0. his leads t the fllwing fact: Fr autnmus system, unifrm (asympttic) stability is the same as (asympttic) stability. Definitin [Psitive Definite (Semi-Definite) Functin (PDF)] A cntinuusly differentiable functin V: R n R is called psitive definite in a regin U R n cntaining the rigin if a. V(0)=0 b. V(x)>0, x U and x 0 A functin is called psitive semi-definite if Cnditin b is replaced by V(x) 0. herem L.1 [Ref1] [Lyapunv herem] Fr autnmus systems, let D R n be a dmain cntaining the equilibrium pint f rigin. If there exists a cntinuusly differentiable psitive definite functin V: D R such that V = V dx V = ( ) ( ) x dt x f x = W x (L.6) is negative semi-definite in D, then, the equilibrium pint 0 is stable. Mrever, if W(x) is psitive definite, then, the equilibrium is asympttically stable. In additin, if D=R n and V is radially unbunded, i.e., x V( x) (L.7) then, the rigin is glbally asympttically stable. 2

3 Fr autnmus systems, when W(x) in the abve therem is nly psitive semi-definite, asympttic stability may still be btained by applying the fllwing simplified versin f LaSalle s herem. herem L.2 [Ref1] [LaSalle s Invariance Principle herem] Fr autnmus systems, let D R n be a dmain cntaining the equilibrium pint f rigin. If there exists a cntinuusly differentiable psitive definite functin V: D R such that V = V ( ) ( ) x f x = W x 0 (L.8) in D. Let m r (L.9) S= x D V ( x ) = 0 and suppse that n slutin can stay identically in S, ther than the rigin. hen, the rigin is asympttically stable. In additin, if D=R n and V is radially unbunded, the rigin is glbally asympttically stable. 2.2 Linear ime Invariant System herem L.3 he fllwing cnditins are equivalent: (a) he equilibrium 0 f the nth rder system x = Ax (L.10) is glbally asympttically stable (expnentially stable). (b) All eigenvalues f A have negative real parts. (c) Fr any psitive definite symmetric matrix Q, there exists a unique psitive definite symmetric matrix P which is the slutin f the fllwing Lyapunv equatin 1 PA + A P = Q. (L.11) Nte: (c) indicates that the PDF functin V (x) = x Px is a Lyapunv functin fr the system. Prf: We will demnstrate that (c) is a necessary and sufficient cnditin fr (a) and (b). Sufficiency: Assume that given a psitive definite symmetric matrix Q there exists a psitive definite symmetric matrix P which satisfies (L.11). Define the PDF functin V (x) = x Px. aking the time derivative f V alng the trajectries f (L.10), we btain n s (L.12) V ( x) x A P PA x x = + = Qx hus, V and V are bth PDF, and the system is glbally asympttically stable. prve expnential stability, we ntice that min max x Qx λ ( Q) x x, x Px λ ( P) x x, 1 he MALAB cmmand fr slving Lyapunv equatin is lyap in cntinuus time and dlyap in discrete time. 3

4 where λ min( Q) and λ max( P) are respectively the minimum eigenvalue f Q and the maximum eigenvalue f P, bth f which are psitive. hus, defining α = λ min ( Q) / λ max( P) > 0, and using the ntatin V( t) = V( x( t)), we btain ( ) Vx λ min( Qx ) x = α Vx ( ) λ max( Px ) x hus, Vt () α Vt (). (L.13) Integrating (L.13), we btain Vt () e αt V( 0 ) (L.14) which implies that V 0 expnentially. Since Vx ( ) λ ( Pxx ) = λ ( P) x, 2 min min where λ min ( P ) > 0 is the minimum eigenvalue f P, x must als cnverge t zer expnentially. Necessity: We first define the unit ball: n s, n 1 1 B = = v R v v= and the vectr induce 2 nrm f a matrix M R n n : M = max Mv = max v M Mv = ( M M) = ( M) 2 v B 2 1 v B1 { } λmax σ. max Assume that the system given by (L.10) is asympttically stable. hus, all eigenvalues f A have negative real parts and, as a cnsequence, A is nnsingular and s is the slutin matrix e At,fr0 t <. Since Q is psitive definite, there exists a nnsingular matrix, Q 1/ 2, such that A t At 1/ 2 1/ Q= ( Q ) ( Q 2 ). hus, the matrix e Qe is psitive definite fr 0 t <. Since all eigenvalues f A have negative real parts, the matrix A t At P= e Qe dt (L.15) z0 λt m λt exists, is unique and P 2 <. (Remember that e, t e L 1 L fr any λ C such that Re(λ) < 0 and any 0 < m < ). hus, P, as defined by (L.15), is psitive definite. We nw prve that P, as defined by (L.15), satisfies the Lyapunv equatin (L.11). since lim t z { } A t At A t At A P+ PA= A e Qe + e Qe A dt z0 A t At A t At = d/ dt e Qe dt = lim e Qe Q= Q 0 { } t { }, At e = 2 0. Q.E.D. 2.3 LI Discrete ime Systems Definitin [Change f V (k, x) relative t a state trajectry] Cnsider the system (L.2). he change f V (k + 1, x) relative t (L.2) is given by 4

5 V( k + 1, x) = V( k + 1, x( k + 1 )) V( k, x( k)) (L.16) = V( k + 1, f ( k, x( k))) V( k, x( k)). herem L.4 he fllwing cnditins are equivalent: (a) he equilibrium 0 f the nth rder system x(k + 1) = Ax(k) is glbally asympttically stable (expnentially stable). (L.17) (b) All eigenvalues f A have magnitudes less than 1. (c) Fr any psitive definite symmetric matrix Q, there exist a unique psitive definite symmetric matrix P which is the slutin f the fllwing Discrete ime Lyapunv equatin A PA P = Q. (L.18) Prf: he prf is very similar t the cntinuus time case and it s left as an exercise. Q.E.D. 2.4 Lyapunv s Indirect Methd herem L.5 [Ref1] Cnsider the autnmus system (L.4) with the rigin as an equilibrium pint. If f: D R n is cntinuusly differentiable and D is a neighbrhd f the rigin. Let f A = x ( x) x= 0 (L.19) hen, (a) he rigin is lcally asympttically stable if A is asympttically stable r all eigenvalues f A have negative real parts. (b) he rigin is unstable if ne r mre f the eigenvalues f A has psitive real part. Nte: (1). Bth the Lyapunv s indirect methd (herem L.5) and direct methd (herem L.1) can be used t judge the lcal stability f an equilibrium pint when the linearized system matrix A is either asympttically stable r unstable. Hwever, the indirect methd des nt tell anything abut the regin f attractin 2 (r dmain f attractin) while the direct methd gives at least sme cnservative estimate f the dmain f attractin. Fr example, if cnditins fr asympttic stability in herem 1 are satisfied and Ω c ={x R n V(x) c} is bunded and cntained in D, then, every trajectry starting in Ω c remains in Ω c and appraches the rigin as t. hus Ω c is an estimate f the regin f attractin. (2). When sme f the eigenvalues f A have zer real parts and all the rest eigenvalues have negative real parts, the lcal stability f the rigin cannt be cncluded frm the abve therem. In such a case, the lcal stability f the rigin depends n higher-rder nnlinear terms als. Advanced stability therems such as Center Manifld herem may be used t judge the lcal stability f the rigin. 2 he regin in which all trajectries cnverge t the equilibrium pint as t appraches. 5

6 Lyapunv Stability herems Fr Nn-autnmus (r ime-varying) Systems Cnsider the nn-autnmus system (L.1) where f: [0, ) D R n is piecewise cntinuus in t and lcally Lipschitz in x n [0, ) D, and D R n is a dmain that cntains the equilibrium pint f rigin x=0. Nte that an equilibrium at rigin f a nn-autnmus system culd be a translatin f a nn-zer time-varying slutin f an autnmus system (e.g., trajectry tracking f an autnmus system). he slutin f a nn-autnmus system may depend n bth t-t 0 and t 0, and the Lyapunv functin V(x,t) in general depends n t als. characterize the psitive definiteness f a time functin, we intrduce the fllwing additin definitins. Definitin [Ref.1] [Class-K Functin] A cntinuus functin α: [0,a) R + is said t be a class-k functin if, (a) α (0) = 0. (b) α is strictly increasing. It is said t belng t class K if a= and α(r) as r. Definitin [Ref.1] [Class-KL Functin] A cntinuus functin β: [0,a) [0, ) R + is said t belng t class-kl if, fr each fixed t, the mapping β(r,t) belngs t class K with respect t r and, fr each fixed r, the mapping β(r,t) is decreasing with respect t t and β(r,t) 0 as t. Definitin [Lcally Psitive Definite Functin (LPDF)] A cntinuus functin V: R + D R + is said t be a Lcally Psitive Definite Functin (LPDF) if there exists a class K functin α such that (a) V (t, x) α ( x ) fr all t 0 and fr all x r, fr sme r > 0. (b) V (t, 0) = 0 Definitin [Psitive Definite Functin (PDF)] A cntinuus functin V: R + R n R + is said t be a Psitive Definite Functin (PDF) if there exists a class K functin α such that (a) V (t, x) α ( x ) fr all t 0 and fr all x R n. (b) V (t, 0) = 0 Definitin [Decrescent Functin] A cntinuus V: R + D R + is said t be lcally decrescent if there exists a class K functin α such that V (t, x) α ( x ) fr all t 0 and fr all x r, fr sme r > 0. It is decrescent if α is a class K functin and the abve inequality is valid fr all x in R n. 6

7 Examples 1) V (x 1, x 2 ) = x1 2 + x2 2 is a PDF and decrescent. 2) V (t, x 1, x 2 ) = (t + 1) (x1 2 + x2 2 ) is a PDF but nt decrescent. t 3) V (t, x 1, x 2 ) = e ( x1 2 + x2 2 ) is nt a PDF. 4) V (x 1, x 2 ) = x1 2 + sin 2 ( x2) is a LPDF and lcally decrescent (but nt PDF and decrescent). 3.1 Cntinuus ime Systems Definitin [Derivative f V (t, x) relative t a state trajectry] Cnsider the system (L.1). he Derivative f V(t, x) relative t (L.1) is given by V( t, x) V( t, x) Vtx (, ) = + f(, t x), t x where L NM Vtx (, ) Vtx (, ) Vtx (, ) Vtx (, ) x = x x 1 2 x n O QP (L.20) herem L.6 [Ref2] he equilibrium pint 0 f (L.1) is lcally stable in the sense f Lyapunv if there exists a LPDF V(t,x) such that Vtx (, ) 0 fr all t t and all x such that x < r fr sme r > 0. herem L.7 [Ref2] he equilibrium pint 0 f (L.1) is lcally unifrmly stable in the sense f Lyapunv if there exists a lcally decrescent LPDF V(t,x) such that Vtx (, ) 0 fr all t 0 and all x such that x < r fr sme r > 0. herem L.8 [Ref2] he equilibrium pint 0 f (L.1) is lcally unifrmly asympttically stable if there exists a lcally decrescent LPDF V(t,x) such that Vtx (, ) is a LPDF. herem L.9 [Ref2] he equilibrium pint 0 f (L.1) is glbally unifrmly asympttically stable if there exists a decrescent PDF V(t,x) such that Vtx (, ) is a PDF. In adaptive cntrl prblems, it is ften the case that Vtx (, ) is nly negative semi-definite, i.e., Vtx (, ) 0. If the system is autnmus, then, LaSalle s Invariance Principle herem L.2 may be applied t btain asympttic tracking. Fr nn-autnmus system, LaSalle s Invariance herem L.2 cannt be applied. Instead, the fllwing Barbalat s lemma shuld be used. 7

8 Lemma L.1 [Ref1] Barbalat s Lemma t z φτ Let φ(t) be a unifrmly cntinuus real functin f t defined fr t 0. Suppse that lim t 0 ( ) d exists and is finite. hen, φ( t) 0 as t Using Barbalat s lemma, the fllwing Lyapunv-like lemma can be btained. Lemma L.2 [Ref2] Lyapunv-Like Lemma If a scalar functin V(t,x) satisfies the fllwing cnditins V(t,x) is lwer bunded Vtx (, ) is negative semi-definite. Vtx (, ) is unifrmly cntinuus in time (A sufficient cnditin is that Vtx (, ) is bunded) then, Vtx (, ) 0 as t 3.2 Discrete ime Systems Definitin [Change f V (k, x) relative t a state trajectry] Cnsider the system (L.2). he change f V(k+1,x) relative t (L.2) is given by V( k + 1, x) = V( k + 1, x( k + 1 )) V( k, x( k)) (L.21) = V( k + 1, f ( k, x( k))) V( k, x( k)). herem L.10 he equilibrium pint 0 f (L.2) is lcally stable in the sense f Lyapunv if there exists a LPDF V(k,x) such that V( k + 1, x) 0 fr all k k and all x such that x < r fr sme r > 0. herem L.11 he equilibrium pint 0 f (L.2) is glbally unifrmly asympttically if there exists a decrescent PDF V(k,x) such that V( k +1, x) is negative definite. 3.3 Linear ime-varying Systems he slutin f the linear time-varying system described by x = A( t) x (L.22) is given by x( t) =Φ( t, t 0 ) x( t 0 ) where Φ( t, t 0 ) is the state transitin matrix. 8

9 herem L.12 [Ref1] he equilibrium pint 0 f (L.22) is (glbally) unifrmly asympttically stable if and nly if the state transitin matrix satisfies the inequality γ ( t t0 ) Φ( tt, ) ke, t t 0 (L.23) 0 0 fr sme psitive cnstant k and γ. herem L.12 shws that, fr linear systems, unifrm asympttic stability f the rigin is equivalent t expnential stability. Nte that, fr linear time-varying system, in general, unifrm asympttic stability cannt be characterized by the lcatin f the eigenvalues f the matrix A herem L.13 [Ref1] Suppse that the equilibrium pint 0 f (L.22) is unifrmly asympttically stable, and A(t) is cntinuus and bunded. Let Q(t) be a cntinuus, bunded, symmetric psitive definite matrix. hen, there is a cntinuusly differentiable, bunded, symmetric psitive definite matrix P(t) such that Pt () = PtAt () () + A () tpt () + Qt () (L.24) Hence, V(t,x)=x P(t)x is a Lyapunv functin fr the system that satisfies the cnditins f herem L.9. Prf: Let z Pt () = Φ ( τ,) tq( τ ) Φ( τ,) td τ (L.25) t It can be verified that P(t) given abve satisfies (L.24). Details are mitted. 3.4 Linearizatin (Lyapunv s Indirect Methd) herem L.14 [Ref1] Cnsider the nn-autnmus system (L.1) with the rigin as an equilibrium pint. Suppse that f: R + D R n is cntinuusly differentiable, and the Jacbian matrix [ f/ x] is bunded and Lipschitz n D, unifrmly in t. Let f At () = (, x tx ) (L.26) x =0 hen, the rigin is an expnentially stable equilibrium pint fr the nnlinear system (L.1) if it is an expnentially stable equilibrium pint fr the linearized linear system (L.22). References [Ref1] Khalil, H. K. (1996), Nnlinear Systems, Secnd editin, Prentice-Hall. [Ref2] Sltine, J.J.E. and Li, Weiping (1991), Applied Nnlinear Cntrl, Prentice-Hall. 9