Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written as: M k Mgsin 0 (4.1) or, equivalently in state space form: 1 (4.) asin 1b where 1, g k, a, and b. We study stability of the origin e 0. M Note that the latter is equivalent to studying stability of all the equilibrium points in the form: e l 0, l 0, 1,, Consider the total energy of the pendulum as a Lyapunov function candidate. 1 V asin ydy a1cos 1 (4.3) 0 Potential Kinetic It is clear that V is a positive definite function, (locally, around the origin). Its time derivative along the system trajectories is: V asin 11 b 0 (4.4) So, the time derivative is negative semidefinite. It is not strictly negative definite because V 0 for 0, irrespective of the value of 1. herefore, we can only conclude that the origin is a stable equilibrium, but not necessarily asymptotically stable. 10
However, using the phase portrait of the pendulum equation (or just common sense), we epect the origin to be asymptotically stable. he Lyapunov energy function argument fails to show it. On the other hand, we notice that for the system to maintain V 0 condition, the trajectory must be confined to the line 0. Using the system dynamics (4.) yields: 0 0sin1 0 1 0 Hence on the segment 1 of the line 0 the system can maintain the V 0 condition only at the origin 0. herefore, V t must decrease to toward 0 and, consequently, t 0 as t, which is consistent with the fact that, due to friction, energy cannot remain constant while the system is in motion. he forgoing argument shows that if in a domain about the origin we can find a Lyapunov function whose derivative along the system trajectories is negative semidefinite, and we can establish that no trajectory can stay identically at points where V 0, ecept at the origin, then the origin is asymptotically stable. his argument follows from the LaSalle s Invariance Principle which is applicable to autonomous systems of the form f, f 0 0 (4.5) Definition 4.1 n A set M is said to be an invariant set with respect to (4.5) if: a positively invariant set with respect to if: 0 M t M, t 0 M t M, t 0 heorem 4.1 (LaSalle s heorem) n Let D be a compact positively invariant set with respect to the system dynamics (4.5). Let V : D be a continuously differentiable function such that V t in. Let E be the set of all points in where V 0. Let 0 M E be the largest invariant set in E. hen every solution starting in approaches M as t, that is liminf t z 0 t zm dist t, M Notice that the inclusion of the sets in the LaSalle s theorem is: n M E D 11
In fact, a formal proof of the theorem reveals that all trajectories t are bounded and approach a positive limit set L M as t. he latter may contain asymptotically stable equilibriums and stable limit cycles. emark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the function be positive definite. V to Most often, our interest will be to show that t 0 as t. For that we will M 0. his is need to establish that the largest invariant set in E is the origin, that is: done by showing that no solution can stay identically in E other than the trivial solution t 0. heorem 4.1 (Barbashin-Krasovskii theorem) Let 0 be an equilibrium point for (4.5). Let V : D be a continuously n differentiable positive definite function on a domain D containing the origin, such that V t S D: V 0 and suppose that no other solution 0 in D. Let can stay in S, other than the trivial solution t 0. hen the origin is locally asymptotically stable. If, in addition, V is radially unbounded then the origin is globally asymptotically stable. Note that if V is negative definite then 0 S and the above theorem coincides with the Lyapunov nd theorem. Also note that the LaSalle s invariant set theorems are applicable to autonomous system only. Eample 4. Consider the 1 st order system a u together with its adaptive control law u kˆ t he dynamics of the adaptive gain ˆk t is ˆk where 0 is called the adaptation rate. hen the closed-loop system becomes: ˆ k t a ˆ k he line 0 represents the system equilibrium set. We want to show that the trajectories approach this equilibrium set, as t, which means that the adaptive 1
controller regulates t to zero in the presence of constant uncertainty in a. Consider the Lyapunov function candidate V, kˆ 1 1 k ˆ b where b a. he time derivative of V along the trajectories of the system is given by 1 V ˆ ˆ, k kbkˆ kˆa kˆb ba0 V, k ˆ is positive definite and radially unbounded function, whose derivative Since V, ˆ k 0 is semi-negative, then, ˆ :, ˆ c k V k c is compact, positively invariant set. hus, taking c, all the conditions of LaSalle s heorem are satisfied. E, kˆ : 0. Since any point on the line 0 is an he set E is given by c equilibrium point, E is an invariant set. herefore, in this eample M E. From LaSalle s heorem we conclude that every trajectory starting in approaches E, as t, that is 0 t as t. Moreover, since, ˆ c V k is radially unbounded, the 0, k ˆ 0 because the 0, kˆ 0 c. conclusion is global, that is it holds for all initial conditions constant c in the definition of can be chosen large enough that Homework: Simulate the closed-loop system from Eample 4.. 0, k ˆ 0. est different initial conditions c un simulations of the system while increasing the rate of adaptation 0 until high frequency oscillations and / or system departure occurs. ry to quantify imum allowable as a function of the initial conditions. un simulations of the system while increasing the control time delay 0, that is using control in the form ut kˆ t t. ry to quantify imum allowable time delay, (as a function of the initial conditions and rate of adaptation), before the system starts to oscillate or departs. 5. Boundedness and Ultimate Boundedness Consider the nonautonomous system f t, (5.1) n where f :0, D is piecewise continuous in t, locally Lipschitz in on n 0, D, and D is a domain that contains the origin 0. If the origin is the equilibrium point for (5.1) then by definition: f t,0 0, t 0. On the other hand, even if there is no equilibrium at the origin, Lyapunov analysis can still be used to show boundedness of the system trajectories. We begin with a motivating eample. 13
Eample 5.1 Consider the IVP with nonautonomous scalar dynamics sin t (5.) t 0 a 0 he system has no equilibrium points. he IVP eplicit solution can be easily found and shown to be bounded for all t t0, uniformly in t 0, that is with a bound b independent of t 0. In this case, the solution is said to be uniformly ultimately bounded (UUB), and b is called the ultimate bound, (Homework: Prove this statement). he UUB property of (5.) can be established via Lyapunov analysis and without using eplicit solutions of the state equation. Starting with V, we calculate the time derivative of V along the system trajectories. V sin t sin t It immediately follows that V 0, In other words, the time derivative of V is negative outside the set B, or equivalently, all solutions that start outside of B will reenter the set within a finite time, and will remain their afterward. Formally, it can be stated as follows. Choose Since V is negative on the set boundary then all solutions starting in the set Bc V c B c c. will remain therein for all future time. Hence the solutions are uniformly bounded. Moreover, an ultimate bound of these solutions can also be found. Choose such that c hen V is negative in the annulus set V c, which implies that in this set V t will decrease monotonically in time until the solution enters the set V. From that time on, the solution cannot leave the set because V is negative on its boundary V. Since V, we can conclude that these solutions are UUB with the ultimate bound. Definition 5.1 he solutions of (5.1) are 14
Uniformly Bounded if there eists a positive constant c, independent of t 0 0, and for every a 0, c, there is a 0, independent of t 0, such that t0 a t, t t0 (5.3) Globally Uniformly Bounded if (5.3) holds for arbitrarily large a. Uniformly Ultimately Bounded with ultimate bound b if there eist positive constants b and c, independent of 0 0 a 0, c, there is a, b, independent of t 0, such that 0 0 t, and for every t a t b, t t (5.4) Globally Uniformly Ultimately Bounded if (5.4) holds for arbitrarily large a. Figure 5.1: UUB Concept In the definition above, the term uniform indicates that the bound b does not depend on t 0. he term ultimate means that boundedness holds after the lapse of a certain time. he constant c defines a neighborhood of the origin, independent of t 0, such that all trajectories starting in the neighborhood will remain bounded in time. If c can be chosen arbitrarily large then the UUB notion becomes global. Basically, UUB can be considered as a milder form of stability in the sense of Lyapunov (SISL). A comparison summary between SISL and UUB concepts is given below. SISL is defined with respect to an equilibrium, while UUB is not. Asymptotic SISL is a strong property that is very difficult to achieve in practical dynamical systems. 15
SISL requires the ability to keep the state arbitrarily close to the system equilibrium by starting sufficiently close to it. his is still too strong a requirement for practical systems operating in the presence of unknown disturbances. he main difference between UUB and SISL is that the UUB bound b cannot be made arbitrarily small by starting closer to the equilibrium or the origin. In practical systems, the bound b depends on disturbances and system uncertainties. o demonstrate how Lyapunov analysis can be used to study UUB, consider a continuously differentiable positive definite function V, choose 0 c, and suppose that the sets V and c c V c are compact. Let V c and suppose that it is known that the time derivative of V t along the trajectories of the nonautonomous dynamical system (5.1) is negative definite inside, that is V t W t 0,, t t 0 where W t is a continuous positive definite function. Since V is negative in, a trajectory starting in must move in the direction of decreasing V t. It can be shown that in the set the trajectory behaves as if the origin was uniformly asymptotically stable, (which it does not have to be in this case). Consequently, the function V t will continue decreasing until the trajectory enters the set in finite time and stays there for all future time. Hence, the solutions of (5.1) are UUB with the ultimate bound b. A sketch of the sets, c, is shown in Figure 5.. Figure 5.: UUB by Lyapunov Analysis In many problems, the relation V t, W is derived and shown to be valid on a domain, which is specified in terms of a Euclidian norm. In such cases, UUB analysis involves finding the corresponding domains of attraction and an ultimate bound. 16
n his analysis will be performed net. Let denote a bounded domain and suppose that the system dynamics are: t, AB (5.5) m where A is Hurwitz. Also suppose that t, is a bounded function on. Let QQ 0 and consider V P (5.6) where P P 0 is the unique positive definite symmetric solution of the algebraic Lyapunov equation. PA A P Q (5.7) hen the time derivative of V evaluated along the system (5.1) trajectories satisfies the following relation: V Q PB t,,, t t (5.8) 0 Suppose that 0 is chosen such that the sphere S is inside the domain, that is: S (5.9) Also suppose that t, for all S formally prove that all the solutions t of (5.1) that start in a subset of S are UUB., uniformly in t. hen one can In order to prove the UUB property, we assume that. hen for any symmetric positive definite matri P and for all vectors, min P P P (5.10) where min P, P denote the smallest and the largest eigenvalues of P, respectively. An upper bound for V in (5.8) can be calculated, for. V min Q PB min Q PB (5.11) Define a sphere. PB Sr r min Q (5.1) It follows from (5.11) that V r S S (5.13) Let b r P r and let b V 0, r b r r. hen Sr b r. In fact, if Sr then using the right hand side of (5.10) yields: P P Pr b r (5.14) Hence, b r and the inclusion Sr b r is proven. 17
Let b min P and define b V b b. hen then using the left hand side of (5.10), yields: Hence, min. In fact, if min P PV b P (5.15), that is S, and the inclusion b S is proven. Net, we need to ensure that br b, that is: br P r min P B (5.16) or, equivalently: r min P (5.17) P he above inequality can be viewed as a restriction placed on the eigenvalues of P and the constants r and. his relation ensures the desired set inclusions: Sr b r b S (5.18) Graphical representation of the four sets is given in Figure 5.. b S Figure 5.: epresentation of the sets S S and the Ultimate Bound M r br b Net, we show that all solutions starting in b will enter b r and remain there afterwards. If t 0 b then since V 0 in r b b, V t is a decreasing r function of time outside of. herefore, solutions that start in will remain there. Suppose that t0. Inequality (5.11) implies: b r b r 18
hus, where V min Q PB min Q PB P min P P V min P V min Q PB V V P min P V t 0 satisfies the following differential inequality, (as a function of time): V av g V min Q PB a and g are positive constants. Let W V P P hen (5.0) is equivalent to Define hen because of (5.1), Solving (5.) for W yields, and, therefore: (5.19) (5.0) min 0. a g W W (5.1) a Z W W (5.) g Z t, t t0 (5.3) a t a 0 t t t t0 W t W t e e Z d (5.4) 0 t t t a g t t 0 g g a t a 0 t0 W t W t e e Z d W t e e d W t e e a t a a a tt0 g t tt0 g tt0 0 0 1 a t0 e W t a a a 0 o1 o1 Consequently, as t : g P PB P a min Q min P min P t P t o1 o1 r o1 r (5.5) (5.6) Choose 0. hen it is easy to see that there eists independent of t 0 such that o1 and, consequently 19
P (5.7) t P t r, t t0, min P Since the above relation is valid for all solutions that start in, it is also valid for the solution which starts in and imizes the left hand side of the inequality. In other words P t P t r (5.8) B min P On the other hand t P t P t (5.9) Hence B P P P B,, (5.30) t M r t t0 min Consequently, all solutions of (5.1) are UUB with the ultimate bound M. he bound is shown in Figure 5.. Summarizing all formally proven results, we state the following theorem. heorem 5.1 Let P P 0 and QQ 0 be symmetric positive definite matrices that satisfy the Lyapunov algebraic equation (5.7), and let V t, (5.5) and suppose that where 0 suppose that is chosen such that r min P P P. Consider the dynamics in, uniformly in t, and for all S, PB S. Let Sr r and min Q. hen those solutions of (5.5) that start in the bounded set b V b min P are UUB, with the ultimate bound M, as it is defined in (5.30). b b 6. Invariance like heorems For autonomous systems, the LaSalle s invariance set theorems allow asymptotic stability conclusions to be drawn even when V is only negative semi-definite in a domain. In that case, the system trajectory approaches the largest invariant set E, which is a subset of all points where V 0. However the invariant set theorems are not applicable to nonautonomous systems. In the case of the latter, it may not even be clear how to define a set E, since V may eplicitly depend on both t and. Even when V V does not eplicitly depend on t the nonautonomous nature of the system dynamics precludes the use of the LaSalle s invariant set theorems. 0
Eample 6.1 he closed-loop error dynamics of an adaptive control system for 1 st order plant with one unknown parameter is e e wt ew t wt is a bounded function of time t. Due to where e represents the tracking error and the presence of wt, the system dynamics is nonautonomous. Consider the Lyapunov function candidate V e, e Its time derivative along the system trajectories is V e, ee ee wt ewt e 0 his implies that V is a decreasing function of time, and therefore, both et and t are bounded signals of time. But due to the nonautonomous nature of the system dynamics, the LaSalle s invariance set theorems cannot be used to conclude the convergence of et to the origin., 0 then we may epect that the trajectory of the system approaches the set 0, as t. Before we formulate main results, we state a lemma that is interesting in its own sake. he lemma is an important result about asymptotic properties of functions and their derivatives and it is known as the Barbalat s lemma. We begin with the definition of a uniform continuity. In general, if V t W W Definition 6.1 (uniform continuity) A function f t is said to be uniformly continuous if : 0 0 t t1 f t f t1 Note that t 1 and t play a symmetric role in the definition of the uniform continuity. Lemma 6.1 (Barbalat) Let f t : is be differentiable and has a finite limit, as t. If f t uniformly continuous then f t 0, as t. Lemma 6. If f t is bounded then f t is uniformly continuous. An immediate and a very practical corollary of Barbalat s lemma can now be stated. Corollary 6.1 1
If f : t is twice differentiable, has a finite limit, and its nd derivative is bounded then f t 0 as t. In general, the fact that derivative tends to zero does not imply that the function has a limit. Also, the converse is not true. In other words: f t C f t Eample 6. As t, f t sin ln t 0 cos ln t does not have a limit, while f t 0 t. t t f t e sin e 0 f t e t sin e t e t cos e t. As t,, while