3. Fundamentals of Lyapunov Theory

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1 Applied Nonlinear Control Nguyen an ien -.. Fundamentals of Lyapunov heory he objective of this chapter is to present Lyapunov stability theorem and illustrate its use in the analysis and the design of nonlinear systems..1 Nonlinear Systems and Equilibrium Points Nonlinear systems A nonlinear dynamic system can usually be presented by the set of nonlinear differential equations in the form & = f (, (.1 where n f R : nonlinear vector function n R : state vectors n : order of the system he form (.1 can represent both closed-loop dynamics of a feedback control system and the dynamic systems where no control signals are involved. A special class of nonlinear systems is linear system. he dynamics of linear systems are of the from & = A( with n n A R. Autonomous and non-autonomous systems Linear systems are classified as either time-varying or timeinvariant. For nonlinear systems, these adjectives are replaced by autonomous and non-autonomous. Definition.1 he nonlinear system (.1 is said to be autonomous if f does not depend eplicitly on time, i.e., if the system s state equation can be written & = f ( (. Otherwise, the system is called non-autonomous. Equilibrium points It is possible for a system trajectory to only a single point. Such a point is called an equilibrium point. As we shall see later, many stability problems are naturally formulated with respect to equilibrium points. Definition. A state is an equilibrium state (or equilibrium points of the system if once ( is equal to, it remains equal to for all future time. has a single equilibrium point (the origin if A is nonsingular. If A is singular, it has an infinity of equilibrium points, which contained in the null-space of the matri A, i.e., the subspace defined by A =. A nonlinear system can have several (or infinitely many isolated equilibrium points. Eample.1 he pendulum θ R Fig..1 Pendulum Consider the pendulum of Fig..1, whose dynamics is given by the following nonlinear autonomous equation MR & θ + bθ& + MgRsinθ = (.5 where R is the pendulum s length, M its mass, b the friction coefficient at the hinge, and g the gravity constant. Leting 1 = θ, = θ &, the corresponding state-space equation is & 1 = (.6a b g = sin 1 MR R (.6b herefore the equilibrium points are given by =, sin( 1 =, which leads to the points ( [π ], and ( π [π ],. Physically, these points correspond to the pendulum resting eactly at the vertical up and down points. In linear system analysis and design, for notational and analytical simplicity, we often transform the linear system equations in such a way that the equilibrium point is the origin of the state-space. Nominal motion Let ( be the solution of & = f (, i.e., the nominal motion trajectory, corresponding to initial condition ( =. Let us now perturb the initial condition to be ( = + δ, and study the associated variation of the motion error e( = ( ( as illustrated in Fig... Mathematically, this means that the constant vector satisfies = f ( (. ( e( ( Equilibrium points can be found using (.. A linear time-invariant system n 1 & = A (.4 Fig.. Nominal and perturbed motions Chapter Fundamentals of Lyapunov heory 7

2 Applied Nonlinear Control Nguyen an ien -. Since both ( and ( are solutions of (.: & = f (, we have & = f ( ( = & = f ( ( = + δ then e( satisfies the following non-autonomous differential equation e & = f ( + e, f (, = g( e, (.8 with initial condition e( = δ (. Since g (, =, the new dynamic system, with e as state and g in place of f, has an equilibrium point at the origin of the state space. herefore, instead of studying the deviation of ( from ( for the original system, we may simply study the stability of the perturbation dynamics (.8 with respect to the equilibrium point. However, the perturbation dynamics non-autonomous system, due to the presence of the nominal trajectory ( on the right hand side. Eample. Consider the autonomous mass-spring system m & + k1 + k = which contains a nonlinear term reflecting the hardening effect of the spring. Let us study the stability of the motion ( which starts from initial point. Assume that we slightly perturb the initial position to be ( = + δ. he resulting system trajectory is denoted as (. Proceeding as before, the equivalent differential equation governing the motion error e is m & e + k1 e + k[ e + e ( + e ( ] = Clearly, this is a non-autonomous system.. Concepts of Stability Notation B R : spherical region (or ball defined by R S R : spherical itself defined by = R : for any : there eist : in the set : implies that : equivalent Stability and instability Definition. he equilibrium state = is said to be stable if, for any R >, there eist r >, such that if ( r then ( R for all t. Otherwise, the equilibrium point is unstable. R >, r >, ( < r t, ( < R or, equivalently R >, r >, ( Br t, ( Br Essentially, stability (also called stability in the sense of Lyapunov, or Lyapunov stability means that the system trajectory can be kept arbitrarily close to the origin by starting sufficiently close to it. More formally, the definition states that the origin is stable, if, given that we do not want the state trajectory ( to get out of a ball of arbitrarily specified radius B R. he geometrical implication of stability is indicated in Fig.. 1 ( Sr curve 1 - asymptotically stable curve - marginally stable curve - unstable S R Fig.. Concepts of stability Asymptotic stability and eponential stability In many engineering applications, Lyapunov stability is not enough. For eample, when a satellite s attitude is disturbed from its nominal position, we not only want the satellite to maintain its attitude in a range determined by the magnitude of the disturbance, i.e., Lyapunov stability, but also required that the attitude gradually go back to its original value. his type of engineering requirement is captured by the concept of asymptotic stability. Definition.4 An equilibrium points is asymptotically stable if it is stable, and if in addition there eist some r > such that ( r implies that ( ast. Asymptotic stability means that the equilibrium is stable, and in addition, states start close to actually converge to as time goes to infinity. Fig.. shows that the system trajectories starting form within the ball Br converge to the origin. he ball Br is called a domain of attraction of the equilibrium point. In many engineering applications, it is still not sufficient to know that a system will converge to the equilibrium point after infinite time. here is a need to estimate how fast the system trajectory approaches. he concept of eponential stability can be used for this purpose. Definition.5 An equilibrium points is eponential stable if there eist two strictly positive number α and λ such that λt t >, ( α ( e (.9 in some ball Br around the origin. Using the above symbols, Definition. can be written in the (.9 means that the state vector of an eponentially stable form system converges to the origin faster than an eponential Chapter Fundamentals of Lyapunov heory 8

3 Applied Nonlinear Control Nguyen an ien -. function. he positive number λ is called the rate of eponential convergence. For eample, the system & = (1 + sin is eponentially convergent to = with the rate λ = 1. Indeed, its solution is t + τ dτ t = [1 sin ( ] ( ( e, and therefore t ( ( e. Note that eponential stability implies asymptotic stability. But asymptotic stability does not implies guarantee eponential stability, as can be seen from the system =, ( = 1 (.1 whose solution is = 1/(1 +, a function slower than any eponential function e λt. Local and global stability Definition.6 If asymptotic (or eponential stability holds for any initial states, the equilibrium point is said to be asymptotically (or eponentially stable in the large. It is also called globally asymptotically (or eponentially stable.. Linearization and Local Stability Lyapunov s linearization method is concerned with the local stability of a nonlinear system. It is a formalization of the intuition that a nonlinear system should behave similarly to its linearized approimation for small range motions. Consider the autonomous system in (., and assumed that f( is continuously differentiable. hen the system dynamics can be written as f = + fh. o. t.( (.11 = A similar procedure can be applied for a controlled system. Consider the system & ( + 1 u =. he system can be linearly approimated about = as & + + ( + 1 u = or & = u. Assume that the control law for the original nonlinear system has been selected to be u = sin + + cos, then the linearized closed-loop dynamics is & + + =. he following result makes precise the relationship between the stability of the linear system (. and that of the original nonlinear system (.. heorem.1 (Lyapunov s linearization method If the linearized system is strictly stable (i.e., if all eigenvalues of A are strictly in the left-half comple plane, then the equilibrium point is asymptotically stable (for the actual nonlinear system. If the linearizad system is un stable (i.e., if at least one eigenvalue of A is strictly in the right-half comple plane, then the equilibrium point is unstablle (for the nonlinear system. If the linearized system is marginally stable (i.e., if all eigenvalues of A are in the left-half comple plane but at least one of them is on the jω ais, then one cannot conclude anything from the linear approimation (the equilibrium point may be stable, asymptotically stable, or unstable for the nonlinear system. Eample.5 Consider the equilibrium point ( θ = π, θ& = of the pendulum in the eample.1. Since the neighborhood ofθ = π, we can write where fh. o. t. stands for higher-order terms in. Let us use the constant matri A denote the Jacobian matri of f with respect f to at = : A =. hen, the system = & = A (.1 is called the linearization (or linear approimation of the original system at the equilibrium point. In practice, finding a system s linearization is often most easily done simply neglecting any term of order higher than 1 in the dynamics, as we now illustrate. Eample.4 Consider the nonlinear system & 1 = + 1 cos & = + ( sin Its linearized approimation about = is & 1 = & = sinθ = sinπ + cosπ ( θ π + h. o. t. = π θ + h. o. t. ~ thus letting θ = θ π, the system s linearization about the equilibrium point ( θ = π, & θ = is ~ && b ~ & g ~ θ + θ θ = MR R Hence its linear approimation is unstable, and therefore so is the nonlinear system at this equilibrium point. Eample.5 5 Consider the first-order system & = a + b. he origin is one of the two equilibrium of this system. he linearization of this system around the origin is & = a. he application of Lyapunov s linearization method indicate the following stability properties of the nonlinear system a < : asymptotically stable a > : unstable a = : cannot tell from the linearization In the third case, the nonlinear system is 5 & = b 1 linearization method fails while, as we shall see, the direct he linearized system can thus be written =. method to be described can easily solve this problem. 1 1 Chapter Fundamentals of Lyapunov heory 9. he

4 Applied Nonlinear Control Nguyen an ien -..4 Lyapunov s Direct Method he basic philosophy of Lyapunov s direct method is the mathematical etension of a fundamental physical observation: if the total energy of a mechanical (or electrical system is continuous dissipated, then the system, whether linear or nonlinear, must eventually settle down to an equilibrium point. hus, we may conclude the stability of a system by eamining the variation of a single scalar function. Let consider the nonlinear mass-damper-spring system in Fig..6, whose dynamic equation is m & + b + k + k1 = (.1 with b : nonlinear dissipation or damping k + k1 : nonlinear spring term nonlinear spring and damper Fig..6 A nonlinear mass-damper-spring system otal mechanical energy = kinetic energy + potential energy V ( = m + ( k + k1 d = m + k + k1 4 (.14 Comparing the definitions of stability and mechanical energy, we can see some relations between the mechanical energy and the concepts described earlier: zero energy corresponds to the equilibrium point ( =, = assymptotic stability implies the convergence of mechanical energy to zero instability is related to the growth of mechanical energy he relations indicate that the value of a scalar quantity, the mechanical energy, indirectly reflects the magnitude of the state vector, and furthermore, that the stability properties of the system can be characterized by the variation of the mechanical energy of the system. he rate of energy variation during the system s motion is obtained by differentiating the first equality in (.14 and using (.1 m.4.1. Positive definite functions and Lyapunov functions Definition.7 A scalar continuous function V ( is said to be locally positive definite ifv ( = and, in a ball B R V ( > IfV ( = and the above property holds over the whole state space, then V ( is said to be globally positive definite. 1 For instance, the function V ( = MR + MR(1 cos 1 which is the mechanical energy of the pendulum in Eample.1, is locally positive definite. Let us describe the geometrical meaning of locally positive definite functions. Consider a positive definite function V ( of two state variables 1 and. In -dimensional space, V ( typically corresponds to a surface looking like an upward cup as shown in Fig..7. he lowest point of the cup is located at the origin. 1 V V = V V = V 1 V > V > V1 V = V Fig..7 ypical shape of a positive definite function V ( 1, he -dimesional geometrical representation can be made as follows. aking 1 and as Cartesian coordinates, the level curves V ( 1, = Vα typically present a set of ovals surrounding the origin, with each oval corresponding to a positive value ofv α.hese ovals often called contour curves may be thought as the section of the cup by horizontal planes, projected on the ( 1, plane as shown in Fig..8. V = V V = V V = V 1 1 V > V > V1 V &( = m && + ( k + k1 = ( b = b (.15 (.15 implies that the energy of the system, starting from some initial value, is continuously dissipated by the damper until the mass is settled down, i.e., =. he direct method of Lyapunov is based on generalization of the concepts in the above mass-spring-damper system to more comple systems. Fig..8 Interpreting positive definite functions using contour curves Definition.8 If, in a ball B R, the function V ( is positive definite and has continuous partial derivatives, and if its time derivative along any state trajectory of system (. is negative semi-definite, i.e., V & ( then, V ( is said to be a Lyapunov function for the system (.. Chapter Fundamentals of Lyapunov heory 1

5 Applied Nonlinear Control Nguyen an ien -. A Lyapunov function can be given simple geometrical interpretations. In Fig..9, the point denoting the value of V ( 1, is seen always point down an inverted cup. In Fig..1, the state point is seen to move across contour curves corresponding to lower and lower value ofv. 1 V V ( Fig..9 Illustrating Definition.8 for n= Obviously, this function is locally positive definite. As a mater of fact, this function represents the total energy of the pendulum, composed of the sum of the potential energy and the kinetic energy. Its time derivative yields V &( = θ & sinθ + θθ &&& = θ& herefore, by involving the above theorem, we can conclude that the origin is a stable equilibrium point. In fact, using physical meaning, we can see the reason why V & (, namely that the damping term absorbs energy. Actually, V & ( is precisely the power dissipated in the pendulum. However, with this Lyapunov function, we cannot draw conclusion on the asymptotic stability of the system, because V & ( is only negative semi-definite. Eample.8 Asymptotic stability Let us study the stability of the nonlinear system defined by V = V V = V 1 1 = 1 ( = 41 + ( around its equilibrium point at the origin. V = V V > V > V1 V ( 1, = 1 + Fig..1 Illustrating Definition.8 for n= using contour curves.4. Equilibrium point theorems Lyapunov s theorem for local stability heorem. (Local stability If, in a ball B R, there eists a scalar function V ( with continuous first partial derivatives such that V ( V & ( is positive definite (locally in B is negative semi-definite (locally in B then the equilibrium point is stable. If, actually, the derivative V & ( is locally negative definite in B R, then the stability is asymptotic. In applying the above theorem for analysis of a nonlinear system, we must go through two steps: choosing a positive Lyapunov function, and then determining its derivative along the path of the nonlinear systems. Eample.7 Local stability A simple pendulum with viscous damping is described as & θ + θ& + sin θ = R R its derivativev & along any system trajectory is V & = ( 1 + ( 1 + hus, is locally negative definite in the -dimensional ball B, i.e., in the region defined by ( 1 + <. herefore, the above theorem indicates that the origin is asymptotically stable. Lyapunov theorem for global stability heorem. (Global Stability Assume that there eists a scalar function V of the state, with continuous first order derivatives such that V ( is positive definite V & ( is negative definite V( as then the equilibrium at the origin is globally asymptotically stable. Eample.9 A class of first-order systems Consider the nonlinear system + c( = Consider the following scalar function where c is any continuous function of the same sign as its scalar argument, i.e., such as c( >. Intuitively, 1 ( (1 cosθ θ& this condition indicates that c( pushes the system back V = + towards its rest position =, but is otherwise arbitrary. Chapter Fundamentals of Lyapunov heory 11

6 Applied Nonlinear Control Nguyen an ien -. Since c is continuous, it also implies that c ( = (Fig..1. Consider as the Lyapunov function candidate the square of distance to the origin V =. he function V is radially unbounded, since it tends to infinity as. Its derivative is V & = = c(. hus V & < as long as, so that = is a globally asymptotically stable equilibrium point. c( Fig..1 he function c ( For instance, the system = sin is globally convergent to =, since for, sin sin. Similarly, the system = is globally asymptotically convergent to =. Notice that while this system s linear approimation ( is inconclusive, even about local stability, the actual nonlinear system enjoys a strong stability property (global asymptotic stability. Eample.1 Consider the nonlinear system & 1 = 1( 1 + & = 1 ( 1 + he origin of the state-space is an equilibrium point for this system. Let V be the positive definite function V = 1 +. Its derivative along any system trajectory is V & = ( 1 + which is negative definite. herefore, the origin is a globally asymptotically stable equilibrium point. Note that the globalness of this stability result also implies that the origin is the only equilibrium point of the system. Note that: - Many Lyapunov function may eist for the same system. - For a given system, specific choices of Lyapunov functions may yield more precise results than others. - Along the same line, the theorems in Lyapunov analysis are all sufficiency theorems. If for a particular choice of Lyapunov function candidate V, the condition on V & are not met, we cannot draw any conclusions on the stability or instability of the system the only conclusion we should draw is that a different Lyapunov function candidate should be tried..4. Invariant set theorem Definition.9 A set G is an invariant set for a dynamic system if every system trajectory which starts from a point in G remains in G for all future time. Local invariant set theorem he invariant set theorem reflect the intuition that the decrease of a Lyapunov function V has to graduate vanish (i.e., V & has to converge to zero because V is lower bounded. A precise statement of this result is as follows. heorem.4 (Local Invariant Set heorem Consider an autonomous system of the form (., with f continuous, and let V ( be a scalar function with continuous first partial derivatives. Assume that for some l >, the region Ω l defined by V ( < l is bounded V & ( for all in Ω l Let R be the set of all points within Ωl where V & ( =, and M be the largest invariant set in R. hen, every solution ( originating in Ωl tends to M as t. Note that: - M is the union of all invariant sets (e.g., equilibrium points or limit cycles within R - In particular, if the set R is itself invariant (i.e., if once V & =, then for all future time, then M=R he geometrical meaning of the theorem is illustrated in Fig..14, where a trajectory starting from within the bounded region Ωl is seen to converge to the largest invariant set M. Note that the set R is not necessarily connected, nor is the set M. he asymptotic stability result in the local Lyapunov theorem can be viewed a special case of the above invariant set theorem, where the set M consists only of the origin. V = l 1 V Fig..14 Convergence to the largest invariant set M Let us illustrate applications of the invariant set theorem using some eamples. Eample.11 Asymptotic stability of the mass-damper-spring system For the system (.1, we can only draw conclusion of marginal stability using the energy function (.14 in the local equilibrium point theorem, because V & is only negative semidefinite according to (.15. Using the invariant set theorem, however, we can show that the system is actually asymptotically stable. O do this, we only have to show that the set M contains only one point. Ω l R M Chapter Fundamentals of Lyapunov heory 1

7 Applied Nonlinear Control Nguyen an ien -. he set R defined by =, i.e., the collection of states with zero velocity, or the whole horizontal ais in the phase plane (,. Let us show that the largest invariant set M in this set R contains only the origin. Assume that M contains a point with a non-zero position 1, then, the acceleration at that point is & = ( k / m ( k1 / m. his implies that the trajectory will immediately move out of the set R and thus also out of the set M, a contradiction to the definition. Eample.1 Domain of attraction limit cycle 1 Consider again the system in Eample.8. For l = 1, the region Ω l, defined by V ( 1, = 1 + < 1, is bounded. he set R is simply the origin, which is an invariant set (since it is an equilibrium poin. All the conditions of the local invariant set theorem are satisfied and, therefore, any trajectory starting within the circle converges to the origin. hus, a domain of attraction is eplicitly determined by the invariant set theorem. Eample.1 Attractive limit cycle Consider again the system Fig..15 Convergence to a limit circle Moreover, the equilibrium point at the origin can actually be shown to be unstable. Any state trajectory starting from the region within the limit cycle, ecluding the origin, actually converges to the limit cycle. Eample.11 actually represents a very common application of the invariant set theorem: conclude asymptotic stability of an equilibrium point for systems with negative semi-definite V &. he following corollary of the invariant set theorem is more specifically tailored to such applications = 1 ( = 1 ( Note that the set defined by 1 + = 1 is invariant, since d ( = ( ( dt which is zero on the set. he motion on this invariant set is described (equivalently by either of the equations & 1 = = 1 herefore, we see that the invariant set actually represents a limit circle, along which the state vector moves clockwise. Is this limit circle actually attractive? Let us define a Luapunov 4 function candidate V = ( which represents a measure of the distance to the limit circle. For any arbitrary positive number l, the region Ω l, which surrounds the limit circle, is bounded. Its derivative V & = 8( 1 + ( hus V & 4 is strictly negative, ecept if 1 + = 1 or =, in which cases V & =. he first equation is simply that defining the limit cycle, while the second equation is verified only at the origin. Since both the limit circle and the origin are invariant sets, the set M simply consists of their union. hus, all system trajectories starting in Ωl converge either to the limit cycle or the origin (Fig..15 Corollary: Consider the autonomous system (., with f continuous, and let V ( be a scalar function with continuous partial derivatives. Assume that in a certain neighborhood Ω of the origin is locally positive definite V & ( is negative semi-definite the set R defined byv & ( = contains no trajectories of (. other than the trivial trajectory hen, the equilibrium point is asymptotically stable. Furthermore, the largest connected region of the form (defined byv ( < l within Ω is a domain of attraction of the equilibrium point. Indeed, the largest invariant set M in R then contains only the equilibrium point. Note that: - he above corollary replaces the negative definiteness condition on V & in Lyapunov s local asymptotic stability theorem by a negative semi-definiteness condition on V &, combined with a third condition on the trajectories within R. - he largest connected region of the form Ω l within Ω is a domain of attraction of the equilibrium point, but not necessarily the whole domain of attraction, because the function V is not unique. - he set Ω itself is not necessarily a domain of attraction. Actually, the above theorem does not guarantee that Ω is invariant: some trajectories starting in Ω but outside of the largest Ωl may actually end up outside Ω. Global invariant set theorem he above invariant set theorem and its corollary can be simply etended to a global result, by enlarging the involved region to be the whole space and requiring the radial unboundedness of the scalar functionv. Chapter Fundamentals of Lyapunov heory 1

8 Applied Nonlinear Control Nguyen an ien -. heorem.5 (Global Invariant Set heorem Consider an autonomous system of the form (., with f continuous, and let V ( be a scalar function with continuous first partial derivatives. Assume that V & ( over the whole state space V( as Let R be the set of all points where V & ( =, and M be the largest invariant set in R. hen all solutions globally asymptotically converge to M as t For instance, the above theorem shows that the limit cycle convergence in Eample.1 is actually global: all system trajectories converge to the limit cycle (unless they start eactly at the origin, which is an unstable equilibrium poin. Because of the importance of this theorem, let us present an additional (and very useful eample. Eample.14 A class of second-order nonlinear systems Consider a second-order system of the form & + b( + c( = where b and c are continuous functions verifying the sign conditions & b( > for and c( > for. he dynamics of a mass-damper-spring system with nonlinear damper and spring can be described by the equation of this form, with the above sign conditions simply indicating that the otherwise arbitrary function b and c actually present damping and spring effects. A nonlinear R-L-C (resistorinductor-capacitor electrical circuit can also be represented by the above dynamic equation (Fig..16 v C = c( v L = & v R = b( Fig..16 A nonlinear R-L-C circuit Note that if the function b and c are actually linear ( b( = α 1, c( = α, the above sign conditions are simply the necessary and sufficient conditions for the system s stability (since they are equivalent to the conditions α 1 >, α >. ogether with the continuity assumptions, the sign conditions b and c are simply that b( = and c = (Fig..17. A positive definite function for this system is V = 1 & + c( y dy, which can be thought of as the sum of the kinetic and potential energy of the system. Differentiating V, we obtain V & = && + c( = b & ( c & ( + c( = b & ( as long as. hus the system cannot get stuck at an equilibrium value other than = ; in other words, with R being the set defined by =, the largest invariant set M in R contains only one point, namely [ =, = ]. Use of the local invariant set theorem indicates that the origin is a locally asymptotically stable point. b ( & c( Fig..17 he functions b ( and c ( Furthermore, if the integral c( r dr is unbounded as, then V is a radially unbounded function and the equilibrium point at the origin is globally asymptotically stable, according to the global invariant set theorem. Eample.15 Multimodal Lyapunov Function Consider the system & + 1 π + = sin y Chose the Lyapunov function V = + y sin dy. his function has two minima, at = ±1, =, and a local maimum in (a saddle point in the state-space at =, 4 =. Its derivative V & = 1, i.e., the virtual power dissipated by the system. Now V & = = or = ±1. Let us consider each of cases: = π & = sin ecept if = or = ± 1 = ±1 & = hus the invariant set theorem indicates that the system converges globally to or ( = 1, =. he first two of these equilibrium points are stable, since they correspond to local minima of V (note again that linearization is inconclusive about their stability. By contrast, the equilibrium point ( =, = is unstable, as can be shown from linearization (& = ( π / 1, or simply by noticing that because that point is a local maimum of V along the ais, any small deviation in the direction will drive the trajectory away from it. Note that: Several Lyapunov function may eist for a given system and therefore several associated invariant sets may be derived. which can be thought of as representing the power dissipated.5 System Analysis Based on Lyapunov s Direct Method in the system. Furthermore, by hypothesis, & b( = only if How to find a Lyapunov function for a specific problem? =. Now = implies that & = c(, which is non-zero here is no general way of finding Lyapunov function for Chapter Fundamentals of Lyapunov heory 14

9 Applied Nonlinear Control Nguyen an ien -. nonlinear system. Faced with specific systems, we have to use eperience, intuition, and physical insights to search for an appropriate Lyapunov function. In this section, we discuss a number of techniques which can facilitate the otherwise blind of Lyapunov functions. Lyapunov functions for linear time-invariant systems Given a linear system of the form & = A, let us consider a quadratic Lyapunov function candidate V & = P, where P is a given symmetric positive definite matri. Its derivative yields.5.1 Lyapunov analysis of linear time-invariant systems Symmetric, skew-symmetric, and positive definite matrices Definition.1 A square matri M is symmetric if M=M (in other words, if i, j M ij = M ji. A square matri M is skewsymmetric if M = M (i.e., i, j M ij = M ji. Note that: - Any square n n matri can be represented as the sum of a symmetric and a skew-symmetric matri. his can be shown in the following decomposition M + M M - M M = symmetric skew symmetric - he quadratic function associated with a skew-symmetric matri is always zero. Let M be a n n skew-symmetric matri and is an arbitrary n 1 vector. he definition of skew-symmetric matri implies that M = M. Since M is a scalar, M = M which yields, M = (.16 In the designing some tracking control systems for robot, this fact is very useful because it can simplify the control law. - (.16 is a necessary and sufficient condition for a matri M to be skew-symmetric. Definition.11 A square matri M is positive definite (p.d. if M >. Note that: - A necessary condition for a square matri M to be p.d. is that its diagonal elements be strictly positive. - A necessary and sufficient condition for a symmetric matri M to be p.d. is that all its eigenvalues be strictly positive. - A p.d. matri is invertible. - A.d. matri M can always be decomposed as M = U ΛU (.7 where U U = I, Λ is a diagonal matri containing the eigenvalues of M - here are some following facts λmin ( M M λma ( M M = U ΛU = z Λz where U = z λmin ( M I Λ λma ( M I z z = he concepts of positive semi-definite, negative definite, and negative semi-definite can be defined similarly. For instance, a square n n matri M is said to be positive semi-definite (p.s.d. if, M. A time-varying matri M( is uniformly positive definite if α >, t, M( α I. V & = P + P = - Q (.18 where A P + P A = -Q (.19 (.19 is so-called Lyapunov equation. Note that Q may be not p.d. even for stable systems. Eample.17 Consider the second order linear system with A = If we take P = I, then - Q = P A + A P = 4. he 4 4 matri Q is not p.d.. herefore, no conclusion can be draw from the Lyapunov function on whether the system is stable or not. A more useful way of studying a given linear system using quadratic functions is, instead, to derive a p.d. matri P from a given p.d. matri Q, i.e., choose a positive definite matri Q solve for P from the Lyapunov equation check whether P id p.d. If P is p.d., then ( 1/ P is a Lyapunov function for the linear system. And the global asymptotical stability is guaranteed. heorem.6 A necessary and sufficient condition for a LI system & = A to be strictly stable is that, for any symmetric p.d. matri Q, the unique matri P solution of the Lyapunov equation (.19 be symmetric positive definite. Eample.18 Consider again the second order linear system in Eample p.18. Let us take Q = I and denote P by = 11 p P 1, p1 p where due to the symmetry of P, p 1 = p1. hen the Lyapunov equation is p1 p p11 1 p1 p1 = 1 p Chapter Fundamentals of Lyapunov heory 15 p11 p1 1 whose solution is p 11 = 5, p 1 = p = 1. he corresponding matri P = is p.d., and therefore the linear system is globally asymptotically stable..5. Krasovskii s method Krasovskii s method suggests a simplest form of Lyapunov function candidate for autonomous nonlinear systems of the

10 Applied Nonlinear Control Nguyen an ien -. form (., namely, V = f f. he basic idea of the method is simply to check whether this particular choice indeed leads to a Lyapunov function. heorem.7 (Krasovkii Consider the autonomous system defined by (., with the equilibrium point of interest being the origin. Let A( denote the Jacobian matri of the system, i.e., f A( = If the matri F = A + A is negative definite in a neighborhood Ω, then the equilibrium point at the origin is asymptotically stable. A Lyapunov function for this system is V ( = f ( f ( If Ω is the entire state space and, in addition, V( as, then the equilibrium point is globally asymptotically stable. Eample.19 Consider the nonlinear system & 1 = 61 + = 1 6 We have function for this system. If the region Ω is the whole state space, and if in addition, V( as, then the system is globally asymptotically stable..5. he Variable Gradient Method he variable gradient method is a formal approach to constructing Lyapunov functions. o start with, let us note that a scalar function V ( is related to its gradient V by the integral relation V ( = V d where V = { V / 1, K, V / n }. In order to recover a unique scalar function V from the gradient V, the gradient function has to satisfy the so-called curl conditions V V i j = j i ( i, j = 1,, K, n Note that the i th component Vi is simply the directional derivative V / i. For instance, in the case n =, the above simply means that V1 V = 1 = f 6 A 6 6 = 1 4 F A + A = he matri F is easily shown to be negative definite. herefore, the origin is asymptotically stable. According to the theorem, a Lyapunov function candidate is V ( = ( (1 6 Since V ( as, the equilibrium state at the origin is globally asymptotically stable. he applicability of the above theorem is limited in practice, because the Jcobians of many systems do not satisfy the negative definiteness requirement. In addition, for systems of higher order, it is difficult to check the negative definiteness of the matri F for all. heorem.7 (Generalized Krasovkii heorem Consider the autonomous system defined by (., with the equilibrium point of interest being the origin, and let A( denote the Jacobian matri of the system. hen a sufficient condition for the origin to be asymptotically stable is that there eist two symmetric positive definite matrices P and Q, such that, the matri F ( = A P + PA + Q is negative semi-definite in some neighborhood Ω of the origin. he function V ( = f ( f ( is then a Lyapunov he principle of the variable gradient method is to assume a specific form for the gradient V, instead of assuming a specific form for a Lyapunov function V itself. A simple way is to assume that the gradient function is of the form n Vi = aij j j= 1 Chapter Fundamentals of Lyapunov heory 16 (.1 where the a ij s are coefficients to be determined. his leads to the following procedure for seeking a Lyapunov functionv assume that V is given by (.1 (or another form solve for the coefficients a ij so as to sastify the curl equations assume restrict the coefficients in (.1 so that V & is negative semi-definite (at least locally computev from V by integration check whetherv is positive definite Since satisfaction of the curl conditions implies that the above integration result is independent of the integration path, it is usually convenient to obtain V by integrating along a path which is parallel to each ais in turn, i.e., 1 V ( = V K + K + K+ 1 ( 1,,, d1 V ( 1,,, d n Vn ( 1,, K, d n

11 Applied Nonlinear Control Nguyen an ien -. Eample. Let us use the variable gradient method top find a Lyapunov function for the nonlinear system 1 = 1 & = + 1 We assume that the gradient of the undetermined Lyapunov function has the following form V 1 = a111 + a1 V = a11 + a he curl equation is V1 V = 1 a1 a1 a1 + = a If the coefficients are chosen to be a 11 = a = 1, a1 = a1 = which leads to V 1 = 1, V = then V & can be computed as V ( = 1 d1 + d = (. his is indeed p.d., and therefore, the asymptotic stability is guaranteed. If the coefficients are chosen to be a 11 = 1, a1 =, a 1 =, a =, we obtain the p.d. function 1 V ( = (. whose derivative is V & = 1 6 ( 1 1. We can verify that V & is a locally negative definite function (noting that the quadratic terms are dominant near the origin, and therefore, (. represents another Lyapunov function for the system..5.4 Physically motivated Lyapunov functions.5.5 Performance analysis Lyapunov analysis can be used to determine the convergence rates of linear and nonlinear systems. A simple convergence lemma Lemma: If a real function W ( satisfies the inequality W& ( + α W ( (.6 α where α is a real number. hen W ( W ( e he above Lemma implies that, if W is a non-negative function, the satisfaction of (.6 guarantees the eponential convergence of W to zero. t Estimating convergence rates for linear system Let denote the largest eigenvalue of the matri P by λ ma ( P, the smallest eigenvalue of the matri Q by λ min ( Q, and their ratio λma ( P / λmin ( Q by γ. he p.d. of P and Q implies that these scalars are all strictly positive. Since matri theory shows that P λ ma ( P I and λ min ( Q I Q, we have λmin ( Q Q [ λma ( P I ] γ V λma ( P his and (.18 implies that V& γ V.his, according to t lemma, means that Q V( e γ. his together with the fact P λmin ( P (, implies that the state converges to the origin with a rate of at least γ /. he convergence rate estimate is largest for Q = I. Indeed, let P be the solution of the Lyapunov equation corresponding to Q = I is A P + P A = I and let P the solution corresponding to some other choice of Q A P + PA = Q 1 Without loss of generality, we can assume that λ min ( Q1 = 1 since rescaling Q 1 will rescale P by the same factor, and therefore will not affect the value of the corresponding γ. Subtract the above two equations yields A ( P - P + ( P - P A = ( Q1 - I Now since λ min ( Q1 = 1 = λma ( I, the matri ( Q1 - I is positive semi-definite, and hence the above equation implies that ( P - P is positive semi-definite. herefore λma ( P λma ( P Since λ min ( Q1 = 1 = λmin ( I, the convergence rate estimate γ = λmin ( Q / λma ( P corresponding to Q = I the larger than (or equal to that corresponding to Q = Q1. Estimating convergence rates for nonlinear systems he estimation convergence rate for nonlinear systems also involves manipulating the epression of V & so as to obtain an eplicit estimate of V. he difference lies in that, for nonlinear systems, V and V & are not necessarily quadratic function of the states. Eample. Consider again the system in Eample.8 1 = 1 ( = 41 + ( 1 + Choose the Lyapunov function candidate V =, its derivative is V & dv = V ( V 1. hat is = dt. he V (1 V solution of this equation is easily found to be Chapter Fundamentals of Lyapunov heory 17

12 Applied Nonlinear Control Nguyen an ien -. dt α e V ( = dt 1+ α e V (, where α =. 1 V ( If ( = V ( < 1, i.e., if the trajectory starts inside the t unit circle, then α >, and V ( < α e. his implies that the norm ( of the state vector converges to zero eponentially, with a rate of 1. However, if the trajectory starts outside the unit circle, i.e., if V ( > 1, then α <, so that V ( and therefore tend to infinity in a finite time (the system is said to ehibit finite escape time, or eplosion..6 Control Design Based on Lyapunov s Direct Method here are basically two ways of using Lyapunov s direct method for control design, and both have a trial and error flavor: Hypothesize one form of control law and then finding a Lyapunov function to justify the choice Hypothesize a Lyapunov function candidate and then finding a control law to make this candidate a real Lyapunov function Eample. Regulator design Consider the problem of stabilizing the system & + = u. Chapter Fundamentals of Lyapunov heory 18