MCE693/793: Analysis and Control of Nonlinear Systems

MCE693/793: Analysis and Control of Nonlinear Systems Lyapunov Stability - I Hanz Richter Mechanical Engineering Department Cleveland State University

Definition of Stability - Lyapunov Sense Lyapunov stability is related to the tendency of system trajectories to remain in prescribed regions surrounding an equilibrium point. This includes the possibility (but does not require) convergence to the equilibrium point, where the stability is asymptotic. The harmonic oscillator as an autonomous LTI system ẋ 1 = x 2 ẋ 2 = w 2 x 1 gives circular orbits passing through any initial condition (x 10,x 20 ). Each circle of radius x 2 10 +x2 20 determines a confinement region Ω, so that all initial conditions within Ω result in trajectories remaining in Ω. But no nonzero initial condition converges to the origin. The origin of the autonomous system is stable in the sense of Lyapunov, but not asymptotically stable. Warning: The forced undamped oscillator is unstable in the bounded input - bounded output sense. 2 / 33

Mathematical Definition of Lyapunov Stability Consider the autonomous nonlinear system ẋ = f(x) and let x eq be an equilibrium point. The system is stable about the equilibrium point if the following property holds: Fo any R > 0 there always exists r > 0 so that if x 0 x eq < r, then x(t) x eq < R for all t > 0. R defines a desired confinement region, while r defines the neighborhood of x eq where x 0 must belong so that x(t) does not exit the confinement region. R x eq r x 0 x(t) 3 / 33

Asymptotic Stability Instead of saying the nonlinear system is stable relative to equilibrium point x 0, we can just say x 0 is stable in context. The equilibrium point x 0 is asymptotically stable if it is stable and also there exists r > 0 such that the following implication holds: x(0) x 0 < r = x(t) x 0 as t That is, x 0 is the only positive limit point of all solutions with initial conditions in the ball B(x 0,r). This ball is a region of attraction of x 0. The maximal region of attraction does not have to be a ball. In general, regions of attraction have more complex shapes. 4 / 33

Exponential Stability The equilibrium point x 0 is exponentially stable if α,λ such that x(t) x 0 α x(0) x 0 e λt Checking whether a system is stable is in general impossible by using only the definition (a large number of initial conditions would have to be tested numerically, or the explicit solution could be used for very simple systems). Examine the example in S&L: ẋ = (1+sin 2 x)x Here exponential stability (and therefore asymptotic stability) can be established with the solution. 5 / 33

Local and Global (in the large) Stability The radius r involved in the definitions of stability and asymptotic stability provide a domain of attraction. When r is finite, stability is local. If r can be extended to infinity (if in the definition, r can always be chosen equal to R), stability is global. Immediate observations: If a system has multiple equilibrium points, can any one of them be globally stable? Asymptotic stability implies stability trivially, by definition, but: Why is stability included in the definition of asymptotic stability in the first place? Does the convergence condition alone imply stability? As an exercise, we show that the system given by Vinograd (S&L Figure 3.5) is unstable, but still satisfies the convergence property. 6 / 33

Lyapunov s Linearization Method This method allows us to determine the stability of the nonlinear system about the equilibrium point on the basis of the linearized system. Simply: If the eigenvalues of the A matrix in the linearized system have negative real parts, the nonlinear system is asymptotically stable about the equilibrium point. If at least one eigenvalue of the A matrix in the linearized system has positive real part, the nonlinear system is unstable about the equilibrium point. If at least one eigenvalue ofthe A matrix in the linearized system has zero real part, the test is inconclusive. The linear approximation is insufficient to determine stability. However, methods exist to include higher-order terms (center manifold technique). 7 / 33

Positive-Definite Functions and Matrices The function f : R n R is positive semidefinite if f(0) = 0 and f(x) 0 for all x R n. If f(x) > 0 for all nonzero x R n, then f is positive definite. Example: f(x 1,x 2 ) = x 2 1 +x 2 2 +sin 2 (x 1 ) is positive-definite. If f(x) 0 (resp. f(x) > 0) in a subset A R n, we say that f is positive-semidefinite (resp. positive definite) in A. When f(x) is a quadratic function defined through a symmetric matrix P as f(x) = x T Px then we say that P is positive-semidefinite (resp. positive definite) when the associated function is so. Finally, if f is positive-semidefinite (resp. positive definite), we say that f is negative-semidefinite (resp. negative definite). If f changes sign, it is sign-indefinite. 8 / 33

Testing for Positive-Definite Matrices Suppose f(x) = x 2 1 +2x2 2 0.1x2 3 x 1x 2 +8x 2 x 3 5x 1 x 3. How can we tell if f is sign-definite? The symmetric matrix associated with f is 1 1 2 5 2 P = 1 2 2 4 5 2 4 0.1 (1) The eigenvalue test can be applied. If all eigenvalues are nonnegative, we have a p.s.d. matrix. If they are all positive, we have a p.d. matrix. The same idea applies for n.s.d. and n.d. If the eigenvalues have mixed signs, the matrix is sign-indefinite. 9 / 33

Lyapunov s Direct (Second) Method For simplicity, suppose the origin is an equilibrium point. Suppose we find a function V : R n R which has continuous first partial derivatives and is positive-definite in a region surrounding the origin. We call this function a candidate Lyapunov function. If the function V is negative-semidefinite in the same region, then the origin is a stable equilibrium point. If V is negative-definite, the origin is asymptotically stable. We use Lyapunov s method with a quadratic function to prove that the van der Pol equation with µ = 2 is stable relative to the origin. Is the proven stability global? Lyapunov s direct method is only a sufficient condition. Hence the difficulties in finding Lyapunov functions 10 / 33

Global Asymptotic Stability Stronger conditions on V and its derivative are required for global asymptotic stability. Suppose V(x) has continuous first partial derivatives and satisfies: 1. V is positive definite in R n 2. V is negative-definite in R n 3. V(x) as x (V is radially unbounded) then the origin is globally asymptotically stable. 11 / 33

Examples The function V(x) = x is positive-definite, but cannot be used with the above results because it does not have a partial derivative at zero. However, results are available for piecewise-continuous Lyapunov functions. The system ẋ = x 3 is G.A.S. at zero, but the linearization method is inconclusive. Systems of the form ẋ = c(x), where c(x) is a continuous odd function with c(0) = 0 are always G.A.S. at x = 0. All stable linear systems ẋ = Ax admit a quadratic Lyapunov function V(x) = 1 2 xt Px. For linear stable systems and V(x) = x T Px, V = x T (A T P +PA)x. Admissible P s can be found by solving Lyapunov s equation: with any Q > 0. A T P +PA = Q 12 / 33

Constructive Controller Design with Lyapunov Functions Many techniques for control design are based on the idea of requiring a given Lyapunov function candidate to be negative (semi)definite as a way to obtain a control law. This is commonly done in robust and adaptive control, as well as the backstepping and sliding mode techniques. As an introductory example, consider the second-order control system ẋ 1 = x 2 ẋ 2 = x 3 1 +x 2 +u Examine system stability about the origin without control. Then use a quadratic Lyapunov function to find a stabilizing control law. Is the stability global or asymptotic? 13 / 33

Results for semi-definite V Frequently, Lyapunov s direct method goes only as far as showing stability, even when the system is actually asymptotically stable. This can happen with systems as simple as the linear mass-spring-damper with a quadratic V. Invariant set theorems, including the LaSalle-Krasovskii s theorem (known as LaSalle s invariance principle, 1959-1960) and Barbalat s lemma (1959) are useful to analyze long-term system behavior when the Lyapunov function derivative is only negative semi-definite. The LaSalle invariance principle is used to prove asymptotic stability when the Lyapunov function derivative is only negative semi-definite. Barbǎlat s result is used very often in adaptive control (and in the proofs of the invariant set theorems). 14 / 33

Invariant Sets Let a nonlinear system be defined by ẋ = f(x) A set I R n is invariant if the following is true: x(t 0 ) I x(t) I t > t 0 Example: In the damped pendulum with states θ and θ, let E represent the total energy of the pendulum. For any l > 0, the set I = {(θ, θ) : E(θ, θ) l} is invariant. What is the shape of the boundary of I? 15 / 33

Local Invariant Set Theorem Refer to S&L, Th. 3.4. Examine Fig. 3.14 Let a nonlinear system be defined by ẋ = f(x) with f : R n R n continuous and f(0) = 0. Let V(x) be a scalar function with continuous first partial derivatives. For some l > 0 define sets Ω l and R by Ω l = {x R n : V(x) < l and V(x) 0} R = {x Ω l : V(x) = 0} Let M be the largest invariant set contained in R. Then every trajectory x(t) with x(0) Ω l is such that x(t) x M as t for some point x M M. X is the largest invariant set contained in Y means: if Z Y and Z is invariant, then Z X 16 / 33

LaSalle s Invariance Principle This is a special case of the local invariant set theorem. Suppose the conditions of the local invariant set theorem hold, and in addition, suppose V(x) is positive-definite in a set Ω containing the origin, and that V(x) 0 in Ω. Define R = {x Ω : V(x) = 0} If R = {0} (a singleton), then 0 is asymptotically stable. Moreover, let Ω l be the largest connected subset of Ω that can be defined by Ω l = {x R n : V(x) < l and V(x) 0} for some l > 0. Then Ω l is a domain of attraction of the origin. 17 / 33

LaSalle s Principle... Note that if R is invariant, then M = R. To use LaSalle s result, show that the only possible trajectory belonging to R is the zero trajectory: x(t) = 0 for t 0. Classical example: pendulum with viscous damping. We do this in class. Example 3.13 in S&L is very illustrative. We add details in class. 18 / 33

Global Invariance Theorem If Ω l is extended to R n and radial unboundedness is required, the invariance theorem can be applied to solutions starting from anywhere in R n. Consider the autonomous system ẋ = f(x), with f continuous. Let V(x) be a scalar function with continuous first partial derivatives and assume V(x) 0 x R n and V(x) as x. Define R = {x : V(x) = 0} and let M be the largest invariant set in R. Then every trajectory x(t) satisfies x(t) x M as t for some point x M M. 19 / 33

Barbǎlat s Lemma A function f : R n R is square integrable if: 0 f T (t)qf(t)dt for some matrix Q = Q T > 0. A function g : R n R is uniformly continuous if for any ǫ > 0 there is a δ > 0 such that: x y < δ g(x) g(y) < ǫ for all x,y R n. Note that δ can be a function of ǫ but not on the points x and y (that would be plain continuity for g; uniform continuity is a stronger property). Barbǎlat s lemma: If f(t) is square integrable and df(t) dt is uniformly continuous then df(t) dt 0 as t. 20 / 33

Barbǎlat s Lemma in Lyapunov Theory Let a nonlinear system be defined by ẋ = f(x,ψ(t)) where ψ(t) is some input function. Suppose the origin is an equilibrium point (f(0,ψ) = 0) for any ψ. Suppose we found a function V(x,t) which is lower-bounded in a set D containing the origin, and that V(x, t) 0 in D. If V(x, t) is uniformly continuous with respect to time, then V(x,t) 0 as t. Note that uniform continuity of V can be satisfied by verifying that V is bounded. 21 / 33

Variants of Barbǎlat s Lemma Some alternatives have surfaced over the years. For instance, Tao, G. A simple alternative to the Barbǎlat Lemma, IEEE Trans. Aut. Ctrl. V42,N5., 1997. If f is square integrable and has a bounded derivative, then f itself converges to zero asymptotically. 22 / 33

Lyapunov Function Construction - Krasovskii s Method When considering a system ẋ = f(x) suspected to be stable, Lyapunov theory provides a powerful method to verify stability. Note that f may be the closed-loop system resulting of applying a new control design method. Now we want to prove that the design is indeed stabilizing. Krasovskii s method involves using V(x) = f T (x)f(x) as a candidate Lyapunov function. Krasovskii s Theorem: Suppose f(0) = 0 and let A(x) be the Jacobian matrix function. If F(x) = A(x)+A T (x) is negative definite in some neighborhood Ω of 0, then 0 is asymptotically stable and V(x) = f T (x)f(x) is a Lyapunov function. If Ω = R n and V(x) is radially unbounded, 0 is globally asymptotically stable. Examine example 3.19. Additional example in class. 23 / 33

Generalized Krasovskii s Method It is very likely that F(x) will not be negative definite, or that checking this might be really hard for high-order systems. Some flexibility is gained by extending the Lyapunov function candidate to a general quadratic function. Suppose f(0) = 0 and let A(x) be the Jacobian matrix function. Suppose P = P T > 0 and Q = Q T > 0 such that F(x) = A T (x)p +PA(x)+Q is negative semidefinite in some deleted neighborhood Ω of 0. Then 0 is asymptotically stable and V(x) = f T (x)pf(x) is a Lyapunov function. If Ω = R n and V(x) is radially unbounded, 0 is globally asymptotically stable. Although more flexibility has been afforded, the search for P and Q will be computationally intensive. If we shrink Ω to just the origin and obtain F < 0, what can we conclude? 24 / 33

Vector Calculus Refresher: Potential Functions This material is necessary for full understanding of the Variable Gradient Method for constructing Lyapunov functions. Refer to Hass, Weir and Thomas, Multivariable University Calculus: Early Transcendentals, 2nd Ed., Addison-Wesley, 2012. Definition: If F : R n R n is a vector field defined on a set D and Φ such that Φ = F on D, Φ is called a potential function for F. Examples: Φ(h) = mg(h 0 h) is a potential function for F = mg. Φ(x) = 1 2 kx2 is a potential function for F = kx. Φ(x 1,x 2 ) = sin(x 1 )x 2 +1 is a potential function for F = [x 2 cos(x 1 ) sin(x 1 )] In physics, a force field F is conservative if it is the gradient of some potential. 25 / 33

Curl Condition for the Existence of a Potential Function If a vector function F is given, how do we know if it has a potential function? Theorem: Let F(x) = [F 1 (x),f 2 (x),...f n (x)] be a vector field defined on a simply connected domain (a closed curve can be continuously shrunk to a point without leaving the domain). Assume the F i have continuous first partial derivatives. Then Φ such that Φ = F if and only if for all i,j = 1,2...n. F i x j = F j x i For n = 3, the condition is simply curl(f) = 0, hence the name. In fluid flow, we call a velocity field F satisfying this condition irrotational. 26 / 33

Conservative Fields and Path Independence Fundamental Theorem of Line Integrals: Let Φ be a scalar function defined on a domain D and let C be a curve joining two points x,y D R n. Then Φ(y) Φ(x) = Φ(r).dr In physics, we evaluate the work done by a conservative force between two points by evaluating the change in kinetic energy, instead of computing a line integral. C 27 / 33

Variable Gradient Method The idea is to propose the gradient of a Lyapunov function instead of the function directly. In doing this, we have the opportunity to enforce negative-definiteness of V, but we then have to check positive-definiteness of V (the method is often unsuccessful, but worth trying). 1. Start by assuming a simple form for V, for instance ( V) i = n j=1 a ijx j, where the a ij could be variable. 2. Impose the curl conditions 3. Impose conditions on a ij so that V 0 in a domain D surrounding the equilibrium point that is as large as possible. 4. Integrate to find V. Here we must perform the line integrals, but path independence allows us to use a convenient rectangular path. 5. Check if V is positive definite, at least in a subset of D. Follow S&L Example 3.20 carefully. 28 / 33

Example Schultz G. and Gibson, J.E., The variable gradient method for generating Liapunov functions, Trans. AIEE, Pt II 81 pp. 203-210, 1962. Consider ẋ 1 = x 2 ẋ 2 = x 3 1 x 2 29 / 33

Regions of Attraction and Zubov s Construction Method Often, a controller design results in non-global asymptotic stability. For safe operation, it is important to obtain an estimate of the region of attraction, so the system is not allowed to operated away from its guaranteed stability region. Zubov s Theorem: Consider the autonomous system ẋ = f(x), with f(0) = 0 and let Ω be a domain containing the origin. Suppose there exist a function V : Ω R and φ : R n R such that: 1. V is positive definite and continously differentiable in Ω 2. φ is positive definite and continuous in R n 3. 0 V(x) < 1 for all x Ω and V(x) = 1 as x approaches the boundary of Ω (or x if Ω is unbounded). 4. V = (V 1)φ. Then the origin is asymptotically stable and Ω is the (exact) region of attraction. 30 / 33

Example (Hahn) ẋ 1 = x 1 +2x 2 1x 2 ẋ 2 = x 2 Take φ(x 1,x 2 ) = x 2 1 +x2 2 and compute the region of attraction. We must assume an arbitrary form for φ(x), usually a sum of squares. When an exact solution for V is not possible, we can obtain an estimate by approximating the nonlinear system. 31 / 33

Zubov s Approximation Method See Zubov, V.I., Methods of A.M. Lyapunov and Their Application, P. Noordhoff LTD, Groningen 1964 for full details (requires fairly advanced math to follow) For instance if we can assume ẋ = f(x) Ax+g(x) where A is Hurwitz and g(x) has second and higher-order terms, then we assume V = V 2 +V 3 +...V j, where the subindex indicates quadratic, cubic, etc. homogeneous functions (sums of monomials of the same degree). Equate and match coefficients. V = T V.f = (V 1)φ 32 / 33

Zubov s Approximation... Zubov s result: for every approximation ˆV = V 2 +V 3 +...V m, the boundary of the true region of attraction lies in the region defined by α ˆV(x) β, where: Notes: α = max ˆV=0ˆV β = min ˆV=0ˆV 1. The exact contour is obtained in the limit, with ˆV an infinite power series. 2. Convergence is not monotonic. Using 4 terms could be worse than using 3. But in the long run, more terms lead to a better approximation. Example: Damped pendulum ẍ + sin(x) + kẋ = 0. We know stability of the origin can t be global because other equilibrium points exist. We use Zubov s method to find the region of attraction of the origin. 33 / 33