Georgia Institute of Technology Nonlinear Controls Theory Primer ME 6402

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1 Georgia Institute of Technology Nonlinear Controls Theory Primer ME 640 Ajeya Karajgikar April 6, 011 Definition Stability (Lyapunov): The equilibrium state x = 0 is said to be stable if, for any R > 0, there exists r > 0, such that, if x(0) < r, then x(t) < R for all t 0. Otherwise, the equilibrium point is unstable. R > 0, r > 0, x(0) < r t 0, x(t) < R Equivalently, R > 0, r > 0, x(0) B r t 0, x(t) B R Definition Asymptotical Stability: The equilibrium state x is asymptotically stable if it is stable, and if in addition there exists some r > 0 such that x(0) < r implies that x(t) x as t. An equilibrium point which is Lyapunov stable but not asymptotically stable is called marginally stable. Definition Exponential Stability: There is a need to estimate how fast the system trajectory approaches the equilibrium state x. The equilibrium state x is exponentially stable if there exist two strictly positive numbers α and λ such that t > 0, x(t) α x(0) e λt in some ball B r around the origin. Definition Global Stability: If asymptotic (or exponential) stability holds for any initial states, the equilibrium point is said to be asymptotically (or exponentially) stable in the large. It is also called globally asymptotically (or exponentially) stable. Theorem 0.1 Lyapunov s Linearization method: If the linearized system is strictly stable (i.e, if all eigenvalues of A are strictly in the left-half complex plane), then the equilibrium point is asymptotically stable (for the actual nonlinear system). If the linearized system is unstable (i.e, if at least one eigenvalue of A is strictly in the right-half complex plane), then the equilibrium point is unstable (for the nonlinear system). If the linearized system is marginally stable (i.e, all eigenvalues of A are in the left-half complex plane, but at least one of them is on the jω axis), then one cannot conclude anything from the linear approximation (the equilibrium point may be stable, asymptotically stable, or unstable for the nonlinear system). Theorem 0. Lyapunov s theorem for stability: There exists a scalar function V (x) with continuous first partial derivatives such that in some ball B RO V (x) is lpdf V (x) 0 (nsdf) (along the system s trajectories) Then the equilibrium point x is stable. If, in addition, V (x) is ndf ( V (x) < 0) then x is asymptotically stable. Note that V (x) = dv dt = V dx x dt = V x ẋ = V x f(x) = n V i=1 f i (x) x i Note: If V (x) > 0, it does not necessarily mean that the system is unstable. It could have been a poor choice of V, the Lyapunov candidate. 1

2 Theorem 0.3 Lyapunov s theorem for global stability: There exists a scalar function V (x) with continuous first partial derivatives such that in some ball B RO V (x) is pdf V (x) < 0 (ndf) (along the system s trajectories) Then the equilibrium point x is globally stable. If, in addition, V (x) as x (radially unbounded) then x is globally asymptotically stable. 3 1 Curve 1 - asymptotically stable Curve - marginally stable Curve 3 - unstable Curve 4 - globally asymptotically stable 4 0 x(0) S r S R Figure 1: Concepts of Stability Definition Invariant Set: 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. Theorem 0.4 Local Invariant Set Theorem: Consider an autonomous system with f continuous and let V (x) be a scalar function with continuous first partial derivatives. Assume that: For some l > 0, the region Ω l = {x R n : V (x) < l} defined by V (x) < l is bounded V (x) 0 (nsdf) for all x in Ω l dv (x) Let R be the set of all points within Ω l so that R = {x Ω l : = 0} where dt V (x) = 0 and M be the largest invariant set in R. Then every solution x(t) originating in Ω l tends to M as t. Corollary 0.5 Local Invariant Set Theorem: Consider an autonomous system with f continuous and let V (x) be a scalar function with continuous first partial derivatives. Assume that in a certain neighbourhood Ω of the origin: V (x) is lpdf V (x) 0 (nsdf) for all x in Ω The set R be the set of all points within Ω so that R = {x Ω l : dv (x) dt = 0} contains no trajectory other than x 0 Then, the equilibrium point 0 is asymptotically stable. Furthermore, the largest connected region of the form Ω l (defined by V (x) < l) within Ω is a DOA of the equilibrium point.

3 Theorem 0.6 Global Invariant Set Theorem: Consider an autonomous system with f continuous and let V (x) be a scalar function with continuous first partial derivatives. Assume that: V (x) as x V (x) 0 (nsdf) over the whole state space (for all x) dv (x) Let R be the set of all points so that R = {x Ω l : = 0} where dt V (x) = 0 and M be the largest invariant set in R. Then all solutions globally asymptotically converge to M as t. If R = 0, then the system is globally asymptotically stable. Theorem 0.7 Lyapunov function for LTI system: Consider the LTI system ẋ = Ax. The following statements are equivalent: The system is asympotically stable All the eigenvalues of A have negative real parts For any symmetric pdf Q, the Lyapunov equation has a unique symmetric pdf solution for P A T P + P A + Q = 0 Theorem 0.8 : A necessary and sufficient condition for a LTI system ẋ = Ax to be strictly stable is that, for any symmetric p.d. matrix Q, the unique matrix P solution of the Lyapunov equation A T P +P A+Q = 0 be symmetric p.d. Theorem 0.9 Krasovskii: Consider a autonomous system with the equilibrium point of interest being the origin. Let A(x) denote the Jacobian matrix of the system, i.e. A(x) = f. If the matrix F = A + AT x is n.d. in a neighbourhood Ω, then the equilibrium point at the origin is asymptotically stable. A Lyapunov function for this system is V (x) = f T (x)f(x). If Ω is the entire state space and, in addition, V (x) as x, then the equilibrium point is globally asymptotically stable. Theorem 0.10 Generalized Krasovskii Theorem: Consider a autonomous system with the equilibrium point of interest being the origin. Let A(x) denote the Jacobian matrix of the system, i.e. A(x) = f x. Then, a sufficient condition for the origin to be asymptotically stable is that there exist two symmetric p.d. matrices P and Q, s.t. x 0, the matrix F (x) = A T P + P A + Q is n.s.d. in some neighbourhood Ω of the origin. The function V (x) = f T (x)p f(x) is then a Lyapunov function for this system. If Ω is the entire state space and, in addition, V (x) as x, then the system is globally asymptotically stable. Non-Autonomous Systems Definition Stability: The equilibrium point 0 is stable at t o if for any R > 0, there exists a positive scalar r(r, t o ) s.t. x(t o ) < r x(t) < R t t o Otherwise, the equilibrium point 0 is unstable. Definition Asymptotic Stability: The equilibrium point 0 is asymptotically stable at t o if It is stable r(t o ) > 0, s.t. x(t o ) < r(t o ) x(t) 0 as t Definition Exponential Stability: The equilibrium point 0 is exponentially stable if there exist two positive numbers, α and λ, s.t. for sufficiently small x(t o ), x(t) α x(t o ) e λ(t to) t t o Definition Globally Asympototically Stable: The equilibrium point 0 is globally asymptotically stable if x(t o ), x(t) 0 as t 3

4 Definition Uniform Stability: The equilibrium point 0 is locally uniformly stable if the scalar r can be chosen independently of t o, i.e. if r = r(r) Definition Asymptotical Stability: The equilibrium point at the origin is locally uniformly asymptotically stable if It is uniformly stable There exists a ball of attraction B Ro, whose radius is independent of t o, s.t. any system trajectory with initial states in B Ro converges to 0 uniformly in t o Definition Decrescent: A scalar continuous function V (x, t) is said to be decrescent if V (0, t) = 0 and if there exists a time-invariant p.d.f. V 1 (x) s.t. t 0, V (x, t) V 1 (x) In other words, a scalar function V (x, t) is decrescent if it is dominated by a time-invariant p.d.f. Theorem 0.11 Lyapunov theorem for non-autonomous systems: Stability: If in a ball B Ro around the equilibrium point 0, there exists a scalar function V (x, t) with continuous partial derivatives s.t. V is p.d. V is n.s.d. then the equilibrium point 0 is stable in the sense of Lyapunov. Theorem 0.1 Lyapunov theorem for non-autonomous systems: Uniform Stability: If, furthermore, V is decrescent then the origin is uniformly stable. Theorem 0.13 Lyapunov theorem for non-autonomous systems: Uniform Asymptotic Stability: If, furthermore, V is n.d. instead of being n.s.d. then the equilibrium point 0 is uniformly asymptotically stable. Theorem 0.14 Lyapunov theorem for non-autonomous systems: Global Uniform Asymptotic Stability: If the ball B Ro is replaced by the whole state space and all of the above conditions are satified along with, V (x, t) is radially unbounded are all satisfied, then the equilibrium point at 0 is globally uniformly asymptotically stable. Usually it is difficult to find the asymptotic stability of time-varying systems because it is very difficult to find Lyapunov functions with a nd derivative. We know in case of autonomous systems, if V is nsdf, then it is possible to know the asymptotic behaviors by invoking invariant-set theorems. However, the flexibility is not available for time-varying systems. This is where Babarlats lemma comes into picture. Lemma 0.15 Barbalat s Lemma: If the differentiable function f(t) has a finite limit as t, and if f is uniformly continuous, then f(t) 0 as t Application of Lemma is on Page 15 of [1]. Corollary 0.16 Corollary to Barbalat s Lemma: If the differentiable function f(t) has a finite limit as t, and is such that f exists and is bounded, then f(t) 0 as t Lemma 0.17 Lyapunov-Like Lemma: If a scalar function V (x, t) satisfies the following conditions V (x, t) is lower bounded V (x, t) is n.s.d. V (x, t) is uniformly continuous in time then, V (x, t) 0 as t 4

5 Lemma 0.18 Kalman-Yakubovich Lemma: Consider a controllable LTI system ẋ = Ax + bu y = c T x The transfer function h(p) = c T [pi A] 1 b is SPR if and only if there exist p.d. matrices P and Q s.t. A T P + P A + Q = 0 and P b = c In the KY Lemma, the involved system is required to be asymptotically stable and completely controllable. The KYM lemma relaxes the controllability condition. Lemma 0.19 Kalman-Yakubovich-Meyer Lemma: Given a scalar γ 0, vectors b and c, an asymptotically stable matrix A, and a symmetric p.d. matrix L, if the transfer function, H(p) = γ + ct [pi A] 1 b is SPR, then there exists a scalar ε > 0, a vector q, and a symmetric p.d. matrix P s.t. A T P + P A = qq T εl P b = c + γq The system is only required to be stabilizable (but not necessarily controllable) Illustrative Examples Consider a second order model as shown in Fig. with a constant positive spring constant and constant positive damping coefficient. Prove that this system is stable. Show that 0 is asymptotically stable. Can global exponential stablility be guaranteed? What about global asymptotical stability? Shown that the origin is an equilibrium point for this system. Figure : Mass spring damper time-invariant system The equation of motion of this system is given by mẍ + bẋ + kx = 0. This can be split up into two equations to represent it in a state-space form. Let x 1 = x and x = ẋ, therefore, ẋ 1 = x ẋ = k m x 1 b m x Choosing the pdf Lyapunov function V (x) to be equal to the total mechanical energy, we have, V (x) = 1 kx mx Taking the derivative of the Lyanpunov function, we get, V (x) = kx 1 ẋ 1 + mx ẋ = bx 0 Hence, we have a pdf Lyapunov function which has V (0) = 0. Furthermore, V (x) is nsdf which means 5

6 that this system is stable. We need to use Lasalle s principle to make sure that 0 is asymptotically stable. We can easily show that the origin is an equilibrium point of this system using IST. R = {x R : V (x) 0} = {x R : bx 0} = {x R : x 0} ẋ 1 = x 0, which implies x 1 is some constant 0 ẋ = k m x 1 b m x, which implies that x 1 = 0 Hence we have (x 1, x ) = (0, 0). Now using Lasalle s principle, we have 0 to be asymptotically stable since as t the trajectory tends towards the largest invariant set which is just 0 in this case. The function V is quadratic but not lpdf, since it does not depend on x 1, and hence we cannot conclude exponential stability. Instead, we can find the Jacobian matrix for the system and see if the poles fall on the LHP. The Jacobian matrix for this system is A = ( 0 1 k m b m ) which has the characteristic equation λ + b m λ + k m = 0 The solutions of the characteristic equation are λ = b ± b 4km m which always have negative real parts and hence the system is globally exponentially stable. One can further argue that since the system is globally exponentially stable, it should be globally asymptotically stable. Now, let us consider a second order non-autonomous model as shown in Fig. 3 with a time-varying spring constant k(t) = 1 + asin(t) where a < 1 and a constant damping coefficient b(t) = b with mass m = 1 kg. Under what conditions is this system uniformly asympotically stable. Figure 3: Mass spring damper time-varying system The equation of motion of this system is given by mẍ + bẋ + k(t)x = 0. This can be split up into two equations to represent it in a state-space form. Let x 1 = x and x = ẋ, therefore, 6

7 ẋ 1 = x ẋ = k(t)x 1 bx Choosing the pdf Lyapunov function V (x) to be equal to the total mechanical energy, we have, V (x, t) = 1 k(t)x x For simplification purposes, we take V (x, t) to be V (x, t) = 1 k(t) x 1 + x This function is decrescent because V (0, t) = 0 and there exists a time-invariant pdf V o (x) s.t. V o (x) V (x, t) so that, a (x 1 + x ) V (x, t) 1 1 a (x 1 + x ) Taking the derivative of the Lyapunov function, we get, V (x) = kx 1 ẋ 1 + x ẋ k(t) x k(t) k(t) = x 1 x + x k(t) ( k(t)x 1 bx ) x k(t) k(t) = (bk(t) + k(t)) x k(t) 0 Thus, uniform stability is guaranteed if, k(t) < bk(t) In order to prove uniform asymptotical stability, we need to show that V is nd and not just nsd. can use Barbalat s Lemma for this. We We know that v 0 and V (x, t) is lower bounded. This implies that V (x, t) approaches a limit as t Taking the second derivative of the Lyapunov function, we have, where ẋ = k(t)x 1 bx V (x) = (bk(t) + k(t)) x ẋ k(t) (bk(t) + k(t)) x k(t) k(t) 3 (b k(t) + k(t)) x k(t) d V dt is radially bounded for all t Therefore, dv dt 0 = x (t) 0 ẋ (t) 0 if ẍ is bounded or ẋ (t) is uniformly continuous. Thus, by Barbalat s Lemma, lim x = 0 t ẋ 1 = x ẋ = k(t)x 1 bx ẍ = k(t)x 1 k(t)ẋ 1 bẋ = k(t)x 1 k(t)x + bk(t)x 1 + b x From this we get x 1 (t) 0. Thus, (x 1 (t), x (t)) (0, 0) as t The system is uniformly asymptotically stable. 7

8 For a more detailed explanation and for good problems please refer to [1]. An excellent primer is available online on Caltech s website []. References [1] J.-J. E. Slotine and W. Li, Applied Nonlinear Control. Prentice Hall, [] R. M. Murray, Z. Li, and S. S. Sastry. A mathematical introduction to robotic manipulation. [Online]. Available: murray/courses/primer-f01/mls-lyap.pdf 8