# LMI Methods in Optimal and Robust Control

Save this PDF as:
Size: px
Start display at page:

## Transcription

1 LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 15: Nonlinear Systems and Lyapunov Functions

2 Overview Our next goal is to extend LMI s and optimization to nonlinear systems analysis. Today we will discuss 1. Nonlinear Systems Theory 1.1 Existence and Uniqueness 1.2 Contractions and Iterations 1.3 Gronwall-Bellman Inequality 2. Stability Theory 2.1 Lyapunov Stability 2.2 Lyapunov s Direct Method 2.3 A Collection of Converse Lyapunov Results The purpose of this lecture is to show that Lyapunov stability can be solved Exactly via optimization of polynomials. M. Peet Lecture 15: 1 / 44

3 Ordinary Nonlinear Differential Equations Computing Stability and Domain of Attraction Consider: A System of Nonlinear Ordinary Differential Equations ẋ(t) = f(x(t)) Problem: Stability Given a specific polynomial f : R n R n, find the largest X R n such that for any x(0) X, lim t x(t) = 0. M. Peet Lecture 15: 2 / 44

4 Nonlinear Dynamical Systems Long-Range Weather Forecasting and the Lorentz Attractor A model of atmospheric convection analyzed by E.N. Lorenz, Journal of Atmospheric Sciences, ẋ = σ(y x) ẏ = rx y xz ż = xy bz M. Peet Lecture 15: 3 / 44

5 Stability and Periodic Orbits The Poincaré-Bendixson Theorem and van der Pol Oscillator 3 An oscillating circuit model: ẏ = x (x 2 1)y ẋ = y y Domain of attraction x Figure: The van der Pol oscillator in reverse Theorem 1 (Poincaré-Bendixson). Invariant sets in R 2 always contain a limit cycle or fixed point. M. Peet Lecture 15: 4 / 44

6 Stability of Ordinary Differential Equations Consider ẋ(t) = f(x(t)) with x(0) R n. Theorem 2 (Lyapunov Stability). Suppose there exists a continuous V and α, β, γ > 0 where β x 2 V (x) α x 2 V (x) T f(x) γ x 2 for all x X. Then any sub-level set of V in X is a Domain of Attraction. A Sublevel Set: Has the form V δ = {x : V (x) δ}. M. Peet Lecture 15: 5 / 44

7 Do The Equations Have a Solution? The Cauchy Problem The first question people ask is the Cauchy problem: For Autonomous (Uncontrolled) Systems: Definition 3 (Cauchy Problem). The Cauchy problem is to find a unique, continuous x : [0, t f ] R n for some t f such that ẋ is defined and ẋ(t) = f(t, x(t)) for all t [0, t f ]. If f is continuous, the solution must be continuously differentiable. Controlled Systems: For a controlled system, we have ẋ(t) = f(x(t), u(t)) and assume u(t) is given. This precludes feedback In this lecture, we focus on the autonomous system. Including t complicates the analysis. However, results are almost all the same. M. Peet Lecture 15: 6 / 44

8 Ordinary Differential Equations Existence of Solutions There exist many systems for which no solution exists or for which a solution lems onlyof exists solutions over a finite time interval. Finite Escape Time exponential stability quadratic stability time-varying systems invariant sets center manifold theorem Even for something as simple as ation ẋ(t) = x(t) 2 x(0) = x 0 x(0)= x 0 has the solution t, 0 t< 1 x 0 x(t) = x 0 1 x 0 t which clearly has escape time = 1 x 0 x(t) Finite escape time of dx/dt = x 2 t e = 1 x Time t ssproblems Figure: Simulation of ẋ = x 2 for several x(0) Previous problem is like the water-tank problem in backwar time M. Peet Lecture 15: 7 / 44

9 Ordinary Differential Equations Non-Uniqueness A classical example of a system without a unique solution is ẋ(t) = x(t) 1/3 x(0) = 0 For the given initial condition, it is easy to verify that ( 2t x(t) = 0 and x(t) = 3 both satisfy the differential equation. ) 3/ Figure: Matlab simulation of ẋ(t) = x(t) 1/3 with x(0) = 0 Figure: Matlab simulation of ẋ(t) = x(t) 1/3 with x(0) = M. Peet Lecture 15: 8 / 44

10 Ordinary Differential Equations Non-Uniqueness Uniqueness Problems Example: The equation ẋ= x, x(0)=0has man An Example of a system with several solutions is given by ẋ(t) = { (t C) x(t) x(t) x(0) = 2 /4 t> C = 0 0 t C For the given initial condition, it is easy to verify that for any C, { (t C) 2 x(t) = 4 t > C 0 t C satisfies the differential equation. x(t) Time t Compare with water tank: Figure: Several solutions of ẋ = x M. Peet Lecture 15: 9 / 44

11 Continuity of a Function Customary Notions of Continuity Nonlinear Stability requires some additional Math Definitions. Definition 4 (Continuity at a Point). For normed spaces X, Y, a function f : X Y is continuous at the point x 0 X if for any ɛ > 0, there exists a δ > 0 such that x x 0 < δ (U) implies f(x) f(x 0 ) < ɛ (V ). M. Peet Lecture 15: 10 / 44

12 Ordinary Differential Equations Customary Notions of Continuity Definition 5 (Continuity on a Set of Points (B)). For normed spaces X, Y, a function f : A X Y is continuous on B if it is continuous at any point x 0 B. A function is simply continuous if B = A. Dropping some of the notation, Definition 6 (Uniform Continuity on a Set of Points (B)). f : A X Y is uniformly continuous on B if for any ɛ > 0, there exists a δ > 0 such that for x, y B, x y < δ implies f(x) f(y) < ɛ. A function is simply uniformly continuous if B = A. Example: f(x) = x 3 is uniformly continuous on B = [0, 1], but not B = R f (x) = 3x 2 < 3 for x [0, 1] hence f(x) f(y) 3 x y. So given δ, choose ɛ < 1 3 δ. M. Peet Lecture 15: 11 / 44

13 Lipschitz Continuity A Quantitative Notion of Continuity Definition 7 (Lipschitz Continuity). The function f is Lipschitz continuous on X if there exists some L > 0 such that f(x) f(y) L x y for any x, y X. The constant L is referred to as the Lipschitz constant for f on X. Definition 8 (Local Lipschitz Continuity). The function f is Locally Lipschitz continuous on X if for every x X, there exists a neighborhood, B of x such that f is Lipschitz continuous on B. Definition 9. The function f is Globally Lipschitz if it is Lipschitz on its entire domain. Example: f(x) = x 3 is Locally Lipschitz on [ 1, 1] with L = 3. But f(x) = x 3 is NOT Globally Lipschitz on R L is typically just a bound on the derivative. M. Peet Lecture 15: 12 / 44

14 A Theorem on Existence of Solutions Existence and Uniqueness Let B(x 0, r) be the unit ball, centered at x 0 of radius r. Theorem 10 (A Typical Existence Theorem). Suppose x 0 R n, f : R n R n and there exist L, r such that for any x, y B(x 0, r), f(x) f(y) L x y and f(x) c for x B(x 0, r). Let t f < min{ 1 L, r c }. Then there exists a unique differentiable x : [0, t f ] R n, such that x(0) = x 0, x(t) B(x 0, r) and ẋ(t) = f(x(t)). M. Peet Lecture 15: 13 / 44

15 Examples on Existence of Solutions Theorem 11 (A Typical Existence Theorem). Suppose x 0 R n, f : R n R n and there exist L, r such that peaking for any x, y B(x 0, r), f(x) f(y) L x y and f(x) c for x B(x 0, r). Let t f < min{ 1 L, r c }. Then quadratic there stability time-varying systems exists a invariant sets unique differentiable solution on interval [0, t f ]. center manifold theorem Recall: has the solution Nonlinear Control Theory 2006 Existence problems of solutions ẋ(t) Example: = x(t) The differential 2 equation x(0) = x 0 has the solution dx dt = x2, x(0)= x0 x(t) = x 0 x(t)= x0 1 x 0 t 1 x0t, 0 t< 1 x0 Lets take r = 1, xfinite 0 = escape 1. time Then L = sup x [0,2] f (x) = 4. t f= c = sup x [0,2] f(x) = 4. Then we 1 x0have a solution for t f < min{ 1 L, r c } = min{ 1 4, 1 4 } = 1 4 and where x(t) < 2 for t [0, t f ]. We can verify that the solution Uniqueness x(t) Problems = 1 1 t < 4 3 for t < t f. Lecture 1++, 2006 Nonlinear Phenomena and Stability theory Nonlinear phenomena [Khalil Ch 3.1] existence and uniqueness finite escape time Linear system theory revisited Second order systems [Khalil Ch 2.4, 2.6] periodic solutions / limit cycles Stability theory [Khalil Ch. 4] Lyapunov Theory revisited exponential stability x(t) Finite Escape Time Finite escape time of dx/dt = x Time t Previous problem is like the water-tank problem in backw time M. Peet Example: The equation ẋ= x, x(0)=0has many solutions: Lecture 15: (Substitute τ = t in differential equation). 14 / 44

16 Examples on Existence of Solutions Non-Uniqueness Theorem 12 (A Typical Existence Theorem). Suppose x 0 R n, f : R n R n and there exist L, r such that for any x, y B(x 0, r), f(x) f(y) L x y and f(x) c for x B(x 0, r). Let t f < min{ 1 L, r c }. Then there exists a unique differentiable solution on interval [0, t f ]. Recall the system without a unique solution is ẋ(t) = x(t) 1/3 x(0) = 0 The problem here is that f (x) = 1 3 x 2 3 = 1 L = 3x 2 3 sup f (x) = sup x [0,2] x [0,2] Since 1 0 =. So there is no Lipschitz Bound.. 1 3x 2 3 = M. Peet Lecture 15: 15 / 44

17 Typical Existence of Solutions Proof Approach Step 1: Contraction Mapping Theorem Theorem 13 (Contraction Mapping Principle). Let (X, ) be a complete normed space and let P : X X. Suppose there exists a ρ < 1 such that P x P y ρ x y for all x, y X. Then there is a unique x X such that P x = x. Furthermore for y X, define the sequence {x i } as x 1 = y and x i = P x i 1 for i > 2. Then lim i x i = x. Some Observations: Proof of CM: Show that P k y is a Cauchy sequence for any y X. For existence of solutions, X is the space of solutions. The contraction is and x := sup t [0,tf ] x(t). (P x)(t) = x 0 + t 0 f(x(s))ds, M. Peet Lecture 15: 16 / 44

18 Ordinary Differential Equations Contraction Mapping Theorem Key Point: P x = x if and only if x is a solution! (P x)(t) = x 0 + t 0 f(x(s))ds, Theorem 14 (Fundamental Theorem of Calculus). Suppose x and f : R n R n are continuous. Then the following are equivalent. 1. x is differentiable at any t [0, t f ] and ẋ(t) = f(x(t)) at all t [0, t f ] x(0) = x 0 2. x(t) = x 0 + t 0 f(x(s))ds for all t [0, t f ] M. Peet Lecture 15: 17 / 44

19 Ordinary Differential Equations Existence and Uniqueness First recall what we are trying to prove: Theorem 15 (A Typical Existence Theorem). Suppose x 0 R n, f : R n R n and there exist L, r such that for any x, y B(x 0, r), f(x) f(y) L x y and f(x) c for x B(x 0, r). Let t f < min{ 1 L, r c }. Then there exists a unique differentiable solution on interval [0, t f ]. We will show that (P x)(t) = x 0 + t 0 f(x(s))ds is a contraction if t f is sufficiently small. Then P x = x has a solution! M. Peet Lecture 15: 18 / 44

20 Proof of Existence Theorem Proof. For given x 0, define the complete normed space B = {x C[0, t f ] : x(0) = x 0, x(t) B(x 0, r)} with norm sup t [0,tf ] x(t) (Proof of Completeness omitted). Recall we define P as P x(t) = x 0 + t 0 f(x(s))ds Step 1: We first show that if x B, then P x B. Suppose x B. To see that P x is continuous in t, we note that P x(t 2 ) P x(t 1 ) = t2 t 1 f(x(s))ds t2 t 1 f(x(s)) ds c(t 2 t 1 ) Which is the definition of uniform continuity. Clearly P x(0) = x 0. Finally, we need to show (P x)(t) B(x 0, r). Since f(x) c when x B(x 0, r), we have sup P x(t) x 0 = sup t [0,t f ] t [0,t f ] t 0 f(x(s))ds tf where the final inequality is from the constraint t f < min{ 1 L, r c }. 0 f(x(s)) ds t f c < r M. Peet Lecture 15: 19 / 44

21 Proof of Existence Theorem Proof. We have shown that P : B B. Step 2: Now show that P is a contraction. For any two x, y B, we have P x P y = sup t [0,t f ] tf L 0 sup t [0,t f ] t ( 0 t 0 f(x(s)) f(y(s))ds f(x(s)) f(y(s)) ds) = tf x(s) y(s) ds Lt f x y < x y 0 f(x(s)) f(y(s)) ds Where the last inequality comes from the constraint t f < min{ 1 L, r c }. Thus P is a contraction which means there exists a solution x B such that ẋ = f(x(t)) M. Peet Lecture 15: 20 / 44

22 Picard Iteration Constructing the Solution This proof provides a way of actually constructing the solution. Picard-Lindelöf Iteration: From the proof, unique solution of P x = x is a solution of ẋ = f(x ), where t (P x)(t) = x 0 + f(x(s))ds From the contraction mapping theorem, the solution P x = x can be found as x = lim P k z for ANY z B k Note that this existence theorem only guarantees existence on the interval [ t 0, 1 ] [ or t 0, r ] L c Where L is a Lipschitz constant for f in the ball of radius r centered at x 0 c is a bound for f in the ball of radius r centered at x 0. Q: What does a Lyapunov function prove if there is no guaranteed solution? A: Nothing. (Particularly problematic for PDEs) 0 M. Peet Lecture 15: 21 / 44

23 Illustration of Picard Iteration This is a plot of Picard iterations for the solution map of ẋ = x 3. z(t, x) = 0; P z(t, x) = x; P 2 z(t, x) = x tx 3 ; P 3 z(t, x) = x tx 3 + 3t 2 x 5 3t 3 x 7 + t 4 x Figure: The solution for x 0 = 1 Convergence is only guaranteed on interval t [0,.333]. M. Peet Lecture 15: 22 / 44

24 Extension of Existence Theorem For Lyapunov Functions to work, we need to extend the solution Indefinitely! Step 1: Prove that solutions exist for a fixed r (yielding L, c, and t f ). Step 2: Use a Lyapunov function to show that there exists a δ < r, such that any solution which begins in B(0, δ) will stay in B(0, r). Step 3: Since x(0) < δ < r, a solution exists on [0, t f ]. Further, x(t f ) r. Step 4: Since x(t f ) < r, a solution exists on [0, 2t f ]. Further, Step 2 proves x(2t f ) r. Repeat Step 4 Indefinitely For a given set W, define W r := {x : x y r, y W }. Theorem 16 (Formal Solution Extension Theorem). Suppose that there exists a domain D and L > 0 such that f(x) f(y) L x y for all x, y D R n and t > 0. Suppose there exists a compact set W and r > 0 such that W r D. Furthermore suppose that it has been proven that for x 0 W, any solution to ẋ(t) = f(x), x(0) = x 0 must lie entirely in W. Then, for x 0 W, there exists a unique solution x with x(0) = x 0 such that x lies entirely in W. M. Peet Lecture 15: 23 / 44

25 Illustration of Extended Picard Iteration We take the previous approximation to the solution map and extend it. x t t Figure: The Solution map φ and the functions G k i for k = 1, 2, 3, 4, 5 and i = 1, 2, 3 for the system ẋ(t) = x(t) 3. The interval of convergence of the Picard Iteration is T = 1 3. M. Peet Lecture 15: 24 / 44

26 Stability Definitions Whenever you are trying to prove stability, Please define your notion of stability! Denote the set of bounded continuous functions by C := {x C : x(t) r, r 0} with norm x = sup t x(t). We define g : D C to be the solution map: g(x 0, t) if t g(x 0, t) = f(g(x 0, t)) and g(x 0, 0) = x 0 x 0 D Definition 17. The system is locally Lyapunov stable on D where D contains an open neighborhood of the origin if it defines a unique map g : D C which is continuous at the origin (x 0 = 0). The system is locally Lyapunov stable on D if for any ɛ > 0, there exists a δ(ɛ) such that for x(0) δ(ɛ), x(0) D we have x(t) ɛ for all t 0 M. Peet Lecture 15: 25 / 44

27 Stability Definitions Definition 18. The system is globally Lyapunov stable if it defines a unique map g : R n C which is continuous at the origin. We define the subspace of bounded continuous functions which tend to the origin by G := {x C : lim t x(t) = 0} with norm x = sup t x(t). Definition 19. The system is locally asymptotically stable on D where D contains an open neighborhood of the origin if it defines a map g : D G which is continuous at the origin. M. Peet Lecture 15: 26 / 44

28 Stability Definitions Definition 20. The system is globally asymptotically stable if it defines a map g : R n G which is continuous at the origin. Definition 21. The system is locally exponentially stable on D if it defines a map g : D G where g(x, t) Ke γt x for some positive constants K, γ > 0 and any x D. Definition 22. The system is globally exponentially stable if it defines a map g : R n G where g(x, t) Ke γt x for some positive constants K, γ > 0 and any x R n. M. Peet Lecture 15: 27 / 44

29 Lyapunov Theorem Consider the system: ẋ = f(x), f(0) = 0 Theorem 23. Let V : D R be a continuously differentiable function and D compact such that V (0) = 0 V (x) > 0 for x D, x 0 V (x) T f(x) 0 for x D. Then ẋ = f(x) is well-posed and locally Lyapunov stable on the largest sublevel set V γ = {x : V (x) γ} of V contained in D. Furthermore, if V (x) T f(x) < 0 for x D, x 0, then ẋ = f(x) is locally asymptotically stable on the largest sublevel set V γ = {x : V (x) γ} contained in D. M. Peet Lecture 15: 28 / 44

30 Lyapunov Theorem Sublevel Set: For a given Lyapunov function V and positive constant γ, we denote the set V γ = {x : V (x) γ}. Proof. Existence: Denote the largest bounded sublevel set of V contained in the interior of D by V γ. Because V (x(t)) = V (x(t)) T f(x(t)) 0, if x(0) V γ, then x(t) V γ for all t 0. Therefore since f is locally Lipschitz continuous on the compact V γ, by the extension theorem, there is a unique solution for any initial condition x(0) V γ. Lyapunov Stability: Given any ɛ > 0, choose ɛ < ɛ with B(ɛ) V γ, choose γ i such that V γi B(ɛ). Now, choose δ > 0 such that B(δ) V γi. Then B(δ) V γi B(ɛ) and hence if x(0) B(δ), we have x(0) V γi B(ɛ) B(ɛ ). Asymptotic Stability: V monotone decreasing implies lim t V (x(t)) = 0. V (x) = 0 implies x = 0. Proof omitted. M. Peet Lecture 15: 29 / 44

31 Lyapunov Theorem Exponential Stability Theorem 24. Suppose there exists a continuously differentiable function V and constants c 1, c 2, c 3 > and radius r > 0 such that the following holds for all x B(r). c 1 x p V (x) c 2 x p V (x) T f(x) c 3 x p Then ẋ = f(x) is exponentially stable on any ball contained in the largest sublevel set contained in B(r). Exponential Stability allows a quantitative prediction of system behavior. M. Peet Lecture 15: 30 / 44

32 The Gronwall-Bellman Inequality Exponential Stability Lemma 25 (Gronwall-Bellman). Let λ be continuous and µ be continuous and nonnegative. Let y be continuous and satisfy for t b, Then y(t) λ(t) + If λ and µ are constants, then y(t) λ(t) + t a t a µ(s)y(s)ds. [ t ] λ(s)µ(s) exp µ(τ)dτ ds s y(t) λe µt. For λ(t) = y 0, the condition is equivalent to ẏ(t) µ(t)y(t), y(0) = y(t). M. Peet Lecture 15: 31 / 44

33 Lyapunov Theorem Exponential Stability Proof. We begin by noting that we already satisfy the conditions for existence, uniqueness and asymptotic stability and that x(t) B(r). Now, observe that V (x(t)) c 3 x(t) 2 c 3 c 2 V (x(t)) Which implies by the Gronwall-Bellman inequality (µ = c3 c 2, λ = V (x(0))) that V (x(t)) V (x(0))e c 3 t c2. Hence x(t) 2 1 V (x(t)) 1 e c 3 t c2 V (x(0)) c 2 e c t 2 x(0) 2. c 1 c 1 c 1 c 3 M. Peet Lecture 15: 32 / 44

34 Lyapunov Theorem Invariance Sometimes, we want to prove convergence to a set. Recall Definition 26. V γ = {x, V (x) γ} A set, X, is Positively Invariant if x(0) X implies x(t) X for all t 0. Theorem 27. Suppose that there exists some continuously differentiable function V such that V (x) > 0 for x D, x 0 V (x) T f(x) 0 for x D. for all x D. Then for any γ such that the level set X = {x : V (x) = γ} D, we have that V γ is positively invariant. Furthermore, if V (x) T f(x) 0 for x D, then for any δ such that X V δ D, we have that any trajectory starting in V δ will approach the sublevel set V γ. M. Peet Lecture 15: 33 / 44

35 Problem Statement 1: Global Lypunov Stability Given: Vector field, f(x) Find: function V, scalars α, β such that i α i =.01, i β i =.01 and V (x) i α i (x T x) i for all x V (x) i β i (x T x) i for all x V (x) T f(x) 0 for all x Conclusion: Lyapunov stability for any x(0) R n. Can replace V (x) i β i(x T x) i with V (0) = 0 if it is well-behaved. M. Peet Lecture 15: 34 / 44

36 Examples of Lyapunov Functions Mass-Spring: Pendulum: ẍ = c mẋ k m x ẋ 2 = g l sin x 1 ẋ 1 = x 2 V (x) = 1 2 mẋ kx2 V (x) = (1 cos x 1 )gl l2 x 2 2 V (x) = ẋ( cẋ kx) + kxẋ = cẋ 2 kẋx + kxẋ V (x) = glx 2 sin x 1 glx 2 sin x 1 = 0 = cẋ 2 0 M. Peet Lecture 15: 35 / 44

37 Problem Statement 2: Global Exponential Stability Given: Vector field, f(x), exponent, p Find: function V, scalars α, β, δ such that i α i =.01, i β i =.01 and V (x) i V (x) i α i x 2p i β i x 2p i for all x for all x V (x) T f(x) δv (x) for all x Conclusion: Exponential stability for any x(0) R n. Convergence Rate: x(t) 2p βmax α min x(0) e δ 2p t M. Peet Lecture 15: 36 / 44

38 Problem Statement 2: Global Exponential Stability Example Consider: Attitude Dynamics of a rotating Spacecraft: What about: V (x) T f(x) = ω 1 ω 2 ω 3 J 1 ω 1 = (J 2 J 3 )ω 2 ω 3 J 2 ω 2 = (J 3 J 1 )ω 3 ω 1 J 3 ω 3 = (J 1 J 2 )ω 1 ω 2 V (x) = ω ω ω 2 3? T J 2 J 3 J 1 ω 2 ω 3 J 3 J 1 J 2 ω 3 ω 1 J 1 J 2 J 3 ω 1 ω 2 OK, maybe not. Try u i = k i ω i. ( J2 J 3 = + J 3 J 1 + J ) 1 J 2 ω 1 ω 2 ω 3 J 1 J 2 J 3 ( J 2 = 2 J 3 J3 2 J 2 + J3 2 J 1 J1 2 J 3 + J 2 J1 2 J2 2 J 1 J 1 J 2 J 3 ) ω 1 ω 2 ω 3 M. Peet Lecture 15: 37 / 44

39 Problem Statement 3: Local Exponential Stability Given: Vector field, f(x), exponent, p Ball of radius r, Find: function V, scalars α, β, δ such that i α i =.01, i β i =.01 and V (x) i V (x) i α i x 2p i for all x : x T x r 2 β i x 2p i for all x : x T x r 2 V (x) T f(x) δv (x) for all x : x T x r 2 Conclusion: A Domain of Attraction! of the origin Exponential stability for x(0) {x : V (x) γ} if V γ B r. Sub-Problem: Given, V, r, max γ γ such that V (x) γ for all x {x T x r} M. Peet Lecture 15: 38 / 44

40 Domain of Attraction The van der Pol Oscillator An oscillating circuit: (in reverse time) ẋ = y Choose: Derivative V (x) T f(x) = ẏ = x + (x 2 1)y V (x) = x 2 + y 2, r = 1 = xy + xy + (x 2 1)y 2 0 for x 2 1 [ ] T [ ] 2x y 2y x (x 2 1)y Level Set: V γ=1 = {(x, y) : x 2 + y 2 1} = B 1 y Domain of attraction x Figure: The van der Pol oscillator in reverse So B 1 = V γ=1 is a Domain of Attraction! M. Peet Lecture 15: 39 / 44

41 Problem Statement 4: Invariant Regions/Manifolds Given: Vector field, f(x), exponent, p Ball of radius r, Find: function V, scalars α, β, δ such that i α i =.01, i β i =.01 and V (x) i V (x) i α i x 2p i for all x : x T x r 2 β i x 2p i for all x : x T x r 2 V (x) T f(x) δv (x) for all x : x T x r 2 Conclusion: There exist a T such that x(t) {x : V (x) γ} for all t T if B r V γ. Sub-Problem: Given, V, r, min γ γ such that x T x r 2 for all x {V (x) γ} M. Peet Lecture 15: 40 / 44

42 Nonlinear Dynamical Systems Long-Range Weather Forecasting and the Lorentz Attractor A model of atmospheric convection analyzed by E.N. Lorenz, Journal of Atmospheric Sciences, ẋ = σ(y x) ẏ = rx y xz ż = xy bz M. Peet Lecture 15: 41 / 44

43 Problem Statement 5: Controller Synthesis (Local) Suppose ẋ(t) = f(x) + g(x)u(t) Given: Vector fields, f(x), g(x), exponent, p u(t) = k(x) Find: function functions, k, V, scalars α 0, β 0 such that i α i =.01, i β i =.01 and V (x) i V (x) i α i x 2p i for all x : x T x r 2 β i x 2p i for all x : x T x r 2 V (x) T f(x) + V (x) T g(x)k(x) 0 for all x : x T x r 2 Conclusion: Controller u(u) = k(x(t)) stabilizes the system for x(0) {x : V (x) γ} if V γ B r. M. Peet Lecture 15: 42 / 44

44 Problem Statement 6: Output Feedback Controller Synthesis (Global Exponential ) Suppose ẋ(t) = f(x) + g(x)u(t) y(t) = h(x(t)) Given: Vector fields, f(x), g(x), h(x) exponent, p u(t) = k(y(t)) Find: function functions, k, V, scalars α 0, β 0, δ > 0 such that i α i =.01, i β i =.01 and V (x) α i x 2p i for all x i V (x) i β i x 2p i for all x V (x) T f(x) + V (x) T g(x)k(h(x)) δv (x) for all x Conclusion: Controller u(t) = k(y(t)) exponentially stabilizes the system for any x(0) R n. M. Peet Lecture 15: 43 / 44

45 How to Solve these Problems? General Framework for solving these problems Convex Optimization of Functions: Variables V C[R n ] and γ R max V,γ γ subject to V (x) x T x 0 x X V (x) T f(x) + γx T x 0 x X Going Forward Assume all functions are polynomials or rationals. Assume X := {x : g i (x) 0} (Semialgebraic) M. Peet Lecture 15: 44 / 44

### Nonlinear Systems Theory

Nonlinear Systems Theory Matthew M. Peet Arizona State University Lecture 2: Nonlinear Systems Theory Overview Our next goal is to extend LMI s and optimization to nonlinear systems analysis. Today we

### Half of Final Exam Name: Practice Problems October 28, 2014

Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half

### LMI Methods in Optimal and Robust Control

LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 4: LMIs for State-Space Internal Stability Solving the Equations Find the output given the input State-Space:

### Nonlinear Control Systems

Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 3. Fundamental properties IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Example Consider the system ẋ = f

### EE222 - Spring 16 - Lecture 2 Notes 1

EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued

### System Control Engineering 0

System Control Engineering 0 Koichi Hashimoto Graduate School of Information Sciences Text: Nonlinear Control Systems Analysis and Design, Wiley Author: Horacio J. Marquez Web: http://www.ic.is.tohoku.ac.jp/~koichi/system_control/

### Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1

Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium

### Converse Lyapunov theorem and Input-to-State Stability

Converse Lyapunov theorem and Input-to-State Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts

### Lyapunov Stability Theory

Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous

### ẋ = f(x, y), ẏ = g(x, y), (x, y) D, can only have periodic solutions if (f,g) changes sign in D or if (f,g)=0in D.

4 Periodic Solutions We have shown that in the case of an autonomous equation the periodic solutions correspond with closed orbits in phase-space. Autonomous two-dimensional systems with phase-space R

### Prof. Krstic Nonlinear Systems MAE281A Homework set 1 Linearization & phase portrait

Prof. Krstic Nonlinear Systems MAE28A Homework set Linearization & phase portrait. For each of the following systems, find all equilibrium points and determine the type of each isolated equilibrium. Use

### 3 Stability and Lyapunov Functions

CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such

### 1 Lyapunov theory of stability

M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability

### Optimization of Polynomials

Optimization of Polynomials Matthew M. Peet Arizona State University Thanks to S. Lall and P. Parrilo for guidance and supporting material Lecture 03: Optimization of Polynomials Overview In this lecture,

### Modern Optimal Control

Modern Optimal Control Matthew M. Peet Arizona State University Lecture 19: Stabilization via LMIs Optimization Optimization can be posed in functional form: min x F objective function : inequality constraints

### Existence and uniqueness of solutions for nonlinear ODEs

Chapter 4 Existence and uniqueness of solutions for nonlinear ODEs In this chapter we consider the existence and uniqueness of solutions for the initial value problem for general nonlinear ODEs. Recall

### Topic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis

Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of

### Nonlinear Control. Nonlinear Control Lecture # 2 Stability of Equilibrium Points

Nonlinear Control Lecture # 2 Stability of Equilibrium Points Basic Concepts ẋ = f(x) f is locally Lipschitz over a domain D R n Suppose x D is an equilibrium point; that is, f( x) = 0 Characterize and

### Lecture 4. Chapter 4: Lyapunov Stability. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.

Lecture 4 Chapter 4: Lyapunov Stability Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 4 p. 1/86 Autonomous Systems Consider the autonomous system ẋ

### Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems

p. 1/5 Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems p. 2/5 Time-varying Systems ẋ = f(t, x) f(t, x) is piecewise continuous in t and locally Lipschitz in x for all t

### Dynamical Systems as Solutions of Ordinary Differential Equations

CHAPTER 3 Dynamical Systems as Solutions of Ordinary Differential Equations Chapter 1 defined a dynamical system as a type of mathematical system, S =(X, G, U, ), where X is a normed linear space, G is

### An introduction to Mathematical Theory of Control

An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018

### THE INVERSE FUNCTION THEOREM

THE INVERSE FUNCTION THEOREM W. PATRICK HOOPER The implicit function theorem is the following result: Theorem 1. Let f be a C 1 function from a neighborhood of a point a R n into R n. Suppose A = Df(a)

### Polynomial level-set methods for nonlinear dynamical systems analysis

Proceedings of the Allerton Conference on Communication, Control and Computing pages 64 649, 8-3 September 5. 5.7..4 Polynomial level-set methods for nonlinear dynamical systems analysis Ta-Chung Wang,4

### ODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class

ODE Homework Due Wed. 9 August 2009; At the beginning of the class. (a) Solve Lẏ + Ry = E sin(ωt) with y(0) = k () L, R, E, ω are positive constants. (b) What is the limit of the solution as ω 0? (c) Is

### 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

### Passivity-based Stabilization of Non-Compact Sets

Passivity-based Stabilization of Non-Compact Sets Mohamed I. El-Hawwary and Manfredi Maggiore Abstract We investigate the stabilization of closed sets for passive nonlinear systems which are contained

### Nonlinear equations. Norms for R n. Convergence orders for iterative methods

Nonlinear equations Norms for R n Assume that X is a vector space. A norm is a mapping X R with x such that for all x, y X, α R x = = x = αx = α x x + y x + y We define the following norms on the vector

### Handout 2: Invariant Sets and Stability

Engineering Tripos Part IIB Nonlinear Systems and Control Module 4F2 1 Invariant Sets Handout 2: Invariant Sets and Stability Consider again the autonomous dynamical system ẋ = f(x), x() = x (1) with state

### Main topics and some repetition exercises for the course MMG511/MVE161 ODE and mathematical modeling in year Main topics in the course:

Main topics and some repetition exercises for the course MMG5/MVE6 ODE and mathematical modeling in year 04. Main topics in the course:. Banach fixed point principle. Picard- Lindelöf theorem. Lipschitz

### Problem List MATH 5173 Spring, 2014

Problem List MATH 5173 Spring, 2014 The notation p/n means the problem with number n on page p of Perko. 1. 5/3 [Due Wednesday, January 15] 2. 6/5 and describe the relationship of the phase portraits [Due

### MCE693/793: Analysis and Control of Nonlinear Systems

MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear

### Nonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems

Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Time-varying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin

### CDS 101/110a: Lecture 2.1 Dynamic Behavior

CDS 11/11a: Lecture 2.1 Dynamic Behavior Richard M. Murray 6 October 28 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium

### d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N

Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x

### 1 The Observability Canonical Form

NONLINEAR OBSERVERS AND SEPARATION PRINCIPLE 1 The Observability Canonical Form In this Chapter we discuss the design of observers for nonlinear systems modelled by equations of the form ẋ = f(x, u) (1)

### Applied Dynamical Systems

Applied Dynamical Systems Recommended Reading: (1) Morris W. Hirsch, Stephen Smale, and Robert L. Devaney. Differential equations, dynamical systems, and an introduction to chaos. Elsevier/Academic Press,

### SOLVING DIFFERENTIAL EQUATIONS

SOLVING DIFFERENTIAL EQUATIONS ARICK SHAO 1. Solving Ordinary Differential Equations Consider first basic algebraic equations with an unknown, e.g., 3(x + 2) = 18, 3x 2 = cos x. Such equations model some

### Solution of Additional Exercises for Chapter 4

1 1. (1) Try V (x) = 1 (x 1 + x ). Solution of Additional Exercises for Chapter 4 V (x) = x 1 ( x 1 + x ) x = x 1 x + x 1 x In the neighborhood of the origin, the term (x 1 + x ) dominates. Hence, the

### CDS 101/110a: Lecture 2.1 Dynamic Behavior

CDS 11/11a: Lecture.1 Dynamic Behavior Richard M. Murray 6 October 8 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium

### Complex Dynamic Systems: Qualitative vs Quantitative analysis

Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems Chiara Mocenni Department of Information Engineering and Mathematics University of Siena (mocenni@diism.unisi.it) Dynamic

### Existence and Uniqueness

Chapter 3 Existence and Uniqueness An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect

### 2.1 Dynamical systems, phase flows, and differential equations

Chapter 2 Fundamental theorems 2.1 Dynamical systems, phase flows, and differential equations A dynamical system is a mathematical formalization of an evolutionary deterministic process. An evolutionary

### EN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015

EN530.678 Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 Prof: Marin Kobilarov 0.1 Model prerequisites Consider ẋ = f(t, x). We will make the following basic assumptions

### Lecture 7. Please note. Additional tutorial. Please note that there is no lecture on Tuesday, 15 November 2011.

Lecture 7 3 Ordinary differential equations (ODEs) (continued) 6 Linear equations of second order 7 Systems of differential equations Please note Please note that there is no lecture on Tuesday, 15 November

### Measure and Integration: Solutions of CW2

Measure and Integration: s of CW2 Fall 206 [G. Holzegel] December 9, 206 Problem of Sheet 5 a) Left (f n ) and (g n ) be sequences of integrable functions with f n (x) f (x) and g n (x) g (x) for almost

### BIBO STABILITY AND ASYMPTOTIC STABILITY

BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to

### Nonlinear Dynamical Systems Lecture - 01

Nonlinear Dynamical Systems Lecture - 01 Alexandre Nolasco de Carvalho August 08, 2017 Presentation Course contents Aims and purpose of the course Bibliography Motivation To explain what is a dynamical

### Chapter III. Stability of Linear Systems

1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,

### Lecture 8. Chapter 5: Input-Output Stability Chapter 6: Passivity Chapter 14: Passivity-Based Control. Eugenio Schuster.

Lecture 8 Chapter 5: Input-Output Stability Chapter 6: Passivity Chapter 14: Passivity-Based Control Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture

### THREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations

THREE DIMENSIONAL SYSTEMS Lecture 6: The Lorenz Equations 6. The Lorenz (1963) Equations The Lorenz equations were originally derived by Saltzman (1962) as a minimalist model of thermal convection in a

### Chapter 13 Internal (Lyapunov) Stability 13.1 Introduction We have already seen some examples of both stable and unstable systems. The objective of th

Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter

### Part II. Dynamical Systems. Year

Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2

### LMI Methods in Optimal and Robust Control

LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 20: LMI/SOS Tools for the Study of Hybrid Systems Stability Concepts There are several classes of problems for

### Solution of Linear State-space Systems

Solution of Linear State-space Systems Homogeneous (u=0) LTV systems first Theorem (Peano-Baker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state

### Math 328 Course Notes

Math 328 Course Notes Ian Robertson March 3, 2006 3 Properties of C[0, 1]: Sup-norm and Completeness In this chapter we are going to examine the vector space of all continuous functions defined on the

### Ordinary Differential Equation Theory

Part I Ordinary Differential Equation Theory 1 Introductory Theory An n th order ODE for y = y(t) has the form Usually it can be written F (t, y, y,.., y (n) ) = y (n) = f(t, y, y,.., y (n 1) ) (Implicit

### McGill University Math 354: Honors Analysis 3

Practice problems McGill University Math 354: Honors Analysis 3 not for credit Problem 1. Determine whether the family of F = {f n } functions f n (x) = x n is uniformly equicontinuous. 1st Solution: The

### Metric Spaces and Topology

Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies

### Stability and Robustness Analysis of Nonlinear Systems via Contraction Metrics and SOS Programming

arxiv:math/0603313v1 [math.oc 13 Mar 2006 Stability and Robustness Analysis of Nonlinear Systems via Contraction Metrics and SOS Programming Erin M. Aylward 1 Pablo A. Parrilo 1 Jean-Jacques E. Slotine

### Applied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.

Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R

### 6.2 Brief review of fundamental concepts about chaotic systems

6.2 Brief review of fundamental concepts about chaotic systems Lorenz (1963) introduced a 3-variable model that is a prototypical example of chaos theory. These equations were derived as a simplification

### ACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016

ACM/CMS 17 Linear Analysis & Applications Fall 216 Assignment 4: Linear ODEs and Control Theory Due: 5th December 216 Introduction Systems of ordinary differential equations (ODEs) can be used to describe

### Nonlinear Control. Nonlinear Control Lecture # 3 Stability of Equilibrium Points

Nonlinear Control Lecture # 3 Stability of Equilibrium Points The Invariance Principle Definitions Let x(t) be a solution of ẋ = f(x) A point p is a positive limit point of x(t) if there is a sequence

### Stability in the sense of Lyapunov

CHAPTER 5 Stability in the sense of Lyapunov Stability is one of the most important properties characterizing a system s qualitative behavior. There are a number of stability concepts used in the study

### On Stability of Linear Complementarity Systems

On Stability of Linear Complementarity Systems Alen Agheksanterian University of Maryland Baltimore County Math 710A: Special Topics in Optimization On Stability of Linear Complementarity Systems Alen

### Lyapunov Theory for Discrete Time Systems

Università degli studi di Padova Dipartimento di Ingegneria dell Informazione Nicoletta Bof, Ruggero Carli, Luca Schenato Technical Report Lyapunov Theory for Discrete Time Systems This work contains a

### Modeling and Analysis of Dynamic Systems

Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System - Stability

### 1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0

Numerical Analysis 1 1. Nonlinear Equations This lecture note excerpted parts from Michael Heath and Max Gunzburger. Given function f, we seek value x for which where f : D R n R n is nonlinear. f(x) =

### MAS3706 Topology. Revision Lectures, May I do not answer enquiries as to what material will be in the exam.

MAS3706 Topology Revision Lectures, May 208 Z.A.Lykova It is essential that you read and try to understand the lecture notes from the beginning to the end. Many questions from the exam paper will be similar

### In essence, Dynamical Systems is a science which studies differential equations. A differential equation here is the equation

Lecture I In essence, Dynamical Systems is a science which studies differential equations. A differential equation here is the equation ẋ(t) = f(x(t), t) where f is a given function, and x(t) is an unknown

### Nonlinear Control Lecture 5: Stability Analysis II

Nonlinear Control Lecture 5: Stability Analysis II Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 5 1/41

### Applied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books.

Applied Analysis APPM 44: Final exam 1:3pm 4:pm, Dec. 14, 29. Closed books. Problem 1: 2p Set I = [, 1]. Prove that there is a continuous function u on I such that 1 ux 1 x sin ut 2 dt = cosx, x I. Define

### Lorenz like flows. Maria José Pacifico. IM-UFRJ Rio de Janeiro - Brasil. Lorenz like flows p. 1

Lorenz like flows Maria José Pacifico pacifico@im.ufrj.br IM-UFRJ Rio de Janeiro - Brasil Lorenz like flows p. 1 Main goals The main goal is to explain the results (Galatolo-P) Theorem A. (decay of correlation

### Nonlinear Control Lecture # 1 Introduction. Nonlinear Control

Nonlinear Control Lecture # 1 Introduction Nonlinear State Model ẋ 1 = f 1 (t,x 1,...,x n,u 1,...,u m ) ẋ 2 = f 2 (t,x 1,...,x n,u 1,...,u m ).. ẋ n = f n (t,x 1,...,x n,u 1,...,u m ) ẋ i denotes the derivative

### FIXED POINT METHODS IN NONLINEAR ANALYSIS

FIXED POINT METHODS IN NONLINEAR ANALYSIS ZACHARY SMITH Abstract. In this paper we present a selection of fixed point theorems with applications in nonlinear analysis. We begin with the Banach fixed point

### Solutions: Problem Set 4 Math 201B, Winter 2007

Solutions: Problem Set 4 Math 2B, Winter 27 Problem. (a Define f : by { x /2 if < x

### Stability lectures. Stability of Linear Systems. Stability of Linear Systems. Stability of Continuous Systems. EECE 571M/491M, Spring 2008 Lecture 5

EECE 571M/491M, Spring 2008 Lecture 5 Stability of Continuous Systems http://courses.ece.ubc.ca/491m moishi@ece.ubc.ca Dr. Meeko Oishi Electrical and Computer Engineering University of British Columbia,

### Convergence Rate of Nonlinear Switched Systems

Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the

### 11 Chaos in Continuous Dynamical Systems.

11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 47 11 Chaos in Continuous Dynamical Systems. Let s consider a system of differential equations given by where x(t) : R R and f : R R. ẋ = f(x), The linearization

### Continuous Functions on Metric Spaces

Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0

### DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.

EE 533 Homeworks Spring 07 Updated: Saturday, April 08, 07 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. Some homework assignments refer to the textbooks: Slotine

### Chapter #4 EEE8086-EEE8115. Robust and Adaptive Control Systems

Chapter #4 Robust and Adaptive Control Systems Nonlinear Dynamics.... Linear Combination.... Equilibrium points... 3 3. Linearisation... 5 4. Limit cycles... 3 5. Bifurcations... 4 6. Stability... 6 7.

### Nonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems

Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Time-varying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin

### Nonlinear Control Systems

Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 7. Feedback Linearization IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs1/ 1 1 Feedback Linearization Given a nonlinear

### Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Section 2.6 (cont.) Properties of Real Functions Here we first study properties of functions from R to R, making use of the additional structure

### Lecture 25: Subgradient Method and Bundle Methods April 24

IE 51: Convex Optimization Spring 017, UIUC Lecture 5: Subgradient Method and Bundle Methods April 4 Instructor: Niao He Scribe: Shuanglong Wang Courtesy warning: hese notes do not necessarily cover everything

### MA5510 Ordinary Differential Equation, Fall, 2014

MA551 Ordinary Differential Equation, Fall, 214 9/3/214 Linear systems of ordinary differential equations: Consider the first order linear equation for some constant a. The solution is given by such that

### Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10)

Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10) Mason A. Porter 15/05/2010 1 Question 1 i. (6 points) Define a saddle-node bifurcation and show that the first order system dx dt = r x e x

### Georgia Institute of Technology Nonlinear Controls Theory Primer ME 6402

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,

### LECTURE 8: DYNAMICAL SYSTEMS 7

15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin

### Economics 204 Fall 2011 Problem Set 2 Suggested Solutions

Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit

### CDS 101 Precourse Phase Plane Analysis and Stability

CDS 101 Precourse Phase Plane Analysis and Stability Melvin Leok Control and Dynamical Systems California Institute of Technology Pasadena, CA, 26 September, 2002. mleok@cds.caltech.edu http://www.cds.caltech.edu/

### Lecture 4: Numerical solution of ordinary differential equations

Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor

### 2.152 Course Notes Contraction Analysis MIT, 2005

2.152 Course Notes Contraction Analysis MIT, 2005 Jean-Jacques Slotine Contraction Theory ẋ = f(x, t) If Θ(x, t) such that, uniformly x, t 0, F = ( Θ + Θ f x )Θ 1 < 0 Θ(x, t) T Θ(x, t) > 0 then all solutions

### Formal Groups. Niki Myrto Mavraki

Formal Groups Niki Myrto Mavraki Contents 1. Introduction 1 2. Some preliminaries 2 3. Formal Groups (1 dimensional) 2 4. Groups associated to formal groups 9 5. The Invariant Differential 11 6. The Formal

### Nonlinear systems. Lyapunov stability theory. G. Ferrari Trecate

Nonlinear systems Lyapunov stability theory G. Ferrari Trecate Dipartimento di Ingegneria Industriale e dell Informazione Università degli Studi di Pavia Advanced automation and control Ferrari Trecate

### An idea how to solve some of the problems. diverges the same must hold for the original series. T 1 p T 1 p + 1 p 1 = 1. dt = lim

An idea how to solve some of the problems 5.2-2. (a) Does not converge: By multiplying across we get Hence 2k 2k 2 /2 k 2k2 k 2 /2 k 2 /2 2k 2k 2 /2 k. As the series diverges the same must hold for the

### Numerical Methods for Differential Equations Mathematical and Computational Tools

Numerical Methods for Differential Equations Mathematical and Computational Tools Gustaf Söderlind Numerical Analysis, Lund University Contents V4.16 Part 1. Vector norms, matrix norms and logarithmic

### Econ Lecture 14. Outline

Econ 204 2010 Lecture 14 Outline 1. Differential Equations and Solutions 2. Existence and Uniqueness of Solutions 3. Autonomous Differential Equations 4. Complex Exponentials 5. Linear Differential Equations