# WUSTL ESE 543 - Lecture 5 and 6 stab Lyap theory - Copy (56 pages)

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## Lecture 5 and 6 stab Lyap theory - Copy

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- Washington University in St. Louis
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- Ese 543 - Control Systems Design by State Space Methods

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Lecture 5 Nonlinear Systems Reference Texts J J Slotine W Li Applied Nonlinear Control Prentice Hall 1995 Khalil H K Nonlinear Systems Prentice Hall 2002 Linear Systems x t Ax t x R Unique equilibrium point if Stable equilibrium if A nx t R is nonsingular Re A 0 System response is a composition of natural modes If x Ax Bu x R nx u R nu t R Principle of superposition If Re A 0 BIBO stability in presence of u Sinusoidal input produces sinusoidal output of same frequency These same properties do not hold for nonlinear systems K A Wise 2 Nonlinear Systems General representation of nonlinear dynamics nx x f t x x R The solution t R x x t x t0 x0 is called state trajectory Definition The nonlinear system is called autonomous if the dynamics is not explicitly depending upon time Definition A state once x t x f x x is called an equilibrium state point of the system if is equal to x it remains equal to x for all future time It can be found by solving the nonlinear algebraic equations 0 f x 3 K A Wise Nonlinear Systems cont General representation of nonlinear dynamics nx x f t x x R t R x f x g x u u R nu x f x u Multiple equilibrium points are possible Stability depends upon initial conditions Can exhibit limit cycles self excited oscillations Exhibit bifurcations Changes in parameters can cause stability and the number of equilibrium points to change 4 K A Wise Equilibrium Points for Linear Systems A linear time invariant system x A x has a single equilibrium point at the origin if A is non singular If A is singular it has an infinity of equilibrium points which are contained in the null space of the matrix A i e by the subspace defined by Ax 0 This implies that the equilibrium points are not isolated Example x x 0 All the points on the x 1 x2 x 2 x2 x2 0 i e x axis are equilibrium points 5 K A Wise Nonlinear Example Duffing Equation 3 x x x 0 Undamped Solve for equilibriums 3 x x 0 0 x 0 i i Is a critical bifurcation value x Called a pitchfork bifurcation 6 K A Wise Pendulum Example Equations of motion M R b R M g sin 0 2 R State space representation x 1 x2 M Mg b g x 2 x2 2 sin x1 MR R Equilibrium points x2 0 sin x1 0 K A Wise x1 x2 0 0 0 2 7 0 Reducing Non zero Equilibrium Points to the Origin Let x be the equilibrium point of interest for the dynamics x f x f x 0 Introduce new variable Notice that Then because Thus y 0 y x x x y x x 0 x y f y x will be the equilibrium point in the new variables 8 K A Wise Other Characteristics of Nonlinear Systems Hopf Bifurcation Changes in system parameters can cause limit cycles Chaos For linear systems small changes to initial conditions produces small changes in the systems response For chaotic system response is very sensitive to initial condition This prevents one from predicting the system output Chaotic systems are deterministic systems not random motion 9 K A Wise Nominal Motion Let x t be the solution of the equation x f x x 0 x0 Perturb the initial condition corresponding solution Define x 0 x0 x0 Let x t be the e t x t x t Then e t f x t e t f x t g e t e 0 x0 g 0 t 0 Thus the error dynamics are non autonomous x2 x t e t x t x1 K A Wise 10 Example Using the Duffing Equation Consider the following autonomous system 3 mx k1 x k 2 x 0 Let x t be the solution with the initial condition Perturb the initial condition corresponding solution Define x 0 x0 x0 x t0 x0 Let x t be the e t x t x t Then 2 me k1e k 2 e 3e x t 3e x t 0 3 2 Thus we have non autonomous error equation It becomes non autonomous due to the nonlinearity 11 K A Wise Lyapunov Stability Theory Alexander Michailovich Lyapunov 1857 1918 Russian mathematician and engineer who laid out the foundation of the Stability Theory Results published in 1892 Russia Translated into French 1907 Reprinted by Princeton University 1947 American Control Engineering Community Interest 1960 s 12 K A Wise Stability Theory Definitions Equilibrium State Stability Nonlinear Dynamics x f x x t0 x0 f xe 0 Equilibrium Condition Definition Stable In the Sense of Lyapunov SISL An equilibrium state xe is said to be stable ISL if R 0 a r R if x0 xe r then x t xe R t t0 Definition Asymptotically Stable AS An equilibrium state xe is said to be AS if it is stable ISL and lim x t xe t Definition Asymptotically Stable in the Large ASL An equilibrium state xe is said to be ASL if it is AS and holds x0 13 K A Wise Concepts of Stability Stability and Instability Definition The equilibrium point is said to be stable if for all R 0 there exists r R 0 such that if x 0 r then x t R for all t 0 Definition The equilibrium point is called unstable if it is not stable as defined above x2 limit cycle Van der Pol Oscillator x 1 x2 x 2 x1 1 x x2 2 1 R x0 x1 The origin is unstable equilibrium 14 K A Wise Stability and Asymptotic Stability Definition The equilibrium point x 0 is said to be asymptotically stable if it is stable and in addition there exists r 0 such that x 0 r implies that x t 0 as t The ball Br x x r is called domain of attraction of the equilibrium point Stable x2 x0 x0 R 1 x1 The state converges to the origin but the origin is unstable K A Wise R Asymptotically stable Stable Unstable 0 r AS Unstable 15 Rate of Convergence Exponential Stability Definition The equilibrium point x 0 is said to be exponentially stable if there exist two strictly positive numbers and such that x t x 0 e t t 0 in some ball Br around the origin x 1 sin 2 x x Example Solution Example t 2 x t x 0 exp 1 sin x d 0 x t x 0 e t x x 2 x 0 1 Solution x t K A Wise 1 1 1 t Asymptotically Stable but NOT exponentially 16 Local and Global Stability Definition If asymptotic exponential stability holds for any initial state the equilibrium point is said to be asymptotically exponentially stable in the large or otherwise saying it is globally asymptotically exponentially stable Easy to check facts Linear time invariant systems are either asymptotically stable or marginally stable or unstable Linear asymptotic …

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