Lecture 33 Stability Analysis of Nonlinear Systems Using Lyapunov heory I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
Outline Motivation Definitions Lyapunov Stability heorems Analysis of LI System Stability Instability heorem Examples 2
References H. J. Marquez: Nonlinear Control Systems Analysis and Design, Wiley, 2003. J-J. E. Slotine and W. Li: Applied Nonlinear Control, Prentice Hall, 1991. H. K. Khalil: Nonlinear Systems, Prentice Hall, 1996. 3
echniques of Nonlinear Control Systems Analysis and Design Phase plane analysis Differential geometry (Feedback linearization) Lyapunov theory Intelligent techniques: Neural networks, Fuzzy logic, Genetic algorithm etc. Describing functions Optimization theory (variational optimization, dynamic programming etc.) 4
Motivation Eigenvalue analysis concept does not hold good for nonlinear systems. Nonlinear systems can have multiple equilibrium points and limit cycles. Stability behaviour of nonlinear systems need not be always global (unlike linear systems). Need of a systematic approach that can be exploited for control design as well. 5
Definitions System Dynamics X = f X f D n : R (a locally Lipschitz map) D : an open and connected subset of R n Equilibrium Point X e X = f X = e 0 e 6
Definitions Open Set n A set A is open if for every p A, B p A r Connected Set A connected set is a set which cannot be represented as the union of two or more disjoint nonempty open subsets. Intuitively, a set with only one piece. Space A is connected, B is not. 7
Definitions Stable Equilibrium X e is stable, provided for each ε > 0, δε > 0 : X (0) X < δε X( t) X < ε t t e e 0 Unstable Equilibrium If the above condition is not satisfied, then the equilibrium point is said to be unstable 8
Definitions Convergent Equilibrium If δ : X 0 X < δ lim X t = X e t e Asymptotically Stable If an equilibrium point is both stable and convergent, then it is said to be asymptotically stable. 9
Definitions Exponentially Stable λt αλ, > 0: X t X α X 0 X e t > 0 whenever X 0 Convention e X < δ e he equilibrium point X e = 0 (without loss of generality) e 10
Definitions A function V : D R is said to be positive semi definite in if it satisfies the following conditions: D ()0 i D and V (0) = 0 ( ii) V X 0, X D V : D R is said to be positive definite in if condition (ii) is replaced by V X > in D 0 {0} V : D R is said to be negative definite (semi definite) in D if V ( X) is positive definite. D 11
Lyapunov Stability heorems heorem 1 (Stability) n Let X = 0 be an equilibrium point of X = f X, f : D R. Let V : D R be a continuously differentiable function such that: () i V 0 = 0 ( ii) V X > 0, in D {0} ( iii) V X 0, in D {0} hen X = 0 is "stable". 12
Lyapunov Stability heorems heorem 2 (Asymptotically stable) n Let X = 0 be an equilibrium point of X = f X, f : D R. Let V : D R be a continuously differentiable function such that: () i V 0 = 0 ( ii) V X > 0, in D {0} ( iii) V X < 0, in D {0} hen X = 0 is "asymptotically stable". 13
Lyapunov Stability heorems heorem 3 (Globally asymptotically stable) n Let X = 0 be an equilibrium point of X = f X, f : D R. Let V : D R be a continuously differentiable function such that: () i V 0 = 0 ( ii) V X > 0, in D {0} ( iii) V X is "radially unbounded" ( iv) V X < 0, in D {0} hen X = 0 is "globally asymptotically stable". 14
Lyapunov Stability heorems heorem 3 (Exponentially stable) Suppose all conditions for asymptotic stability are satisfied. In addition to it, suppose constants k, k, k, p: () i k X V X k X 1 2 ( ii) V X k X p 3 p p 1 2 3 hen the origin X = 0 is "exponentially stable". Moreover, if these conditions hold globally, then the origin X = 0 is "globally exponentially stable". 15
Example: Pendulum Without Friction x 1 x 2 θ, x θ x = ( g / l) sin x 2 1 System dynamics x 1 2 Lyapunov function V = KE+ PE 1 = m( ωl) 2 + mgh 2 1 2 2 = ml x2 + mg(1 cos x1) 2 16
Pendulum Without Friction = ( ) V X V f X V X V V = f1 X f2 X x1 x 2 2 g = mgl sinx1 ml x 2 x2 sinx1 l = mglx sin x mglx sin x = 0 0 (nsdf) 2 1 2 1 Hence, it is a stable system. 17
Pendulum With Friction Modify the previous example by adding the friction force klθ ma = mg sin θ kl θ Defining the same state variables as above x = x 1 2 g k x = sin x x l m 2 1 2 18
Pendulum With Friction = ( ) V X V f X V V = f1 X f2 X x1 x 2 2 g k = mgl sinx1 ml x 2 x2 sinx1 x2 l m = kl x 2 2 2 0 ( nsdf) V X Hence, it is also just a stable system. (A frustrating result..!) 19
Analysis of Linear ime Invariant System System dynamics: X = AX, A R n n Lyapunov function: Derivative analysis: V X = X PX, P> 0 pdf = + V X PX X PX = X APX+ ( ) X PAX X AP PAX = + 20
Analysis of Linear ime Invariant System For stability, we aim for V X QX Q ( 0) = > By comparing X AP + PAX= XQX For a non-trivial solution PA + A P + Q = 0 (Lyapunov Equation) 21
Analysis of Linear ime Invariant System heorem : ( λ ) n n he eigenvalues λ of a matrix A satisfy Re < 0 i if and only if for any given symmetric pdf matrix Q, a unique pdf matrix P satisfying the Lyapunov equation. i Proof: Please see Marquez book, pp.98-99. Note : P and Q are related to each other by the following relationship: At At P e Qe dt = 0 However, the above equation is seldom used to compute P. Instead P is directly solved from the Lyapunov equation. 22
Analysis of Linear ime Invariant Systems Choose an arbitrary symmetric positive definite matrix Q Q= I Solve for the matrix P form the Lyapunov equation and verify whether it is positive definite P V ( X ) < 0 Result: If is positive definite, then and hence the origin is asymptotically stable. 23
Lyapunov s Indirect heorem ( λ ) ( i = n) Let the linearized system about X = 0 be Δ X = A ΔX. he theorem says that if all the eigenvalues λ 1,, of the matrix A satisfy Re < 0 (i.e. the linearized system is exponentially stable), then for the nonlinear system the origin is locally exponentially stable. i i 24
Instability theorem Consider the autonomous dynamical system and assume X =0 is an equilibrium point. Let V : D R have the following properties: () i V(0) = 0 Under these conditions, is unstable ( ii) X R, arbitrarily close to X = 0, such that V X > 0 n 0 0 ( iii) V > 0 X U, where the set U is defined as follows U = { X D: X ε and V X > 0} X =0 25
Summary Motivation Notions of Stability Lyapunov Stability heorems Stability Analysis of LI Systems Instability heorem Examples 26
References H. J. Marquez: Nonlinear Control Systems Analysis and Design, Wiley, 2003. J-J. E. Slotine and W. Li: Applied Nonlinear Control, Prentice Hall, 1991. H. K. Khalil: Nonlinear Systems, Prentice Hall, 1996. 27
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