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Nonlinear System Analysis Lyapunov Based Approach Lecture 4 Module 1 Dr. Laxmidhar Behera Department of Electrical Engineering, Indian Institute of Technology, Kanpur. January 4, 2003 Intelligent Control Lecture Series Page 1

Overview Non-linear Systems : An Introduction Linearization Lyapunov Stability Theory Examples Summary January 4, 2003 Intelligent Control Lecture Series Page 2

Model: Nonlinear Systems: An Introduction Properties: Don t follow the principle of superposition, i.e, January 4, 2003 Intelligent Control Lecture Series Page 3

Multiple equilibrium points Limit cycles : oscillations of constant amplitude and frequency Subharmonic, harmonic oscillations for constant frequency inputs Chaos: randomness, complicated steady state behaviours Multiple modes of behaviour January 4, 2003 Intelligent Control Lecture Series Page 4

Nonlinear Systems: An Introduction Autonomous Systems: the nonlinear function does not explicitly depend on time. Affine System: Unforced System: input, January 4, 2003 Intelligent Control Lecture Series Page 5

Nonlinear Systems: An Introduction Example: Pendulum: The state equations are January 4, 2003 Intelligent Control Lecture Series Page 6

Nonlinear Systems: An Introduction Exercise Identify the category to which the following differential equations belong to? Why? 1. 2. 3. where is an external input. 4. 5. January 4, 2003 Intelligent Control Lecture Series Page 7

Linearization Concept of Equilibrium Point: Consider a system where functions! are continuously differentiable. The equilibrium point this system is defined as for January 4, 2003 Intelligent Control Lecture Series Page 8

What is linearization? Linearization is the process of replacing the nonlinear system model by its linear counterpart in a small region about its equilibrium point. Why do we need it? We have well stablished tools to analyze and stabilize linear systems. January 4, 2003 Intelligent Control Lecture Series Page 9

# $ # $ # $ $ % # $ % # % # Linearization The method: Let us write the the general form of nonlinear system as: (1). January 4, 2003 Intelligent Control Lecture Series Page 10

$ % Linearization Let & that forces the system constant equilibrium state & holds true. ' ( ' ( be a constant input such that to settle into a January 4, 2003 Intelligent Control Lecture Series Page 11

Linearization We now perturb the equilibrium state by allowing: ) ) *)and * *. T aylor s expansion yields # ) # ) ) * * ) * + + ) ) * ) + + * ) * * January 4, 2003 Intelligent Control Lecture Series Page 12

Linearization where + + ) ) *,,, -./ -0 / 1 1 1 -./ -0 2.. -.2-0 / 1 1 1 -. 2-0 2 3 3 3 4 4 4 4 4 4 4 4 4 0 5 687 5 + + * ) *,,, -./ - 7 / 1 1 1 -./ - 7 9.. -. 2-7 / 1 1 1 -.2-7 9 3 3 3 4 4 4 4 4 4 4 4 4 0 5 687 5 are the Jacobian matrices of January 4, 2003 Intelligent Control Lecture Series Page 13

# ) * # ) + ) + * * * ) ) + + ) * ) # # # # # ) # ) # # ) ) Linearization Note that because Let is constant. Furthermore,. and Neglecting higher order terms, we arrive at the linear approximation January 4, 2003 Intelligent Control Lecture Series Page 14

; < ) * : $ $ $ : % % % Linearization Similarly, if the outputs of the nonlinear system model are of the form. or in vector notation January 4, 2003 Intelligent Control Lecture Series Page 15

; ) * ; ; ; Linearization Taylor s series expansion can again be used to yield the linear approximation of the above output equations. Indeed, if we let then we obtain January 4, 2003 Intelligent Control Lecture Series Page 16

= A = Linearization Example Consider a first order system: where Linearize it about origin Its solution is : initial state =?> @, the state will settle at. Whatever may be the as B, which is the only equilibrium point that this linearized system has. January 4, 2003 Intelligent Control Lecture Series Page 17

Linearization However, the solution of actual nonlinear system is =D>E@ =?> @ = C A A For various initial conditions, the system has two equilibrium points: in the following figure. and Cas can be seen January 4, 2003 Intelligent Control Lecture Series Page 18

Linearization 2 1.5 1 0.5 0 0 F F Stable and Unstable Equilibrium Points x 0 =1.1 x 0 =1.01 x 0 =0.5 x(t)=x 0 e -t /(1-x 0 +x 0 e -t ) X(t) 0-0.5 x 0 =-0.5-1 x 0 =1.5-1.5-2 0 1 2 3 4 5 6 7 8 Time(seconds) =is a Stable while is an unstable equilibrium point. January 4, 2003 Intelligent Control Lecture Series Page 19

Lyapunov Stability Theory The concept of stability: Consider the nonlinear system G G Let an equilibrium point of the system be, January 4, 2003 Intelligent Control Lecture Series Page 20

I I G We say that is stable in the sense of Lyapunov if there exists positive quantity Hsuch that for every J JLK J JLK H we have G J G = B G = H I G G for all if it is stable and. we say that is asymptotically stable J as We call unstable if it is not stable. January 4, 2003 Intelligent Control Lecture Series Page 21

G Lyapunov Stability Theory How to determine the stability or instability of without explicitly solving the dynamic equations? Lyapunov s first or indirect method: Start with a nonlinear system G G Expand in Taylor series around redefine ) (we also January 4, 2003 Intelligent Control Lecture Series Page 22

R = 0 QPO J G 0 + 4 4 + 4 Lyapunov Stability Theory where is the Jacobian matrix of 4NM evaluated at contains the higher order terms, i.e., and RTS J J J Then the nonlinear system asymptotically stable if and only if the linear system is stable, i.e., if all eigenvalues of have negative real parts. is January 4, 2003 Intelligent Control Lecture Series Page 23

G Lyapunov Stability Theory Advantage: Easy to apply Disadvantages: If some eigenvalues of are zero, then we cannot draw any conclusion about stability of the nonlinear system. It is valid only if initial conditions are close to the equilibrium. This method provides no indication as to how close is close. January 4, 2003 Intelligent Control Lecture Series Page 24

Lyapunov Stability Theory ULyapunov s second or direct method: Consider the nonlinear system B G G Suppose that there exists a function, called Lyapunov function, properties: with following 1. 2., for :Positive definite 3. K along trajectories of : Negative January 4, 2003 Intelligent Control Lecture Series Page 25

G Lyapunov Stability Theory Definite Then, is asymptotically stable. The method hinges on the existence of a Lyapunov function, which is an energy-like function. $ + + + $ 1 1 1 + + + + + + # 1 + # January 4, 2003 Intelligent Control Lecture Series Page 26

Lyapunov Stability Theory Advantages: answers stability of nonlinear systems without explicitly solving dynamic equations can easily handle time varying systems can determine asymptotic stability as well as plain stability can determine the region of asymtotic stability or the domain of attraction of an equilibrium January 4, 2003 Intelligent Control Lecture Series Page 27

V Lyapunov Stability Theory Example Oscillator with a nonlinear spring: W V V Linearize this system, The characteristic equation of linearized system is. January 4, 2003 Intelligent Control Lecture Series Page 28

Lyapunov Stability Theory The Vcharacteristic root corresponds to the damping term but notice the existence of aroot from the lack of a linear term in the spring restoring force. The linearized version of the system cannot recognize the existence of a nonlinear spring term and it fails to produce a non-zero characteristic root related to the restoring force. January 4, 2003 Intelligent Control Lecture Series Page 29

V Lyapunov Stability Theory Let s look at Lyapunov based approach. Consider the state space model W G G with equilibrium. Let s try for a Lyapunov function Y B X C C We can see that for all,. January 4, 2003 Intelligent Control Lecture Series Page 30

Lyapunov Stability Theory The time derivative of is + + + + W V W V K It follows then that G is asymptotically stable. January 4, 2003 Intelligent Control Lecture Series Page 31

Lyapunov Stability Theory Disadvantage of Lyapunov based Approach There is no systematic way of obtaining Lyapunov functions Lyapunov stability criterion provides only sufficient condition for stability. January 4, 2003 Intelligent Control Lecture Series Page 32

References 1. Nonlinear Systems, Hassan K. Khalil, Third Edition, Prentice Hall. 2. Applied Nonlinear Control, J. J. E. Slotine and W. Li, Prentice Hall 3. Nonlinear System Analysis, M. Vidyasagar, Second Edition, Prentice Hall January 4, 2003 Intelligent Control Lecture Series Page 33

Nonlinear System Analysis Lyapunov Based Approach Lecture 5 Module 1 Dr. Laxmidhar Behera Department of Electrical Engineering, Indian Institute of Technology, Kanpur. January 4, 2003 Intelligent Control Lecture Series Page 34

Lyapunov Stability Theory Stability of Linear Systems Consider a linear system in the form Choose as Lyapunov function the quadratic form ( where is a symmetric positive definite matrix. January 4, 2003 Intelligent Control Lecture Series Page 35

Lyapunov Stability Theory Then we have ( ( ( ( ( ( ( ( ( ( ( where Lyapunov Matrix Equation January 4, 2003 Intelligent Control Lecture Series Page 36

Z ( Lyapunov Stability Theory If the matrix is positive definite, then the system is asymptotically stable. Therefore, we could pick Z, the identity matrix and solve for and see if is positive definite January 4, 2003 Intelligent Control Lecture Series Page 37

Lyapunov Stability Theory Note: The usefulness of Lyapunov s matrix equation for linear systems is that it can provide an initial estimate for a Lyapunov function for a nonlinear system in cases where this is done computationally. Furthermore, it can be used to show stability of the linear quadratic regulator design. January 4, 2003 Intelligent Control Lecture Series Page 38

+ + Let Let Lyapunov Stability Theory LaSalle-Yoshizawa Theorem be an equilibrium point of. be a continuously differentiable, positive definite and radially unbounded function such that [SA QPO where solutions of is a continuous function. Then, all bounded and satisfy are globally uniformly January 4, 2003 Intelligent Control Lecture Series Page 39

Lyapunov Stability Theory In addition, if equilibrium asymptotically stable. is positive definite, then the is globally uniformly January 4, 2003 Intelligent Control Lecture Series Page 40

Lyapunov Stability Theory More Examples We will discuss three examples that demonstrate applications of Lyapunov s method, namely 1. How to assess the importance of nonlinear terms in stability or instability. 2. How to estimate the domain of attraction of an equilibrium point. 3. How to design a control law that guarantees global asymptotic stability January 4, 2003 Intelligent Control Lecture Series Page 41

Lyapunov Stability Theory: Examples Example.1 Consider the system with. Find the equilibrium of the system by solving following equations: G G G G G G January 4, 2003 Intelligent Control Lecture Series Page 42

Lyapunov Stability Theory: Examples Multiply the first equation by G, the second by G and add them to get G G Hence, the equilibrium point is G G. The linearized system is C C January 4, 2003 Intelligent Control Lecture Series Page 43

># Lyapunov Stability Theory: Examples The characteristic equation is 4 4 4 4 4 4 C 4 4 4 \ C 4 4 C 4 Since the characteristic roots are purely imaginary, we can not draw any conclusion on the stability of the nonlinear system. Now we resort to Lyapunov based approach. Choose the Lyapunov function to be the sum of the kinetic and potential energy of the linear system January 4, 2003 Intelligent Control Lecture Series Page 44

Lyapunov Stability Theory: Examples (this does not work always!): C C We see that B for all,. Then January 4, 2003 Intelligent Control Lecture Series Page 45

B K Lyapunov Stability Theory: Examples Therefore, we see that b if, the system is asymptotically stable if, the system is unstable b this result can not be obtained obtained by linearization January 4, 2003 Intelligent Control Lecture Series Page 46

Lyapunov Stability Theory: Examples Example.2 Suppose we want to determine the stability of the origin of the nonlinear system (Show that this is the equilibrium of the system.), January 4, 2003 Intelligent Control Lecture Series Page 47

C C C C Lyapunov Stability Theory: Examples The easiest way to show the stability is by linearization. The linearized form of the system is The characteristic equation is We can see that the system is stable. Wait! since this result is based on linearization, it says that if initial condition is close to the equilibrium point January 4, 2003 Intelligent Control Lecture Series Page 48

Lyapunov Stability Theory: Examples then the solution will tend to the equilibrium as. To find how close is close we need to get an estimate of the domain of attraction. We can do this by using Lyapunov theory. Let s try a Lyapunov function candidate C C January 4, 2003 Intelligent Control Lecture Series Page 49

Lyapunov Stability Theory: Examples Take its time derivative W Y Y Y Y C We can see, therefore, that stability is guaranteed if K or K C January 4, 2003 Intelligent Control Lecture Series Page 50

Lyapunov Stability Theory: Examples This means that the domain of attraction of the equilibrium is a circular disk of radius 1. As long as the initial conditions are inside the disk, it is guaranteed that the solution will end up at the stable equilibrium. In case, the initial conditions lie outside the disk then convergence is not guaranteed. January 4, 2003 Intelligent Control Lecture Series Page 51

Lyapunov Stability Theory: Examples Note: It should be mentioned that the above disk is an estimate of the domain attraction based on the particular Lyapunov function we selected. A different Lyapunov function could have produced a different estimate of the domain of attraction. January 4, 2003 Intelligent Control Lecture Series Page 52

Lyapunov Stability Theory: Examples Example.3 Trajectory Tracking Consider a single link manipulator > > ^] ` ` > ^] _ Now we want to find a control so that tracks a desired trajectory. Define. Then above equation may be written as _error dynamic equation January 4, 2003 Intelligent Control Lecture Series Page 53

> C C Lyapunov Stability Theory: Examples Choose a Lyapunov Function Candidate : > > > Take its time derivative : > : > > ^] > > January 4, 2003 Intelligent Control Lecture Series Page 54

> > > K a a B > a ^] Lyapunov Stability Theory: Examples We can choose our control input as : > >. Then assume This implies that imply that and dynamic equation, we get. But, it does not. Substituting the control into error : > > a > > This implies that : >. January 4, 2003 Intelligent Control Lecture Series Page 55

Lyapunov Stability Theory: Examples Consider the system Example.4 Back-Stepping W Start with the scalar system W Design the feedback control the origin. to stabilize January 4, 2003 Intelligent Control Lecture Series Page 56

Lyapunov Stability Theory: Examples We cancel the nonlinear term by taking Thus we obtain W K Taking Lyapunov function candidate as, its time derivative satisfies Y dc b Hence, the origin is globally asymptotically stable. January 4, 2003 Intelligent Control Lecture Series Page 57

Lyapunov Stability Theory: Examples Now to back step, we use the change of variable e So the transformed system W e e C W e January 4, 2003 Intelligent Control Lecture Series Page 58

Lyapunov Stability Theory: Examples Taking f e as a composite Lyapunov Function, we obtain f W e e & C W e ' Y e & C W e ' January 4, 2003 Intelligent Control Lecture Series Page 59

Lyapunov Stability Theory: Examples Choosing control input as C W e e C W we get f Y e Hence with this control law, the origin is globally asymptotically stable. January 4, 2003 Intelligent Control Lecture Series Page 60