Lyapunov Stability Analysis: Open Loop

Copyright F.L. Lewis 008 All rights reserved Updated: hursday, August 8, 008 Lyapunov Stability Analysis: Open Loop We know that the stability of linear time-invariant (LI) dynamical systems can be determined by examining the system poles. Continuous-time systems are: marginally stable (MS) if : Poles in the left half of the s-plane, with any jω-axis poles nonrepeated. asymptotically stable (AS) if : Poles strictly in the open left half of the s-plane. Discrete-time systems are: marginally stable (MS) if : Poles in or on the unit circle of the z-plane, with any poles on the unit circle nonrepeated. asymptotically stable (AS) if : Poles strictly inside the unit circle of the z-plane. he Routh est gives an elegant method of determining stability of polynomials without actually finding the roots. Now we will show another method of determining stability of dynamical systems that does not involve the poles. It is based on energy concepts, and can be extended to time-varying and nonlinear systems. Alexsandr Lyapunov was a Russian scientist who in 189 published his work defining the Lyapunov function and using it in the stability analysis of dynamical systems. Lyapunov stability analysis is a general method that can be used for nonlinear systems. It is noteworthy that the first satellite Sputnik was launched by Russia in 1957. Nonlinear Systems Analysis was in large part responsible for the control systems on this spacecraft. In the western world at that time, control systems analysis was primarily being carried out in the frequency domain and mainly for linear systems. his was a serious limitation, which became more clear as we advanced into the space age after 1957. Aerospace systems are fundamentally nonlinear systems. 1

Aleksandr Mikhailovich Lyapunov (from Wikipedia) Born June 6, 1857 Yaroslavl, Imperial Russia Died November 3, 1918 (aged 61) Residence Nationality Fields Institutions Alma mater Known for Russia Russian Applied mathematics Saint Petersburg State University Russian Academy of Sciences Kharkov University Saint Petersburg State University Lyapunov function Lyapunov Analysis for Continuous-ime Systems Stability Analysis Using Energy Consider the continuous-time linear time invariant autonomous (i.e. no control input) system x = Ax, x(0)

n with state x( t) R. It is desired to find a way of studying stability that does not involve the system poles or determining the characteristic polynomial Λ () s = si A. Example 1. Stability Analysis Using Energy R u(t) i(t) C x(t) Electric Circuit A circuit is shown. One has u x i = cx = R so that a state variable description is given by 1 1 x = x+ u, x(0) RC RC with x(0) the initial voltage across the capacitor. We are focusing here on open-loop stability, so set u=0 to get the open-loop undriven (or autonomous) dynamics 1 x = x RC We know this is asymptotically stable since the pole is at s = 1/ RC. he natural mode is t/ RC e and the time constant is τ = RC. he state is given by t/ RC x() t = e x(0). o determine stability by another method not involving the poles, consider the energy stored in the capacitor 1 V( x) = Cx 1. Asymptotic Stability. Clearly, since V(x) is quadratic in x(t), it is positive definite, i.e. V( x) 0 V ( x) = 0if and only if x = 0 he energy is a function of time, Vt () = Vxt ( ()). Its derivative is given by 3

dv V V ( x) = = x dt x 1 1 V = Cxx = Cx x = x RC R where in the last line we substituted the system dynamics 1 x = x. RC Now note that the derivative of the energy, evaluated thus along the system dynamics, is negative definite, i.e. V ( x) 0 V ( x) = 0if and only if x = 0 hus we are faced with a positive energy (V>0) which is always decreasing ( V < 0 ) as long as the state x(t) is not equal to zero. Consequently, the energy V(t) decreases until it is equal to zero, i.e. until x(t)=0. herefore, the state must go to zero for all initial conditions x(0). his means that the system is asymptotically stable.. Exponential Stability. o find the rate of decrease of the energy, we perform some extra analysis to relate V(x) and V ( x). Note that x = V( x) C so that 1 V = x = V R RC his is a dynamical equation with solution given by / t RC Vt () = e V(0) 1 with initial energy given by Vt ( = 0) = Cx0. herefore the energy decreases exponentially with time constant τ V = RC /. Note that x() t = V() t C x(0) = V(0) C so that 4

t/ RC t/ RC xt () = V(0) e = x(0) e. C his exponentially decaying state response was found without determining the poles or solving the state equation. Note that the energy time constant is one half the state time constant since energy is a quadratic function of x(t). Definition of Continuous-ime Lyapunov Function We now formalize these concepts for general continuous-time (C) systems. n Let there be given x( t) R and a C autonomous (i.e. no input) nonlinear dynamic system x = f( x), x(0) Let f(x) be continuously differentiable so that there exist unique solutions, and suppose f(x)=0 so that x=0 is an equilibrium point, i.e. when x=0 one has x = f(0) = 0. his is always the case for linear systems x = Ax. Stability Definitions for Nonlinear Systems 1. he origin of the system is said to be stable in the sense of Lyapunov (SISL), if, for every ε > 0, there exists a δ = δ(ε) > 0 such that, if x(0) < δ then xt () < ε for t 0.. he origin of the system is said to be asymptotically stable (AS) if it is Lyapunov stable and if there exists δ > 0 such that if x(0) < δ then lim xt ( ) = 0. 3. he origin of the system is said to be exponentially stable (ES) if it is asymptotically stable and if there exist α,β,δ > 0 such that if x(0) < δ then x() t α x(0) e βt for t 0. For linear time-invariant systems, SISL is equivalent to marginal stability, AS is the usual asymptotic stability we have discussed before, and AS and ES are equivalent. SISL means that we can keep the state as small as desired for all times t 0 (i.e. xt () any choice of ε) by selecting the initial state small enough (i.e. x(0) < δ ( ε ) ). Definiteness of Functions and Matrices t < ε for Since Lyapunov techniques are extensions of energy concepts, the following definitions are germane. 5

n Definition: Definiteness of Real Functions. We say a real function x ( ) : R Ris: positive semidefinite, x ( ) 0 if x ( ) 0, x positive definite, (x)>0, if x ( ) 0, x ( x) = 0if and only if x = 0 negative semidefinite, x ( ) 0if x ( ) 0, x negative definite, (x)<0, if x ( ) 0, x ( x) = 0if and only if x = 0 Let (x) be the quadratic form x ( ) = xx with an n n real matrix. uadratic forms are very convenient for Lyapunov analysis. We define the following properties for square matrices. Definition: Definiteness of Square Matrices. We say a real square n n matrix is: positive semidefinite, 0 if x x 0, x positive definite, >0, if xx 0, x x x = 0if and only if x = 0 negative semidefinite, 0 if x x 0, x negative definite, <0, if xx 0, x x x = 0if and only if x = 0 If is a symmetric matrix, then the definiteness properties of (x) can be determined in terms of the eigenvalues of : 6

Fact. Definiteness of Real uadratic Functions. A real square symmetric matrix = is: positive semidefinite, if all the eigenvalues of are greater than or equal to zero positive definite, if all the eigenvalues of are greater than zero negative semidefinite, if all the eigenvalues of are less than or equal to zero negative definite, if all the eigenvalues of are less than zero he next result gives an alternative test for determining definiteness of matrices that is n n often easier than finding the eigenvalues. Let = [ q ] R. he leading minors of are q11 q1 q13 q11 q1 m1 = q11, m =, m3 = q1 q q3,, mn = q1 q q q q In terms of the leading minors, we have ij 31 3 33. > 0 if m i > 0, all i < 0 if mi mi < 0, > 0, all odd i all even i. Lyapunov heorems for Continuous-ime Systems Now we formalize the results of Example 1. Definition: Continuous-time (C) Lyapunov Function V(x) is said to be a C :Lyapunov Function if: 1. V(x) is a continuous real-valued function. V( x ) > 0, i.e. V(x) is positive definite 3. V = dv 0, i.e. V ( x) is negative semi-definite dt Condition 3 must be verified along the system trajectories. hat is, compute dv/dt and substitute in the system dynamics x = f( x). 7

he following results were proven by A.M. Lyapunov. Lyapunov heorem. SISL. Suppose that for a given system there exists a Lyapunov function. hen the system is SISL. Lyapunov heorem. AS. Suppose that for a given system there exists a Lyapunov function which also satisfies the stronger third condition V = dv < 0, i.e. V ( x) is negative definite. hen dt the system is AS. As shown in Example 1, exponential stability can be verified by extra analysis to relate V(x) and V ( x) and show that the Lyapunov function decreases exponentially. hese theorems are powerful and elegant. However, they do not say how to find a Lyapunov function. his may be difficult for nonlinear systems. Often, energies can be used. he next definition is useful in finding Lyapunov functions for specific systems. Definition. Lyapunov Function Candidate (LFC). V(x) is said to be a C :Lyapunov Function Candidate (LFC) if: 1. V(x) is a continuous real-valued function. V( x ) > 0, i.e. V(x) is positive definite Having selected a reasonable LFC, one computes its time derivative (along the system trajectories) to determine whether condition 3. holds, i.e. whether the LFC is in fact a Lyapunov function that verifies the stability of the system. Lyapunov Functions for C Linear systems: C Lyapunov Equation. Consider the continuous-time (C) linear time invariant (LI) autonomous system x = Ax, x(0) n with state x( t) R. For LI systems it is easy to find a Lyapunov function. In fact, quadratic energy always works. herefore define the energy quadratic function 1 V( x) = x Px with constant kernel matrix P symmetric and positive definite, i.e. P= P > 0. Clearly 1. V(x) is a continuous real-valued function 8

. V( x ) > 0, i.e. V(x) is positive definite herefore, V(x) is a Lyapunov function candidate LFC. If we can prove that V = dv 0 along the system trajectories, then V(x) is a Lyapunov dt function and the system is SISL. Compute the Lyapunov derivative d 1 1 V = ( x Px) = ( x Px + x Px + x Px ) dt Now P is constant, and substituting the system dynamics x = Ax yields 1 1 V = ( ( Ax) Px + x PAx) = x ( A P + PA) x along the system trajectories. Now suppose that there exists a positive semidefinite matrix such that AP+ PA= hen 1 1 V = x ( A P+ PA) x= x x 0 hen, V(x) is a Lyapunov function and the system is SISL. We call AP+ PA= the continuous-time Lyapunov Equation. Note that this equation is the same if it is transposed. herefore, if is symmetric, so is the solution P. herefore, we see that if there exists a symmetric positive definite solution P to the C Lyapunov equation with a symmetric positive semidefinite matrix, then V(x) is a Lyapunov function and the system is SISL. In fact, this condition turns out to be necessary and sufficient for LI C systems, as formulated in the next result. heorem. SISL for LI C Systems. Let be a symmetric positive semidefinite matrix. hen the system x = Ax is SISL (e.g. marginally stable) if and only if the (symmetric) matrix P which solves the C Lyapunov equation AP+ PA= is positive definite. If = is in fact positive definite, we have the next stronger result. heorem. AS for LI C Systems. Let be a symmetric positive definite matrix. hen the system x = Ax is AS if and only if the (symmetric) matrix P which solves the C Lyapunov equation 9

AP+ PA= is positive definite. In fact, it turns out that can always be selected as the n n identity matrix to test a system for AS using this method. Example. Stability by Lyapunov Equation Solution for C LI Systems. Consider the LI C autonomous system 0 1 x = Ax = x 3 Let us test for stability using the Lyapunov equation. First, let us try for AS. Select =I. he Lyapunov equation becomes PA + A P = p1 p 0 1 0 p1 p 1 0 p p + = 3 3 1 3 p p 3 0 1 where symmetry of P has been used. his yields 4p p1 3p p3 1 0 = p1 3p p3 p 6p 3 0 1 Now the top left (e.g. (1,1) element) yields 4p = 1 p = 1/4 and then the bottom left (.) equation yields p 6p3 = 1 p 3 = 1/4 Finally, p1 3p p3 = 0 so that p 1 = 5/4 herefore, 5 1 p1 p 4 4 P = 1 1 p p = 3 4 4 he leading minors of P are 10

p = > 0 5 1 4 P = > 0 1 4 herefore P>0. hus the system is AS. Notes: 1. In this case one could solve for the elements of P sequentially because of the easy structure of matrix A. Generally this is not possible. hen, write 4p p1 3p p3 1 0 = p1 3p p3 p 6p 3 0 1 as 0 4 0 p1 1 1 3 p 0 = 0 6 p 3 1 and invert the coefficient matrix to find [ p p p ] 1 3. If in some example one selects =I and the solution P of the Lyapunov equation is not positive definite, the system is not AS. However, it may still be SISL. o verify this, select a that is positive semidefinite and see if the resulting Lyapunov 1 0 solution P is positive definite. If so, the system is MS. For instance, one could try = 0 0 0 0 or = 0 1. Lyapunov Analysis for Discrete-ime Systems Consider now the discrete-time autonomous nonlinear system x f( x ), x = k+ 1 k 0 Suppose f(x)=0 so that x=0 is an equilibrium point. his is always the case for linear D systems. Given a real function V( x k ) define the first difference as Δ V( xk) = V( xk+ 1) V( xk) Definition: Discrete-ime (D) Lyapunov Function V(x) is said to be a D :Lyapunov Function if: 11

1. V(x) is a continuous real-valued function. V( x ) > 0, i.e. V(x) is positive definite 3. Δ V( xk) = V( xk+ 1) V( xk) 0, i.e. the first difference Δ V( x k ) is negative semidefinite he following results were proven by A.M. Lyapunov. SISL and AS are defined for D systems in terms of time trajectories the same as for C systems. D Lyapunov heorem. SISL. Suppose that for a given system there exists a Lyapunov function. hen the system is SISL. D Lyapunov heorem. AS. Suppose that for a given system there exists a Lyapunov function which also satisfies the stronger third condition Δ V( xk) = V( xk+ 1) V( xk) < 0, i.e. Δ V( x k ) is negative definite. hen the system is AS. In the D case stability is referred to the unit circle in the z-plane. he next definition is useful in finding Lyapunov functions for specific systems. Definition. Lyapunov Function Candidate (LFC). V(x) is said to be a C :Lyapunov Function Candidate (LFC) if: 1. V(x) is a continuous real-valued function. V( x ) > 0, i.e. V(x) is positive definite Having selected a reasonable LFC, one computes its first difference (along the system trajectories) to determine whether condition 3. holds, i.e. whether the LFC is in fact a Lyapunov function that verifies the stability of the system. Lyapunov Functions for D Linear systems: D Lyapunov Equation. Consider the discrete-time (D) linear time invariant (LI) autonomous system x = k+ 1 Axk, x0 n with state R. xk For LI systems it is easy to find a Lyapunov function. In fact, quadratic energy always works. herefore define the energy quadratic function 1

1 V( xk) = xkpxk with constant kernel matrix P symmetric and positive definite, i.e. P= P > 0. Clearly 1. V(x) is a continuous real-valued function. V( x ) > 0, i.e. V(x) is positive definite herefore, V(x) is a Lyapunov function candidate LFC. If we can prove that Δ V( xk) = V( xk+ 1) V( xk) 0 along the system trajectories, then V(x) is a D Lyapunov function and the system is SISL. o study this, compute the Lyapunov function first difference Δ V( xk) = V( xk+ 1) V( xk) = xk+ 1Pxk+ 1 xkpxk Now substituting the system dynamics x = k+ 1 Ax yields k Δ V( xk) = xk ( A PA P) xk along the system trajectories. Now suppose that there exists a positive semidefinite matrix such that APA P= hen 1 1 Δ V( x ) = x ( A PA P) x = x x 0 k k k k k hen, V(x k ) is a Lyapunov function and the system is SISL. We call APA P= the discrete-time Lyapunov Equation. Note that this equation is the same if it is transposed. herefore, if is symmetric, so is the solution P. herefore, we see that if there exists a symmetric positive definite solution P to the D Lyapunov equation with a symmetric positive semidefinite matrix, then V(x k ) is a D Lyapunov function and the system is SISL. In fact, this condition turns out to be necessary and sufficient for LI C systems, as formulated in the next result. heorem. SISL for LI D Systems. Let be a symmetric positive semidefinite matrix. hen the system xk+ 1 = Axk is SISL (e.g. marginally stable) if and only if the (symmetric) matrix P which solves the D Lyapunov equation APA P= is positive definite. If = is in fact positive definite, we have the next stronger result. 13

heorem. AS for LI C Systems. Let be a symmetric positive definite matrix. hen the system xk+ 1 = Axk is AS if and only if the (symmetric) matrix P which solves the D Lyapunov equation APA P= is positive definite. In fact, it turns out that can always be selected as the n n identity matrix to test a system for AS using this method. Note that the solution properties of the C Lyapunov equation AP+ PA= refer to the locations of the poles of system matrix A with respect to the left half of the complex plane. On the other hand, the solution properties of the D Lyapunov equation APA P= refer to the locations of the poles of system matrix A with respect to the unit circle of the complex plane. 14