$SPAD/src/input lyapunov.input

Maya Ramakrishnan and Timothy Daly

December 3, 2016

Abstract

Maya Ramakrishnan “A Gentle Introduction to Lyapunov Functions” ORSUM August 2003

www.or.ms.unimelb.edu.au/handouts/lyaptalk.1.pdf

Zoran Gajić and Muhammad Tahir Javed Qureshi “Lyapunov Matrix Equation in System Stability and Control” ISBN 0-486-46668-X Dover 2008

Contents

1 Motivation

Often systems can be represented or modelled using systems of differential equations. Given a system, questions which naturally arise include

• What are the equilibrium points or orbits

• Is the system stable?

• What are the dominant features of the portrait of the orbit? Some common applications of Lyapunov functions are in the area of

• assessing the importance of non-linear terms in stability and instability

• estimating the domain of attraction of an equilibrium point

• designing control laws that guarantee global asymptotic stability Rate control for communications networks Kelly et al (1998)

• Proposed optimization framework – Larger system problem decomposed into a User and a Network problem

• Proposed rate control algorithm – System of differential equations in the rate/capacity allocated to each user

• With the correct choice of Lyapunov function, it is shown that: – the found equilibrium point is asymptotically stable

– the lyapunov function closely approximates the Network’s optimization problem. Lyapunov equations have been used to determine the stability of a:

• F-8 aircraft

• steam power system

• voltage regulator

• synchronous machine connected to an infinite bus

• distillation column

• catalytic cracker,

• gas absorber

• industrial reactor • L-1011 fighter aircraft

• hydroturbine governors

• DC motor

• magnetic tape control system

• one-dimensional diffusion process

• aircraft under wind disturbances

• chemical plant

2 Definitions

2.1 The continuous case

The general differential Lyapunov matrix equation corresponding to time varying continuous systems is given by

A^{T} (t)P (t) + P (t)A(t) + Q(t) = Ṗ (t), P (t_{0}) = P_{0}  (1)

where P (t) is an n × n solution matrix of unknown entries, A(t) is an n × n system matrix, and Q(t) is an n × n symmetric matrix chosen according to computational convenience and in most cases is positive semi-definite. If the system is time invariant then Ṗ = 0 (steady state) so that the above equation becomes the algebraic Lyapunov matrix equation

A^{T} P + P A + Q = 0

The necessary and sufficient condition for the existence of a unique solution is that no two eigenvalues of A add up to 0.

λ_{i} + λ_{j} ≠ 0, i, j = 1, 2, …, n

This condition is satisfied if A is an asymptotically stable matrix in the continuous-time domain (all eigenvalues of A are located in the closed left half-plane).

2.1.1 Continuous, time-invariant linear systems

Assuming a continous, time-invariant linear system of the form

ẋ(t) = Ax(t), x(t_{0}) = x_{0}

where x(t) ∈ ℝ^{n} is the system state vector and A ∈ ℝ^{n×x} is the system matrix. Definition 2.1.1: The linear system is stable if all eigenvalues of the matrix A are in the closed left half of the complex plane and those on the imaginary axis are distinct roots of the minimal polynomial of A. Definition 2.1.2: The linear system is asymptotically stable if all eigenvalues of A are in the open left half of the complex plane.

2.2 The discrete case

The corresponding Lyapunov matrix difference equation for discrete-time varying systems is given by

A^{T} (k)P (k)A(k) + Q(k) = P (k + 1), P (k_{0}) = P_{0}

and for time invariant discrete systems at the steady state, (P (k + 1) = P (k) = P), we have the discrete version of the algebraic Lyapunov matrix equation

A^{T} P A + Q = P

The condition for the existence of a unique solution is that no two eigenvalues have a product equal to 1.

λ_{i}λ_{j} ≠ 1, i, j = 1, 2, …, n

This conditions is satisfied if A is asymptotically stable in the discrete-time domain (all eigenvalues of A are strictly inside of the unit circle.

2.3 Basic Definitions

Consider the system

ẋ = f(x)

where f is not necessarily a linear function of x. Definition 2.3.1: If \overline{x} is an equilibrium point of the system, then f(\overline{x}) = 0.

A system is stable at an equilibrium point if it responds to small changes from the equilibrium with only small changes in its subsequent states. Definition 2.3.2: An equilibrium point, \overline{x}, is said to be stable if for all ϵ > 0, there exists a δ = δ(ϵ), such that

|x(t_{0}) − \overline{x}| < δ ⟹ |x(t) − \overline{x}| < ϵ, ∀t > t_{0}

Note that a system can be locally stable. Definition 2.3.3: We say that \overline{x} is asymptotically stable if it is stable and

|x(t) − \overline{x}| → 0, as t → ∞

Definition 2.3.4: A system is neutrally stable at \overline{x} if it is stable at \overline{x}, but not asymptotically stable. Definition 2.3.5: A system that is not stable at \overline{x} is said to be unstable.

3 Methods of determining stability

3.1 Indirect Method

Consider the system above, where f is not a linear function. Using Taylor expanson, we have

ẋ = Ax + g(x)

where A is the Jacobian matrix of f(x) evaluated at \overline{x}, i.e.

A = \frac{∂f}{∂x} |\overline{x}

and g(x) is a function of the higher order terms.

The resulting linear system is given by

ẋ = Ax

The stability of the system can then be assessed using the following rules:

1. The equilibrium point is asymptotically stable if the linearised system is strictly stable.

2. The equilibrium point is unstable if the linearised system is strictly unstable.

3. If the linearised system is marginally stable, then we cannot conclude anything from the linear approximation.

Consider rule 1. If all the eigenvalues of A have negative real parts, then the linearised system is asymptotically stable. Problems include

• If some of the eigenvalues are zero, we cannot conclude anything about the system.

• The linear approximation is valid when the initial conditions are close to \overline{x}, however how “close” is close?

3.2 Direct Method

Stability is determined by observing how a scalar function of the state variables of the system changes in time.

Consider the system

ẋ = f(x)

where • \overline{x} is the equilibrium point of the system;

• f ∈ C^{1}(R), where C^{k}(R) defines the class of all continuous functions on region R with kth derivatives also continuous on R; and

• x(t^{0}) = x^{0}, x^{0} ∈ R Definition 2.3.6: Let V (x) be a real valued scalar function belonging to C^{0}(S) for some region S of R^{n}. Suppose V (\overline{x}) = 0. Then,

1. V is positive definite on S if V (x) > 0, for x ≠ \overline{x}

2. V is negative definite on S if V (x) < 0, for x ≠ \overline{x}

3. If the equalities are not strict, then V is considered positive and negative semi-definite respectively.

Definition 2.3.7: Let V (x) be a real valued scalar function belonging to C^{1}(R). The total derivative along the solution curves of ẋ = f(x), i.e. with respect to the orbits x = x(t, x^{0}) is

V̇ = \frac{d}{dt}V (x(t, x^{0}))
= ∑_{i=1}^{n} \frac{∂V}{∂x_{i}} ẋ_{i}(t, x^{0})
= ∇V (x)^{T} f(x)

Theorem 1: Suppose there exists a function V (x) with the following properties:

1. V (\overline{x}) = 0

2. V (x) is positive definite for x ≠ \overline{x}

3. V̇ (x) is negative semi-definite along trajectories of ẋ = f(x)

Then V (x) is known as a Lyapunov function and \overline{x} is stable. Note that if V̇ (x) is negative definite, then \overline{x} is asymptotically stable. Proof: This proof will assume that the equilibrium point \overline{x} = 0 for simplicity. It is easily extendable to the situation where this is not the case.

Let

• J_{ϵ} be the interior of a sphere of radius ϵ, whose center is at point 0. Further, assume ϵ is such that J_{ϵ} lies in the region R.

• S_{ϵ} be the spherical surface of the sphere

Let V be a function that satisfies properties 1-3 from above.

Since V (x) is continuous and positive on the closed, bounded surface S_{ϵ}, by the Minimum Value Theorem for continuous functions, V (x) has a minimum on S_{ϵ}. Suppose this minimum is L. Then:

0 < L ≤ V (x), ∀x ∈ S_{ϵ}

Since V (x) is continous and has value 0 at x = 0, there exists an open ball J_{d}elta centered at 0, which lies entirely inside J_{ϵ}.

• Choose δ such that V (x) < L, ∀x ∈ J_{δ}

Let x^{0} be a point inside J_{δ}.

• Consider the trajectory x(x^{0}, t) coming from point x^{0} and satisfying the system of differential equations under consideration

• Assume the trajectory travels outside the sphere S_{ϵ}.

Since the trajectory is continuous, it will intersect the boundary S_{ϵ} at some point x^{∗}. Recall that V (x) satisfies the property

V̇ = ∑_{i=0}^{n} \frac{∂V}{∂x_{i}} ẋ_{i} ≤ 0

Hence, the function V is not increasing along the trajectory of x(x^{0}, t). This means that

V (x^{∗}) ≤ V (x^{0}) < L

But x^{∗} ∈ S_{ϵ} and therefore L ≤ V (x^{∗}). This yields a contradiction. Hence as t increases, the trajectory x(x^{0}, t) does not pass outside the limits of S_{ϵ}.

We have shown, with \overline{x} = 0, that for all ϵ > 0, there exists δ(ϵ) > 0, such that

|x^{0} − \overline{x}| < δ ⟹ |x(x^{0}, t) − \overline{x}| < ϵ

as required.

4 Example

Consider the system

\begin{aligned}ẋ_{1} & = & −x_{2} + ax_{1}x_{2}^{2} \\ ẋ_{2} & = & x_{1} − bx_{1}^{2}x_{2} \end{aligned}

where a ≠ b.

To find the equilibrium points, set

\begin{aligned}ẋ_{1} & = & 0 \\ ẋ_{2} & = & 0 \end{aligned}

Basic manipulation shows that the equilibrium point is \overline{x}_{1} = \overline{x}_{2} = 0 Using the indirect method, consider the linearised system:

\begin{aligned}ẋ_{1} & = & Ax \\ [ ẋ_{1} ; ẋ_{2} ] & = & [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] [ \begin{matrix}x_{1} \\ x_{2} \end{matrix} ] \end{aligned}

Next consider the eigenvalues, λ, of A

det [ \begin{matrix}−λ & −1 \\ 1 & λ \end{matrix} ] = λ^{2} + 1 = 0

Recall for asymptotic stability of the linear system, we require all eigenvalues to have negative real parts. The roots of this equation are purely imaginary. Hence we cannot draw any conclusions about the stability of the non-linear system.

Now using the direct method, consider the function

V (x) = \frac{1}{2}x_{1}^{2} + \frac{1}{2}x_{2}^{2}

Clearly, V (\overline{x}) = 0. When x ≠ \overline{x}, V (x) > 0, i.e. it is positive definite. Next consider V̇ (x):

V̇ (x) = x_{1}ẋ_{1} + x_{2}ẋ_{2}
= x_{1}(−x_{2} + ax_{1}x_{2}^{2}) + x_{2}(x_{1} − bx_{1}^{2}x_{2})
= x_{1}^{2}x_{2}^{2}(a − b)

For asymptotic stability, we need V̇ (x) to be negative definite. Hence, if a < b, then the system is asymptotically stable.

5 Conclusion

Lyapunov functions are useful in assessing the stability of systems and in particular the method can be used

• for exploring non-linear systems

• for time varying systems, i.e. ẋ = f(x, t)

• to determine both stability and asymptotic stability The drawback is that to find a Lyapunov function is often more of an art than a science.

— * —

)set break resume
)sys rm -f lyapunov.output
)spool lyapunov.output
)set message test on
)set message auto off
)clear all
)spool
)lisp (bye)

———-

References^{[1] nothing}

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