Lyapunov



1. Literature


# Title Category Description
1 MATLAB/Simulinkによる現代制御入門 Universal Lyapunov in Chap. 7, pp. 140
2 Multi-objective filter design for uncertain stochastic time-delay systems.pdf Exponentially stable Exponentially stable lemma in page 3, pp. 151, lemma 1
3 Observers for Nonlinear Stochastic Systems.pdf Exponentially stable The origin of lemma 1 of LIT#2
4 Basic Lyapunov theory.pdf Properties GAS, stability theorems (exponential), Lasalle's theorem, candidate
5 Lyapunov Stability Theory.pdf Exponentially stable Exponential stability based on state in page 3, Exponential stability theorem based on Lyapunov functional in page 6
6 Basic Concepts of Stability Theory Stability A tutorial help understanding Lyapunov stability
7 Equilibrium Points of Linear Autonomous Systems Universal Types of equilibrium points, phase portrait, especially for singular transition matrix
8 Method of Lyapunov Functions Trajectory * help understanding phase trajectory, derivatives, monotonicity
9 Nonlinear Systems (3rd Edition).pdf ROA Region of Attraction in pp.312
10 MATLAB/Simulinkと実機で学ぶ制御工学 Experiment Experiment platform based on lego NXT
11 Making an Inverted Pendulum using LEGO MINDSTORMS EV3 Experiment Another experiment platform based on lego EV3
12 台車の振幅制限を考慮した倒立振子の安定化制御 ROA Calculation of ROA boundary
13 Exact Asymptotic Stability Analysis and Region-of-Attraction Estimation for Nonlinear Systems ROA
14 Nonlinear Systems and Control Lecture # 11 Exponential Stability & Region of Attraction ROA Curriculum slides of Michigan State University
15 Lasalle 不変原理による 2 次系の漸近安定性の証明 Asymptotical stability
16 Riccati & Lyapunov equations LQR Riccati LQR and Lyapunov theory
17 Markov過程の量子化を用いたLyapunov関数の構築 Quantum
18 Quantum stochastic calculus Quantum
19 Itô's lemma Quantum


2. Point


• Lypunov Stability: Asymptotical, Exponential


Figure 1 left: Lyapunov stable , right: asymptotically stable
For general non-linear zero input system ([LIT 1] pp. 141)

x˙(t)=f(x(t)), x(0)=x0                                                  (1)


Figure 2 Lyapunov stability
The equilibrium point of equation (1) is xe=0 . As is shown in Figure 2(a) , for any ε>0 , there exists δ(ε)>0 (no matter how small it is), when x0<δ(ε) , for any time t , x(t)<ε , equilibrium point xe=0 is stable. More generally [ref] , as is shown in Figure 3 ,

Figure 3 Lyapunov stability in 2-dimensional phase diagram
when x0-xe<δ(ε) , there is x(t)-xe<ε, t , furthermore, if limtx(t)-xe=0 , then equilibrium point xe is asymptotically stable ( Figure 2(b) ) . Even furthermore, if limtx(t)-xe=0 , for x0n , system equation (1) is called asymptotically stable in the large or globally asymptotically stable (GAS) at equilibrium point xe .
Even even furthermore, if envelop ( red curve as is shown in Figure 1(b) ) of x(t) converge is at exponential rate, then the system equation (1) is called exponentially stable . (Lemma 1 [LIT#2] , [LIT#3] )

Figure 4 Relation of Lyapunov stability, asymptotically stable and exponentially stable
EXAMPLE: Energy and stability of pendulum system

Figure 5 Pendulum system with parameters

Jθ¨(t)=-μθ˙(t)-Mglsinθ(t)                                                  (2)

Let x(t)=x1(t)x2(t)T=θ(t)θ˙(t)T , denote the state variables. Then equation (2) is rewritten as

x˙1(t)x˙2(t)=x2(t)-MglJsinx1(t)-μJx2(t)                                               (3)

Equilibrium point is obtained as

x˙1(t)x˙2(t)=000=x2e0=-MglJsinx1e-μJx2ex2e=0sinx1e=0                     (4)

Therefore, xe=0 is one of the Equilibrium point . Besides, the mechanical energy of pendulum system equation (3) is

ϕ(x(t))=12Jx22(t)+Mgl(1-cosx1(t))                                                (5)

where the first term at right hand is called kinetic energy , the second one is called potential energy as is shown in Figure 6

Figure 6 Mechanical energy of pendulum
When x(t)=0 , that the position of pendulum is six o'clock, and velocity of it is 0, then ϕ(x(t))=0 . For any other state x(t) there is ϕ(x(t))>0 . The derivative of ϕ(x(t)) with equation (3) as simultaneous equations in order to cancel derivatives of states, the following formula holds as

ϕ˙(x(t))=Jx2(t)x˙2(t)+Mglx˙1(t)sinx1(t)=(Jx˙2(t)+Mglsinx1(t)x2(t)=-μx22(t)                 (6)

Assume
M l J=Ml2 ( moment of inertia ) μ
1 1 1 1
Matlab code of machanical energy is:
    [X,Y] = meshgrid(-7:0.1:7,-10:0.1:10);
    Z = Y.^2 + 9.8*(1-cos(X));
    surf(X,Y,Z)
    xlabel('angle')
    ylabel('angular velocity')

Figure 7 Mechanical energy of pendulum (Matlab chart)
Matlab code of ϕ˙(x(t)) is:
    [X,Y] = meshgrid(-7:0.1:7,-10:0.1:10);
    Z = -Y.^2;
    surf(X,Y,Z)
    xlabel('angle')
    ylabel('angular velocity')

Figure 8 Derivative of pendulum mechanical energy (Matlab chart)
Note that ϕ(x(t)) is a positive-definite function (PDF), ( ϕ(x(t))>0, x(t)02 ), while ϕ˙(x(t))0 . Unlike single-variable functions, whose time-derivatives indicate monotonicity, multi-variable function as shown in equation (5) , whose time-derivative is utilized to exploit the stability of an equilibrium point. [ref] Generally,

dVdt=Vx1dx1dt+Vx2dx2dt++Vxndxndt                               (7)

which in a scalar (dot) product can be written as

dVdt=grad V, dXdt, grad V=Vx1, Vx2, , Vxn , dXdt=dx1dt, dx2dt, , dxndt          (8)

Here, the first vector is the gradient of V(X) , id est, it’s always directed toward the greatest increase in V(X) . The second vector is velocity vector, it is always tagent to the phase trajectory .

(a)

(b)
Figure 9 (a) Lyapunov function. (b) Phase trajectory.
Take pendulum system equation (3) again for example, the gradient and phase trajectory of which is obtained by Matlab code as below [ref] :
    x=-pi:pi/20:pi;
    y=x';
    z = y.^2 + 9.8*(1-cos(x));
    [px,py] = gradient(z);
    figure
    contour(x,y,z)
    hold on
    quiver(x,y,px,py)
    hold off
    xlabel('angle')
    ylabel('angular velocity')
    x=-pi:pi/20:pi;
    y=x';
    z = y.^2 + 9.8*(1-cos(x));
    vx = y*ones(1,41);
    vy=-9.8*sin(x)-y*ones(1,41);
    figure
    contour(x,y,z)
    hold on
    quiver(x,y,vx,vy)
    hold off
    xlabel('angle')
    ylabel('angular velocity')

(a)

(b)
Figure 10 (a) Gradient chart. (b) Phase trajectory chart.
For phase trajectory and state trajectory of system (2) [ref] [ref] [ref] [ref]
    syms x(t)
    [V] = odeToVectorField(diff(x, 2) == -diff(x,1) - 9.8*sin(x));
    M = matlabFunction(V,'vars', {'t','Y'});
    sol = ode45(M,[0 20],[2 2]);
    z = linspace(0,20,1000);
    x1 = deval(sol,z,1);
    x2 = deval(sol,z,2);
    figure
    hold on;
    plot(x1,x2);
    xlabel('angle')
    ylabel('angular velocity')
    hold off;
    figure
    hold on;
    fplot(@(x)deval(sol,x,1), [0, 20])
    xlabel('time')
    ylabel('angle')
    hold off;
    figure
    hold on;
    fplot(@(x)deval(sol,x,2), [0, 20])
    xlabel('time')
    ylabel('angular velocity')
    hold off;

Figure 11 Phase portrait

Figure 12(a) angle trajectory

Figure 12(b) angular velocity trajectory
Test initial condition for [5,5],[3.14,5],[3.14,0],[3.14,-5],[pi,0].
Back to equation (6) , when x1(t)0 , if x2(t)=0 , there are chances that x(t)=0 is stable. Whatsoever,

x2(T)=x˙1(T)=0x˙2(T)=-MglJsinx1(T)                        (9)

When x1(T)0 , then sinx1(T)0x˙2(T)0 . Such that, ϕ˙(x(t))=-μx22(t)0 , when t>T . Therefore, such "equilibriums" are not stable.
Figure (13) and figure (14) represent mechnical energy of pendulum system (2) and its time derivatives respectively. The matlab code is
    ...
    fplot(@(x) 0.5*deval(sol,x,2).^2+9.8-9.8*cos(deval(sol,x,1)), [0, 20])
    fplot(@(x) -deval(sol,x,2).^2, [0, 20])

Figure 13 θ(x(t)), x(0)=[2,5]

Figure 14 θ˙(x(t)), x(0)=[2,5]

• Lyapunov stability theory