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MSc in Computer Engineering, Cybersecurity and Artificial Intelligence Course FDE , a.a. 2019/2020, Lecture 8 Analysis of nonlinear dynamical systems: The first Lyapunov method

• Prof. Mauro Franceschelli
• Dept. of Electrical and Electronic Engineering

University of Cagliari, Italy

Wednsday, 8th April 2020

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Outline

Introduction Linearization of a dynamical system The first Lyapunov method Linearization and first Lyapunov method (Discrete time)

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Introduction

Nonlinear dynamical systems

• Most physical systems (mechanical, electrical, hydraulic, biological etc.) if

modeled with sufficient detail are nonlinear dynamical systems

• For nonlinear systems the principle of superposition of effects does not hold: we

can not separate the state response into natural and forced response

• In general, the analytic solution of the state trajectory of a nonlinear system is

unknown (except particular cases).

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Introduction

Nonlinear dynamical systems

• Most physical systems (mechanical, electrical, hydraulic, biological etc.) if

modeled with sufficient detail are nonlinear dynamical systems

• For nonlinear systems the principle of superposition of effects does not hold: we

can not separate the state response into natural and forced response

• In general, the analytic solution of the state trajectory of a nonlinear system is

unknown (except particular cases).

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Introduction

Nonlinear dynamical systems

• Most physical systems (mechanical, electrical, hydraulic, biological etc.) if

modeled with sufficient detail are nonlinear dynamical systems

• For nonlinear systems the principle of superposition of effects does not hold: we

can not separate the state response into natural and forced response

• In general, the analytic solution of the state trajectory of a nonlinear system is

unknown (except particular cases).

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Introduction

Nonlinear systems

• It is of interest to determine the behavior of the state trajectory without

computing it: will the state trajectory remain close to an equilibrium point or not?

• If the the nonlinear system is differentiable at a particular equilibrium point in

its state space it can be linearized.

• The local stability of an equilibrium point of a nonlinear system which can be

linearized can be studied with the first Lyapunov method (also called the indirect method).

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Introduction

Nonlinear systems

• It is of interest to determine the behavior of the state trajectory without

computing it: will the state trajectory remain close to an equilibrium point or not?

• If the the nonlinear system is differentiable at a particular equilibrium point in

its state space it can be linearized.

• The local stability of an equilibrium point of a nonlinear system which can be

linearized can be studied with the first Lyapunov method (also called the indirect method).

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Introduction

Nonlinear systems

• It is of interest to determine the behavior of the state trajectory without

computing it: will the state trajectory remain close to an equilibrium point or not?

• If the the nonlinear system is differentiable at a particular equilibrium point in

its state space it can be linearized.

• The local stability of an equilibrium point of a nonlinear system which can be

linearized can be studied with the first Lyapunov method (also called the indirect method).

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Introduction

A classic example: the pendulum

Consider the dynamics of the pendulum depicted below, actuated by a torque (u) provided by an electric DC motor. I: moment of inertia of the pendulum around the pivot point; M: mass of the pendulum; l: length of the pendulum; g: gravity acceleration; θ: Angular position of the pendulum; b: friction at the joint; u: constant input torque provided by a DC electric motor.

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Introduction

A classic example: the pendulum

The equation of motion of this system is: I ¨ θ(t) + b ˙ θ + Mgl sin(θ(t)) = u

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Introduction

A classic example: the pendulum

Let u(t) = u: constant input torque provided by a DC electric motor; x1(t) = θ(t): angular position of the pendulum; x2(t) = ˙ θ(t): angular speed of the pendulum; y(t) = θ(t): output of the system. The SV model becomes: ˙ x1(t) = x2(t) ˙ x2(t) = − Mgl

I sin (x1(t)) − bx2(t) + 1 I u(t)

y(t) = x1(t)

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Introduction

A classic example: the pendulum

The equilibrium points of the system with constant input u are the solutions of f (xe) = 0: = xe2 = − Mgl

I sin (xe1) − b I xe2 + 1 I u

Thus xe = xe1 xe2

• =

x⋆

• where x⋆ is the solution of

sin (x⋆) = 1 Mgl u For u = 0 it holds xe1 = x⋆ = 0 ± kπ for an integer k.

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Introduction

A classic example: the pendulum

The equilibrium points of the system with constant input u are the solutions of f (xe) = 0: = xe2 = − Mgl

I sin (xe1) − b I xe2 + 1 I u

Thus xe = xe1 xe2

• =

x⋆

• where x⋆ is the solution of

sin (x⋆) = 1 Mgl u For u = 0 it holds xe1 = x⋆ = 0 ± kπ for an integer k.

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Introduction

A classic example: the pendulum

The equilibrium points of the system with constant input u are the solutions of f (xe) = 0: = xe2 = − Mgl

I sin (xe1) − b I xe2 + 1 I u

Thus xe = xe1 xe2

• =

x⋆

• where x⋆ is the solution of

sin (x⋆) = 1 Mgl u For u = 0 it holds xe1 = x⋆ = 0 ± kπ for an integer k.

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Introduction

A classic example: the pendulum

• The equilibrium points of the system with constant input u are the solutions of

f (xe) = 0:

• Physically, this means that the pendulum is at equilibrium whenever the angle

x1 = θ is either 0 (pendulum pointing downward) or π (pendulum pointing upward), and the angular velocity x2 = ˙ θ is zero.

• Qualitatively, the equilibrium xe = [0

0]T is stable, while the equilibrium xe = [0 π]T is unstable.

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Introduction

A classic example: the pendulum

• The equilibrium points of the system with constant input u are the solutions of

f (xe) = 0:

• Physically, this means that the pendulum is at equilibrium whenever the angle

x1 = θ is either 0 (pendulum pointing downward) or π (pendulum pointing upward), and the angular velocity x2 = ˙ θ is zero.

• Qualitatively, the equilibrium xe = [0

0]T is stable, while the equilibrium xe = [0 π]T is unstable.

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Introduction

A classic example: the pendulum

• The equilibrium points of the system with constant input u are the solutions of

f (xe) = 0:

• Physically, this means that the pendulum is at equilibrium whenever the angle

x1 = θ is either 0 (pendulum pointing downward) or π (pendulum pointing upward), and the angular velocity x2 = ˙ θ is zero.

• Qualitatively, the equilibrium xe = [0

0]T is stable, while the equilibrium xe = [0 π]T is unstable.

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Introduction

A classic example: the pendulum

• Note that, setting the torque of the electric motor to u = Mglsin (x⋆) can make

any desired angular position x1 = x⋆ an equilibrium point of the system.

• For instance, by imparting a torque u = Mgl, the configuration xe1 = π/2, and

xe2 = 0 is an equilibrium of the pendulum.

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Introduction

A classic example: the pendulum

• Note that, setting the torque of the electric motor to u = Mglsin (x⋆) can make

any desired angular position x1 = x⋆ an equilibrium point of the system.

• For instance, by imparting a torque u = Mgl, the configuration xe1 = π/2, and

xe2 = 0 is an equilibrium of the pendulum.

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Outline

Introduction Linearization of a dynamical system The first Lyapunov method Linearization and first Lyapunov method (Discrete time)

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Linearization of a dynamical system

Linearization

• Although almost every physical system contains nonlinearities,
• ftentimes its behavior within a certain operating range of an equilibrium

point can be reasonably approximated by that of a linear model.

• A reason to approximate a nonlinear system by a linear model is that, by

so doing, one can apply linear control and estimation methods.

• IMPORTANT: A linearized model is valid only when the system operates

in a sufficiently small range around an equilibrium point.

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Linearization of a dynamical system

Linearization

• Although almost every physical system contains nonlinearities,
• ftentimes its behavior within a certain operating range of an equilibrium

point can be reasonably approximated by that of a linear model.

• A reason to approximate a nonlinear system by a linear model is that, by

so doing, one can apply linear control and estimation methods.

• IMPORTANT: A linearized model is valid only when the system operates

in a sufficiently small range around an equilibrium point.

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Linearization of a dynamical system

Linearization

• Although almost every physical system contains nonlinearities,
• ftentimes its behavior within a certain operating range of an equilibrium

point can be reasonably approximated by that of a linear model.

• A reason to approximate a nonlinear system by a linear model is that, by

so doing, one can apply linear control and estimation methods.

• IMPORTANT: A linearized model is valid only when the system operates

in a sufficiently small range around an equilibrium point.

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Linearization of a dynamical system

Linearization around an equilibrium point

• Consider the nonlinear system

˙ x(t) = f (x(t), u(t)) (1) y(t) = g(x(t), u(t))

• Let xe = [xe1, xe2, . . . , xen] be an equilibrium point of the system in eq. (2) for u(t) = u⋆.
• Racall that f (x, u) = [f1(x, u), f2(x, u), . . . , fn(x, u)]T and

g(x, u) = [g1(x, u), g2(x, u), . . . , gr(x, u)]T are vector functions where n is the number of state variables, p is the number of outputs and r is the number of inputs.

• Define:

a new state vector δx(t) = x(t) − xe; a new input δu(t) = u(t) − u⋆; a new output δy(t) = y(t) − g(xe, u⋆).

• The new coordinates represent the variations of the state, input and output variables with

respect to their value at the equilibrium/operating point of the dynamical system.

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Linearization of a dynamical system

Linearization around an equilibrium point

• Consider the nonlinear system

˙ x(t) = f (x(t), u(t)) (1) y(t) = g(x(t), u(t))

• Let xe = [xe1, xe2, . . . , xen] be an equilibrium point of the system in eq. (2) for u(t) = u⋆.
• Racall that f (x, u) = [f1(x, u), f2(x, u), . . . , fn(x, u)]T and

g(x, u) = [g1(x, u), g2(x, u), . . . , gr(x, u)]T are vector functions where n is the number of state variables, p is the number of outputs and r is the number of inputs.

• Define:

a new state vector δx(t) = x(t) − xe; a new input δu(t) = u(t) − u⋆; a new output δy(t) = y(t) − g(xe, u⋆).

• The new coordinates represent the variations of the state, input and output variables with

respect to their value at the equilibrium/operating point of the dynamical system.

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Linearization of a dynamical system

Linearization around an equilibrium point

• By making the suggested change of variables, it holds:

d dt (δx(t) + xe) = f (δx(t) + xe, δu(t) + u⋆) δy(t) + g(xe, u⋆) = g(δx(t) + xe, δu(t) + u⋆)

• Since xe is a constant it holds d

dt xe = 0, thus

d dt δx(t)) = f (δx(t) + xe, δu(t) + u⋆) δy(t) = g(δx(t) + xe, δu(t) + u⋆) − g(xe, u⋆)

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Linearization of a dynamical system

Linearization around an equilibrium point

• By making the suggested change of variables, it holds:

d dt (δx(t) + xe) = f (δx(t) + xe, δu(t) + u⋆) δy(t) + g(xe, u⋆) = g(δx(t) + xe, δu(t) + u⋆)

• Since xe is a constant it holds d

dt xe = 0, thus

d dt δx(t)) = f (δx(t) + xe, δu(t) + u⋆) δy(t) = g(δx(t) + xe, δu(t) + u⋆) − g(xe, u⋆)

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Linearization of a dynamical system

Linearization around an equilibrium point

• Expanding the vector function f (x(t), u(t)) by Taylor series around the point

xe, u⋆, it holds: f (x(t), u(t))|x(t)=xe,u(t)=u⋆ = f (xe, u⋆) + ∂

∂x f (xe, u⋆) (x(t) − xe)

+ ∂

∂uf (xe, u⋆) (u(t) − u⋆)

+O(x(t) − xe2) + O(u(t) − u⋆2) + . . . which is equal to f (x(t), u(t))|x(t)=xe,u(t)=u⋆ =

∂ ∂x f (xe, u⋆)δx(t)

+ ∂

∂uf (xe, u⋆)δu(t)

+O(x(t) − xe2) + O(u(t) − u⋆2) + . . .

• If x(t) and u(t) are sufficiently close to, respectively, xe and u⋆, then the first
• rder terms dominate the second order terms.

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Linearization of a dynamical system

Linearization around an equilibrium point

• Expanding the vector function f (x(t), u(t)) by Taylor series around the point

xe, u⋆, it holds: f (x(t), u(t))|x(t)=xe,u(t)=u⋆ = f (xe, u⋆) + ∂

∂x f (xe, u⋆) (x(t) − xe)

+ ∂

∂uf (xe, u⋆) (u(t) − u⋆)

+O(x(t) − xe2) + O(u(t) − u⋆2) + . . . which is equal to f (x(t), u(t))|x(t)=xe,u(t)=u⋆ =

∂ ∂x f (xe, u⋆)δx(t)

+ ∂

∂uf (xe, u⋆)δu(t)

+O(x(t) − xe2) + O(u(t) − u⋆2) + . . .

• If x(t) and u(t) are sufficiently close to, respectively, xe and u⋆, then the first
• rder terms dominate the second order terms.

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Linearization of a dynamical system

Linearization around an equilibrium point

• Expanding the vector function f (x(t), u(t)) by Taylor series around the point

xe, u⋆, it holds: f (x(t), u(t))|x(t)=xe,u(t)=u⋆ = f (xe, u⋆) + ∂

∂x f (xe, u⋆) (x(t) − xe)

+ ∂

∂uf (xe, u⋆) (u(t) − u⋆)

+O(x(t) − xe2) + O(u(t) − u⋆2) + . . . which is equal to f (x(t), u(t))|x(t)=xe,u(t)=u⋆ =

∂ ∂x f (xe, u⋆)δx(t)

+ ∂

∂uf (xe, u⋆)δu(t)

+O(x(t) − xe2) + O(u(t) − u⋆2) + . . .

• If x(t) and u(t) are sufficiently close to, respectively, xe and u⋆, then the first
• rder terms dominate the second order terms.

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Linearization of a dynamical system

Linearization around an equilibrium point

• The linearized dynamical system is:

˙ δx(t) = ❆δ①(t) + ❇δ✉(t) δ②(t) = ❈δ①(t) + ❉δ✉(t) where A = ∂f ∂x

• xe,u⋆

=   

∂ ∂x1 (f1(xe, u⋆) ∂ ∂x2 f1(xe, u⋆)

. . .

∂ ∂xn f1(xe, u⋆)

. . . . . . . . . . . .

∂ ∂x1 (fn(xe, u⋆) ∂ ∂x2 fn(xe, u⋆)

. . .

∂ ∂xn fn(xe, u⋆)

   B = ∂f ∂u

• xe,u⋆ =

  

∂ ∂u1 f1(xe, u⋆) ∂ ∂u2 f1(xe, u⋆)

. . .

∂ ∂ur f1(xe, u⋆)

. . . . . . . . . . . .

∂ ∂u1 fn(xe, u⋆) ∂ ∂u2 fn(xe, u⋆)

. . .

∂ ∂ur fn(xe, u⋆)

  

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Linearization of a dynamical system

Linearization around an equilibrium point

while for the output map it holds g(x(t), u(t))|x(t)=xe,u(t)=u⋆ = g(xe, u⋆) + ∂

∂x g(xe, u⋆)δx(t)

+ ∂

∂ug(xe, u⋆)δu(t)

+O(x(t) − xe2) + O(u(t) − u⋆2) + . . . Now, g(xe, u⋆) = 0 but δy(t) = g(δx(t) + xe, δu(t) + u⋆) − g(xe, u⋆) thus the term g(xe, u⋆) cancels out and it follows C = ∂g ∂x

• xe,u⋆ =

  

∂ ∂x1 g1(xe, u⋆) ∂ ∂x2 g1(xe, u⋆)

. . .

∂ ∂xn (g1(xe, u⋆)

. . . . . . . . . . . .

∂ ∂x1 gp(xe, u⋆) ∂ ∂x2 gp(xe, u⋆)

. . .

∂ ∂xn gp(xe, u⋆)

  

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Linearization of a dynamical system

Linearization around an equilibrium point

while for the output map it holds g(x(t), u(t))|x(t)=xe,u(t)=u⋆ = g(xe, u⋆) + ∂

∂x g(xe, u⋆)δx(t)

+ ∂

∂ug(xe, u⋆)δu(t)

+O(x(t) − xe2) + O(u(t) − u⋆2) + . . . Now, g(xe, u⋆) = 0 but δy(t) = g(δx(t) + xe, δu(t) + u⋆) − g(xe, u⋆) thus the term g(xe, u⋆) cancels out and it follows C = ∂g ∂x

• xe,u⋆

=   

∂ ∂x1 g1(xe, u⋆) ∂ ∂x2 g1(xe, u⋆)

. . .

∂ ∂xn (g1(xe, u⋆)

. . . . . . . . . . . .

∂ ∂x1 gp(xe, u⋆) ∂ ∂x2 gp(xe, u⋆)

. . .

∂ ∂xn gp(xe, u⋆)

  

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Linearization of a dynamical system

Linearization around an equilibrium point

Finally, D = ∂g ∂x

• xe,u⋆ =

  

∂ ∂u1 g1(xe, u⋆) ∂ ∂u2 g1(xe, u⋆)

. . .

∂ ∂ur (g1(xe, u⋆)

. . . . . . . . . . . .

∂ ∂u1 gp(xe, u⋆) ∂ ∂u2 gp(xe, u⋆)

. . .

∂ ∂ur gp(xe, u⋆)

  

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Linearization of a dynamical system

Example: linearization of the pendulum model

Let u(t) = u: constant input torque provided by a DC electric motor; x1(t) = θ(t): angular position of the pendulum; x2(t) = ˙ θ(t): angular speed of the pendulum; y(t) = θ(t): output of the system. The model of the pendulum is: ˙ x1(t) = x2(t) ˙ x2(t) = − Mgl

I sin (x1(t)) − b I x2(t) + 1 I u(t)

y(t) = x1(t) with equilibirum points xe = xe1 xe2

• =

x⋆

• and x⋆ is the solution of

sin (x⋆) = 1 Mgl u

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Linearization of a dynamical system

Example: linearization of the pendulum model

The model of the pendulum is: ˙ x1(t) = x2(t) ˙ x2(t) = − Mgl

I sin (x1(t)) − b I x2(t) + 1 I u(t)

y(t) = x1(t)

• To linearize the model we first compute the partial derivatives

∂f ∂x

• =
• 1

− Mgl

I cos(x1(t))

− b

I

• ,

∂f ∂u

• =
• − 1

I

• ∂g

∂x

• =

1 , ∂g ∂u

• =
• Then, we compute the value of the partial derivatives at the equilibrium point:

A = ∂f ∂x

• xe,u⋆ =
• 1

− Mgl

I cos(x⋆)

− b

I

• ,

B = ∂f ∂u

• xe,u⋆ =
• − 1

I

• C =

∂g ∂x

• xe,u⋆ =

1 , D = ∂g ∂u

• xe,u⋆ =

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Linearization of a dynamical system

Example: linearization of the pendulum model

The model of the pendulum is: ˙ x1(t) = x2(t) ˙ x2(t) = − Mgl

I sin (x1(t)) − b I x2(t) + 1 I u(t)

y(t) = x1(t)

• To linearize the model we first compute the partial derivatives

∂f ∂x

• =
• 1

− Mgl

I cos(x1(t))

− b

I

• ,

∂f ∂u

• =
• − 1

I

• ∂g

∂x

• =

1 , ∂g ∂u

• =
• Then, we compute the value of the partial derivatives at the equilibrium point:

A = ∂f ∂x

• xe,u⋆ =
• 1

− Mgl

I cos(x⋆)

− b

I

• ,

B = ∂f ∂u

• xe,u⋆ =
• − 1

I

• C =

∂g ∂x

• xe,u⋆ =

1 , D = ∂g ∂u

• xe,u⋆ =

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Linearization of a dynamical system

Example: linearization of the pendulum model

• Finally, the linearized model around the equilibrium point is

˙ δx(t) =

• 1

− Mgl

I cos(x⋆)

− b

I

• δ①(t) +
• − 1

I

• δ✉(t)

δ②(t) =

• ∂g

∂x

• xe,u⋆ =

1 δ①(t) with δx(t) = x(t) − xe; δu(t) = u(t) − u⋆; δy(t) = y(t) − g(xe, u⋆).

• Notice how the linearized model is expressed in terms of a new state, a new

control input, and new output. Recall that these represent variations of x(t), u(t), and y(t) from their equilibrium/operating values. The linearized model above is only valid in a small neighborhood of the equilibrium, that is, it is only valid when the components δx1 and δx2 of the vector δx, are small.

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Linearization of a dynamical system

Example: linearization of the pendulum model

• Finally, the linearized model around the equilibrium point is

˙ δx(t) =

• 1

− Mgl

I cos(x⋆)

− b

I

• δ①(t) +
• − 1

I

• δ✉(t)

δ②(t) =

• ∂g

∂x

• xe,u⋆ =

1 δ①(t) with δx(t) = x(t) − xe; δu(t) = u(t) − u⋆; δy(t) = y(t) − g(xe, u⋆).

• Notice how the linearized model is expressed in terms of a new state, a new

control input, and new output. Recall that these represent variations of x(t), u(t), and y(t) from their equilibrium/operating values. The linearized model above is only valid in a small neighborhood of the equilibrium, that is, it is only valid when the components δx1 and δx2 of the vector δx, are small.

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Outline

Introduction Linearization of a dynamical system The first Lyapunov method Linearization and first Lyapunov method (Discrete time)

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The first Lyapunov method

The first Lyapunov method

• The first Lyapunov method is a systematic approach to determine the

local stability properties of equilibrium points for nonlinear systems based

• n their linearization around the equilibrium point under analysis.
• Consider a nonlinear system ˙

x = f (x, u), linearized at the equilibrium point xe,u⋆. Let ˙ δx(t) = ❆δ①(t) + ❇δ✉(t) δ②(t) = ❈δ①(t) + ❉δ✉(t) be the linearized model and let δ✉(t) = u(t) − u⋆ = 0 for all t ≥ 0, i.e., consider an autonomous linear system.

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The first Lyapunov method

The first Lyapunov method

• The first Lyapunov method is a systematic approach to determine the

local stability properties of equilibrium points for nonlinear systems based

• n their linearization around the equilibrium point under analysis.
• Consider a nonlinear system ˙

x = f (x, u), linearized at the equilibrium point xe,u⋆. Let ˙ δx(t) = ❆δ①(t) + ❇δ✉(t) δ②(t) = ❈δ①(t) + ❉δ✉(t) be the linearized model and let δ✉(t) = u(t) − u⋆ = 0 for all t ≥ 0, i.e., consider an autonomous linear system.

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The first Lyapunov method

The first Lyapunov method

Theorem (First Lyapunov method) Let ˙ δx(t) = ❆(xe, u⋆)δ①(t) be a linear system obtained by linearization of ˙ ①(t) = ❢ (①(t), u(t)) around the equilibrium ①e, u⋆. If matrix ❆(xe, u⋆) has all its eignevalues with negative real part, then the equilibrium point ①e, u⋆ is asymptotically stable. If matrix ❆(xe, u⋆) has one or more eigenvalues with positive real part, then the equilibrium point ①e, u⋆ is unstable.

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The first Lyapunov method

The first Lyapunov method

• Notice that the First Lyapunov method says nothing for the case in

which one or more eigenvalues of the linearized model have zero real part.

• In such a case it is necessary to exploit other tools to establish the

stability property of the equilibrium point. A possibility is to applied the second Lyapunov method, which will be covered in the next lecture.

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The first Lyapunov method

The first Lyapunov method

• Notice that the First Lyapunov method says nothing for the case in

which one or more eigenvalues of the linearized model have zero real part.

• In such a case it is necessary to exploit other tools to establish the

stability property of the equilibrium point. A possibility is to applied the second Lyapunov method, which will be covered in the next lecture.

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The first Lyapunov method

Example: the first Lyapunov method applied to the pendulum system

• The linearized pendulum model around the equilibrium point xe = [x⋆

0]T with δu(t) = 0 is ˙ δx(t) =

• 1

− Mgl

I cos(x⋆)

− b

I

• δ①(t)

where x⋆ satisfies sin (x⋆) = 1 Mgl u⋆ Let M = 1, l = 1, g = 10, I = 1

3Ml2 = 1 3, b = 1

• 3. It holds

˙ δx(t) =

• 1

−30cos(x⋆) −1

• δ①(t)

and sin (x⋆) = 1 10u⋆

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The first Lyapunov method

Example: the first Lyapunov method applied to the pendulum system

• The linearized pendulum model around the equilibrium point xe = [x⋆

0]T with δu(t) = 0 is ˙ δx(t) =

• 1

− Mgl

I cos(x⋆)

− b

I

• δ①(t)

where x⋆ satisfies sin (x⋆) = 1 Mgl u⋆ Let M = 1, l = 1, g = 10, I = 1

3Ml2 = 1 3, b = 1

• 3. It holds

˙ δx(t) =

• 1

−30cos(x⋆) −1

• δ①(t)

and sin (x⋆) = 1 10u⋆

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The first Lyapunov method

Example: the first Lyapunov method applied to the pendulum system

1) - Consider the case where zero torque is applied by the electric motor, i.e., u⋆ = 0, and the pendulum is pointing downward, i.e, x⋆ = 0. It follows ˙ δx(t) =

• 1

−30 −1

• δ①(t)

The eigenvalues are λ = −0.5000 ± 5.4544i. All eigenvalues have strictly negative real part. Therefore the equilibrium point xe = [0 0]T with u⋆ = 0 is asymptotically stable for the first Lyapunov method.

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The first Lyapunov method

Example: the first Lyapunov method applied to the pendulum system

2) - Consider the case where zero torque is applied by the electric motor, i.e., u⋆ = 0, and the pendulum is pointing upward, i.e, x⋆ = π. It follows ˙ δx(t) =

• 1

30 −1

• δ①(t)

The eigenvalues are λ1 = 5 and λ2 = −6. There is one eigenvalue with positive real part. Therefore the equilibrium point xe = [π 0]T with u⋆ = 0 is unstable for the first Lyapunov method.

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The first Lyapunov method

Example: the first Lyapunov method applied to the pendulum system

3) - Consider the case where a constant torque is applied by the electric motor, i.e., u⋆ = Mglsin(π/3) = 8.6603, and the pendulum is pointing at π/3 radians, i.e, x⋆ = π

3 . It follows

˙ δx(t) =

• 1

−30cos( π

3 )

−1

• δ①(t)

The eigenvalues are λ = −0.5000 ± 3.8406i and λ2 = −6. All eigenvalues have negative real part. Therefore the equilibrium point xe = [ π

3

0]T with u⋆ = 8.6603 is asymptotically stable for the first Lyapunov method.

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The first Lyapunov method

Example: the first Lyapunov method applied to the pendulum system

4) - Consider the case where a constant torque is applied by the electric motor, i.e., u⋆ = Mglsin(x⋆) = 10sin(x⋆), and the pendulum is pointing at x⋆ radians. It follows ˙ δx(t) =

• 1

−30cos(x⋆) −1

• δ①(t)

The eigenvalues are λ = −1/2 ± 1

2

• 1 − 120cos(x⋆).

For cos(x⋆) >

1 120 all eigenvalues have negative real part and the equilibrium

point is asymptotically stable. For cos(x⋆) =

1 120 the eigenvalues are λ1 = 0 and λ2 = −1. Nothing can

said based on the first Lyapunov method. For cos(x⋆) <

1 120 the eigenvalues there is at least one eigenvalue with

positive real part. Therefore those equilibrium points are unstable.

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Outline

Introduction Linearization of a dynamical system The first Lyapunov method Linearization and first Lyapunov method (Discrete time)

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point in DT

• Consider the nonlinear system

x(k + 1) = f (x(k), u(k)) (2) y(k) = g(x(k), u(k))

• Let xe = [xe1, xe2, . . . , xen] be an equilibrium point of the system in eq. (2) for

u(k) = u⋆ for all k = 0, 1, . . . , ∞.

• Define:

a new state vector δx(k) = x(k) − xe; a new input δu(k) = u(k) − u⋆; a new output δy(k) = y(k) − g(xe, u⋆).

• The new coordinates represent, similarly as in continuous-time, the variations of

the state, input and output variables with respect to their value at the equilibrium/operating point of the dynamical system.

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point in DT

• Consider the nonlinear system

x(k + 1) = f (x(k), u(k)) (2) y(k) = g(x(k), u(k))

• Let xe = [xe1, xe2, . . . , xen] be an equilibrium point of the system in eq. (2) for

u(k) = u⋆ for all k = 0, 1, . . . , ∞.

• Define:

a new state vector δx(k) = x(k) − xe; a new input δu(k) = u(k) − u⋆; a new output δy(k) = y(k) − g(xe, u⋆).

• The new coordinates represent, similarly as in continuous-time, the variations of

the state, input and output variables with respect to their value at the equilibrium/operating point of the dynamical system.

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point in DT

• By making the suggested change of variables, it holds:

δx(k + 1) + xe = f (δx(k) + xe, δu(k) + u⋆) δy(k) + g(xe, u⋆) = g(δx(k) + xe, δu(k) + u⋆)

• Thus

δx(k + 1) = −xe + f (δx(k) + xe, δu(k) + u⋆) δy(k) = g(δx(k) + xe, δu(k) + u⋆) − g(xe, u⋆)

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point in DT

• By making the suggested change of variables, it holds:

δx(k + 1) + xe = f (δx(k) + xe, δu(k) + u⋆) δy(k) + g(xe, u⋆) = g(δx(k) + xe, δu(k) + u⋆)

• Thus

δx(k + 1) = −xe + f (δx(k) + xe, δu(k) + u⋆) δy(k) = g(δx(k) + xe, δu(k) + u⋆) − g(xe, u⋆)

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point

• Expanding the vector function f (x(k), u(k)) by Taylor series around the point xe, u⋆, it holds:

f (x(k), u(k))|x(k)=xe,u(k)=u⋆ = f (xe, u⋆) +

∂ ∂x f (xe, u⋆) (x(t) − xe)

+ ∂

∂u f (xe, u⋆) (u(k) − u⋆)

+O(x(k) − xe2) + O(u(k) − u⋆2) + . . . since f (xe, u⋆) = xe for an equilibrium point in DT, it holds f (x(k), u(k))|x(k)=xe,u(k)=u⋆ = xe +

∂ ∂x f (xe, u⋆)δx(k)

+ ∂

∂u f (xe, u⋆)δu(k)

+O(x(k) − xe2) + O(u(k) − u⋆2) + . . .

• Thus, xe cancels out in

δx(k + 1) = −xe + xe + ∂ ∂x f (xe, u⋆)δx(k) + ∂ ∂u f (xe, u⋆)δu(k)

• If x(k) and u(k) are sufficiently close to, respectively, xe and u⋆, then the first order terms

dominate the second order terms.

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point

• Expanding the vector function f (x(k), u(k)) by Taylor series around the point xe, u⋆, it holds:

f (x(k), u(k))|x(k)=xe,u(k)=u⋆ = f (xe, u⋆) +

∂ ∂x f (xe, u⋆) (x(t) − xe)

+ ∂

∂u f (xe, u⋆) (u(k) − u⋆)

+O(x(k) − xe2) + O(u(k) − u⋆2) + . . . since f (xe, u⋆) = xe for an equilibrium point in DT, it holds f (x(k), u(k))|x(k)=xe,u(k)=u⋆ = xe +

∂ ∂x f (xe, u⋆)δx(k)

+ ∂

∂u f (xe, u⋆)δu(k)

+O(x(k) − xe2) + O(u(k) − u⋆2) + . . .

• Thus, xe cancels out in

δx(k + 1) = −xe + xe + ∂ ∂x f (xe, u⋆)δx(k) + ∂ ∂u f (xe, u⋆)δu(k)

• If x(k) and u(k) are sufficiently close to, respectively, xe and u⋆, then the first order terms

dominate the second order terms.

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point

• Expanding the vector function f (x(k), u(k)) by Taylor series around the point xe, u⋆, it holds:

f (x(k), u(k))|x(k)=xe,u(k)=u⋆ = f (xe, u⋆) +

∂ ∂x f (xe, u⋆) (x(t) − xe)

+ ∂

∂u f (xe, u⋆) (u(k) − u⋆)

+O(x(k) − xe2) + O(u(k) − u⋆2) + . . . since f (xe, u⋆) = xe for an equilibrium point in DT, it holds f (x(k), u(k))|x(k)=xe,u(k)=u⋆ = xe +

∂ ∂x f (xe, u⋆)δx(k)

+ ∂

∂u f (xe, u⋆)δu(k)

+O(x(k) − xe2) + O(u(k) − u⋆2) + . . .

• Thus, xe cancels out in

δx(k + 1) = −xe + xe + ∂ ∂x f (xe, u⋆)δx(k) + ∂ ∂u f (xe, u⋆)δu(k)

• If x(k) and u(k) are sufficiently close to, respectively, xe and u⋆, then the first order terms

dominate the second order terms.

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point

• Expanding the vector function f (x(k), u(k)) by Taylor series around the point xe, u⋆, it holds:

f (x(k), u(k))|x(k)=xe,u(k)=u⋆ = f (xe, u⋆) +

∂ ∂x f (xe, u⋆) (x(t) − xe)

+ ∂

∂u f (xe, u⋆) (u(k) − u⋆)

+O(x(k) − xe2) + O(u(k) − u⋆2) + . . . since f (xe, u⋆) = xe for an equilibrium point in DT, it holds f (x(k), u(k))|x(k)=xe,u(k)=u⋆ = xe +

∂ ∂x f (xe, u⋆)δx(k)

+ ∂

∂u f (xe, u⋆)δu(k)

+O(x(k) − xe2) + O(u(k) − u⋆2) + . . .

• Thus, xe cancels out in

δx(k + 1) = −xe + xe + ∂ ∂x f (xe, u⋆)δx(k) + ∂ ∂u f (xe, u⋆)δu(k)

• If x(k) and u(k) are sufficiently close to, respectively, xe and u⋆, then the first order terms

dominate the second order terms.

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point in DT

• The discrete time linearized dynamical system is:

δx(k + 1) = ❆δx(k) + ❇δu(k) δy(k) = ❈δx(k) + ❉δu(k) where A = ∂f ∂x

• xe,u⋆

=   

∂ ∂x1 (f1(xe, u⋆) ∂ ∂x2 f1(xe, u⋆)

. . .

∂ ∂xn f1(xe, u⋆)

. . . . . . . . . . . .

∂ ∂x1 (fn(xe, u⋆) ∂ ∂x2 fn(xe, u⋆)

. . .

∂ ∂xn fn(xe, u⋆)

   B = ∂f ∂u

• xe,u⋆

=   

∂ ∂u1 f1(xe, u⋆) ∂ ∂u2 f1(xe, u⋆)

. . .

∂ ∂un f1(xe, u⋆)

. . . . . . . . . . . .

∂ ∂u1 fn(xe, u⋆) ∂ ∂u2 fn(xe, u⋆)

. . .

∂ ∂un fn(xe, u⋆)

  

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Linearization and first Lyapunov method (Discrete time)

Linearization around an equilibrium point

• For the output map the same computations as for continuous-time systems

hold, thus C = ∂g ∂x

• xe,u⋆

=   

∂ ∂x1 g1(xe, u⋆) ∂ ∂x2 g1(xe, u⋆)

. . .

∂ ∂xn (g1(xe, u⋆)

. . . . . . . . . . . .

∂ ∂x1 gn(xe, u⋆) ∂ ∂x2 gn(xe, u⋆)

. . .

∂ ∂xn gn(xe, u⋆)

   and D = ∂g ∂x

• xe,u⋆

=   

∂ ∂u1 g1(xe, u⋆) ∂ ∂u2 g1(xe, u⋆)

. . .

∂ ∂un (g1(xe, u⋆)

. . . . . . . . . . . .

∂ ∂u1 gn(xe, u⋆) ∂ ∂u2 gn(xe, u⋆)

. . .

∂ ∂un gn(xe, u⋆)

  

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Linearization and first Lyapunov method (Discrete time)

The first Lyapunov method (Discrete time)

Theorem (First Lyapunov method (Discrete time)) Let δx(k + 1) = ❆(xe, u⋆)δx(k) be a linear DT system obtained by linearization of ①(k + 1) = ❢ (①(k), u(k)) around the equilibrium ①e, u⋆. If matrix ❆(xe, u⋆) has all its eigenvalues with magnitude strictly less then one, then the equilibrium point ①e, u⋆ is asymptotically stable. If matrix ❆(xe, u⋆) has one or more eigenvalues with magnitude greater than one, then the equilibrium point ①e, u⋆ is unstable.

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Linearization and first Lyapunov method (Discrete time)

The first Lyapunov method (Discrete Time)

• Notice that the First Lyapunov method in discrete time says nothing for

the case in which one or more eigenvalues of the linearized model have unitary magnitude.

• In such a case it is necessary to exploit other tools to establish the

stability property of the equilibrium point. A possibility is to applied the second Lyapunov method, which will be covered in the next lecture.

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Linearization and first Lyapunov method (Discrete time)

The first Lyapunov method (Discrete Time)

• Notice that the First Lyapunov method in discrete time says nothing for

the case in which one or more eigenvalues of the linearized model have unitary magnitude.

• In such a case it is necessary to exploit other tools to establish the

stability property of the equilibrium point. A possibility is to applied the second Lyapunov method, which will be covered in the next lecture.

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