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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 1 / 38

Outline Introduction Linearization of a dynamical system The first Lyapunov method Linearization and first Lyapunov method (Discrete time) 2 / 38

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). 3 / 38

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). 3 / 38

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). 3 / 38

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). 4 / 38

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). 4 / 38

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). 4 / 38

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. 5 / 38

Introduction A classic example: the pendulum The equation of motion of this system is: I ¨ θ ( t ) + b ˙ θ + Mgl sin ( θ ( t )) = u 6 / 38

Introduction A classic example: the pendulum Let u ( t ) = u : constant input torque provided by a DC electric motor; x 1 ( t ) = θ ( t ): angular position of the pendulum; x 2 ( t ) = ˙ θ ( t ): angular speed of the pendulum; y ( t ) = θ ( t ): output of the system. The SV model becomes: x 1 ( t ) ˙ = x 2 ( t ) − Mgl I sin ( x 1 ( t )) − bx 2 ( t ) + 1 x 2 ( t ) ˙ = I u ( t ) y ( t ) = x 1 ( t ) 7 / 38

Introduction A classic example: the pendulum The equilibrium points of the system with constant input u are the solutions of f ( x e ) = 0: 0 = x e 2 − Mgl I sin ( x e 1 ) − b I x e 2 + 1 0 = I u Thus � x e 1 � x ⋆ � � x e = = x e 2 0 where x ⋆ is the solution of 1 sin ( x ⋆ ) = Mgl u For u = 0 it holds x e 1 = x ⋆ = 0 ± k π for an integer k . 8 / 38

Introduction A classic example: the pendulum The equilibrium points of the system with constant input u are the solutions of f ( x e ) = 0: 0 = x e 2 − Mgl I sin ( x e 1 ) − b I x e 2 + 1 0 = I u Thus � x e 1 � x ⋆ � � x e = = x e 2 0 where x ⋆ is the solution of 1 sin ( x ⋆ ) = Mgl u For u = 0 it holds x e 1 = x ⋆ = 0 ± k π for an integer k . 8 / 38

Introduction A classic example: the pendulum The equilibrium points of the system with constant input u are the solutions of f ( x e ) = 0: 0 = x e 2 − Mgl I sin ( x e 1 ) − b I x e 2 + 1 0 = I u Thus � x e 1 � x ⋆ � � x e = = x e 2 0 where x ⋆ is the solution of 1 sin ( x ⋆ ) = Mgl u For u = 0 it holds x e 1 = x ⋆ = 0 ± k π for an integer k . 8 / 38

Introduction A classic example: the pendulum • The equilibrium points of the system with constant input u are the solutions of f ( x e ) = 0: • Physically, this means that the pendulum is at equilibrium whenever the angle x 1 = θ is either 0 (pendulum pointing downward) or π (pendulum pointing upward), and the angular velocity x 2 = ˙ θ is zero. 0] T is stable, while the equilibrium • Qualitatively, the equilibrium x e = [0 π ] T is unstable. x e = [0 9 / 38

Introduction A classic example: the pendulum • The equilibrium points of the system with constant input u are the solutions of f ( x e ) = 0: • Physically, this means that the pendulum is at equilibrium whenever the angle x 1 = θ is either 0 (pendulum pointing downward) or π (pendulum pointing upward), and the angular velocity x 2 = ˙ θ is zero. 0] T is stable, while the equilibrium • Qualitatively, the equilibrium x e = [0 π ] T is unstable. x e = [0 9 / 38

Introduction A classic example: the pendulum • The equilibrium points of the system with constant input u are the solutions of f ( x e ) = 0: • Physically, this means that the pendulum is at equilibrium whenever the angle x 1 = θ is either 0 (pendulum pointing downward) or π (pendulum pointing upward), and the angular velocity x 2 = ˙ θ is zero. 0] T is stable, while the equilibrium • Qualitatively, the equilibrium x e = [0 π ] T is unstable. x e = [0 9 / 38

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 x 1 = x ⋆ an equilibrium point of the system. • For instance, by imparting a torque u = Mgl , the configuration x e 1 = π/ 2, and x e 2 = 0 is an equilibrium of the pendulum. 10 / 38

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 x 1 = x ⋆ an equilibrium point of the system. • For instance, by imparting a torque u = Mgl , the configuration x e 1 = π/ 2, and x e 2 = 0 is an equilibrium of the pendulum. 10 / 38

Outline Introduction Linearization of a dynamical system The first Lyapunov method Linearization and first Lyapunov method (Discrete time) 11 / 38

Linearization of a dynamical system Linearization • Although almost every physical system contains nonlinearities, oftentimes 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. 12 / 38

Linearization of a dynamical system Linearization • Although almost every physical system contains nonlinearities, oftentimes 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. 12 / 38

Linearization of a dynamical system Linearization • Although almost every physical system contains nonlinearities, oftentimes 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. 12 / 38