Nonlinear Systems and Control - Lecture 1

Syllabus

Motivation

Solution of Differential Equations

Nonlinear Systems and Control Lecture 1 Assistant Prof. Dr. Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

Master Course in Electronic and Communication Engineering Credits (3/0/3)

Klaus Schmidt

Department

Department of Electronic and Communication Engineering – Cankaya University Syllabus

Motivation

Solution of Differential Equations

Outline

1

Syllabus

2

Motivation

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Solution of Differential Equations

Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

Department

Syllabus

Motivation

Solution of Differential Equations

Content and Structure Content Properties of nonlinear systems Stability analysis Passivity Feedback linearization Flatness-based control Backstepping Passivity-based control Structure 3 lecture hours: Exercises sheets during the lecture Office hours: Klaus Schmidt

Department

Department of Electronic and Communication Engineering – Cankaya University Syllabus

Motivation

Solution of Differential Equations

Grading and Literature Grading 5 Quizzes (10%) 1 Midterm Exam (35%) 1 Final Exam (55%) Literature Horacio J. Marquez: “Nonlinear Control Systems: Analysis and Design”, Wiley-Interscience, 2003 (ISBN: 0-471-42799-3) (Main Textbook) Alberto Isidori: “Nonlinear Control Systems”, Springer, 1995 (ISBN: 3-54-019916-0) (Main Textbook) Hassan K. Khalil: “Nonlinear Systems”, Prentice Hall, 2002 (ISBN: 0-13-067389-7) Eduardo D. Sontag: “Mathematical Control Theory: Deterministic Finite Dimensional Systems”, Second Edition, Springer, New York, 1998 (online: http://www.math.rutgers.edu/∼sontag/FTP DIR/sontag mathematical control theory springer98.pdf) Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

Department

Syllabus

Motivation

Solution of Differential Equations

Linear System Description State Equation mass

x˙ = A x + B u x: state vector

spring constant c

u: input vector A: dynamics matrix

position

m

y d damper constant

B: input matrix Mass-Spring-Damper Example

Gap 1

Klaus Schmidt

Department

Department of Electronic and Communication Engineering – Cankaya University Syllabus

Motivation

Solution of Differential Equations

Linear System Description Mass-Spring-Damper Example Gap 2

Tools for Linear Systems Solution of the state equation

√

Analysis: Stability, controllability, dynamics

√

Controller design: State feedback control, PID design, etc. Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

√ Department

Syllabus

Motivation

Solution of Differential Equations

Nonlinear Systems: Examples Magnetic Suspension Gap 3

magnet

current ± iM voltage u sphere

m

position y

Gap 4

Klaus Schmidt

Department

Department of Electronic and Communication Engineering – Cankaya University Syllabus

Motivation

Solution of Differential Equations

Nonlinear Systems: Examples State Vector x1 = y

Gap 5

x2 = y˙ x3 = i M State equations x˙ 1 = x2 k λµx32 x˙ 2 = g − x2 − m 2m(1 + µx1 )2
λµx3 x2 1 + µx1 − Rx3 + + u x˙ 3 = λ (1 + µx1 )2 ⇒ Nonlinear state equation x˙ = f (x, u) Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

Department

Syllabus

Motivation

Solution of Differential Equations

Nonlinear Systems: Examples Differential Drive

v2

Gap 6

v θ

y2

v1 l y1 ϕ1 r

wheel Klaus Schmidt

Department

Department of Electronic and Communication Engineering – Cankaya University Syllabus

Motivation

Solution of Differential Equations

Nonlinear Systems: Examples State Vector x1 = y1 x2 = y2 x3 = θ

State equations   r  r x˙ 1 cos θ cos θ 2 2 x˙ 2  =  r sin θ r sin θ  2 2 r x˙ 3 − 2lr 2l | {z } | {z } x˙

⇒ Nonlinear state equation x˙ = f (x, u) = g (x)u

g (x)





ϕ˙ 1 ϕ˙ 2 | {z } u

Aim of this Lecture: Investigation of Nonlinear Control Systems Solution of the state equation Analysis: Stability, controllability, dynamics Controller design: Feedforward and feedback control, modern design methods Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

Department

Syllabus

Motivation

Solution of Differential Equations

Notation General Model x˙ = f (t, x, u) x ∈ Rn : state vector

f : R × Rn × Rm : right hand side (rhs) u ∈ Rm

n: state space order; m: number of inputs Convention Autonomous system: x˙ = f (x) Non-autonomous system: x˙ = f (t, x) Autonomous system with inputs: x˙ = f (x, u) For analysis, consider autonomous and non-autonomous systems Klaus Schmidt

Department

Department of Electronic and Communication Engineering – Cankaya University Syllabus

Motivation

Solution of Differential Equations

Solution Definition Definition Consider a non-autonomous system x˙ = f (t, x).

(1)

For an interval I ⊆ R, the time function x : t 7→ x(t) : I → Rn is a solution of (1) if x is differentiable on I and fulfills (1) (i.e., x(t) ˙ = f (t, x(t))). If (1) has an initial condition x(t0 ) = x0 , then the combination of (1) and x(t0 ) = x0 is denoted as an initial value problem (IVP). Remark: Direct solution for linear time-invariant systems x˙ = f (t, x) = A x,

x(0) = x0 ⇒ x(t) = e At x0

Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

Department

Syllabus

Motivation

Solution of Differential Equations

Existence of Solutions: Peano Illustration of Solution

Gap 7

Gap 8

Theorem (Peano) Consider the non-autonomous system in (1). If f is continuous in a neighborhood R ⊆ R × Rn of (t0 , x0 ), then there is at least one solution of the IVP. Remark Neighborhood can be very small Klaus Schmidt

Department

Department of Electronic and Communication Engineering – Cankaya University Syllabus

Motivation

Solution of Differential Equations

Existence of Solutions: Example Gap 9

Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

Department

Syllabus

Motivation

Solution of Differential Equations

Uniqueness of Solutions: Lipschitz-continuity Definition (Lipschitz-continuity) A function f : (t, x) 7→ f (t, x) : R × Rn → Rn is said to be locally Lipschitz-continuous with respect to x in the region R ⊆ R × Rn if there is a constant L > 0 (Lipschitz-constant) such that ||f (t, xˆ) − f (t, x˜)|| ≤ L||ˆ x − x˜||, ∀(t, xˆ), (t, x˜) ∈ R If R = R × Rn , then f is globally Lipschitz-continuous. Remark

Gap 10

∂f is continuous in x If ∂x ⇒ f is locally Lipschitz-continuous

∂f is bounded on R × Rn If ∂x ⇒ f is globally Lipschitz-continuous

Klaus Schmidt

Department

Department of Electronic and Communication Engineering – Cankaya University Syllabus

Motivation

Solution of Differential Equations

Uniqueness of Solutions: Picard-Lindel¨of Theorem (Picard-Lindel¨of) Consider a non-autonomous system with initial condition x(t0 ) = x0 and let R ⊆ R × Rn with (t0 , x0 ) ∈ R. If f is continuous in t on R and f is Lipschitz-continuous in x on R, then there is an interval I ⊆ R with t0 ∈ I such that the IVP has a unique solution on I. Remark If f is globally Lipschitz-continuous ⇒ There is a unique solution of the IVP for all times

If f is locally Lipschitz-continuous everywhere in R × Rn ⇒ Either there is a unique solution of the IVP for all times or IVP has finite escape time (see example) Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

Department

Syllabus

Motivation

Solution of Differential Equations

Uniqueness of Solutions: Example Gap 11

Klaus Schmidt

Department

Department of Electronic and Communication Engineering – Cankaya University Syllabus

Motivation

Solution of Differential Equations

Equilibrium Points Definition Let x˙ = f (t, x) be a non-autonomous system that is defined over a region R ∈ R × Rn . A point x = xe ∈ R is called an equilibrium point of the system if f (xe ) = 0. ⇒ If the system state is xe , then it remains at xe for all future times

Gap 12

Nonlinear Systems expected in this Lecture Unique solution that exists for all times Potentially multiple equilibrium points Klaus Schmidt Department of Electronic and Communication Engineering – Cankaya University

Department