AC2 06 Nonlinear

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Lecture: Nonlinear systems

Automatic Control 2

Nonlinear systems

Prof. Alberto Bemporad

University of Trento

Academic year 2010-2011

Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 1 / 18


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Lecture: Nonlinear systems Stability analysis

Nonlinear dynamical systems

x

(t

)u

(t

)

!"!#$!%&'()*!&+$,(-'",%..

x = f(x, u)

Most existing processes in practical applications are described bynonlinear

dynamics x= f(x, u)

Often the dynamics of the system can be linearized around an operating point

and a linear controller designed for the linearized process

Question #1: will the closed-loop system composed by the nonlinear process+ linear controller be asymptotically stable ? (nonlinear stability analysis)

Question #2: can we design a stabilizing nonlinear controller based on the

nonlinear open-loop process ? (nonlinear control design)

This lecture is based on the book Applied Nonlinear Control by J.J.E. Slotine and W. Li,

1991



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Positive definite functions

Key idea: if the energy of a system dissipates over time, the systemasymptotically reaches a minimum-energy configuration

Assumptions: consider the autonomous nonlinear system x= f(x), with f()differentiable, and let x= 0 be an equilibrium (f(0) = 0)

Some definitions of positive definiteness of a function V: n V is called locally positive definite ifV(0) = 0 and there exists a ball

B =x: x2

around the origin such that V(x) > 0 xB \ 0

V is called globally positive definite ifB = n (i.e. )

V is called negative definite ifV is positive definiteV is called positive semi-definite ifV(x) 0 xB, x= 0V is called positive semi-negative if

V is positive semi-definite

Example: let x= [x1 x2], V : 2

V(x) = x21

+x22

is globally positive definite

V(x) = x21

+x22x3

1is locally positive definite

V(x) = x41

+ sin2(x2) is locally positive definite and globally positive semi-definite



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Lyapunovs direct method

Theorem

Given the nonlinear system x= f(x), f(0) = 0, let V : n be positive definitein a ball B around the origin, > 0, V C1(). If the function

V(x) = V(x)x= V(x)f(x)

is negative definite on B, then the origin is an asymptotically stable equilibrium

point with domain of attraction B (limt+x(t) = 0 for all x(0) B). IfV(x) isonly negative semi-definite on B, then the the origin is a stable equilibrium point.

Such a function V : n is called a Lyapunov function for the system x= f(x)Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 4 / 18

L N li S bili l i


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Example of Lyapunovs direct method

Consider the following autonomous dynamical system

x1 = x1(x21

+x22 2) 4x1x22

x2 = 4x21x2 +x2(x

21

+x22 2)

as f1(0, 0) = f2(0, 0) = 0, x= 0 is an equilibrium

consider then the candidate Lyapunov function

V(x1,x2) = x21

+x22

which is globally positive definite. Its time derivative V is

V(x1,x2) = 2(x21

+x22

)(x21

+x22

2)

It is easy to check that V(x1,x2) is negative definite ifx22 = x21 +x22 < 2.Then for anyB with 0 < 0

leading to the asymptotic convergence of the tracking error e(t) = q(t)

qd(t)

and its derivative e(t) = q(t) q(t) to zeroe(t)

e(t)

= et

1 +t t

2t 1t

e(0)

e(0)

MATLAB syms lam t

A=[0 1;-lam^2 -2*lam];

factor(expm(A*t))

In robotics, feedback linearization is also known as computed torque, and can

be applied to robots with an arbitrary number of joints


Lecture: Nonlinear systems Feedback linearization


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English-Italian Vocabulary

nonlinear system sistema non lineare

Lyapunov function funzione di Lyapunov

feedback linearization feedback linearization

Translation is obvious otherwise.


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AC2 06 Nonlinear