1 In this lecture we will compare two linearizing controller for a single-link robot: Linearization...

ECE 893 Industrial Applications of Nonlinear Control Dr. Ugur HasirciClemson University, Electrical and Computer Engineering Department Spring 2013

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In this lecture we will compare two linearizing controller for a single-link robot:• Linearization via Taylor Series Expansion• Feedback Linearization


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Linear control theory has been predominantly concerned with Linear Time Invariant (LTI) systems of the form

with x being a vector of states and A being the system matrix. LTI systems have quaite simple properties such as

• A linear system has a unique equilibrium point if A is nonsingular;• The equilibrium point is stable if all eigenvalues of A have negative real parts,

regardless of initial conditions;• In the presence of an external input u(t), i.e., with

the system has a number of interesting properties. For example a sinusoidal input leads to a sinusoidal output of the same frequency.

x Ax

x Ax Bu

Slotine, Li, 1993.


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Never forget

THERE IS NO LINEAR SYSTEMS

IN NATURE


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One of the characteristic properties of nonlinear systems is “Multiple Equilibrium Points”

Nonlinear systems frequently have more than one equilibrium point (an equilibrium point is a point where the system can stay forever without moving). This can be seen by the following simple examples.

Consider the first order linear system x x

Solution of this differential equation is 0( ) tx t x e

Following figure shows the time variation of this solution for various initial conditions. The system clearly has a unique equilibrium point at x=0.

Slotine, Li, 1993.


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Now consider the following nonlinear systems:2x x x

Solution of this differential equation is

0

0 0

( )1

t

t

x ex t

x x e

Following figure shows the time variation of this solution. The system has two equilibrium points, x=0 and x=1, and its qualitative behavior strongly depends on its initial condition.


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1. Linearization of Nonlinear Systems via Taylor Series Expansion

General form of an n-dimensional nonlinear system is

( ) ( )x f x g x u

and of an n-dimensional linear time-invariant system is

x Ax Bu

The linearized form of a nonlinear system can be found as

ex x

fA

x

eu u

gB

u


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Example: Linearize System

22 1 2

2 1 1 1 2

2 2 1 2

1 2 1 2

[0,0]

cos( )

1 sin( )

cos( ) 2 sin( )

2 1 sin( ) 1 cos( )

1 0

1 1Tx

x x xf

x x x x x

x x x xf

x x x xx

f

x

Eigenvalues = 1,1

Origin is unstable


Let’s linearize a single-link robotic manipulator model now.

Dynamic model is as follows:

sin( )Jq Bq N q

: motor inertia

: motor position

: viscous friction coeffient

: load constant

: applied torque.

J

q

B

N


By selecting the state variables as

1

2

x q

x q

we get the state-space representation as follows:

1 2

2 2 1

1sin

x x

B Nx x x

J J J

sin( )Jq Bq N q


By setting the control input signal, u, to zero, let’s find the equilibrium points

1 2

2 2 1s

0

0in

x x

B Nx x x

J J

From the first equation, we get

2 0ex

Finally, from the second equation, we get

1 1sin 0 0 , 0,1,2,.......e ex x k k


Then the equilibrium points are

1

2

0

0e

e

x k

x

Let’s linearize the system around the origin

1

2

0

0e

e

x

x


Remember that the system dynamics is

1 2

2 2 1

1sin

x x

B Nx x x

J J J

Then

2

1

2 2 1

( )

( ) ( ) sin

xf x

B Nf x f x x x

J J


The Jacobian is

1 1

1 21

2 2

1 2

0 1

cos( )

f f

x x N Bx

f f J J

x x

1 2

2 2 1

1sin

x x

B Nx x x

J J J


Since we want to linearize the system around the equilibrium point

1

2

0

0e

e

x

x

then

1

2

1 0

0

0 1 0 1

cos( )e

e

x

x

N B N Bx

J J J J


Then the linearized form of the system is

1 1

2 2

0 1 0

1x x

uN Bx x

J J J

Note that this dynamical model is a general LTI system of the form

x Ax Bu

By using MATLAB Symbolic Toolbox, we find the eigenvalues of A matrix as

2

1,2

4

2

B JNB

J

not viscous friction !

viscous friction !


1 1

2 2

0 1 0

1x x

uN Bx x

J J J

Let’s design a simple linear state feedback controller in the form of

u Kx

x Ax Bu

so that we get

x A BK x

In this way, by properly selecting the entries of K vector, we will be able to locate the eigenvalues of newly-created system matrix, (A-BK), to the left-half plane to get stability.


But this stability result will be valid only around the small neighborhood of the linearization point,

1

2

0

0e

e

x

x

and we will not have a global stability result.


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There are many algorithms in the literature proposed to find the entries of K vector.

The most conventional algorithm can be implemented in MATLAB as follows .

% Desired closed-loop polesp1 = -1; p2 = -2;

% Entries of KK = place(A,B,[p1 p2]);

The control input signal u=-Kx drives the trajectories to the equilibrium point x1=x2=0. If the desired trajectory for position is xd, then the control law is modified as

11 2

2

dx xu k k

x


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2. Linearization of Nonlinear Systems via Feedback Linearization

Feedback linearization is a nonlinear control method and the control input signal to be designed will contain a nonlinear term.

Again consider the general form of a nonlinear system

( ) ( )x f x g x u

If we can rewrite the system in the simplest form, i.e., the form that we will not need a coordinate transformation to transform the system into a linear form as

( ) ( )x Ax B x u x then

1( ) ( )u x x renders the linear time-invariant and controllable system

0. x Ax B

if (A,B) controllable and

Let’s see if we can write single link robot dynamics in this form.


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1 2

2 2 1

1sin

x x

B Nx x x

J J J

( ) ( )x Ax B x u x

1 11

2 2

0 1 0sin1

0

x xN xB

x xJ J

1

2rank B AB


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1 11

2 2

0 1 0sin1

0

x xN xB

x xJ J

Then

1sin( )N x

renders

1 1

2 2

0 1 0

10

x xB

x xJ J


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Compare the performances of these two controllers via simulation.

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1 In this lecture we will compare two linearizing controller for a single-link robot: Linearization...

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