Nonlinear Control TheoryIntroduction

Harry G. KwatnyDepartment of Mechanical Engineering &

MechanicsDrexel University

Outline Course resources Overview What is nonlinear control? Linearization

Why nonlinear control? Some examples

ResourcesProfessor Harry G. KwatnyOffice: 151-A Tel: 895-2356 e-mail: hkwatny@coe.drexel.eduURL: http://www.pages.drexel.edu/faculty/hgk22Text Books & Software:

Kwatny, H. G. and Blankenship, “Nonlinear Control & AnalyticalMechanics,” Birkhauser, 2000 – Obtain update from H. KwatnyMathematica, Student Version 5.0

Requirements:Problem setsFinal: Take-home project

Other References.1. Slotine, J-J. E. and Li, W., “Applied Nonlinear Control,” Prentice-Hall, 1991.2. Vidyasagar, M., “Nonlinear Systems Analysis 2nd edition,” Prentice-Hall, 1993.3. Isidori, Alberto, “Nonlinear Control Systems-3rd edition,” Springer-Verlag, 1995.4. Nijmeijer, H. and H. J. van der Schaft, 1990: Nonlinear Dynamical Control

Systems. Springer–Verlag.5. Khalil, H. K., 1996: Nonlinear Systems-2nd edition. MacMillan.

MEM 636 ~ Part I Introduction

Course Overview, Using Mathematica Nonlinear Dynamics, Stability

The State Space, Equilibria & Stability, Hartman-Grobman Theorem Stability ~ Liapunov Methods

Geometric Foundations Manifolds, Vector Fields & Integral Curves Distributions, Frobenius Theorem & Integral Surfaces Coordinate Transformations

Controllability & Observability Controllability & Observability, Canonical Forms

Stabilization via Feedback Linearization Linearization via Feedback Stabilization using IO Linearization, Gain Scheduling

Robust & Adaptive Control Tracking & Disturbance Rejection

General Model of Nonlinear Systemstate

( )input u t ( )output y t( )x tSystem

( )( )

, state equation, output equation

, ,n m q

x f x uy h x u

x R u R y R

==

∈ ∈ ∈

Special Case: Linear System

Most real systems are nonlinear Sometimes a linear approximation is

satisfactory Linear systems are much easier to analyze

( )( )

,

,

x f x u x Ax Buy Cx Duy h x u

= = += +=

Nonlinear System Linear System

Linear Systems are Nice1. Superposition Principle: A linear combination of any two solutions

for a linear system is also a solution.2. Unique Equilibrium: Recall that an equilibrium is a solution x(t),

with u(t) = 0, for which x is constant. A generic Linear system has a unique (isolated) equilibrium at the origin x(t) = 0 and its stability is easily determined.

3. Controllability: There are known necessary and sufficient conditions under which a control exists to steer the state of a linear system from any initial value to any desired final value in finite time.

4. Observability: There are known necessary and sufficient conditions under which the system’s state history can be determined from its input and output history.

5. Control design tools: A variety of controller and observer design techniques exist for linear systems (including classical techniques, pole placement, LQR/LQG, Hinf, etc.)

Why Nonlinear Control Contemporary control problems require it,

Robotics, ground vehicles, propulsion systems, electric power systems, aircraft & spacecraft, autonomous vehicles, manufacturing processes, chemical & material processing,…

Smooth (soft) nonlinearities the system motion may not remain sufficiently close to an equilibrium point that the

linear approximation is valid. Also, linearization often removes essential physical effects – like Coriolis forces. The optimal control may make effective use of nonlinearities Robotics, process systems, spacecraft

Non-smooth (hard) nonlinearities Saturation, backlash, deadzone, hysteresis, friction, switching

Systems that are not linearly controllable/observable may be controllable/observable in a nonlinear sense Nonholonomic vehicles (try parking a car with a linear controller), underactuated

mechanical systems (have fewer controls than dof), compressors near stall, ground vehicles near directional stability limit

Systems that operate near instability (bifurcation points) Power system voltage collapse, aircraft stall & spin, compressor surge & rotating stall,

auto directional, cornering & roll stability Parameter adaptive & intelligent systems are inherently nonlinear

Linearization 1: linear approximation near an equilibrium point

( )( )

0 0 0

The general nonlinear system in standard form, involves the state vector , input vector , and output vector .

, state equations

, output equationsA set of values ( , , ) is called an

x u yx f x u

y h x ux u y

=

=

eq

( )( )

0 0

0 0 0

if they satisfy:0 ,

,We are interested in motions that remain close to the equilibriumpoint.

f x u

y h x u

=

=

uilibrium point

Linearization 2( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( )( ) ( ) ( )( )

( ) ( ) ( ) ( )

( )

0 0 0

0 0

0 0 0

0 0 0 00 0 0 0

0 0 0

Define: , ,

,The equations become:

,

Now, construct a Taylor series for ,, ,

, ,

, ,

x t x x t u t u u t y t y y t

x f x x t u u t

y y t h x x t u u t

f hf x u f x u

f x x u u f x u x u hotx u

h x x u u h x

δ δ δ

δ δ δ

δ δ δ

δ δ δ δ

δ δ

= + = + = +

= + +

+ = + +

∂ ∂+ + = + + +

∂ ∂

+ + =

( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )

0 0 0 00

0 0 0 0 0

0 0 0 0

0 0 0 0

, ,t

Notice that , 0 and , , so

, ,

, ,

h x u h x uu x u ho

x uf x u h x u y

f x u f x ux x u x A x B ux u

y C x D uh x u h x uy x u

x u

δ δ

δ δ δ δ δ δδ δ δ

δ δ δ

∂ ∂+ + +

∂ ∂= =

∂ ∂= + = +∂ ∂ ⇒

= +∂ ∂= +

∂ ∂

Some Examples of Nonlinear Systems

Example 1: Fully-Actuated Robotic System

( )( ) ( ) ( )

( )

( ) ( ) ( )

Models always look like:kinematics:

dynamics: , ,

det 0If we want to regulate velocity, choose

, , , a linear system!

If we want to regulate position, we c

q V q pM q p C q p p F q p u

M q q

u C q p p F q p M q v p v

=+ + =

≠ ∀

= + + ⇒ =

an - more algebra. This is called the computed 'torque method.' We will generalize it.

Decoupling Stabilizing

coordinatesquasi-velocities

qp

Example 2

, velocityV

, headingψ

, rudderδ

, sideslipβ

x

y

1 2

1 2

cos sin 0kinematics: sin cos 0

0 0 1Consider and to be control inputs and suppose 0.

cos 0sin 0

0 1

Lineari

x uy v

ru u u r

vxy u u

ψ ψψ ψ

ψ

ψψ

ψ

− =

= ==

= +

( ) ( )ze about the origin , , = 0,0,0 to get

1 0, 0, 0 0 The

0 1linear system is not controllable! In particular,

0, so it is not possible to steer from the origin along the -axis

x y

x Ax Bu A B

yy

Ψ

= + = = ⇒

=

Velocities in inertial coordinates

Velocities in body coordinates

If the boat has a keel, typically, we have sideslip v=0.

Example 2, cont’d

( ) ( )1 2

But, of course, if you allow large motions you can steer to a point 0, , 0. How?

Try this: cos , sin

x y yt tu t u t

ψπ πε ε

= = =

= =

The system appears not to be linearly uncontrollable but nonlinearly controllable!!

By coordinating the rudder and forward speed, we can cause the vehicle to move along the y-axis.

Example 3: Drive Motor & Load with Non-smooth Friction

K 1s

1

1J s

ϕ ω1 1( )

1s

1

2J s

ϕ ω2 2( )

u-

- ω 1ω 2 θ 1θ 2

Drive Motor Inertial LoadShaft

• The problem is to accurately position the load.

• The problem becomes even more interesting if the friction parameters are uncertain.

motor

50:1

5.4e4 Nm/rad(output)

6.72e4 Nm/rad

Gearbox

Coupling

Load

5.9e-5kgm2

1.69e-4kgm2

1.14e-3kgm2

1.116kgm2

Example 4: Automobile – Multiple Equilibria

θ•

x

Y

XSpace Frame

θ

y

VVs = V

Body Frameu

β

v

rF

lF

a

b

δ

m J,

131.5 132 132.5 133Vs

-2

-1

1

2

131.5 132 132.5 133Vs

-0.75

-0.5

-0.25

0.25

0.5

0.75

( )

( )

, , , ,

equilibria: set 0, (a parameter)solve for , ,

0 , , ,0,

s s d

s s

d

s d

d V F V Fdt

V VF

F V F

ωω β δ

β

δω β

ω β

=

= =

=

Example 5 – Simple 1 dof Rotation

2 2

cosreplace by ,

sin

1A stabilizing controller is easily obtained viaLyapunov design sin

ux

x yy

x yy x

ux y

u y u

θ ωω

θθ

θω

ωω

ω θ ω

==

=→ =

= −==

+ =

= − − ⇒ = − −

Example 6 - GTM

( )( )

( )( )

( ) ( )( )( )

212

212

2 21 12 2

cossin

1 cos , , sin

1 sin , , cos

, , , , ,

D e

L e

m e Z e cgref cg cg ty

qx Vz V

V T V SC q mgm

T V SC q mgmVMq M V Sc C q V Sc C q x x mg x l TI

θγγ

α ρ α δ γ

γ α ρ α δ γ

ρ α δ ρ α δ

α θ γ

===

= − −

= + −

= = + − − +

= −

GTM Equilibrium Surface

19

Straight and Level Flight Analysis

Nonlinear Equilibrium Structure Analysis of Upset Using GTM (1)

Coordinated Turn Analysis

Coordinated turn of GTM @ 85 fps

Coordinated turn of GTM @ 87 fps

Coordinated turn of GTM @ 90 fps

20

Nonlinear Equilibrium Structure Analysis of Upset Using GTM (2)

Coordinated Turn Analysis

21Techno-Sciences, Inc. Proprietary