control of nonlinear dynamic systems

8/12/2019 Control of Nonlinear Dynamic Systems

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Control of Nonlinear Dynamic Systems:Theory and Applications

J. K. Hedrick and A. Girard


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Control of Nonlinear Dynamic Systems: Theory and Applications

J. K. Hedrick and A. Girard 2010

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1 `Introduction

We consider systems that can be written in the following general form, where x is the

state of the system, u is the control input, w is a disturbance, and f is a nonlinear function.

We are considering dynamical systems that are modeled by a finite number of coupled,

first-order ordinary differential equations. The notation above is a vector notation, whichallows us to represent the system in a compact form.

Key points

Few physical systems are truly linear. The most common method to analyze and design controllers for system is to

start with linearizing the system about some point, which yields a linear model,

and then to use linear control techniques.

There are systems for which the nonlinearities are important and cannot beignored. For these systems, nonlinear analysis and design techniques exist and

can be used. These techniques are the focus of this textbook.


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In many cases, the disturbance is not considered explicitly in the system analysis, that is,

we consider the system described by the equation . In some cases we will

look at the properties of the system when f does not depend explicitly on u, that is,

. This is called the unforced response of the system. This does not

necessarily mean that the input to the system is zero. It could be that the input has been

specified as a function of time, u = u(t), or as a given feedback function of the state,u =u(x), or both.

When f does not explicitly depend on t, that is, if , the system is said to be

autonomous or time invariant. An autonomous system is invariant to shifts in the time

origin.

We call x the state variables of the system. The state variables represent the minimum

amount of information that needs to be retained at any time t in order to determine the

future behavior of a system. Although the number of state variables is unique (that is, ithas to be the minimum and necessary number of variables), for a given system, the

choice of state variables is not.

Linear Analysis of Physical Systems

The linear analysis approach starts with considering the general nonlinear form for a

dynamic system, and seeking to transform this system into a linear system for thepurposes of analysis and controller design. This transformation is called linearization

and is possible at a selected operating point of the system.

Equilibrium pointsare an important class of solutions of a differential equation. They

are defined as the points xesuch that:

A good place to start the study of a nonlinear system is by finding its equilibrium points.

This in itself might be a formidable task. The system may have more than oneequilibrium point. Linearization is often performed about the equilibrium points of the

system. They allow one to characterize the behavior of the solutions in the neighborhood

of the equilibrium point.

If we write x, u and w as a constant term, followed by a perturbation, in the following

form:

We first seek equilibrium points that satisfy the following property:


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We then perform a multivariable Taylor series expansion about one of the equilibrium

points x0, u0, w0. Without loss of generality, assume the coordinates are transformed so

that x0= 0. HOT designates Higher Order Terms.

We can set:

The dimensions of A are n by n, B is n by m, and !is n by p.

We obtain a linear model for the system about the equilibrium point (x 0, u0, w0) by

neglecting the higher order terms.

Now many powerful techniques exist for controller design, such as optimal linear state

space control design techniques, H!control design techniques, etc This produces a

feedback law of the form:

This yields:

Evaluation and simulation is performed in the following sequence.


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Figure 1.1. Linearized system design framework

Suppose the simulation did not yield the expected results. Then the higher order termsthat were neglected must have been significant. Two types of problems may have arisen.

a. When is the existence of a Taylor series guaranteed?

The function (and the nonlinearities of the system) must be smooth and free of

discontinuities. Hard (non-smooth or discontinuous) nonlinearities may be caused byfriction, gears etc

Figure 1.2. Examples of hard nonlinearities.

b.

Some systems have smooth nonlinearities but wide operating ranges.

Linearizations are only valid in a neighborhood of the equilibrium point.


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Figure 1.3. Smooth nonlinearity over a wide operating range. Which slope should be

pick for the linearization?

The nonlinear design framework is summarized below.

Figure 1.4. Nonlinear system design framework


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2 General Properties of

Linear and Nonlinear

Systems

Aside: A Brief History of Dynamics (Strogatz)

The subject of dynamics began in the mid 1600s, when Newton invented differentialequations, discovered his laws of motion and universal gravitation, and combined them to

explain Keplers laws of planetary motion. Specifically, Newton solved the two-bodyproblem: the problem of calculating the motion of the earth around the sun, given the

inverse square law of gravitational attraction between them. Subsequent generations of

mathematicians and physicists tried to extend Newton analytical methods to the three

body problem (e.g. sun, earth and moon), but curiously the problem turned out to be

Key points

Linear systems satisfy the properties of superposition and homogeneity. Any

system that does not satisfy these properties is nonlinear.

In general, linear systems have one equilibrium point at the origin. Nonlinear

systems may have many equilibrium points. Stability needs to be precisely defined for nonlinear systems. The principle of superposition does not necessarily hold for forced response for

nonlinear systems. Nonlinearities can be broadly classified.


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much more difficult to solve. After decades of effort, it was eventually realized that thethree-body problem was essentially impossible to solve, in the sense of obtaining explicit

formulas for the motions of the three bodies. At this point, the situation seemed hopeless.

The breakthrough came with the work of Poincare in the late 1800s. He introduced a newviewpoint that emphasized qualitative rather than quantitative questions. For example,

instead of asking for the exact positions of the planets at all times, he asked: Is the solarsystem stable forever, or will some planets eventually fly off to infinity. Poincare

developed a powerful geometric approach to analyzing such questions. This approach hasflowered into the modern subject of dynamics, with applications reaching far beyond

celestial mechanics. Poincare was also the first person to glimpse the possibility of chaos,

in which a deterministic system exhibits aperiodic behavior that depends sensitively oninitial conditions, thereby rendering long term prediction impossible.

But chaos remained in the background for the first half of this century. Instead, dynamicswas largely concerned with nonlinear oscillators and their applications in physics and

engineering. Nonlinear oscillators played a vital role in the development of such

technologies as radio, radar, phase-locked loops, and lasers. On the theoretical side,nonlinear oscillators also stimulated the invention of new mathematical techniques

pioneers in this area include van der Pol, Andropov, Littlewood, Cartwright, Levinson,

and Smale. Meanwhile, in a separate development, Poincares geometric methods were

being extended to yield a much deeper understanding of classical mecahsnics, thanks tothe work of Birkhoff and later Kolmogorov, Arnold, and Moser.

The invention of the high-speed computer in the 1950s was a watershed in the history ofdynamics. The computer allowed one to experiment with equations in a way that was

impossible before, and therefore to develop some intuition about nonlinear systems. Such

experiments led to Lorenzs discovery in 1963 of chaotic motion on a strange attractor.

He studied a simplified model of convection rolls in the atmosphere to gain insight intothe notorious unpredictability of the weather. Lorenz found that the solutions to his

equations never settled down to an equilibrium or periodic state instead, they continued

to oscillate in an irregular, aperiodic fashion. Moreover, if he started his simulations fromtwo slightly different initial conditions, the resulting behaviors would soon become

totally different. The implication was that the system was inherently unpredictable tiny

errors in measuring the current state of the atmosphere (or any other chaotic system)would be amplified rapidly, eventually leading to embarrassing forecasts. But Lorenz

also showed that there was structure in the chaos when plotted in three dimensions, the

solutions to his equations fell onto a butterfly shaped set of points. He argued that this sethad to be an infinite complex of surfaces today, we would regard it as an example of

a fractal.

Lorenzs work had little impact until the 1970s, the boom years for chaos. Here are some

of the main developments of that glorious decade. In 1971 Ruelle and Takens proposed anew theory for the onset of turbulence in fluids, based on abstract considerations about

strange attractors. A few years later, May found examples of chaos in iterated mappings

arising in population biology, and wrote an influential review article that stressed the

pedagogical importance of studying simple nonlinear systems, to counterbalance the


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often misleading linear intuition fostered by traditional education. Next came the mostsurprising discovery of all, due to the physicist Feigenbaum. He discovered that there are

certain laws governing the transition from regular to chaotic behavior. Roughly speaking,

completely different systems can go chaotic in the same way. His work established a linkbetween chaos and phase transitions, and enticed a generation of physicists to the study

of dynamics. Finally, experimentalists such as Gollub, Libchaber, Swinney, Linsay,Moon, and Westervelt tested the new ideas about chaos in experiments on fluids,

chemical reactions, electronic circuits, mechanical oscillators, and semiconductors.

Although chaos stole the spotlight, there were two other major developments in dynamics

in the 1970s. Mandelbrot codified and popularized fractals, produced magnificientcomputer graphics of them, and showed how they could be applied to a variety of

subjects. And in the emerging area of mathematical biology, Winfree applied the methods

of dynamics to biological oscillations, especially circadian (roughly 24 hour) rhythms andheart rhythms.

By the 1980s, many people were working on dynamics, with contributions too numerousto list.

Lorenz attractor


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Dynamics A Capsule History

1666 Newton Invention of calculus

Explanation of planetary motion

1700s Flowering of calculus and classical mechanics

1800s Analytical studies of planetary motion

1890s Poincare Geometric approach, nightmares of chaos

1920-1950 Nonlinear oscillators in physics and engineeringInvention of radio, radar, laser

1920-1960 Birkhoff Complex behavior in Hamiltonian mechanicsKolmogorov

Arnold

Moser

1963 Lorenz Strange attractor in a simple model of convection

1970s Ruelle/Takens Turbulence and chaos

May Chaos in logistic map

Feigenbaum Universality and renormalizationConnection between chaos and phase transitions

Expertimental studies of chaos

Winfree Nonlinear oscillators in biology

Mandelbrot Fractals

1980s Widespread interest in chaos, fractals, oscillatorsand their applications.


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Dynamical Systems

There are two main types of dynamical systems: differential equations and iterated maps

(also known as difference equations). Differential equations describe the evolution ofsystems in continuous time, whereas iterated maps arise in problem where time is

discrete. Differential equations are used much more widely in science and engineering,and we shall therefore concentrate on them.

Confining our attention to differential equations, the main distinction is between ordinary

and partial differential equations. Our concern here is purely with temporal behavior, and

so we will deal with ordinary differential equations exclusively.

A Brief Reminder on Properties of Linear Time Invariant Systems

Linear Time Invariant (LTI) systems are commonly described by the equation:

In this equation, x is the vector of n state variables, u is the control input, and A is a

matrix of size (n-by-n), and B is a vector of appropriate dimensions. The equation

determines the dynamics of the response. It is sometimes called a state-space realizationof the system. We assume that the reader is familiar with basic concepts of system

analysis and controller design for LTI systems.

Equilibrium point

An important notion when considering system dynamics is that of equilibrium point.

Equilibrium points are considered for autonomous systems (no explicit control input).

Definition:

A point x0in the state space is an equilibrium point of the autonomous system if

when the state x reaches x0, it stays at x0 for all future time.

That is, for an LTI system, the equilibrium point is the solutions of the equation:

If A has rank n, then x0= 0. Otherwise, the solution lies in the null space of A.


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Stability

The system is stable if .

A more formal statement would talk about the stability of the equilibrium point in thesense of Lyapunov. There are many kinds of stability (for example, bounded input,

bounded output) and many kinds of tests.

Forced response

The analysis of forced response for linear systems is based on the principle of

superposition and the application of convolution.

For example, consider the sinusoidal response of LTIS.

The output sinusoids amplitude is different than that of the input and the signal alsoexhibits a phase shift. The Bode plot is a graphical representation of these changes. For

LTIS, it is unique and single-valued.


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Example of a Bode plot. The horizontal axis is frequency, !. The vertical axis of the top

plot represents the magnitude of |y/u| (in dB, that is, 20 log of), and the lower plot

represents the phase shift.

As another example, consider the Gaussian response of LTIS.

If the input into the system is a Gaussian, then the output is also a Gaussian. This is auseful result.

Why are nonlinear problems so hard?

Why are nonlinear systems so much harder to analyze than linear ones? The essential

difference is that linear systems can be broken down into parts. Then each part can besolved separately and finally recombined to get the answer. This idea allows fantastic

simplification of complex problems, and underlies such methods as normal modes,Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear

system is precisely equal to the sum of its parts.

But many things in nature dont act this way. Whenever parts of a system interfere, or

cooperate, or compete, there are nonlinear interactions going on. Most of everyday life is

nonlinear, and the principle of superposition fails spectacularly. If you listen to your two

favorite songs at the same time, you wont get double the pleasure! Within the realm ofphysics, nonlinearity if vital to the operation of a laser, the formation of turbulence in a

fluid, and the superconductivity of Josephson junctions, for example.


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Nonlinear System Properties

Equilibrium point

Reminder:

A point x0in the state space is an equilibrium point of the autonomous system

if when the state x reaches x0, it stays at x0 for all future time.

That is, for a nonlinear system, the equilibrium point is the solutions of the equation:

One has to solve n nonlinear algebraic equations in n unknowns. There might be between

0 and infinity solutions.

Example: Pendulum

L is the length of the pendulum, g is the acceleration of gravity, and "is the angle of the

pendulum from the vertical.

The equivalent (nonlinear) system is:

x1=x

2

x2=

"

k

mL2 x2"

g

Lsin

x1

#$%

&%

Nonlinearity makes the pendulum equation very difficult to solve analytically. The usualway around this is to fudge, by invoking the small angle approximation for sin x" x

for x


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Example: Mass with Coulomb friction

Stability

One must take special care to define what is meant by stability.

For nonlinear systems, stability is considered about an equilibrium point, in thesense of Lyapunov or in an input-output sense.

Initial conditions can affect stability (this is different than for linear systems), and

so can external inputs. Finally, it is possible to have limit cycles.

Example:

A limit cycle is a unique, self-excited oscillation. It is also a closed trajectory in the state-

space.


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In general, a limit cycle is an unwanted feature in a mechanical system, as it causesfatigue.

Beware: a limit cycle is different from a linear oscillation.

Note that in other application domains, for example in communications, a limit cyclemight be a desirable feature.

In summary, be on the lookout for this kind of behavior in nonlinear systems. Rememberthat in nonlinear systems, stability, about an equilibrium point:

Is dependent on initial conditions

Local vs. global stability is important Possibility of limit cycles


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Forced response

The principle of superposition does not hold in general. For example for initial conditions

x0, the system may be stable, but for initial conditions 2x0, the system could be unstable.


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Classification of Nonlinearities

Single-valued, time invariant

Memory or hysteresis

Example:


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Single-input vs. multiple input nonlinearities


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SUMMARY: General Properties of Linear and Nonlinear Systems

LINEAR SYSTEMS NONLINEAR SYSTEMS

EQUILIBIUM POINTS

A point where the system can stay forever

without moving.

UNIQUE

If A has rank n, then xe=0, otherwise the

solution lies in the null space of A.

MULTIPLE

f(xe)=0

n nonlinear equations in n unknowns0!+"solutions

ESCAPE TIME x!+"as t!+" The state can go to infinity in finite time.

STABILITY The equilibrium point is stable if all

eigenvalues of A have negative real part,

regardless of initial conditions.

About an equilibrium point:

Dependent on IC

Local vs. Global stabilityimportant

Possibility of limit cycles

LIMIT CYCLES

A unique, self-excited

oscillation

A closed trajectory in the state

space Independent of IC

FORCED RESPONSE

The principle of superposition

holds.

I/O stability!bounded input,

bounded output Sinusoidal input!sinusoidal

output of same frequency

The principle of superposition

does not hold in general.

The I/O ratio is not unique in

general, may also not be single

valued.

CHAOS

Complicated steady-state behavior, may

exhibit randomness despite the

deterministic nature of the system.


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A Dynamical View of the World (Strogatz)

One axis tells us the number of variables needed to characterize the state of thesystem. Equivalently, this number is called the dimension of the phase space. The

other dimension tells us whether the system is linear or nonlinear.

Admittedly, some aspects of the picture are debatable. You may think that sometopics should be added, or place differently, or even that more axes are needed. The

point is to think about classifying systems on the basis of their dynamics.

There are some striking patterns in the above figure. All the simplest systems occur in

the upper left hand corner. These are the small linear systems that we learn about in

the first few years of college. Roughly speaking, these linear systems exhibit growth,decay or equilibrium when n = 1, or oscillations when n = 2. For example, an RC

circuit has n = 1 and cannot oscillate, whereas an RLC circuit has n = 2 and can

oscillate.

The next most familiar part of the picture is the upper right hand corner. This is the

domain of classical applied mechanics and mathematical physics where the linearpartial differential equations live. Here we find Maxwells equations of electricity and

magnetism, the heat equation and so on. These partial differential equations involvean infinite continuum of variables because each point in space contributes

additional degrees of freedom. Even though such systems are large, they are tractable,

thanks to such linear techniques as Fourier analysis and transform methods.


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In contrast, the lower half of the figure (the nonlinear half) is often ignored ordeferred to other courses. No more In this class, we will start at the lower left hand

corner and move to the right. As we increase the phase space dimension from n = 1 t

n = 3, we encounter new phenomena at every step, from fixed points and bifurcationswhen n = 1 to nonlinear oscillations when n = 2 to chaos and fractals when n = 3. In

all cases, a geometric approach proves to be powerful and gives us most of theinformation we want, even though we cant usually solve the equations in the

traditional sense of finding a formula for the answer.Youll notice that the figure also contains a region forbiddingly marked The

frontier. Its like in those old maps of the world, where the mapmakers wrote Here

there be dragons on the unexplored parts of the globe. These topics are notcompletely unexplored, but it is fair to say that they lie at the limits of current

understanding. These problems are very hard, because they are both large and

nonlinear. The resulting behavior is typically complicated in both space and time, asin the motion of a turbulent fluid or the patterns of electrical activity in a fibrillating

heart. Towards the end of the course, time permitting, we will touch on some of these

problems.


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3 Phase-Plane Analysis

Phase plane analysis is a technique for the analysis of the qualitative behavior of second-

order systems. It provides physical insights.

Reference: Graham and McRuer, Analysis of Nonlinear Control Systems, Dover Press,

1971.

Consider the second-order system described by the following equations:

x1and x2are states of the system

p and q are nonlinear functions of the states

Key points

Phase plane analysis is limited to second-order systems. For second order systems, solution trajectories can be represented by curves in

the plane, which allows for visualization of the qualitative behavior of the

system. In particular, it is interesting to consider the behavior of systems around

equilibrium points. Under certain conditions, stability information can be

inferred from this.


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phase plane = plane having x1and x2as coordinates!get rid of time

We look for equilibrium points of the system (also called singular points), i.e. points at

which:

Example:

Find the equilibrium point(s) of the system described by the following equation:

Start by putting the system in the standard form by setting :

We have the following equilibrium point:

Looking at the slope of the phase plane trajectory:

Investigate the linear behaviour about a singular point:


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Set

Then

x =Ax witha b

c d

"

#$

%

&'

This is the general form of a second-order linear system.

Such a system is linear in the sense that if x1and x2are solutions, then so is any linear

combination c1x1+c2x2. Notice that x = 0 when x=0, so the origin is always anequilibrium point for any choice of A. The solutions of x =Ax can be visualized as

trajectories moving on the (x1, x2) plane, in this context called the phase plane.

Phase Plane Example: Simple Harmonic Oscillator

As discussed in elementary physics courses, the vibrations of a mass hanging from alinear spring are governed by the linear differential equation: mx + kx = 0 where m is the

mass, k is the spring constant, and x is the displacement of the mass from equilibrium.

As youll probably recall, it is easy to solve the equation in terms of sines and cosines.This is what makes linear systems so special. For the nonlinear equations of ultimate

interest to us, its usually impossible to find an analytic solution. We want to develop

methods to deduce the behaviour of ODEs without actually solving them.

A vector field that comes from the original differential equation determines the motion in

the phase plane. To find this vector field, we note that the state of the system ischaracterized by its current position x and velocity v. If we know the values of both x and

v, then the equation above uniquely determines the future states of the system. We can

rewrite the ODE in terms of the state variables, as follows:

x = v

v = "k

mx = "#

2x


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This system assigns a vector ( x,v)to each point (x,v) and therefore represents a vector

field on the phase plane.

For example, lets see what the vector field looks like when were on the x-axis. Then, v

= 0, so ( x,v) = (0,"#2x). The vectors point vertically downward for positive x and

vertically upward for negative x. As x gets larger in magnitude, the vectors get longer.Similarly, on the v axis, the vector field is ( x,v) = (v,0) , which points to the right when

v>0 and to the left when v


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What do fixed points and closed orbits have to do with the problem of a mass on aspring? The answers are beautifully simple. The fixed point (x,v) =(0,0)corresponds to a

static equilibrium of the system: the mass is at rest at its equilibrium position and will

remain there forever, since the spring is relaxed. The closed orbits have a more

interesting interpretation: they correspond to periodic motion, that is, oscillations of the

mass. To see this, we can look at some points on a closed orbit. When the displacement xis most negative, the velocity v is zero. This corresponds to one extreme of the

oscillation, when the spring is most compressed. In the next instant, as the phase pointflows along the orbit, it is carried to points where x has increased and v is now positive;

the mass is being pushed back towards its equilibrium position. But by the time the mass

has reached x=0, it has a large positive velocity, and it overshoots x=0. The masseventually comes to rest at the other end of the swing, where x is most positive and v is

zero again. Then the mass gets pulled up and completes the cycle.

The shape of the closed orbits also has an interesting physical interpretation. The orbits

are actually ellipses given by the equation "2x 2 + v 2 =C, where C is a positive constant.

One can show that this geometric result is equivalent to conservation of energy.

Back to the phase plane method:

Next, we obtain the characteristicequation:

deta" # b

c d" #

$

%&

'

()=0 which yields ("# a)("# d) # bc =0

This equation admits the roots:


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"1,2=

a+ d

2

(a+ d)2# 4(ad# bc)

2

This yields the following possible cases:

!1, !2real and negative Stable node

!1, !2real and positive Unstable node

!1, !2real and opposite signs Saddle point

!1, !2complex and negative real parts Stable focus

!1, !2complex and positive real parts Unstable focus

!1, !2complex and zero real parts Center


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In Class Problem:

Graph the phase portraits for the linear system x =Ax where A =a 0

0 "1

#

$%

&

'(

Solution: the system can be written as:x = ax

y = "y

The equations are uncoupled. In this simple case, each equation may be solved

separately. The solution is:

x(t) = x0e

at

y(t) = y0e" t

The phase portraits for different values of a are shown below. In each case, y decaysexponentially. Name the different cases.


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A complete phase space analysis: Lotka-Volterra Model

We consider here the classic Lotka-Volterra model of competition between two species,

here imagined to be rabbits and sheep. Suppose both species are competing for the samefood supply (grass) and the amount available is limited. Also, lets ignore all other

complications, like predators, seasonal effects, and other sources of food. There are twomain effects that we wish to consider:

1. Either species would grow to its carrying capacity in the absence of the other.

This can be modeled by assuming logistic growth for each species. Rabbits have a

legendary ability to reproduce, so perhaps we should assign them a higherintrinsic growth rate.

2. When rabbits and sheep encounter each other, the trouble starts. Sometimes the

rabbit gets to eat, but more usually the sheep nudges the rabbit aside and startsnibbling (on the grass). Well assume that these conflicts occur at a rate

proportional to the size of each population. (If there are twice as many sheep, the

odds of a rabbit encountering a sheep are twice as great). Also, assume that theconflicts reduce the growth rate for each species, but the effect is more severe for

the rabbits.

A specific model that incorporates these assumptions is:x =x(3 "x " 2y)

y =y(2 "x "y)

where x(t) is the population of rabbits and y(t) is the population of sheep. Of course, x

and y are positive. The coefficients have been chosen to reflect the described scenario,but are otherwise arbitrary.

There are four fixed points for this system: (0,0), (0,2), (3,0) and (1,1). To classify them,we start by computing the Jacobian:

A =3"2x" y "2x

"y 2"x" 2y

#

$%

&

'(

To do the analysis, we have to consider the four points in turn.

(0,0): Then

A =3 0

0 2

"

#$

%

&'

The eigenvalues are both positive at 3 and 2, so this is an unstable node. Trajectories

leave the origin parallel to the eigenvector for !=2, that is, tangential to v = (0,1), which

spans the y-axis. (General rule at a node, the trajectories are tangential to the slow

eigendirection, which is the eigendirection with the smallest |!|.


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(0,2): Then

A ="1 0

"2 "2

#

$%

&

'(

The matrix has eigenvalues -1, -2. The point is a stable node. Trajectories approach along

the eigendirection associated with -1. You can check that this direction is spanned by (1, -2).

(3,0): Then

A = "3

"6

0 "1#

$% &

'(

The matrix has eigenvalues -1, -3. The point is a stable node. Trajectories approach along

the slow eigendirection. You can check that this direction is spanned by (3, -1).

(1,1): Then

A ="1 "2

"1 "1

#

$%

&

'(


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The matrix has eigenvalues "1 2 . This is a saddle point. The phase portrait is as showbelow:

Assembling the figures, we get:

Also, the x and y axes remain straight line trajectories, since x = 0 when x=0 andsimilarly y = 0when y=0.

We can assemble the entire phase portrait:

This phase portrait has an interesting biological interpretation. It shows that one species

generally drives the other to extinction. Trajectories starting below the stable manifold


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lead to the eventual extinction of the sheep, while those starting above lead to theeventual extinction of the rabbits. This dichotomy occurs in other models of competition

and has led biologist to formulate the principle of competitive exclusion, which states that

two species competing for the same limited resource cannot typically co-exist.


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Stability (Lyapunovs First Method)

Consider the system described by the equation:

Write x as :

Then

Lyapunov proved that the eigenvalues of A indicate local stability of the nonlinearsystem about the equilibrium point if:

a) (The linear terms dominate)

b) There are no eigenvalues with zero real part.

Example:

Consider the equation:

If x is small enough, then

Thought question: What if a = 0?

Example: Simplified satellite control problem

Built in the 1960s.


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After about one month, would run out of gas.

How was the controller designed?

Lets pick .

It cold in space: the valves would freeze open. If and are small, there is not enough

torque to break the ice, so the valves get frozen open and all the gas escapes. One

solution is either relay control and / or bang-bang control. (These methods are inelegant).

Pick , and .

Case 1: Pick u = 0. The satellite just floats.


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In the thick black line interval, all trajectories point towards the switching line.


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Bad idea!

On the line, . (a>0).

On average:

On the average, the trajectory goes to the origin.

Introduction to Sliding Mode Control (also called Variable Structure Control)

Consider the system governed by the equation:

Inspired by the previous example, we select a control law of the form:

where . How should we pick the function s?


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Case 1:

This does not yield the performance we want.

Case 2:

This does not yield the performance we want.

Case 3:


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When is ?

Let s>0. Then

That is, if s>0, iff

Example


where d(t) is an unknown disturbance. The disturbance d is bounded, that is,

The goal of the controller is to guarantee the type of response shown below.

1) Is it possible to design a controller that guarantees this response assuming no

bounds on u?


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2) If your answer on question (1) is yes, design the controller.

The desired behavior is a first-order response. Define

If s=0, we have the desired system response. Hence our goal is to drive s to zero.

If u appears in the equation for s, set s=0 and solve for u. Unfortunately, this is not thecase. Keep differentiating the equation for s until u appears.

Look for the condition for .

We therefore select u to be:

The first term dictates that one always approaches zero. The second term is called theswitching term. The parameter "is a tuning parameter that governs how fast one goes to

zero.

Once the trajectory crosses the s=0 line, the goals are met, and the system slidesalong the line. Hence the name sliding mode control.

Does the switching surface s have to be a line?

No, but it keeps the problem analyzable.

Example of a nonlinear switching surface


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For a mechanical system, an analogy would be making a cart reach a given position at

zero velocity in minimal time.

The request for a minimal time solution suggests a bang-bang type of approach.

This can be obtained, for example, with the following expression for s:

The shape of the sliding surface is as shown below.

This corresponds to the following block diagram:


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Logic is missing for the case when s is exactly equal to zero. In practice for acontinuous system such as that shown above this case is never reached.

Classical Phase-Plane Analysis Examples

Reference: GM Chapter 7

Example: Position control servo (rotational)

Case 1: Effect of dry friction

The governing equation is as follows:

For simplicity and without lack of generality, assume that I = 1. Then:

That yields:


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The friction function is given by:

There are an infinite number of singular points, as shown below:

When , we have , that is, we have an undamped linear oscillation

( a center). Similarly, when , we have (another center).

From a controls perspective, dry friction results in an offset, that is, a loss of staticaccuracy.


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To get the accuracy back, it is possible to introduce dither into the system. Dither is ahigh-frequency, low-amplitude disturbance (an analogy would be tapping an offset scale

with ones finger to make it return to the correct value).

On average, the effect of dither pulls you in. Dither is a linearizing agent, that

transforms Coulomb friction into viscous friction.

Example: Servo with saturation

There are three different zones created by the saturation function:


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The effects of saturation do not look destabilizing. However, saturation affects the

performance by slowing it down.

The effect of saturation is to slow down the system.

Note that we are assuming here that the system was stable to start with before we appliedsaturation.

Problems appear if one is not operating in the linear region, which indicates that the gainshould be reduced in the saturated region.

If you increase the gain of a linear system oftentimes it eventually winds up unstable,except if the root locus looks like:

Root locus for a conditionally stable system (for example an inverted pendulum).

So there are systems for which saturation will make you unstable.


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SUMMARY:Second-Order Systems and Phase-Plane Analysis

Graphical Study of Second-Order Autonomous Systems

x1and x2are states of the system

p and q are nonlinear functions of the states

phase plane = plane having x1and x2as coordinates

!get rid of time

As t goes from 0 ! +", and given some initial conditions, the solution x(t) can be

represented geometrically as a curve (a trajectory) in the phase plane. The family ofphase-plane trajectories corresponding to all possible initial conditions is called thephase

portrait.

Due to Henri Poincar

French mathematician, (1854-1912).

Main contributions:

! Algebraic topology! Differential Equations! Theory of complex variables

! Orbits and Gravitation!

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Poincare.html

Poincar conjecture

In 1904 Poincar conjectured that any closed 3-dimensional manifold which is homotopy

equivalent to the 3-sphere must be the 3-sphere. Although higher-dimensional analoguesof this conjecture have been proved, the original conjecture remains open.

Equilibrium (singular point)

Singular point = equilibrium point in the phase plane


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Slope of the phase trajectory

At an equilibrium point, the value of the slope is indeterminate (0/0) !singular point.

Investigate the linear behaviour about a singular point

Set

Then

Which is the general form of a second-order linear system.

Obtain the characteristic equation

This equation admits the roots:

"1,2=

a + d

2

(a + d)2# 4(ad# bc)

2

Possible cases

Pictures are from H. Khalil,Nonlinear Systems, Second Edition.


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#1and #2are real and negative

STABLE NODE

#1and #2are real and positive

UNSTABLE NODE

#1and #2are real and of opposite sign

SADDLE POINT (UNSTABLE)

#1and #2are complex with negative real

parts

STABLE FOCUS


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#1and #2are complex with positive real

parts

UNSTABLE FOCUS

#1and #2are complex with zero real parts

CENTER

Which direction do circles and spirals spin, and what does this mean?

Consider the system:

Let and .

With $page of straightforward algebra, one can show that: (see homework 1 for details)

and

The r equation says that in a Jordan block, the diagonal element, %, determines whether

the equilibrium is stable. Since r is always non-negative, % greater than zero gives agrowing radius (unstable), while %less than zero gives a shrinking radius. &gives the rate

and direction of rotation, but has no effect on stability. For a given physical system,

simply re-assigning the states can get either positive or negative &.

In summary:


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If %> 0, the phase plot spirals outwards.If %< 0, the phase plot spirals inwards.

If &> 0, the arrows on the phase plot are clockwise.If &< 0, the arrows on the phase plot are counter-clockwise.

Stability

x=xe+'x

Lyapunov proved that the eigenvalues of A indicate local stability if:

(a)the linear terms dominate, that is:

(b)there are no eigenvalues with zero real part.


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4 `Equilibrium Finding

We consider systems that can be written in the following general form, where x is the

state of the system, u is the control input, and f is a nonlinear function.

Let u = ue = constant.

At an equilibrium point, .

Key points

Nonlinear systems may have a number of equilibrium points (from zero to

infinity). These are obtained from the solution of n algebraic equations in n

unknowns. The global implicit function theorem states condition for uniqueness of an

equilibrium point.

Numeral solutions to obtain the equilibrium points can be obtained using several

methods, including (but not limited to) the method of Newton-Raphson andsteepest descent techniques.


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To obtain the equilibrium points, one has to solve n algebraic equations in n unknowns.

How can we find out if an equilibrium point is unique? See next section.

Global Implicit Function Theorem

Define the Jacobian of f.

The solution xeof for a fixed ue is unique provided:

1. det[J(x)] !0 for allx

2.

Note: in general these two conditions are hard to evaluate (particularly condition 1).

For peace of mind, check this with linear system theory. Suppose we had a linear system:

. Is xeunique? J=A, which is different from 0 for all x, and f = Ax, so the limit

condition is true as well (good!).

How does one generate numerical solutions to ? (for a fixed ue)

There are many methods to find numerical solutions to this equation, including, but notlimited to:

- Random search methods

- Methods that require analytical gradients (best)- Methods that compute numerical gradients (easiest)

Two popular ways of computing numerical gradients include:- The method of Newton-Raphson

- The steepest descent method

Usually both methods are combined.

The method of Newton-Raphson

We want to find solutions to the equation . We have a value, xi, at the ith

iteration and an error, ei, such that ei= f(xi).


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We want an iteration algorithm so that:

Expand in a first order Taylor series expansion.

We have: .

Suppose that we ask for: (ask for, not get)

Then:

That is, we get an expression for the Newton-Raphson iteration:

Note: One needs to evaluate (OK) and invert (not so good) the Jacobian.Note: Leads to good convergence properties close to xe but causes extreme starting

errors.

Steepest Descent Technique (Hill Climbing)

Define a scalar function of the error, then choose to guarantee a reduction in this

scalar at each step.

Define: which is a guaranteed positive scalar. We attempt to minimize L.

We expand L in a first-order Taylor series expansion.

and

We want to impose the condition: L(i+1) < L(i).

This implies:

where !is a scalar.

This yields:


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and

That is, the steepest descent iteration is given by:

Note: Need to evaluate J but not invert it (good).

Note: this has good starting properties but poor convergence properties.

Note: Usually, the method of Newton-Raphson and the steepest descent method are

combined:

where "1and "2are variable weights.


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6 `Controllability and

Observability of

Nonlinear Systems

Controllability for Nonlinear Systems

The Use of Lie Brackets: Definition

We shall call a vector function f :"

n#"

n

a vector field in !

n

to be consistent withterminology used in differential geometry. The intuitive reason for this term is that toevery vector function f corresponds a field of vectors in an n-dimensional space (one can

think of a vector f(x) emanating from every point x). In the following we shall only be

interested in smooth vector fields/ By smoothness of a vector field, we mean that thefunction f(x) has continuous partial derivatives of any required order.

Key points

Nonlinear observability is intimately tied to the Lie derivative. The Liederivative is the derivative of a scalar function along a vector field.

Nonlinear controllability is intimately tied to the Lie bracket. The Lie bracket

can be thought of as the derivative of a vector field with respect to another. References

o

Slotine and Li, section 6.2 (easiest)

o Sastry, chapter 11 pages 510-516, section 3.9 and chapter 8

o Isidori, chapter 1 and appendix A (hard)


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How this came about

,

So for example:

If we keep going:

x =A3x + A2Biui =A3x + adf

2Bi[ ]uii=1

m

"i=1

m

"

Notice how this time the minus signs cancel out.

x(n ) =dnx

dtn=A

nx + An"1Biui = A

nx + ("1)n"1 adfn"1Bi[ ]ui

i=1

m

#i=1

m

#

Re-writing the controllability condition:

C= B1,...,B

m,ad

fB

1,...ad

fB

m,...ad

f

n"1B1,...ad

f

n"1Bm[ ]

The condition has not changed just the notation.The terms B1through Bmcorrespond to the B term in the original matrix, the terms with

adfcorrespond to the AB terms, the terms with adfn-1

correspond to the An-1

B terms.

Nonlinear Systems

Assume we have an affine system:

The general case is much more involved and is given in Hermann and Krener.If we dont have an affine system, we can sometimes ruse:

Let


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Select a new state: and v is my control "the system is affine in (z,v), and pick#

to be OK.

Theorem

The system defined by:

is locally accessible about x0if the accessibility distribution C spans n space, where n isthe dimension of x and C is defined by:

C= g1,g

2,...,gm, gi,gj[ ],...,adg i

kgj,..., f,gi[ ],...adf

kgi,...[ ]

The giterms are analogous to the B terms, the [gi,gj] terms are new from having a

nonlinear system, the [f,gi] terms correspond to the AB terms, etc

Note:if f(x) = 0 then and if in this case C has rank n, then the system is

controllable.

Example: Kinematics of an Axle


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Basically,$is the yaw angle of the vehicle, and x1and x2are the Cartesian locations of

the wheels. u1is the velocity of the front wheels, in the direction that they are pointing,

and u2is the steering velocity.

We define our state vector to be:

Our dynamics are:

The system is of the form:

f(x) = 0, and

Note:

If I linearize a nonlinear system about x0 and the linearization is controllable, then the

nonlinear system is accessible at x0 (not true the other way if the linearization is

uncontrollable the nonlinear system may still be locally accessible).

Back to the example:

where and in our case,


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So

C has rank 3 everywhere, so the system is locally accessible everywhere, and f(x)=0 (free

dynamics system) so the system is controllable!

Example 2:

Note: if I had the linear system:

, , "

and the linear system is controllable.

Back to the example 2:

Is the nonlinear system controllable? Answer is NO, because x1can only increase.

But lets show it.

In standard form:

,


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So

Accessible everywhere except where x2=0

If we tried [f,[f,g]], would we pick up new directions? It turns out they will also bedependent on x2, and the rank will drop at x2= 0.

Example 3:

where

The system is of the form:

where

, and


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We have:

If C has rank 4, then the system is locally accessible. Have fun

Observability for Nonlinear Systems

Intuition for observability:

From observing the sensor(s) for a finite period of time, can I find the state at previous

times?

Review of Linear Systems

where

where and p


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This does not carry over to nonlinear systems, so we take a local approach.

Local Approach to Observability (Linear Systems)

v(t) is the measurement noise, can cause problems.

z(n"1)

= MAn"1x

" O must have rank n

Lie Derivatives:

The gradient of a smooth scalar function h(x) of the state x is denoted by:

"h =#h

#x

The gradient is represented by a row-vector of elements: ("h) j =#h#x j

.

Similarly, given the vector field f(x), the Jacobian of f is:

"f =#f

#x

It is represented by an nxn matrix of elements: ("f) ij =#fi

#x j

Definition

Let f: !n&!nbe a vector field in !n.

Let h: !n&!be a smooth scalar function.

Then the Lie derivative of h with respect to f is a new scalar defined by:


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Dimensions

f looks like:

h looks like: h(x) with x '!n"associates a scalar to each point in !n

The Lie derivative looks like:

"Lfh is a scalar.

Conventions:

By definition,

We can also define higher-order Lie derivatives:

etc

One can easily see the relevance of Lie derivatives to dynamic systems by consideringthe following single-output system:

x = f(x)

y = h(x)

Then

And


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Etc so

Use of Lie Derivative Notation for Linear Systems

so f(x)=Ax

, Mi is 1xn

"

Define G =

Lf

0(h

1) ... Lf

0(hp )

... ... ...

Lfn"1

(h1) ... Lf

n"1(hp )

#

$

%%%

&

'

(((=

M1x ... Mpx

... ... ...

M1A

n"1x ... MpAn"1x

#

$

%%%

&

'

(((

Now, define a gradient operator:

O must have rank n for the system to be observable.

Nonlinear Systems

Theorem:

Let G denote the set of all finite linear combinations of the Lie derivatives of h1,,hp

with respect to f for various values of u = constant. Let dG denote the set of all theirgradients. If we can find n linearly independent vectors within dG, then the system is

locally observable.


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The system is locally observable, that is distinguishable at a point x0if there exists aneighborhood of x0such that in this neighborhood,

if the states are different, the sensor readings are different

Case of a single measurement:

Look at the derivatives of z:

Let:

Expand in a first-order series about x0for u = u0

Then must have rank n

Example:


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SUMMARY: Controllability and Observability for Nonlinear Systems

Controllability

The system is locally accessibleabout a point x0if and only if

C = [ g1,...,gm, [gi, gj],...[adgik

,gj],..., [f,gi],..., [adfk,gi],...]

has rank n where n is the rank of x. C is the accessibility distribution.

If the system has the form: that is, f(x) = 0, and C has rank n, then the

system is controllable.

Observability

z=h(x)

Two states x0and x1are distinguishableif there exists an input function u* such that:z(x0) (z(x1)

The system is locally observableat x0if there exists a neighbourhood of x0 such that

every x in that neighbourhood other than x0 is distinguishable from x0.

A test for local observability is that:

must have rank n, where n is the rank of x and

For a px1 vector,

z = [h1, ..., hp]T,


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LINEAR SYSTEMS NONLINEAR SYSTEMS

CONTROLLABILITY

AND

ACCESSIBILITY

Intuition: the system is

controllable "you can get

anywhere you want in a finite

amount of time.

LINEAR TIME INVARIANT

SYSTEMS

CONTROLLABILITY

The system is controllable if:C = [ B AB ... An-1B ]

has rank n, where n is the rank of

x.

AFFINE SYSTEMS

ACCESSIBILITY

The system is locally accessibleabout a point x0if and only if

C = [ g1,...,gm, [gi, gj],...

[adgikgj],..., [f,gi],..., [adf

kgi],...]

has rank n where n is the rank of

x. C is the accessibility

distribution.

CONTROLLABILITY

If f(x) = 0 and C has rank n, then

the system is controllable.


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OBSERVABILITY

AND DISTINGUISHABILITY

Intuition: the system is observable"from observing the sensor

measurements for a finite period

of time, I can obtain the state at

previous times.

z=Mx

x has rank n

z has rank pp


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f looks like:

h looks like: h(x) with x '!n"associates a scalar to each point in !n

The Lie derivative looks like:

"Lfh is a scalar.

Physically (time for pictures!)

Picture of f

f associates an n-dimensional vector to each point in !n

In !2:

For example, let f(x) ="1 0

0 "2

#

$%

&

'(

x1

x2

#

$%

&

'(


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)ft(x0) = flow along the vector field for time t, starting at x0

"tangent to the phase plane plot at every single point

Picture of h

For example, in !2, pick h to be the distance to the origin:

Lie derivative picture

Using this example:


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Lfh =

" x2

"x1

" x2

"x2

#

$%

&

'(

)1 0

0 )2

#

$%

&

'(

So, the Lie derivative gives the rate of change in a scalar function h as one flows

along the vector field f.

In a control systems context:

x'!n f: !n&!n

y=h(x) y'! h: !n&!

along the flow of f

How does this tie into observability?

Imagine:x =Ax

y =Cx

"#$

and we can only see y, a scalar, and we wish to find x'!n


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y = Cx

y = Cx = CAx

y

(n-1)= CA

n-1x

and solve for x (n equations)"

if [C CA ... CAn-1

] has rank n, we have n independent equations in nvariables "OK

Using the Lie derivative

f(x) = Ax, h(x) = Cx

and by convention,

The Lie Bracket and Controllability

Definition

Let f: !n&!nbe a smooth vector field in !n.Let g: !n&!nbe a smooth vector field in !n.

Then the Lie bracket of f and g is a third-order vector field given by:

f ,g[ ] ="g

"x.f #

"f

"x.g

Dimensions

f looks like: , g also looks like:


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So [f,g] is a vector field.

How does this tie into controllability?

Consider:

u1, u2are scalar inputs

x'!3

,

What directions can we steer x if we start at some point x0?

Clearly, we can move anywhere in the span of {g1(x0),g2(x0)}.

Lets say that: ,

Can we move in the x3direction?

The directions that we are allowed to move in by infinitesimally small changesare [g1,g2].


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8 `Feedback Linearization

Key points

Feedback linearization = ways of transforming original system models intoequivalent models of a simpler form.

Completely different from conventional (Jacobian) linearization, because

feedback linearization is achieved by exact state transformation and feedback,

rather than by linear approximations of the dynamics.

Input-Output, Input-State

Internal dynamics, zero dynamics, linearized zero dynamics Jacobis identity, the theorem of Frobenius MIMO feedback linearization is also possible.


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If r, the relative degree, is less than n, the order of the system, then there will be internaldynamics. If r = n, then I/O and I/S linearizations are the same.

Input/State Linearization

A control technique where some new output ynew= hnew(x) is chosen so that with respect

to ynew, the relative degree of the system is n. Then the design procedure using this new

output ynewis the same as for I/O linearization.

SISO Systems

Consider a SISO nonlinear system:

Here, u and y are scalars.

y ="h

"xx =Lf

1 h +Lg (h)u =Lf1 h if Lg (h) = 0

If , we keep taking derivatives of y until the output u appears. If the output

doesnt appear, then u does not affect the output! (Big difficulties ahead).

If , we keep going.

We end up with the following set of equalities:


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with

with

with

The letter r designates the relative degree of y=h(x) iff:

That is, r is the smallest integer for which the coefficient of u is non-zero over the spacewhere we want to control the system.

Lets set:

Then , where

v(x) is called the synthetic input or synthetic control. y(r)

=v

We have an r-integrator linear system, of the form: .

We can now design a controller for this system, using any linear controller design

method. We have . The controller that is implemented is obtained through:

Any linear method can be used to design v. For example,


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Problems with this approach:

1. Requires a perfect model, with perfect derivatives (one can anticipate robustness

problems).

2. If the goal is , .

If , and r = 2, there are 18 states for which we dont know what is

happening. That is, if , we have internal dynamics.

Note: There is an ad-hoc approach to the robustness problem, by setting:

Here the first term in the expression is the standard feedback linearization term, and the

second term is tuned online for robustness.

Internal Dynamics

Assume r


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where and

We define:

where z is rx1 and "is (n-r)x1. ( ).

The normal forms theorem tells us that there exists an "such that:

Note that the internal dynamics are not a function of u.

So we have:

The "equation represents internal dynamics; these are not observable because z does

not depend on "at all !internal, and hard to analyze!

We want to analyze the zero dynamics. The system is difficult to analyze. Oftentimes, tomake our lives easier, we analyze the so-called zero dynamics:

and in most cases we even look at the linearized zero dynamics.


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and we look at the eigenvalues of J.

If these are well behaved, perhaps the nonlinear dynamics might be well-behaved. If

these are not well behaved, the control may not be acceptable!

For linear systems:

We have:

The eigenvalues of the zero dynamics are the zeroes of H(s). Therefore if the zeroes of

H(s) are non-minimum phase (in the right-half plane) then the zero dynamics are

unstable.#

By analogy, for nonlinear systems: if is unstable, then the system:

is called a non-minimum phase nonlinear system.

Input/Output Linearization

o Procedure

a) Differentiate y until u appears in one of the equations for the derivatives of y

after r steps, u appears


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b) Choose u to give y(r)

=v, where v is the synthetic input

c) Then the system has the form:

Design a linear control law for this r-integrator liner system.

d) Check internal dynamics.

o

Example

Oral exam question

Design an I/O linearizing controller so that y $0 for the plant:

Follow steps:

a) u appears ! r = 1

b) Choose u so that

!

In our case, and .

c) Choose a control law for the r-integrator system, for example proportional control

Goal: to send y to zero exponentially

! since ydes= 0


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d) Check internal dynamics:

Closed loop system:

If x1$0 as desired, x2is governed by

!Unstable internal dynamics!

There are two possible approaches when faced with this problem:

!

Try and redefine the output: y=h(x1,x2)! Try to linearize the entire system/space !Input/State Linearization

#


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Input/State Linearization (SISO Systems)

Question: does there exist a transformation (x) such that the transformed system is

linear?

Define the transformed states:

I want to find %(x) such that where , with:

! v=v(x,u) is the synthetic control! the system is in Brunowski (controllable) form

and

A is nxn and B is nx1.

We want a 1 to 1 correspondence between z and x such that:

Question: does there exist an output y=z1(x) such that y has relative degree n?

with


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Let

Then: . And the form I need is:

does there exist a scalar z1(x) such that:

for k = 1,,n-2

And ?

z"

z1

z2

...

zn

#

$

%%%%

&

'

((((

=

Lf0(z

1)

Lf1(z

1)

...

Lfn)1

(z1)

#

$

%%%%

&

'

((((

is there a test?

so the test should depend on f and g.

Jacobis identity

Carl Gustav Jacob Jacobi


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Born: 10 Dec 1804 in Potsdam, Prussia (now Germany)Died: 18 Feb 1851 in Berlin, Germany

Famous for his work on:! Orbits and gravitation

!

General relativity! Matrices and determinants

Jacobis Identity

A convenient relationship (S+L) is called Jacobis identity.

Remember:L

f

0 h =h , Lfi h =Lf(Lf

i"1h) =#(Lfi"1h).f

adf0g =g . adfg =[f,g] ="g.f #"f.g , adf

ig =[f,adf

i"1g]

This identity allows us to keep the conditions in first order in z1

Trod through messy algebra

! For k = 0

(first order)

! For k = 1


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!2nd

order (gradient)

Things get messy, but by repeated use of Jacobis identity, we have:

for for (*)

The two conditions above are equivalent. Evaluating the second half:

This leads to conditions of the type:

The Theorem of Frobenius

Ferdinand Georg Frobenius:

Born: 26 Oct 1849 in Berlin-Charlottenburg, Prussia (now Germany)

Died: 3 Aug 1917 in Berlin, Germany


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Famous for his work on:

! Group theory! Fundamental theorem of algebra! Matrices and determinants

Theorem of Frobenius:

A solution to the set of partial differential equations for exists

if and only if:

a) g,adfg,...,adfn"1g[ ] has rank n

b) g,adfg,...,adfn"2g[ ]is involutive

#

Definition of involutive:

A linear independent set of vectors (f1, , fm) is involutive if:

#

i.e. when you take Lie brackets you dont generate new vectors.

Note: this is VERY hard to do.

Reference: George Myers at NASA Ames, in the context of helicopter control.

Example: (same as above)

Question: does there exist a scalar z1(x1,x2) such that the relative degree be 2?

This will be true if:

a) (g, [f, g]) has rank 2b) g is involutive (any Lie bracket on g is zero $OK)


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Setting stuff up to look at (a):

Note:

looks dangerous

Question: how do we find z1?

We get a list of conditions:

!

(simplest)

! (always true for constant, independent vectors) In our case, ok if

x1" 3 3

So lets trod through and check:

(good that u doesnt appear, or r=1!)

(u appears! (good))

Define ,

Hope the problem is far away from

Let


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!z1$z1d

Question: How to pick z1d?

We want: for


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167

Feedback Linearization for MIMO Nonlinear Systems

Consider a square system (where the number of inputs is equal to the number of

outputs = n)

Let rk, the relative degree, be defined as the relative degree of each output, i.e.

For some i,

Let J(x) be an mxm matrix such that:

J(x) is called the invertibility or decoupling matrix.

We will assume that J(x) is non-singular.

Let:

where yris an mx1 vector


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Then we have:

where v is the synthetic input (v is mx1).

We obtain a decoupled set of equations:

so

Design v any way you want to using linear techniques

Problems:

! Need confidence in the model

! Internal dynamics

Internal Dynamics

The linear subspace has dimension (or relative degree) for the whole system:

!we have internal dynamics of order n-rT.


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The superscript notation denotes which output we are considering. We have:

where zTis rx1, "

Tis (n-rT)x1

The representation for x may not be unique!

Can we get a "who isnt directly a function of the controls (like for the SISO case)? NO!

and

Internal dynamics !what is u?

!design v, then solve for u using

The zero dynamics are defined by z = 0.

!

The output is identically equal to zero if we set the control equal to zero (at all times).

Thus the zero dynamics are given by:


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Dynamic Extension - Example

References: Slotine and LiHauser, PhD Dissertation, UCB, 1989 from which this example is taken

Basically, &is the yaw angle of the vehicle, and x1and x2are the Cartesian locations of

the wheels. u1is the velocity of the front wheels, in the direction that they are pointing,and u2is the steering velocity.

We define our state vector to be:

Our dynamics are:

x1 = (cos")u1

x2 = (sin")u1

"

= u2

#

$%

&%

We determined in a previous lecture that the system is controllable (f = 0).

and are defined as outputs.


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y1

y2

"

#$

%

&'=

cos( 0

sin( 0

"

#$

%

&'

u1

u2

"

#$

%

&'

J(x) =cos" 0

sin" 0

#

$

%&

'

( is clearly singular (has rank 1).

Let , where u3is the acceleration of the axle

! the state has been extended.!

x1 = (cos")x3

x2 = (sin")x3

x3= u

1= u

3

"

= u2

#

$

%%

&

%

%

where in the extended state space

Take and .

and the new J(x) matrix: is non-singular for u1'0 (as long as

the axle is moving).

How does one go about designing a controller for this example?


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Given y1d(t), y2d(t):

Let:

!

To obtain the control, u:

and !we have a dynamic feedback controller(the controller has dynamics, not

just gains, in it).#


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and =0 if r


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where (x1,x2) are the coordinates of a point in in the natural basis.

To diagonalize the system, we do a change of coordinates, so we express points like:

1 0

0 1

"

#$

%

&'x = t1 t2[ ]x'

where t1 and t2 are the eigenvectors of A and x represents the coordinates in the newbasis x.

So we get a nice equation in the new coor

control of nonlinear dynamic systems

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