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Control of Nonlinear Dynamic Systems: Theory and Applications

J. K. Hedrick and A. Girard

Copyright © 2010 All rights reserved.

Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

© 2010

1 `Introduction

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 be ignored. For these systems, nonlinear analysis and design techniques exist and can be used. These techniques are the focus of this textbook.

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, which allows us to represent the system in a compact form.

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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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, it has 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 the purposes of analysis and controller design. This transformation is called linearization and is possible at a selected operating point of the system. Equilibrium points are an important class of solutions of a differential equation. They are defined as the points xe such 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 one equilibrium 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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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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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 Γ is n by p. We obtain a linear model for the system about the equilibrium point (x0, 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 terms that 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 by friction, 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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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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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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General Properties of Linear and Nonlinear Systems

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.

Aside: A Brief History of Dynamics (Strogatz) The subject of dynamics began in the mid 1600s, when Newton invented differential equations, discovered his laws of motion and universal gravitation, and combined them to explain Kepler’s laws of planetary motion. Specifically, Newton solved the two-body problem: 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

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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much more difficult to solve. After decades of effort, it was eventually realized that the three-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 new viewpoint 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 solar system stable forever, or will some planets eventually fly off to infinity. Poincare developed a powerful geometric approach to analyzing such questions. This approach has flowered 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 on initial conditions, thereby rendering long term prediction impossible. But chaos remained in the background for the first half of this century. Instead, dynamics was 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, Poincare’s geometric methods were being extended to yield a much deeper understanding of classical mecahsnics, thanks to the 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 of dynamics. 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 Lorenz’s discovery in 1963 of chaotic motion on a strange attractor. He studied a simplified model of convection rolls in the atmosphere to gain insight into the 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 from two 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 set had to be “an infinite complex of surfaces” – today, we would regard it as an example of a fractal. Lorenz’s 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 a new 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 most surprising 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 link between 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 magnificient computer 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 and heart rhythms. By the 1980s, many people were working on dynamics, with contributions too numerous to list.

Lorenz attractor

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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

1920-1950

Geometric approach, nightmares of chaos Nonlinear oscillators in physics and engineering Invention of radio, radar, laser

1920-1960

Birkhoff Complex behavior in Hamiltonian mechanics Kolmogorov Arnol’d Moser

1963

Lorenz

1970s

Ruelle/Takens Turbulence and chaos

Strange attractor in a simple model of convection

May

Chaos in logistic map

Feigenbaum

Universality and renormalization Connection between chaos and phase transitions Expertimental studies of chaos

1980s

Winfree

Nonlinear oscillators in biology

Mandelbrot

Fractals Widespread interest in chaos, fractals, oscillators and their applications.

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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In this chapter, we consider general properties of linear and nonlinear systems. We consider the existence and uniqueness of equilibrium points, stability considerations, and the properties of the forced response. We also present a broad classification of nonlinearities. Where Do Nonlinearities Come From? Before we start a discussion of general properties of linear and nonlinear systems, let’s briefly consider standard sources of nonlinearities. Many physical quantities, such as a vehicle’s velocity, or electrical signals, have an upper bound. When that upper bound is reached, linearity is lost. The differential equations governing some systems, such as some thermal, fluidic, or biological systems, are nonlinear in nature. It is therefore advantageous to consider the nonlinearities directly while analyzing and designing controllers for such systems. Mechanical systems may be designed with backlash – this is so a very small signal will produce no output (for example, in gearboxes). In addition, many mechanical systems are subject to nonlinear friction. Relays, which are part of many practical control systems, are inherently nonlinear. Finally, ferromagnetic cores in electrical machines and transformers are often described with nonlinear magnetization curves and equations. Formal Definition of Linear and Nonlinear Systems Linear systems must verify two properties, superposition and homogeneity. The principle of superposition states that for two different inputs, x and y, in the domain of the function f,

The property of homogeneity states that for a given input, x, in the domain of the function f, and for any real number k,

Any function that does not satisfy superposition and homogeneity is nonlinear. It is worth noting that there is no unifying characteristic of nonlinear systems, except for not satisfying the two above-mentioned properties.

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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 of systems 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 realization of 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 x0 in the state space is an equilibrium point of the autonomous system 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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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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Stability

The system is stable if

.

A more formal statement would talk about the stability of the equilibrium point in the sense 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 sinusoid’s amplitude is different than that of the input and the signal also exhibits 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, ω. 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 a useful 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 be solved 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 don’t 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 won’t get double the pleasure! Within the realm of physics, 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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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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Nonlinear System Properties Equilibrium point Reminder: A point x0 in 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 θ is the angle of the pendulum from the vertical. The equivalent (nonlinear) system is: ⎧⎪ x˙1 = x 2 k g ⎨ ⎪⎩ x˙ 2 = − mL2 x 2 − L sin x1

Nonlinearity makes the pendulum equation very difficult to solve analytically. The usual way around this is to “fudge”, by invoking the small angle approximation for sin x ≈ x € the problem to a linear one, which can then be solved easily. But for x0. Then That is, if s>0,

iff

Example Consider the system governed by the equation:

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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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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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 the case. 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 the switching term. The parameter η 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 “slides” along 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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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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Consider the system governed by the equation:

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 a continuous 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

( a center). Similarly, when

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

(another center).

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

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To get the accuracy back, it is possible to introduce dither into the system. Dither is a high-frequency, low-amplitude disturbance (an analogy would be tapping an offset scale with one’s 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 applied saturation. Problems appear if one is not operating in the linear region, which indicates that the gain should 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

x1 and x2 are states of the system p and q are nonlinear functions of the states phase plane

= plane having x1 and x2 as 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 of phase-plane trajectories corresponding to all possible initial conditions is called the phase portrait. •

Due to Henri Poincaré

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 Poincaré conjecture In 1904 Poincaré conjectured that any closed 3-dimensional manifold which is homotopy equivalent to the 3-sphere must be the 3-sphere. Although higher-dimensional analogues of 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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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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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: a+d (a + d) 2 − 4(ad − bc) λ1,2 = ± 2 2

€ •

Possible cases

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

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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λ1 and λ2 are real and negative

λ1 and λ2 are real and positive

STABLE NODE

UNSTABLE NODE

λ1 and λ2 are real and of opposite sign

λ1 and λ2 are complex with negative real parts

SADDLE POINT (UNSTABLE) STABLE FOCUS

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

λ1 and λ2 are complex with positive real parts

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λ1 and λ2 are complex with zero real parts CENTER

UNSTABLE FOCUS

•

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

Consider the system:

Let

and

.

With ½ 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, σ, determines whether the equilibrium is stable. Since r is always non-negative, σ greater than zero gives a growing radius (unstable), while σ less than zero gives a shrinking radius. ω 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 ω. In summary:

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

If σ > 0, the phase plot spirals outwards. If σ < 0, the phase plot spirals inwards. If ω > 0, the arrows on the phase plot are clockwise. If ω < 0, the arrows on the phase plot are counter-clockwise. •

Stability

x=xe+δx

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

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 and steepest descent techniques.

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,

.

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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 xe of

for a fixed ue is unique provided:

1. det[J(x)] ≠ 0 for all x 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 xe unique? 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 not limited 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 iteration and an error, ei, such that ei = f(xi).

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. We have a value, xi, at the ith

Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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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 scalar at each step. Define:

to guarantee a reduction in this

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 β 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 ρ1 and ρ2 are variable weights.

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6 `Controllability and Observability of Nonlinear Systems Key points • • •

Nonlinear observability is intimately tied to the Lie derivative. The Lie derivative 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)

Controllability for Nonlinear Systems The Use of Lie Brackets: Definition We shall call a vector function f : ℜ n →ℜ n a vector field in ℜn to be consistent with terminology used in differential geometry. The intuitive reason for this term is that to every 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 the function f(x) has continuous partial derivatives of any required order.

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Consider two vector fields f(x) and g(x) on ℜn. Then the Lie bracket operation generates a new vector field defined by:

The Lie bracket is [f,g] is commonly written adfg (where “ad” stands for adjoint). Also, higher order Lie brackets can be defined recursively: (ad 0f ,g) ≡ g (ad1f ,g) ≡ [ f ,g]

(ad 2f ,g) ≡ [ f , [ f ,g]] € … k k −1 (ad € f ,g) ≡ f ,(ad f ,g) for k=1,2,3,… € Recap – Controllability for Linear Systems

[

]

€

C = [ B | AB | ... | A n −1B] Local conditions (linear systems)

€ Let u = constant (otherwise no pb, but you get

etc…)

…

For linear systems, you get nothing new after the nth derivative because of the CayleyHamilton theorem. Re-writing controllability conditions for linear systems using this notation:

, m

m

x˙˙ = A˙x = A 2 x + ∑ ABi ui = A 2 x − ∑ [ f ,Bi ]ui i=1

€

i=1

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

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How this came about… , So for example:

If we keep going: m

m

[

]

˙x˙˙ = A 3 x + ∑ A 2 Bi ui = A 3 x + ∑ ad 2f Bi ui i=1

i=1

Notice how this time the minus signs cancel out. … € x

(n )

m m dn x n n −1 n n −1 = n = A x + ∑ A Bi ui = A x + (−1) ∑ ad nf −1Bi ui dt i=1 i=1

[

]

Re-writing the controllability condition: €

[

C = B1,...,Bm ,ad f B1,...ad f Bm ,...ad nf −1B1,...ad nf −1Bm

]

The condition has not changed – just the notation. The terms B1 through Bm correspond to the B term in the original matrix, the terms with € to the AB terms, the terms with ad n-1 correspond to the An-1B terms. adf correspond f

Nonlinear Systems Assume we have an affine system:

The general case is much more involved and is given in Hermann and Krener. If we don’t have an affine system, we can sometimes ruse:

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

Select a new state:

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and v is my control ⇒ the system is affine in (z,v), and pick τ

to be OK. Theorem The system defined by:

is locally accessible about x0 if the accessibility distribution C spans n space, where n is the dimension of x and C is defined by:

[

[

]

]

C = g1,g2 ,...,gm , gi ,g j ,...,adg i k g j ,..., [ f ,gi ],...ad kf gi ,...

The gi terms 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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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

© 2010

Basically, ψ is the yaw angle of the vehicle, and x1 and x2 are the Cartesian locations of the wheels. u1 is the velocity of the front wheels, in the direction that they are pointing, and u2 is 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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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

© 2010

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 x1 can only increase. But let’s show it. In standard form: ,

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

© 2010

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 be dependent on x2, and the rank will drop at x2 = 0. Example 3:

where The system is of the form:

where

,

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and

Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard

© 2010

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