five more golden rules : knots, codes, chaos, and other great theories of 20th century mathematics

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title:Five More Golden Rules : Knots, Codes, Chaos, and Other Great Theories of 20th Century Mathematicsauthor:Casti, J. L.publisher:John Wiley & Sons, Inc. (US)isbn10 | asin:0471395285print isbn13:9780471395287ebook isbn13:9780471437895language:EnglishsubjectMathematics--Popular works.publication date:2000lcc:QA93.C383 2000ebddc:510subject:Mathematics--Popular works.

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Chapter 1

The Alexander Polynomial:

Knot Theory

Five More Golden Rules Knots, Codes, Chaos, and Other Great Theories of 20th Century Mathematics/files/page_10.html

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The crossing number is thus an invariant of the knot, in the sense that it depends only on the intrinsic "knottiness" of the closed curve and not on the way it happens to be projected onto the plane. Therefore, we can use it as a crude filter to separate some knots from others. For instance, the trefoil with crossing number 3 cannot possibly be deformed into the Figure-8 knot shown in Figure 1.5, which has crossing number 4. Unfortunately, the crossing number is a pretty crude invariant because there are lots of inequivalent knots having the same crossing number. For example, it has been shown that there are exactly 9,988 different knots having at most 13 crossings. Consequently, we need much more subtle invariants if we hope to be able to separate this set of nearly ten thousand knots into distinct classes.

To illustrate the concept of the crossing number, Figure 1.6 shows a tabulation of all inequivalent knots have eight or fewer crossings. It is of some aesthetic interest to note that the famed German Renaissance painter Albrecht Drer created six ornamental knots stimulated by knot designs of Leonardo da Vinci. These six Drer knots are shown in Figure 1.7. But we still need better ways than the crossing number to separate knots.

A knot is a simple, closed curve, so it makes sense to think about starting at some point on the knot, and then traversing it and returning to your starting point. The direction one moves in such a traversal of the knot is called the knot's orientation. This is a basic aspect of a knot that is of great value in finding invariants. We can also attach a sign to each doublepoint of a knot. We do this by using the French traffic rule that cars on the right approaching an intersection have right-of-way. This means that a doublepoint is "positive" if the strand passing under the

Figure1.5

Theoriented

Figure-8knot.


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Chapter 3

The Kalman Filter:

Control Theory


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Looking for Life

In summer 1975, two Viking spacecraft were launched from Cape Canaveral, Florida, on their way to a rendevous with Mars one year later, where their mission was to search for signs of life on the Martian surface. Upon arrival, both spacecraft were put into a predetermined orbit around Mars, from which the search for suitable landing places began. On July 20, 1976, the Viking I lander came to rest in the Chryse Planitia region of Mars, some 23 degrees north of the equator, and six weeks later Viking II settled down onto the Utopia Planitia plain, which is almost exactly 180 degrees of longitude away from the first landing site, thus putting the two landers on opposite sides of the planet.

There were six different instruments on the landers used for the detection of signs of life: two cameras for photographing the landscape, a combined gas chromatograph and mass spectrometer for analyzing the surface for organic material, and three instruments designed to detect the metabolic activities of any microorganisms present in the Martian soil. Sad to say, none of the tests produced unequivocal evidence of life at either of the landing sites. Nevertheless, much data of great scientific value was obtained, including signs of extraordinary chemical reactivity of the Martian soil. But for our story here, what is notable about the entire Viking mission is that it took place at all! Think for a moment what was involved in just getting the two landers to their antipodal locations on the Martian surface.

Leaving aside the mechanical, electrical, and chemical engineering aspects of the Viking spacecraft, the sine qua non of the mission is the obvious point that NASA had to be able to get the landers from Cape Canaveral to their resting points on Mars. This task involved several aspects of guidance and control: determining what kind of propulsion was necessary to transfer a certain kind of vehicle from Earth to a Martian orbit; calculating the various trajectories possible; identifying the trajectories of minimal cost in some combination of time, energy, and other resources; developing the capability to measure where the spacecraft was located at any time during its flight; and being able to exert midcourse corrections if and when the spacecraft drifted off its design trajectory. At the center of all these tasks sits what has come to be called the Kalman filter, a mathematical theory that, along with the simplex method of optimization theory (see Chapter 5 of Five Golden Rules), is


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probably the single most useful piece of mathematics developed in this century. Our story in this chapter is an account of what this gadget is and how it came to occupy its pivotal role in the mathematical theory of control systems.

Can You Get There from Here?

To see what's involved in the problem of guidance and control, let's look at a simple toy version of the problem. Suppose we have a spacecraft whose trajectory is constrained to move along the real line, as shown in Figure 3.1. If the vehicle starts at the origin with an initial velocity in the positive direction, then it's clear that by simple inertia it will just sail along, eventually visiting every location on the positive real axis. But if we want to bring the spacecraft to a soft landing at, say, the point marked X in Figure 3.1, then we would have to turn it around somehow, and then exert a retrothrust to neutralize the inertia from the rocket's initial thrust, thereby reducing the vehicle's velocity to zero at the point X. Under what circumstances can this be done? In other words, when can we transfer the spacecraft from the origin with a given initial velocity to the point X with zero velocity? This is a question that control theorists call a problem of reachability.

To see what's at issue in solving the reachability problem, consider a linear, discrete-time control system moving in the plane. To make things as transparent as possible, suppose the control input at each moment is a simple scalar, and that it enters into the dynamics in a linear fashion. Under these conditions, the system dynamics can be written in the form

Figure3.1

Aspacecraftmovingalongtherealline.


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For ease of writing, let's write the above system in the vector-matrix form

where F and g are given by

Here the quantities fij, gi(i, j = 1, 2) are just real numbers. Furthermore, it turns out to involve no loss of generality to assume that the initial state of the system is x1(0) = x2(0) = 0, that is, the system starts at the origin. The reachability problem can now be stated as: What points (x1, x2) in the plane can be "reached" within, say, T time steps by a suitable choice of the control inputs u(0), u(l),... , u(T 1)?

First of all, let's dispose of the two extreme cases, (1) when there is no control at all (u(t) = 0 for all t), and (2) when the control input cannot influence the system state (if g1 = g2 = 0). In both these cases, the system state remains at the origin forever; hence, no state other than the origin is ever reachable. Trivial as these two cases are, they illustrate the twin points that (1) a control system isn't a control system if you can't exert some influence on the behavior of the system's state, and (2) the only way you can influence the state is when the channel between the control and the system state remains open. So we will henceforth assume without further comment that these conditions are satisfied.

Suppose, then, that the initial control input u(0) is nonzero. Then the system state at time t = 1 is x1(l) = u(0)g1 and x2(1) = u(0)g2. Thus, as long as the quantities g1 and g2 are not both zero, the system moves away from the origin after one time step to the point (g1, g2)u(0). If we introduce the state vector x(t) = (x1 (t), x2 (t))' and the input vector g = (g1, g2)', this can be written more compactly as x(l) = gu(0). (Note: Here the notation v' means simply the transpose of the vector v.) This means that in one time step the system can be transferred to any point on the line from the origin through the point g. By a similar line of argument, the state at the second time step is


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Geometrically, we represent these state transitions in Figure 3.2. These calculations show that by a suitable choice of the control inputs u(0) and u(1), it is possible to transfer the system state from the origin to any point that can be written as a linear combination of the two vectors g and F g. So if these vectors are linearly independent, there is some sequence of control inputs transferring the system state from the origin to any point in the (x1, x2)-plane. We call such a control system completely reachable. Geometrically, these results say that any point that can be reached by application of a control input lies in the subspace generated by the vectors g and Fg.

To summarize, the system is completely reachable if and only if the vectors g and F g are linearly independent (i.e., they generate the entire state space). Moreover, in that case any state can be reached from the origin in not more than two time steps. Now let's look at the general situation.

Figure3.2

Thefirsttwostatetransitionsforthesystem.


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The Problem of Reachability

The reachability results obtained above impose several restrictions on the control system and the admissible control inputs. We now want to examine these assumptions explicitly, looking for how they can be relaxed and still retain the basic conclusions. The main points that need addressing are the following:

Linearity: The system dynamics are linear with constant coefficients (the matrix F and the vector g are have constant entries), while the control input enters additively into the dynamics.

Unbounded, scalar control: No constraints have been imposed on the magnitude of the control inputs; the quantities u(0), u(1),... can be any sequence of real numbers.

Discrete time set: We have assumed that the system changes state only at integer moments in time, that is, the time set of the system is the nonnegative integers t = 0, 1, 2,... .

Planar system: Our results have been obtained only for a two-dimensional system whose state space is the entire plane R2.

Happily, it turns out that the reachability conditions can be generalized to accommodate nonlinear processes, constraints on the control, and continuous time and state spaces of arbitrary, even infinite, dimension. Let's briefly consider these matters in the order opposite to the way things are listed above.

Suppose we have a discrete-time n-dimensional linear control system with m control inputs. This means that the system state is a vector in Rn, whereas the control inputs are now vectors in Rm. Mathematically, we can write such a system in the form

where F and G are n n and n m real matrices, respectively, and x(t) is in Rn and u(t) is in Rm. Note that here we still retain the assumptions of discrete time and unbounded control. We can generalize the earlier reachability result for the system as follows:


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REACHABILITY THEOREM The linear dynamical system is completely reachable if and only if the block matrix

contains n linearly independent vectors, that is, rank C = n.

The proof of this fundamental result follows exactly the same lines as for the planar case given earlier. After the initial step, the system state is x(l) = Gu(0). This means that any state that is a linear combination of the columns of G is reachable in one time step. Similarly, at the second time step we have x(2) = FGu(0) + Gu(1), implying that any state that is a linear combination of the columns of the matrices FG and G can be reached in two time steps. Carrying on in this way, we arrive at the Reachability Theorem. (Note: Just in case you're wondering why we stop after at most n 1 steps, the reason is that the Cayley-Hamilton Theorem of elementary matrix theory implies that for an n n matrix A, the columns of all powers of A higher than n 1 are linear combinations of the columns of the matrices I, A, A2,..., An1. In system-theoretic terms, this means that if you haven't been able to reach a state by n 1 time steps, you'll never get thereit can't be done.)

So we see that there is no problem whatsoever in extending the planar reachability results to the case of n-dimensional systems. The same holds true when we consider continuous rather than discrete time systems. So if the control system is given by the continuous time, n-dimensional process

where the matrices F and G are as before, then the condition for the system to be completely reachable is exactly the same as that given in the Reachability Theorem, namely, that the reachability matrix C be of full rank (i.e., contain n linearly independent vectors). In fact, for continuous time systems, if a state can be reached at all, then it can be reached in an arbitrarily short amount of time. This follows from the fact that we are imposing no finite bound on the magnitude of the control inputs. This will certainly raise the eyebrows of a hard-nosed, practically minded control engineer. But it's theory we're talking about here, not


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practice, and the mathematics says that if you can pump an arbitrarily large amount of energy into a system in a short enough period of time, you can do almost anythingprovided the energy can actually reach the system state.

Now let's illustrate these reachability results with a quasireal-life example from economics.

National Economic Planning

Just about every country in the world is faced with trying to juggle the following two economic objectives: (1) achieving full employment without inflation (internal economic balance), and (2) maintaining a positive, or at least neutral, balance of international payments (external economic balance). These goals have to be achieved through economic policy instruments such as changes in interest rates and in the budget deficit. As a result, the country has a central bank that controls interest rates and a legislative body that controls changes in the government's budget deficit. Although it is politically difficult to combine these institutions into a single controlling agency, it is possible to establish general directives for both of them to follow. Our question here concerns whether the actions of the central bank and the congress taken together can achieve the country's overall economic goals.

To study this problem, let's define the following variables:

Y(t) domestic production (i.e., the income of consumers)X(t) national expendituresC(t) national consumptionS(t) national savingsI(t) domestic investmentsM(t) imports of foreign goods and servicesK(t) net capital outflowT(t) net taxes on transfersA(t) government expenditures on goods and servicesB(t) international balance-of-payments surplusr(t) domestic interest rate

All of these variables are annual rates in time period t and are deflated to a uniform price level. Let us define also the following variables, which are all assumed to be constant:


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Figure1.6

Allknotswitheightorfewercrossings.

doublepoint has the right-of-way, whereas it's "negative" if the strand passing over the doublepoint has priority.

Obviously, a given knot can have two separate orientations, each of which leads to attaching a sign, plus or minus, to the crossing number of the knot. Figure 1.5 shows the Figure-8 knot oriented such that each


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E exports of goods and servicesYF domestic production with full employment and no inflationrF foreign interest rate

Four accounting identities link these variables:

Furthermore, the following linear relationships have been found to be approximately valid:

where the quantities ai, bi, gi, and di (i = 0, 1) are empirically determined constants of proportionality.

If we define the state and control vectors as

where D(t) = A(t) T(t) is the net government deficit, the above relations lead to the linear control system

where

From a given initial state x(0), the government's economic planners want to steer this economy to the target state

in which the international payments are in balance and an internal level YF of outputs is attained that involves full employment and no inflation.

For this economy, it doesn't take too much thought to see that if there are no restrictions on the changes in interest rates and deficits, any desired state of the economy can be reached if the matrix G is nonsingular. Because this will be the case for almost every choice of the quantities d1, b1, g1, and a1, the complete reachability of this economy is ''stable" with respect to small changes in these parameters.


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The foregoing economic example shows how unrealistic it is, in general, to assume that an unlimited amount of influence can be brought to bear on a system. Control costs something, either in time, men, money, energy, or some other currency appropriate for the system at hand. So let's see what can be salvaged from these reachability results when restrictions are imposed on the control inputs.

There are many ways in which control actions can be constrained, so here we just give a flavor of the way in which reachability results are modified by constraints on the control inputs by considering the situation in which the inputs are restricted to be nonnegative. This is probably the most common type of control constraint for the simple reason that many controlling actions make no sense unless the input is positive or zero. For example, if the task is to detoxify a river or lake by putting in chemical nutrients, it's difficult to imagine what it would mean to put in a negative amount of nutrient. A similar situation arises when, for example, we wish to regulate a person's heartbeat by administering medication. Here we can't give the person a negative amount of medicine. For discussions of other types of control constraints, the reader is referred to the material cited in the chapter references.

Suppose we have the scalar control system

From our earlier results, it is clear that this system is completely reachable. But if we impose the condition that u(t) 0 for all t, then the system is not positively reachable; the only states that can be reached by such a system are those lying on the positive real axis. Similarly, the planar system

is also not positively reachable because for this system the only states that can be reached from the origin are those of the form

where

In other words, only poin

five more golden rules : knots, codes, chaos, and other great theories of 20th century mathematics

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