The science of nonlinear dynamics, popularly known as 'Chaos Theory', is now emerging from research laboratories and finding practical uses. Nonlinearity appears everywhere in the real world: gear chatter; stickslip friction; poor lubrication; large deflection of beams; magnetic forces; fluid motion; and on and on. Standard engineering education and practice has been to reduce all problems to the linear regime for the very wellfounded reason that it is possible to derive and understand analytical solutions to linear problems. Nonlinear problems on the other hand, with occasional exceptions, are impossible to solve analytically.
But with the appearance of cheap computer power in the mid 1970's it became possible to make sense of previously untouchable problems resulting in a new understanding of nonlinear dynamics and chaos, and the applications that are now emerging. One of the most promising applications is dynamical feedback control in systems wherenonlinearities cause irregular motion, that is, chaos. Nonlinear dynamical feedback control has several advantages overstandard linear feedback control. The biggest is that: The data is the model, there is no need to know the equations of motion describing the system.
All the information needed to generate control can be extracted from time series data of the system under normal operating conditions. Even more remarkable, data from one system variable is sufficient to build the model. Of course, one can exploit information from known equations or datafrom several independent variables, which can be very helpful but is not essential.
A second advantage is that nonlinear control is achieved by applying smallperturbations at dynamically chosen times. It is not necessary to modify the plant since control can be applied using existing equipment. Furthermore: Control is robust, a 10 to 20 percent error in control parameters still gives good control, the model does not have to be perfect. Dynamical control can be made adaptive and adjust for normal system drifts without redoing the entire control model.
Finally, it is possible to derive multiple control strategies from the same set of data, which is feasible because of the underlying structure of chaotic dynamics.
How dynamical feedback works In order to understand how dynamical feedback control works we need to understand two things. First, we need to understand the origin of chaotic motion and its implications for any control scheme. Second, we need to understand its underlying structure and how it can be used to control what appears to be irregular aperiodic behavior.
Chaos makes its appearance as irregular motion that never repeats itself. But even though the motion is aperiodic it is most definitely not random noise. It is a deterministic consequence of the nonlinearities in the equations of motion that describe a physical system.
The prime example of a chaotic system is the weather. As we all know, except for rough seasonal variation, the weather is unpredictable.
But is it really?
The local weather forecast on the evening news is usually quite goodfor the next two or three days but is untrustworthy a week in advance. What is happening here is due to a hallmark of all chaotic dynamics: Extreme sensitivity to initial conditions. The weatherman has a model which he updates each day with the day's temperature, pressure, the position of the jet stream, and a variety of other variables. The predictions work for a couple of days and then start to differ from what the weather actually does. This is because small errors in the data he puts into his model get amplified by the nonlinearities in the model, in fact, they are amplified exponentially with time, resulting in poor long term forecasts. This exponential amplification of the errors would occur even if his model of the weather were perfect
As a second much simpler example, I have built in my lab a simple double pendulum driven by an electric motor. If the current driving the motor is a sine wave with a fixed frequency and amplitude, the pendulum swings wildly and seems never to do the same thing twice. It will even perform complete 360 degree rotations both clockwise and counterclockwise.
Suppose the pendulum is started from dead vertical one time and one degree from the vertical a second time. The two different initial conditions produce motions that are nearly the same for the first ten or twenty seconds but gradually become different until after a minute or two the motions are completely different, maybe in one case swinging from right to left while in the other doing a 360 counter clockwise rotation. This is another example of sensitive dependence on initial conditions, and illustrates two points: If I know with modest accuracy where the pendulum is now, I can predict what it will do for a short time in the future. This makes control possible since the system will behave reliably if I give it a small perturbation.
A small kick now produces a large and unpredictable change if I wait too long. Consequently, I am going to have to go on making the small control perturbations at the right time and 'place' to maintain control.
Before going on and describing the second piece that makes up control I have to digress on how dynamical systems are modeled. They are usually modeled in phase space or, as it is sometimes called, state space.
Consider a simple pendulum that can be described by two variables, its angle from the vertical and its rotation speed. Normally one would plot the angle or speed versus time. It turns out to be more useful to plot the angle versus the speed and leave the time implicit.
First we would record the angle and speed at a series of points in time giving us a set of anglespeed pairs. We would then plot these pairs with, say, the angle on the horizontal axis and the speed on the vertical axis.
If the simple pendulum happened to be swinging backandforth with the simple harmonic motion of a grandfather clock, the anglespeed pairs would plot out a circle. If the grandfather clock were winding down, the pairs would plot out a decaying spiral that stops at the origin.
The double pendulum: Natural chaos In the case of the double pendulum there are four variables: The angle and speed of the inner arm that is attached to the electric motor, and the angle and speed of the outer arm that pivots on the end of the inner arm. (Of course, the four variables can't be plotted against each other on a piece of paper, but it is easy to do the equivalent in computer memory.) The chaotic motion of the double pendulum is then a very complicated curve in this four dimensional phase space.
We could do the same with the weather model using as variables temperature, pressure, and the many other needed variables. The result would be a trajectory in a very high dimensional phase space. An important feature of nonlinear systems is that we can construct these phase space trajectories without having to measure or even know all the variables in the system. One can construct a 'delay vector' from a single variable by combining measurements taken at different times. For example, instead of a point in phase space being described by two angles and two speeds for the double pendulum, one could use a vector constructed from measurements of the speed of the inner arm now, 5, 10, and 15 seconds ago. A set of such vectors made up of four points separated by 5 seconds in time can be used to model the system phase space.
There is a deep mathematical theorem that proves that the phase space model generated this way is completely equivalent to the usual one created by using all the variables. The time between samples can be calculated in a straightforward way by computing either the autocorrelation function or a quantity called the mutual information. If we don't know ahead of time by physical reasoning how many elements are needed to make up a delay vector, that number also can be evaluated by calculating the fractal dimension of the chaotic time series.
Studying these curves, we notice that only a limited region of phase space is occupied. There are always physical bounds on every variable: for example: the velocity of the pendulum is limited by the energy input from the motor; temperatures in North America range from about 25 F to 110 F; and so on. It is also true that no matter how complicated a chaotic trajectory might become, it never intersects itself. This is because these systems are deterministic, not random.
Repeating an initial condition perfectly always leads to exactly the same results. (Unpredictability came from our imperfect knowledge of the current values of the variables.) The result is that a chaotic trajectory is a tangled curve that folds over on itself in a immensely complicated way but that lies in a bounded region of phase space. In fact, taking a slice through this structure and looking at it 'endon' would reveal that it is a fractal.
But it turns out that this structure has an even more interesting propertyit is composed of an infinite number of periodic orbits. If we find the right initial conditions and don't perturb the system it is possible to make it repeat itself over and over. (These orbits are closed loops in phase space.) So, for the right starting values of angles and speeds, the double pendulum will behave just like a grandfather clock even though it ordinarily swings and twirls in an unpredictable fashion.
Why don't we see these periodic orbits but get chaos instead? Because they are unstable, and unstable in precisely the same way that a marble resting on a saddle is unstable. The stable direction is fronttoback on the saddle making the marble roll to the middle, and the unstable direction is sidetoside making the marble roll off the sides.
Controlling chaos This immediately suggests how to generate control. Move the saddle under the marble so that it rests on the fronttoback line. When it starts to roll off to one side, move the saddle again to reposition the marble once more on the fronttoback line. Clearly this is a never ending process since it is never possible to put the marble exactly at the most stable point on the saddle and even the slightest amount of noise will make the marble roll off in the unstable direction again. How often we make the correction is determined by the shape of the saddle. The marble will roll away from the stable line very slowly if the saddle is broad and flat, requiring only an occasional correction, but very quickly if it is steep and narrow, forcing many quick perturbations.
As I mentioned above, there are an infinite number of unstable periodic orbits. This provides the ability to tailor a control scheme to a particular purpose and even switch among several different ones at will without changing the plant operating parameters. All that is required is to choose the size and timing of the control pulses appropriate forstabilizing a particular orbit. In practice of course, only a modest number of these orbits can be found and are useable, but they still introduce a level of flexibility to dynamic feedback control that is not available with standard linear feedback control.
Controlling with chaos Control of chaos has been successfully demonstrated in the lab in many different areas over the last decade. It is now moving into the industrial world. Motorola is adding this technology to products that will appear in the next few years. Several other Fortune 100 companies are actively incorporating this science into their process technology. Biological applications now under investigation include a smart pacemaker,epilepsy control, and tremor control for diseases like Parkinson's.
The US Navy is applying dynamical feedback control to reduce helicopter vibrations. Onboard ship cranes are nearly useless in even modest seas. The Navy is trying to extend the range over which they can load and unload cargo by adding dynamical control to the cranes.
The coming decade will see expanding use of this technology, limited only by the imagination of engineers. The most obvious application is to rotating machinery in places like paper mills, rolling mills, auto engines, and generators. Other possibilities include petrochemical plants and fluidized beds that are widely used for combustion and chemical catalysis. Even if control is not needed, the techniques can be used to model and monitor machine and plant performance in new ways that will reduce maintenance costs and increase equipment lifetime.
