Optimal control

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Optimal control theory, a generalization of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators.[1]

Contents

[edit] General method

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost functional. The optimal control can be derived using Pontryagin's maximum principle (a necessary condition), or by solving the Hamilton-Jacobi-Bellman equation (a sufficient condition).

We begin with a simple example. Consider a car traveling on a straight line through a hilly road. The question is, how should the driver press the accelerator pedal in order to minimize the total traveling time? Clearly in this example, the term control law refers specifically to the way in which the driver presses the accelerator and shifts the gears. The "system" consists of both the car and the road, and the optimality criterion is the minimization of the total traveling time. Control problems usually include ancillary constraints. For example the amount of available fuel might be limited, the accelerator pedal cannot be pushed through the floor of the car, speed limits, etc.

A proper cost functional is a mathematical expression giving the traveling time as a function of the speed, geometrical considerations, and initial conditions of the system. It is often the case that the constraints are interchangeable with the cost functional.

Another optimal control problem is to find the way to drive the car so as to minimize its fuel consumption, given that it must complete a given course in a time not exceeding some amount. Yet another control problem is to minimize the total monetary cost of completing the trip, given assumed monetary prices for time and fuel.

A more abstract framework goes as follows. Given a dynamical system with time-varying input u(t), time-varying output y(t) and time-varying state x(t), define a cost functional to be minimized. The cost functional is the sum of the path costs, which usually take the form of an integral over time, and the terminal costs, which is a function only of the terminal (i.e., final) state, x(T). Thus, this cost functional typically takes the form

J=\phi(x(T)) + \int_0^T L(x,u,t)\,\mathrm{d}t

where T is the terminal time of the system. It is common, but not required, to have the initial (i.e., starting) time of the system be 0 as shown. The minimization of a functional of this nature is related to the minimization of action in Lagrangian mechanics, in which case L(x,u,t) is called the Lagrangian.

[edit] Linear quadratic control

It is very common, when designing proper control systems, to model reality as a linear system, such as

\dot{x}(t)=A x(t) + B u(t)
y(t) = C x(t) \,.

One common cost functional used together with this system description is

J=\int_0^\infty ( x^T(t)Qx(t) + u^T(t)Ru(t) )\,\mathrm{d}t

where the matrices Q and R are positive-semidefinite and positive-definite, respectively. Note that this cost functional is thought in terms of penalizing the control energy (measured as a quadratic form) and the time it takes the system to reach zero-state.

This functional could seem rather useless since it assumes that the operator is driving the system to zero-state, and hence driving the output of the system to zero. This is indeed correct. However the problem of driving the output to the desired level can be solved after the zero output one is. In fact, it can be proved that this secondary problem can be solved in a very straightforward manner. The optimal control problem defined with the previous functional is usually called the state regulator problem and its solution the linear quadratic regulator (LQR) which is no more than a feedback matrix gain of the form

u(t)=-K(t)\cdot x(t)

where K is a properly dimensioned matrix and solution of the continuous time dynamic Riccati equation. This problem was elegantly solved by Rudolf Kalman.[2]

[edit] Discrete time control

The examples thus far have shown continuous time systems and control solutions. In fact, as optimal control solutions are now often implemented digitally, contemporary control theory is now primarily concerned with discrete time systems and solutions.

[edit] See also

[edit] References

  1. ^ L. S. Pontryagin, 1962. The Mathematical Theory of Optimal Processes.
  2. ^ Kalman, Rudolf. A new approach to linear filtering and prediction problems. Transactions of the ASME, Journal of Basic Engineering, 82:34–45, 1960

[edit] Further reading

  • Bryson, A. E., 1969. Applied Optimal Control: Optimization, Estimation, & Control.
  • Kirk, D. E., 2004. Optimal Control Theory: An Introduction.
  • Lebedev, L. P., and Cloud, M. J., 2003. The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics. World Scientific. Especially chpt. 2.
  • Lewis, F. L., and Syrmos, V. L., 19nn. Optimal Control, 2nd ed. John Wiley & Sons.
  • Stengel, R. F., 1994. Optimal Control and Estimation. Dover.
  • Sethi, S. P., and Thompson, G. L., 2000. Optimal Control Theory: Applications to Management Science and Economics, 2nd ed. Springer (ISBN 0-7923-8608-6)
  • Sontag, Eduardo D. Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition. Springer. (ISBN 0-387-984895) (available free online)
  • Brogan, William L. 1990. Modern Control Theory. ISBN 0135897637
  • Bryson, A.E.; Ho, Y.C. (1975). Applied optimal control. Washington, DC: Hemisphere. 
Journals

[edit] External links

  • Elmer G. Wiens: Optimal Control - Applications of Optimal Control Theory Using the Pontryagin Maximum Principle with interactive models.