Optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in both science and engineering. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy.
Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of variations by Edward J. McShane. Optimal control can be seen as a control strategy in control theory.
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 function. The optimal control can be derived using Pontryagin's maximum principle, or by solving the Hamilton–Jacobi–Bellman equation.We begin with a simple example. Consider a car traveling in a straight line on a hilly road. The question is, how should the driver press the accelerator pedal in order to minimize the total traveling time? 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 function will be a mathematical expression giving the traveling time as a function of the speed, geometrical considerations, and initial conditions of the system. Constraints are often interchangeable with the cost function.
Another related optimal control problem may be 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 related control problem may be 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. Minimize the continuous-time cost functional
subject to the first-order dynamic constraints
the algebraic path constraints
and the boundary conditions
where is the state, is the control, is the independent variable, is the initial time, and is the terminal time. The terms and are called the endpoint cost and Lagrangian, respectively. Furthermore, it is noted that the path constraints are in general inequality constraints and thus may not be active at the optimal solution. It is also noted that the optimal control problem as stated above may have multiple solutions. Thus, it is most often the case that any solution to the optimal control problem is locally minimizing.
Linear quadratic control
A special case of the general nonlinear optimal control problem given in the previous section is the linear quadratic optimal control problem. The LQ problem is stated as follows. Minimize the quadratic continuous-time cost functionalSubject to the linear first-order dynamic constraints
and the initial condition
A particular form of the LQ problem that arises in many control system problems is that of the linear quadratic regulator where all of the matrices are constant, the initial time is arbitrarily set to zero, and the terminal time is taken in the limit . The LQR problem is stated as follows. Minimize the infinite horizon quadratic continuous-time cost functional
Subject to the linear time-invariant first-order dynamic constraints
and the initial condition
In the finite-horizon case the matrices are restricted in that and are positive semi-definite and positive definite, respectively. In the infinite-horizon case, however, the matrices and are not only positive-semidefinite and positive-definite, respectively, but are also constant. These additional restrictions on
and in the infinite-horizon case are enforced to ensure that the cost functional remains positive. Furthermore, in order to ensure that the cost function is bounded, the additional restriction is imposed that the pair is controllable. Note that the LQ or LQR cost functional can be thought of physically as attempting to minimize the control energy.
The infinite horizon problem may seem overly restrictive and essentially useless because 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 a desired nonzero level can be solved after the zero output one is. In fact, it can be proved that this secondary LQR problem can be solved in a very straightforward manner. It has been shown in classical optimal control theory that the LQ optimal control has the feedback form
where is a properly dimensioned matrix, given as
and is the solution of the differential Riccati equation. The differential Riccati equation is given as
For the finite horizon LQ problem, the Riccati equation is integrated backward in time using the terminal boundary condition
For the infinite horizon LQR problem, the differential Riccati equation is replaced with the algebraic Riccati equation given as
Understanding that the ARE arises from infinite horizon problem, the matrices,,, and are all constant. It is noted that there are in general multiple solutions to the algebraic Riccati equation and the positive definite solution is the one that is used to compute the feedback gain. The LQ problem was elegantly solved by Rudolf Kalman.
Numerical methods for optimal control
Optimal control problems are generally nonlinear and therefore, generally do not have analytic solutions. As a result, it is necessary to employ numerical methods to solve optimal control problems. In the early years of optimal control the favored approach for solving optimal control problems was that of indirect methods. In an indirect method, the calculus of variations is employed to obtain the first-order optimality conditions. These conditions result in a two-point boundary-value problem. This boundary-value problem actually has a special structure because it arises from taking the derivative of a Hamiltonian. Thus, the resulting dynamical system is a Hamiltonian system of the formwhere
is the augmented Hamiltonian and in an indirect method, the boundary-value problem is solved. The beauty of using an indirect method is that the state and adjoint are solved for and the resulting solution is readily verified to be an extremal trajectory. The disadvantage of indirect methods is that the boundary-value problem is often extremely difficult to solve. A well-known software program that implements indirect methods is BNDSCO.
The approach that has risen to prominence in numerical optimal control since the 1980s is that of so-called direct methods. In a direct method, the state and/or control are approximated using an appropriate function approximation. Simultaneously, the cost functional is approximated as a cost function. Then, the coefficients of the function approximations are treated as optimization variables and the problem is "transcribed" to a nonlinear optimization problem of the form:
Minimize
subject to the algebraic constraints
Depending upon the type of direct method employed, the size of the nonlinear optimization problem can be quite small, moderate or may be quite large. In the latter case, the nonlinear optimization problem may be literally thousands to tens of thousands of variables and constraints. Given the size of many NLPs arising from a direct method, it may appear somewhat counter-intuitive that solving the nonlinear optimization problem is easier than solving the boundary-value problem. It is, however, the fact that the NLP is easier to solve than the boundary-value problem. The reason for the relative ease of computation, particularly of a direct collocation method, is that the NLP is sparse and many well-known software programs exist to solve large sparse NLPs. As a result, the range of problems that can be solved via direct methods is significantly larger than the range of problems that can be solved via indirect methods. In fact, direct methods have become so popular these days that many people have written elaborate software programs that employ these methods. In particular, many such programs include DIRCOL, SOCS, OTIS, GESOP/ASTOS, DITAN. and PyGMO/PyKEP. In recent years, due to the advent of the MATLAB programming language, optimal control software in MATLAB has become more common. Examples of academically developed MATLAB software tools implementing direct methods include RIOTS,DIDO, DIRECT, FALCON.m, and GPOPS, while an example of an industry developed MATLAB tool is PROPT. These software tools have increased significantly the opportunity for people to explore complex optimal control problems both for academic research and industrial problems. Finally, it is noted that general-purpose MATLAB optimization environments such as TOMLAB have made coding complex optimal control problems significantly easier than was previously possible in languages such as C and FORTRAN.
Discrete-time optimal 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. The Theory of Consistent Approximations provides conditions under which solutions to a series of increasingly accurate discretized optimal control problem converge to the solution of the original, continuous-time problem. Not all discretization methods have this property, even seemingly obvious ones. For instance, using a variable step-size routine to integrate the problem's dynamic equations may generate a gradient which does not converge to zero as the solution is approached. The direct method is based on the Theory of Consistent Approximation.Examples
A common solution strategy in many optimal control problems is to solve for the costate . The costate summarizes in one number the marginal value of expanding or contracting the state variable next turn. The marginal value is not only the gains accruing to it next turn but associated with the duration of the program. It is nice when can be solved analytically, but usually, the most one can do is describe it sufficiently well that the intuition can grasp the character of the solution and an equation solver can solve numerically for the values.Having obtained, the turn-t optimal value for the control can usually be solved as a differential equation conditional on knowledge of. Again it is infrequent, especially in continuous-time problems, that one obtains the value of the control or the state explicitly. Usually, the strategy is to solve for thresholds and regions that characterize the optimal control and use a numerical solver to isolate the actual choice values in time.