Numerical Methods For Optimal Control
Optimal control problems are generally nonlinear and therefore, generally do not have analytic solutions (e.g., like the linear-quadratic optimal control problem). As a result, it is necessary to employ numerical methods to solve optimal control problems. In the early years of optimal control (circa 1950s to 1980s) 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 (or, in the case of a complex problem, a multi-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 form
where
is the augmented Hamiltonian and in an indirect method, the boundary-value problem is solved (using the appropriate boundary or transversality conditions). The beauty of using an indirect method is that the state and adjoint (i.e., ) 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 (particularly for problems that span large time intervals or problems with interior point constraints). A well-known software program that implements indirect methods is BNDSCO.
The approach that has risen to prominence in numerical optimal control over the past two decades (i.e., from the 1980s to the present) is that of so called direct methods. In a direct method, the state and/or control are approximated using an appropriate function approximation (e.g., polynomial approximation or piecewise constant parameterization). 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 (e.g., as in a direct shooting or quasilinearization method) or may be quite large (e.g., a direct collocation method). In the latter case (i.e., a collocation method), 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 (e.g., SNOPT) to solve large sparse NLPs. As a result, the range of problems that can be solved via direct methods (particularly direct collocation methods which are very popular these days) 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 written in FORTRAN include DIRCOL, SOCS, OTIS, GESOP/ASTOS and DITAN. 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, 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-strength 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.
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