Quasi-Newton-Type Continuation Method for Nonlinear Receding Horizon Control

Abstract

A fast computational method of nonlinear receding horizon control is proposed, based on the continuation method combined with the quasi-Newton method. Jacobians in the differential equation of the unknown costate are replaced by their estimates, and the quasi-Newton method is employed to determine time derivatives of those estimates. Several modieed algorithms are obtained by imposing different conditions. The proposed method leads to considerable reduction of computational load compared to the conventional algorithms. Numerical examples of a two-wheeled car and a tethered satellite demonstrate computational time, accuracy, and robustness against noises of the proposed method. controlLyapunovfunctionstostabilizeaclassofnonlinearsystems. A controller designed by backstepping can also be interpreted asan optimal controller in terms of the inverse optimal control problem. 2 However, backstepping does not give a synthesis method to min- imize a performance index specie ed by a designer. Therefore, di- rect optimization methods for a given performance index should be explored. In the direct optimization of a controller, one has to solve the Hamilton- Jacobi-Bellman equation (HJBE). There have been nu- merous efforts to solve the HJBE. Because the HJBE is a partial differential equation for the optimal cost function, its solution has to be given over a some region in the state space. The structure of the solution is usually assumed for the HJBE, for example, in the form of power series, 3 interpolation, 4 or expansion with basis functions. 5 There is also a possibility of an approximate solution technique that does not assume the structure of the solution in the framework of genetic programming. 6 However, those methods can- not avoid the explosive growth of data storage or number of terms for high-dimensional systems and are dife cult to implement for a system whose dimension is higher than two or three. Ontheotherhand, iftheoptimalcontrolinputiscalculatedonline onlyfor thecurrentstate,ahugeamountofdata storageisnotneces- sary, even for a high-dimensional system. In this case, anopen-loop optimalcontrolproblem,whichleadstoatwo-pointboundary-value problem (TPBVP), is solved in place of the HJBE. Receding hori- zon control (model predictive control) is one of such control tech- niques. In receding horizon control, a e nite horizon performance index with a moving initial time and a moving terminal time ismin- imized. Receding horizon control is attractive from the theoretical point of view because it guarantees closed-loop stability, ifthe state isconstrained tobe zero at the end of the horizon, 7;8 or ifother con- ditions are satise ed. 9i13 Those characteristics concerning stability give guidelines for problem setting and selection of a performance index.