H_∞ Preview Control Problem

With the development of sensor technology, there are more and more methods andtools to obtain information. How to deal with and make use of the obtained informationhas become an important and urgent problem. The paper thus concentrates on the topicof using information available in advance to design controller or estimator to achieve thedesired performance, which is fundamental in control theory. The corresponding H_∞problems are very involved and thus listed as one of 53 Open Problems in MathematicalSystems and Control Theory.Consider that there exists a dual relationship between the H_∞fixed-lag smoothingand preview control problem, the conclusions for the preview control problems can beextended to the corresponding fixed-lag smoothing problems. The paper considers H_∞preview control problems for both continuous-time and discrete-time systems, respec-tively, with single-channel preview, multi-channel preview and time-varying preview.Compared with infinite horizon controllers and estimators, the finite horizon onescan offer a better transient performance, which is an important property for applications.In addition, we note that even though the plant is linear time-invariant, it is necessary toconsider time-varying controllers since the natural solutions even for linear time-invariantplants happen to be linear time-varying. Therefore, the paper will focus on the finitehorizon H_∞preview control problem.we investigate the continuous-time H_∞preview control problem so as to clarifythe difference between the standard H_∞control and H_∞preview control. Via differen-tial game theory and classic optimal control theory, we find a pair of state and costatedifferent from the usual one to characterize the game value. Making use of InvariantEmbedding approach, we generalize the linear relationship between initial data and allrelated trajectories and get an H_∞-like Riccati equation with the same dimension as thestate. Associated with the infinite dimensional system theory, the solvable condition andthe desired controller are provided based on the solution to the H_∞-like Riccati equation.A simpler approach to the discrete-time H_∞preview control problem in finite hori-zon is pursued. We consider the relationship between the H_∞preview control and linearquadratic zero-sum game problem, make use of completion of square and method of theundetermined coefficients, and get the explicit controller and the sufficient and neces-sary condition under which the H_∞preview control problem is solvable. The approach consumes less computation and can be generalized to deal with the multi-channel andtime-varying preview case. It also hints an idea for the continuous-time H_∞previewcontrol problem.It considers the H_∞control for systems with multi-channel preview and time-varying preview. The approach proposed for the single-channel case in the above isgeneralized to deal with the multi-channel case. As for the time-varying preview case, itis converted into a single-channel multi-preview problem and then solved.The continuous-time H_∞preview control problem is considered. It is convertedinto a optimization problem for the indefinite quadratic form, which is associated witha differential game problem. According to Dynamic Programming, simple tools suchas completion of square and the method of undetermined coefficients are utilized to getthe coupled ordinary and partial differential equations, which are essential to solve thecontrol and estimate problems for the preview or delay systems. However, there is noaccepted approach to solve the coupled equations. To the end, we present a decouplingidea and thus achieve the analytic solution to the coupled equations by the characteristicapproach.It generalizes the idea for the deterministic problem and considers the H_∞previewcontrol problem for stochastic systems with state and control-dependent noises. Theproblem is converted into a optimization one for the indefinite quadratic form, which isassociated with the differential game problem. Dynamic Programming is then appliedto solve the game problem and provide the solvable condition and the desired controllerbased on the coupled differential and partial differential equations.