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Faux Riccati equation techniques for feedback control of nonlinear and time-varying systems

Prach, Anna
Rapid development of nonlinear control theory for application to challenging and complex problems is motivated by the fast technological development and demand for highly accurate control systems. In infinite-horizon nonlinear optimal control the essential difficulty is that no efficient analytical or numerical algorithm is available to derive exact expressions for optimal controls. This work concerns the numerical investigation of faux Riccati equation methods for control of nonlinear and linear time-varying (LTV) systems. These methods are attractive due to their simplicity and potentially wide applicability. Considered methods include state-dependent Riccati equation (SDRE) control and forward-propagating Riccati equation (FPRE) control. In SDRE control the instantaneous dynamics matrix is used within an algebraic Riccati equation solved at each time step. FPRE control solves the differential algebraic Riccati equation forward in time rather than backward in time as in classical optimal control. While applications and theoretical developments of the SDRE technique are widely reflected in the literature, FPRE is a newly developed approach, which is heuristic and suboptimal in the sense that neither stability nor optimal performance is guaranteed. This approach requires development of a theoretical framework that addresses practical aspects of FPRE design, and provides conditions and guidelines for implementation. This work presents the basic properties of the solution of the FPRE for LTI plants in comparison with the solution of the backward-propagating Riccati equation (BPRE), shows the duality between FPRE and BPRE, and investigates stabilizing properties of FPRE. Pareto performance tradeoff curves are used to illustrate the suboptimality of the FPRE as well as the dependence on the initial condition of the Riccati equation. When applied to nonlinear systems, faux Riccati equation techniques entail pseudolinear models of nonlinear plants that use either a state-dependent coefficient (SDC) or the Jacobian of the vector field. To investigate the strengths and weaknesses of SDRE and FPRE methods, this work presents a numerical study of various nonlinear plants under full-state-feedback and output-feedback control. Within the scope of FPRE, an internal model principle is used for command following and disturbance rejection problems for LTV and nonlinear systems. The performance of this approach is investigated numerically by considering the effect of performance weightings, the initial conditions of the difference Riccati equations, plant initial conditions and domain of attraction, and the choice of SDC. Numerical studies include an inverted pendulum, a two-mass system, Mathieu equation, Van der Pol oscillator, ball and beam, rotational- translational actuator, and a fixed-wing aircraft.