Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method

Abstract

A nonlinear control problem has been posed by Bupp et al. to provide a benchmark for evaluating various nonlinear control design techniques. In this paper, the capabilities of the state‐dependent Riccati equation (SDRE) technique are illustrated in producing two control designs for the benchmark problem. The SDRE technique represents a systematic way of designing nonlinear regulators. The design procedure consists of first using direct parameterization to bring the nonlinear system to a linear structure having state‐dependent coefficients (SDC). A state‐dependent Riccati equation is then solved at each point x along the trajectory to obtain a nonlinear feedback controller of the form u=−R‐1(x)BT(x)P(x)x, where P(x) is the solution of the SDRE. Analysis of the first design shows that in the absence of disturbances and uncertainties, the SDRE nonlinear feedback solution compares very favorably to the optimal open‐loop solution of the posed nonlinear regulator problem, the latter being obtained via numerical optimization. It is also shown via simulation that the closed‐loop system has stability robustness against parametric variations and attenuates sinusoidal disturbances. In the second design it is demonstrated how a hard bound can be imposed on the control magnitude to avoid actuator saturation. © 1998 John Wiley & Sons, Ltd.

Application of the state-dependent Riccati equation method in nonlinear control design for inverted pendulum systems

“…One of the systematic and effective ways for nonlinear state-feedback control design for underactuated systems is the method based on the solution of a state-dependent Riccati equation (SDRE), which relies on the original nonlinear state-space description of the system and creates a separate linear quadratic optimal control problem (LQR) at each time step. As a result, tradeoff between acceptable control accuracy (state error) and control input effort is ensured, which is a property not generally found in other nonlinear control design methods [3].As a significant class of nonlinear unstable underactuated mechanical systems, inverted pendulum systems (IPSs) are well-suited for verification and practice of ideas and techniques emerging in control theory and robotics [4]. Stabilization of a set of interconnected pendulum links in the unstable upright position is considered a benchmark control problem which has been solved by attaching the pendulum links to a base that moves in a controlled linear manner (classical or linear IPSs) or in a rotary manner in a horizontal plane (rotary IPSs) [5].…”

Adaptive SDRE Based Boost-Phase Missile Autopilot Design Using Single and Modular Neural Networks

“…According to the optimal control theory (Mracek et al (1998)), the following state-dependent algebraic Riccati equation (ARE) can be obtained…”

Closed-form solution to finite-horizon suboptimal control of nonlinear systems

“…The result given in Theorem 4 resembles the one given for local asymptotic stability of SDRE in [12], however, besides extending the result to Finite-SDRE and making it stronger, that is, exponential stability of the origin, presenting Lemmas 2 and 3 and the line of proof of Theorem 4 gives a more rigorous proof of the local stability result.…”