A new method for suboptimal control of a class of non-linear systems

Abstract

SUMMARYIn this paper, a new non-linear control synthesis technique (y-D approximation) is discussed. This approach achieves suboptimal solutions to a class of non-linear optimal control problems characterized by a quadratic cost function and a plant model that is affine in control. An approximate solution to the Hamilton-Jacobi-Bellman (HJB) equation is sought by adding perturbations to the cost function. By manipulating the perturbation terms both semi-global asymptotic stability and suboptimality properties are obtained. The new technique overcomes the large-control-for-large-initial-states problem that occurs in some other Taylor series expansion based methods. Also this method does not require excessive online computations like the recently popular state dependent Riccati equation (SDRE) technique. Furthermore, it provides a closed-form non-linear feedback controller if finite number of terms are taken in the series expansion. A scalar problem and a 2-D benchmark problem are investigated to demonstrate the effectiveness of this new technique. Both stability and convergence proofs are given.

Nonlinear Optimal DTC Design and Stability Analysis for Interior Permanent Magnet Synchronous Motor Drives

“…Proof: This Lemma can be proven in a similar way to [29]. Theorem 1: The closed-loop control system obtained by the error dynamics (5) and the nonlinear feedback control law (18) is semi-globally asymptotically stable.…”

“…In [29], [30], an approximation technique called the θ-D method is proposed to efficiently achieve the solution of the HJB equation (7) for a specified class of nonlinear systems.…”

“…According to [29], [30], the SDLEs (14) and (15) can be solved by the Kronecker product. However, by establishing (14) and (15) …”

“…where According to nonlinear control theories [29], [30], we need to find the stabilizing control term u that minimizes the following cost function for the system (5):…”

Explicit Nonlinear Model Predictive Control for a Saucer-Shaped Unmanned Aerial Vehicle

“…The linear model predictive control is used by modeling a discrete linear system [8]. To develop MPC for nonlinear system, many approaches have been proposed, such as the State-Dependent Riccati Equation Technique [9], using second-order model approximation [10], and solving the Hamilton-Jacobi-Bellman equation by power series expansion [11]. However, the methods are mentioned above require a plenty of online computations which is often computationally complex and time consuming and the real-time NMPC implementation is usually limited to slow processes where the sampling time is sufficient to support the computational needs [12].…”