Elsevier

ISA Transactions

Volume 52, Issue 2, March 2013, Pages 285-290
ISA Transactions

Research Article
Nonlinear closed loop optimal control: A modified state-dependent Riccati equation

https://doi.org/10.1016/j.isatra.2012.10.005Get rights and content

Abstract

The state-dependent Riccati equation (SDRE), as a controller, has been introduced and implemented since the 90s. In this article, the other aspects of this controller are declared which shows the capability of this technique. First, a general case which has control nonlinearities and time varying weighting matrix Q is solved with three approaches: exact solution (ES), online control update (OCU) and power series approximation (PSA). The proposed PSA in this paper is able to deal with time varying or state-dependent Q in nonlinear systems. As a result of having the solution of nonlinear systems with complex Q containing constraints, using OCU and proposed PSA, a method is introduced to prevent the collision of an end-effector of a robot and an obstacle which shows the adaptability of the SDRE controller. Two examples to support the idea are presented and conferred. Supplementing constraints to the SDRE via matrix Q, this approach is named a modified SDRE.

Highlights

► A modified SDRE based on supplementing constraints into Q matrix is introduced. ► The PSA for solving the modified SDRE is presented. ► An example with control nonlinearities and time varying Q is solved by three methods. ► Obstacle avoidance via SDRE, as a result, in robotics is expressed. ► A planar and a 3D robot as examples are simulated.

Introduction

The fundamental and base of the SDRE method was initiated in the 1960s by Pearson [1]. Beeler proposed the state-dependent Riccati equation with state and control nonlinearities [2]. Solving SDRE containing control nonlinearities, first proposed by Beeler and explained via solving some examples, and online control update formulation was presented. The SDRE method is one of the approaches for solving the Hamilton–Jacobi–Bellman (HJB) equation as a closed loop nonlinear optimal control method. Beeler et al. presented feedback control methodologies for nonlinear systems. Different methods for solving the HJB equation were presented and the PSA method was also explained for solving the SDRE [3]. In this work, two main purposes are followed. First, the structure of the SDRE with time varying or state-dependent Q was explained. The methods of solving the SDRE were presented. A general example for introducing the ability of the SDRE technique and comparing the online control update, PSA and exact solution is proposed. Next, this paper presents the application of this technique: the idea of adding constraints into Q matrix. The constraint can be the distance between the end-effector of a robot and an obstacle which results obstacle avoidance using the SDRE. Azimi et al. presented the obstacle avoidance in robotics via open loop optimal control (OLOC) approach [4]. In order to solve the SDRE with Q containing constraint, OCU [2] can be applied. Implementing the OCU in practical work is not easy which is discussed in Section 2 and the PSAs proposed in [2], [3] are not peculiar to this type of Q and as a result, a new PSA method for solving the problem is proposed. Structure of the controller and methods of solving the SDRE are developed in Section 2 and an example to illustrate the process is explained. Obstacle avoidance is considered in Section 2.1 and finally, conclusion is expressed in Section 3.

Section snippets

Structure of the controller and methods of solving the SDRE

The design and procedure of the state-dependent Riccati equation method is quiet simple and systematic. The equation of nonlinear system can be written in the form ofẋ(x(t),u(t),t)=f(x(t),t)+g(x(t),u(t),t)where x(t)∈ℝn, f(x(t),t)∈ℝn→ℝn and g(x(t),u(t),t)∈ℝn→ℝn. The state-dependent coefficient (SDC) parameterization needs to be used to form:f(x(t),t)=A(x(t),t)x(t)g(x(t),u(t),t)=B(x(t),u(t),t)u(t)where A(x(t),t):ℝn→ℝn×n,▒ u(t)∈ℝm, and B(x(t),u(t),t):ℝn→ℝn×m. The aim of this method is

Conclusion

In this work, some points of view of the SDRE controller were expressed which presented the capability of this technique. First, a general case which had control nonlinearities and time varying weighting matrix Q was solved with three approaches and the results supported each other. The proposed PSA in this paper is able to deal with time varying or state-dependent Q in nonlinear systems. As a result of having the solution of nonlinear systems with complex Q containing constraints, a method was

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