Conference Paper

# An iterative procedure for optimal nonlinear tracking problem.

Fac. of Eng., Ferdowsi Univ. of Mashhad, Iran

DOI: 10.1109/ICARCV.2002.1234997 Conference: Seventh International Conference on Control, Automation, Robotics and Vision, ICARCV 2002, Singapore, 2-5 December 2002, Proceedings Source: DBLP

**ABSTRACT**

It has been shown that for a class of nonlinear systems x=f(x)+g(x)u, the solution of the infinite horizon optimal regulation problem leads to a state dependent Ricatti equation. Under appropriate assumptions the optimal control may be obtained from the point-wise solution of an algebraic Ricatti equation during state evolution. This method cannot be used in finite horizon optimal regulation and finite or infinite horizon of optimal tracking problem. To solve the nonlinear quadratic regulation or tracking problems, two forward and backward equations corresponding to the state and co-state systems respectively must be solved and then the optimal control can be derived. Since the co-state equation is state-dependent and it develops backward and the state is not accessible in whole time, then the control law cannot be calculated. To overcome this problem, an iterative procedure is proposed. This method can be applied to both finite and infinite horizon optimal regulation and tracking problems. Simulation results are given for the nonlinear benchmark problem introduced in and Lorenz attractor as a chaotic system.

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**ABSTRACT:**In this paper, we address the finite-horizon optimization of a class of nonlinear singularly perturbed systems based on the state-dependent Riccati equation (SDRE) technique and singular perturbation theory. In such systems, both slow and fast variables are nonlinear. Moreover, the performance index for the system states is nonlinearly quadratic. In this study, unlike conventional methods, linearization does not occur around the equilibrium point, and it provides a description of the system as state-dependent coefficients (SDCs) in the form f(x)-A(x)x. One of the advantages of the state-dependent Riccati equation method is that no information about the Jacobian of the nonlinear system, just like the Hamilton-Jacobi-Belman (HJB) equation, is required. Thus, the state-dependent Riccati equation has simplicity of the linear quadratic method. On the other hand, one of the advantages of the singular perturbation theory is that it reduces high-order systems into two lower order subsystems due to the interaction between slow and fast variables. In the proposed method, the singularly perturbed state-dependent Riccati equations are first derived for the system under study. Using the singular perturbation theory, the singularly perturbed state and state-dependent Riccati equations are separated into outer layer, initial, and final layer correction equations. These equations are then solved to obtain the optimal control law. Simulation results in comparison with the previous methods indicate the desirable performance and efficiency of the proposed method. However, it should be noted that due to the dependence of the proposed method on the choice of state-dependent matrices and the presence of a nonlinear optimal control problem, the results are generally suboptimal.Journal of Dynamic Systems Measurement and Control 09/2013; 136(1):011003. DOI:10.1115/1.4024602 · 0.98 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This article investigates finite-time optimal and suboptimal controls for time-varying systems with state and control nonlinearities. The state-dependent Riccati equation (SDRE) controller was the main framework. A finite-time constraint imposed on the equation changes it to a differential equation, known as the state-dependent differential Riccati equation (SDDRE) and this equation was applied to the problem reported in this study that provides general formulation and stability analysis. The following four solution methods were developed for solving the SDDRE; backward integration, state transition matrix (STM) and the Lyapunov based method. In the Lyapunov approach, both positive and negative definite solutions to related SDRE were used to provide suboptimal gain for the SDDRE. Finite-time suboptimal control is applied for robotic manipulator, as finite-time constraint strongly decreases state error and operation time. General state-dependent coefficient (SDC) parameterizations for rigid and flexible joint arms (prismatic or revolute joints) are introduced. By including nonlinear control inputs in the formulation, the actuator׳s limits can be inserted directly to the state-space equation of a manipulator. A finite-time SDRE was implemented on a 6R manipulator both in theory and experimentally. And a reduced 3R arm was modeled and tested as a flexible joint robot (FJR). Evaluations of load carrying capacity and operation time were investigated to assess the capability of this approach, both of which showed significant improvement.ISA Transactions 07/2014; 54. DOI:10.1016/j.isatra.2014.06.006 · 2.98 Impact Factor

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