Transition to Nonlinear H1 Optimal Control From Inverse-Optimal ...
Proceedings of the 2001 IEEE International Conference on Robotics & Automation Seoul, Korea • May 21-26, 2001
Transition to Nonlinear H∞ Optimal Control From Inverse-Optimal Solution for Euler-Lagrange System Jonghoon Park Post Doctoral
Wan Kyun Chung Professor
Robotics and Bio-Mechatronics (RNB) Lab., Department of Mechanical Engineering Pohang University of Science and Technology (POSTECH), Pohang, 790-784, Republic of Korea E-mail: {coolcat,wkchung}@postech.ac.kr Abstract— One of recent achievements in the field of nonlinear H∞ optimal control theories for Euler-Lagrange systems is the analytic solution to the Hamilton-JacobiIsaccs (HJI) equation associated to the so-called nonlinear H∞ inverse-optimal control [1]. In this paper, we address the problem of nonlinear H∞ optimal control design for an Euler-Lagrange system, rather than the inverse-optimal problem. By introducing a technique of control weight loosening and state weight strengthening, we will show that the associated HJI inequality, not the equation, for nonlinear H∞ optimal control can be solved also analytically using the inverse-optimal solution.
1
Introduction
The so-called Euler-Lagrange (EL) system covers a variety of physical systems, so intensive and extensive researches to design controllers for the EL system have been conducted [2], [3]. Difficulties of controlling an EL system, even a given specific system, come from the fact that it should deal with inevitable interferences due to disturbances and uncertainties, therefore robustness issue should be considered. In many situations, robustness requirement should be traded-off as higher performance is needed, e.g. fast and accurate operation spanning wide operation range using limited actuation capability, etc.. Roughly speaking, performance and robustness should both be given equal importance in controller design procedures. The so-called nonlinear H∞ optimal control [4], [5] have attracted many research attentions recently, partly because it is motivated by that philosophy, and partly because it casts the control design problem within a nicely formulated mathematical framework centered at a certain partial differential equation. The nonlinear H∞ optimal control design reflects the performance requirement by optimality of a cost variable for closed-loop system, e.g. the well-known quadratic measure of the state and the control input, while the robustness requirement is tackled by L2 -gain attenuation requirement of the closed-loop system. The stability follows directly as a consequence. Unfortunately such a nonlinear H∞ optimal control design procedure is not so easy to apply. One of the main reasons is that one has to solve a partial differential inequalJonghoon Park is currently a visiting researcher at Faculty of Engineering, Hiroshima University, Japan. E-mail: [email protected].
0-7803-6475-9/01/$10.00© 2001 IEEE
ity or equation, generally called the Hamilton-Jacobi-Isaccs (HJI). However, it is very difficult to solve, even numerically, and even the existence of a solution for a specific HJI equation is not guaranteed [6]. Therefore, many researchers devised various techniques to circumvent the difficulty encountered in dealing with the original nonlinear H∞ optimal control design. Trial to solve a specific nonlinear H∞ optimal control can be found in many literature employing robot manipulators which form one representative class of the EL systems. Specifically, Chen et. al. [7] formulated disturbances including the joint acceleration signal and applied a special coordinate transformation which reduces the HJI expression to an algebraic matrix one. Park et al. [1] aimed at solving a nonlinear H∞ inverse-optimal control analytically. The term inverse-optimal implies that the problem solves a special form of cost variable as well as the control, rather than solving an optimal control when a cost variable is given. Then, there arises one question whether the HJI inequality of equation for EL system is solvable or not, and if solvable what the solution is. This paper is prepared to provide an answer to the question. We will show that every EL system allows a nonlinear H∞ optimal tracking control and arbitrary L2 -gain attenuation is possible for a given quadratic cost function with constant weight matrices, if a specific condition is met. Specifically, we solve the HJI inequality analytically using the solution technique developed in solving the HJI equation for the nonlinear H∞ inverse-optimal control of EL systems [1]. This paper is organized in the following manner. First, we state the nonlinear H∞ optimal control problem which we solve, focusing on the disturbance and cost variable formulation, in Sec. 2. Next, in Sec. 3 we summarize the main system properties of the EL system, especially the nonlinear H∞ inverse-optimality. Then in Sec. 4 we solve the associated HJI inequality using the solution to the HJI equation associated with the inverse-optimal problem, using the tool of state weight strengthening, and discuss the characteristics of the nonlinear H∞ optimality. Sec. 6 will provide one numerical example using a simple EL system.
2
Formulation of Nonlinear H∞ Optimal Control
Many of physical systems can be modeled by the Lagrange equations of motion. Denoting the generalized co-
1161
ordinates of system by q = (q1 , q2 , · · · , qn )T , the equation yields ˙ q˙ + g(q) + d(t), τ = M (q)¨ q + C(q, q)
(1)
where τ ∈
Transition to Nonlinear H∞ Optimal Control From Inverse-Optimal Solution for Euler-Lagrange System Jonghoon Park Post Doctoral
Wan Kyun Chung Professor
Robotics and Bio-Mechatronics (RNB) Lab., Department of Mechanical Engineering Pohang University of Science and Technology (POSTECH), Pohang, 790-784, Republic of Korea E-mail: {coolcat,wkchung}@postech.ac.kr Abstract— One of recent achievements in the field of nonlinear H∞ optimal control theories for Euler-Lagrange systems is the analytic solution to the Hamilton-JacobiIsaccs (HJI) equation associated to the so-called nonlinear H∞ inverse-optimal control [1]. In this paper, we address the problem of nonlinear H∞ optimal control design for an Euler-Lagrange system, rather than the inverse-optimal problem. By introducing a technique of control weight loosening and state weight strengthening, we will show that the associated HJI inequality, not the equation, for nonlinear H∞ optimal control can be solved also analytically using the inverse-optimal solution.
1
Introduction
The so-called Euler-Lagrange (EL) system covers a variety of physical systems, so intensive and extensive researches to design controllers for the EL system have been conducted [2], [3]. Difficulties of controlling an EL system, even a given specific system, come from the fact that it should deal with inevitable interferences due to disturbances and uncertainties, therefore robustness issue should be considered. In many situations, robustness requirement should be traded-off as higher performance is needed, e.g. fast and accurate operation spanning wide operation range using limited actuation capability, etc.. Roughly speaking, performance and robustness should both be given equal importance in controller design procedures. The so-called nonlinear H∞ optimal control [4], [5] have attracted many research attentions recently, partly because it is motivated by that philosophy, and partly because it casts the control design problem within a nicely formulated mathematical framework centered at a certain partial differential equation. The nonlinear H∞ optimal control design reflects the performance requirement by optimality of a cost variable for closed-loop system, e.g. the well-known quadratic measure of the state and the control input, while the robustness requirement is tackled by L2 -gain attenuation requirement of the closed-loop system. The stability follows directly as a consequence. Unfortunately such a nonlinear H∞ optimal control design procedure is not so easy to apply. One of the main reasons is that one has to solve a partial differential inequalJonghoon Park is currently a visiting researcher at Faculty of Engineering, Hiroshima University, Japan. E-mail: [email protected].
0-7803-6475-9/01/$10.00© 2001 IEEE
ity or equation, generally called the Hamilton-Jacobi-Isaccs (HJI). However, it is very difficult to solve, even numerically, and even the existence of a solution for a specific HJI equation is not guaranteed [6]. Therefore, many researchers devised various techniques to circumvent the difficulty encountered in dealing with the original nonlinear H∞ optimal control design. Trial to solve a specific nonlinear H∞ optimal control can be found in many literature employing robot manipulators which form one representative class of the EL systems. Specifically, Chen et. al. [7] formulated disturbances including the joint acceleration signal and applied a special coordinate transformation which reduces the HJI expression to an algebraic matrix one. Park et al. [1] aimed at solving a nonlinear H∞ inverse-optimal control analytically. The term inverse-optimal implies that the problem solves a special form of cost variable as well as the control, rather than solving an optimal control when a cost variable is given. Then, there arises one question whether the HJI inequality of equation for EL system is solvable or not, and if solvable what the solution is. This paper is prepared to provide an answer to the question. We will show that every EL system allows a nonlinear H∞ optimal tracking control and arbitrary L2 -gain attenuation is possible for a given quadratic cost function with constant weight matrices, if a specific condition is met. Specifically, we solve the HJI inequality analytically using the solution technique developed in solving the HJI equation for the nonlinear H∞ inverse-optimal control of EL systems [1]. This paper is organized in the following manner. First, we state the nonlinear H∞ optimal control problem which we solve, focusing on the disturbance and cost variable formulation, in Sec. 2. Next, in Sec. 3 we summarize the main system properties of the EL system, especially the nonlinear H∞ inverse-optimality. Then in Sec. 4 we solve the associated HJI inequality using the solution to the HJI equation associated with the inverse-optimal problem, using the tool of state weight strengthening, and discuss the characteristics of the nonlinear H∞ optimality. Sec. 6 will provide one numerical example using a simple EL system.
2
Formulation of Nonlinear H∞ Optimal Control
Many of physical systems can be modeled by the Lagrange equations of motion. Denoting the generalized co-
1161
ordinates of system by q = (q1 , q2 , · · · , qn )T , the equation yields ˙ q˙ + g(q) + d(t), τ = M (q)¨ q + C(q, q)
(1)
where τ ∈
