Proceedings of the 1998 IEEE International Conference on Control Applications Trieste, Italy 1-4 September 1998
TA05
Design of a Nonlinear H , Controller for the Inverted Pendulum System Bor-Chin Chang" Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA. 19 104,
[email protected]
Shr-Shiung Hu* Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA. 19104,
[email protected] Abstract -- A well-known control application, the inverted pendulum system, is formulated as a noniinear H, control problem. In order to construct a nonlinear H, controller, it is essential to find a solution for the Hamilton-Jacobi equation (HJE). A successive algorithm is employed to obtain an approximate solution of the HJE and then a nonlinear H , controller is constructed. Simulations of the closed-loop pendulum responses for both the nonlinear and the linear H , controllers will be performed and compared. It is found that the nonlinear H , controller has better performance and robustness than the linear controller.
pendulum system. Simulations of the closed-loop system with the nonlinear H , controller will be given and compared to those with linear H, controller. The rest of the paper is organized as follows. In Section 2, we introduce the notations and briefly review the basic concept of the energy dissipation, the nonlinear H, control problem, and the nonlinear H , controller Formulas. In Section 3, the state space representation of the inverted pendulum system is derived and then a nonlinear H , control problem is formulated. In Section 4, a modified successive algorithm for solving the HE is presented. The design procedure in constructing the nonlinear H , controller and the simulations are addressed in Section 5. Section 6 is the Conclusions.
Keywords: nonlinear H, control, energy dissipation, inverted pendulum, Hamilton-Jacobi equation
2. Preliminaries Notations The notations used in this paper are fairly standard. xT is the transpose of vector x. llx112 = x T x is the squared Euclidean norm. IR" is the n-dimensional Euclidean space. 5 is the state of the controller. If we assume E is a differentiable function then the derivative of E with
1. Introduction Since the linear H , control problem can be easily solved by the DGKF [ I ] state-space approach, recently the much more complicated nonlinear H , control problem has drawn attention to many investigators. Although the nonlinear H, controller formulas 12-61 have been successfully derived by the concept of the energy dissipation, not much application is available in the literature. Some questions arise on how to find the solution of the Hamilton-Jacobi equation (IIE) easily and efficiently, how to construct the nonlinear H , controller, and whether the nonlinear H , controller has better performance and robustness than its linear counterpart. In this paper, we will give the design procedure of constructing a nonlinear H- controller for the inverted pendulum control system and address the above questions. The inverted pendulum system [7] used in this paper is quite standard. The pivot of the stick is mounted on the cart which can move only in one-dimension horizontally. The inverted pendulum itself is an unstable nonlinear system. The objective is to design a (nonlinear H,) controller to drive the cart motor so that the cart can move back and forth to maintain the stick at a straightly vertical position. The inverted pendulum system will be formulated as a nonlinear H , control problem. In order to obtain a nonlinear H , controller one needs to solve the HJE. It is very difficult, if not impossible, to find the exact explicit solution for HJE. Some successive approximate solution methods [2,5,9] are available for the HJE. In this paper, we will present a modified successive algorithm based on [2,8] to find approximate solutions for the HJE and then construct a nonlinear H _ controller for the inverted
[e e]
respect to x is defined as E = - . . . . -.*
. I "
represents the t i x n identity matrix. O(x"') means the higher order terms including x"'. (.)('I is the k-th order temi and (.)"' means the accumulated k-th order terms including xi. ARE is the algebraic Riccati equation. HJE and HJI are the Hamilton-Jacobi equation and inequality respectively. Some other notations for the inverted pendulum system will be introduced in Section 3. Concept of Dissipative System Definition 2.1 Consider the following system C .K
G :{
= F ( x , w)
z = H ( x ,w )
where w is the input and z is the output. With y a preassigned tolerance level, the system is said to be Y dissipative i f there exists a nonnegative energy storage fiitiction E with E ( 0 ) = 0 such that
j"r[llzl12 - Y2114Z)df 5 E(x(0))- E ( x ( T ) ) -E(x(T))< 0
(2-2)
The inequality means that the H , n o m of the system is less than or equal to y i f T approaches to infinity. When y = I , the inequality implies that the input energy is greater than or equal to the output energy. In other words,
* This research was supported in part by NASA Langley Research Center under Contract NCC-1-224 and in part by the Boeing Company under Contract NAS 1-20220.
0-7803-4104-X/98/$10.0001998 IEEE 699
Nonlinear H , Controller Formulas
some energy has been dissipated and hence the system is called dissipative. From Definition 2.1, it is easy to see that the system is Y-dissipative if and only if the energy Hamiltonian function H = 1 1 ~ 1 1-~ y2IIwI’
+ E, . F ( x ,w)
The nonlinear H, controller formulas addressed in [2,3,4,6] with the DGKF kind of assumptions are summarized in the following theorem. Theorem 2 , l Consider the nonlinear generalized plant depned in (2-4).If there exists a controller K of the form (2-5) such that the closed-loop system is stable and Ydissipative, then we have the following:
(2-3)
is nonpositive for all x and w in the domain of interest.
The Nonlinear H , Control Problem Consider the following nonlinear generalized plant G : G :
(1) Tlzere exist X(x) and Y , ( x ) such tluzt the following two Hamilton-Jacobi inequalities
input-affine
I
X = f ( x > + g,( x ) w+ g2(x)u z = /z,(x)+ D12(x)u ) I = (x)+ 02, (x)w
HJX(x):=/qT(x)h,(x)+ 2Xr(x)f(x) (2-8a) +x’(x>[Y-2g, (x)g,7(x)- g?(x>g:(x)lx(x>5 0
(2-4)
HJY,(x):= l?;(x)h,(x) - y2/?:(x)/z,(x) + y~(x)[Y-2gl(,~)g:(x)ly/(x) 0
where x E IR” is the state of the system, z E IR”‘ is the controlled output, w E IR”” is the exogenous input including all commands and disturbances, U E IR”” represents the control input, and y E IR’” is the measured output. The problem is to find a controller
are satisfied for all x in the domain of interest.
nonlinear y-dissipative H , controller can be constructed by:
(3) A
= f ( t )Y-’sl(t)C!?T(5)x(h?(5)
+ C!?? (5
such that the closed-loop system is stable and Ydissipative.
cK(5)
= -g,’(
