FP11 zyxwvutsr 16:30 zyxwvuts Proceedingsof the 37th IEEE Conference on Decision zyxw 8. Control Tampa, Florida USA December 1998 zyx Global Stabilization of Unstable Equilibrium Point of Pendulum' zyx A. Shiriaev 0. Egeland H. Ludvigsen Department of Engineering Cybernetics Norwegian University of Science and Technology N-7034 Trondheim NORWAY Halgeir . LudvigsenQi t k.ntnu .no E-mail: Ant on.Shiriaev@i t k.nt nu.no, Olav .Egeland@i tknt nu.no, Keywords: pendulum, energy level stabilization, un- stable equilibrium point stabilization 1 Introduction and Preliminaries During last few years the global (local) stabilization problem of unstable equilibrium point of controlled pendulum has attracted much attention of nonlinear control people and some approaches to the solution of this problem were proposed, see [l, 3, 41 and others. In [3] it was shown that for the controlled pendulum any prescribed energy level can be locally stabilizable by speed-gradient method, see also [5, 61. In [I] the last algorithm was named by energy method and was applied specially for the swinging up from downward position to upright point where it can be caught with different strategy, for instance, by stabi- lization of linearized model of pendulum in some neigh- bourhood of upright point. In this paper we suggest zyxwvutsrqp new solution of pendulum up- per equilibrium point stabilization which also essen- tially based on the energy control approach. An idea which we mainly utilize is that the energy level H, of the unforced pendulum corresponding to the upper equilibrium contains only one w-limit point , namely, the upper equilibrium. So stabilizing the energy level H, one can hope that this desired equilibrium point will be an unique attractive point of closed loop sys- tem. To this end it seems naturally to apply the direct energy control suggested by Fradkov in [3]. But this is not case and energy feedback control does not provide the required convergence rate of the closed loop system solution to H, and, in particular, can lead to infinite rotations with damping energy (proposition 2). It turned out and was very surprizing that the sim- ple modification of the energy control method leads us to the global stabilizability of upper equilibrium point (theorem 3). Moreover, we provide the qualitative analysis of the closed loop system motions. It is shown 'The work was supported by Norwegian Research Council. 0-7803-4394-8198 $1 0.00 zyxwvutsrqp 0 1998 IEEE 4584 that the almost any solution of the closed loop sys- tem achieves the desired energy level H, for finite time and that the set of exceptional initial points, for which the pendulum tends to upright point but achieves the energy level H, for infinite time, is only one smooth curve on the cylindrical phase space. In addition, we prove that this curve does not have any intersections and a-limit points except only the downward equilib- rium point (theorem 4). The equations of motions of the controlled mathemat- ical pendulum are j(t) = -mgl . sin q(t) + ml . U, (1) where y, p are generalized coordinate and moment, U is a control function; m, 1, g are mass, length of pen- dulum and gravity acceleratiofi correspondingly. It is convenient to choose as a phase space of pendulum mo- tions a cylindrical phase space with a unit circle in a base, i. e. -T < q 5 T. Recall that the full energy (or Hamiltonian function) Ho(q, p) of unforced pendulum (with U = 0) has the form and is a conserved quantity of unforced pendulum. Let H, be any nonnegative constant, introduce the scalar functions V(q, p) and y(q, p) as follows V(q7 P) = [Ho(q, P) - H*I2/2, (4) Y(Y , P) = P . [Ho(Q, P) - H*l/I. (5) The functions V, y possess the important property. Namely, the controlled pendulum (l), (2) with the out- put function (5) is a passive system with nonnegative storage function V, see definition 2.4 [a] Proposition zyxw 1 Let [q(t),p(t)] be any motion of un- forced pendulum (l), (2) subjected to constraint: y(t) = 0 for all t zyxwv 2 0. Then iwo cases are only possible: 1.) Ho(q(t),p(t)) = H, for all t 2 0; 2.1 Idt), EO, 01 or [q(t>, dt)l = [r, 01.