Suboptimal Synthesis of a Robust Decentralized Control for Large-Scale Mechanical Systems *

For a large-scale nonlinear mechanical system, a set of nominal trajectories is prescribed together with a region of allowable parameter values. The system is viewed as a set of decoupled linear time-invariant subsystems and a local controller is synthesized for each subsystem neglecting, in the first step, nonlinear coupling among subsystems. An iterative procedure is established to choose unique linear local controllers which can accommodate all nominal trajectories and withstand parameter variation. If such controllers cannot be found a force feedback is implemented as a global control in order to compensate for the influence of nonlinear and variable coupling. An example of a six-degree-of-freedom industrial manipulator is presented. Introduction MODELS of large-scale mechanical systems, such as robotic systems, flying vehicles, hydraulic systems, etc. are usually wellknown. However, these models are often very complex, and include variable parameters. Thus, if we want to synthesize control which should cover all dynamic regimes, we ought to implement a complex nonlinear and/or adaptive control law. Our aim is to synthesize a simple robust control law which could withstand all nonlinearities of the system and all parameter variation. Application of decentralized control to the large-scale mechanical systems leads to a simple, robust and cheap control structure. Theory of decentralized control is highly developed (e.g. Davison, 1976; Sandell and co-workers, 1978; Siljak, 1978, etc.), but its application to the large-scale nonlinear systems is connected with numerous problems. In our previous papers (Vukobratovi~:, 1973; Vukobratovi~ and Stoki6, 1978, 1980a, b, etc.) we have already considered the application of the decentralized control to the large-scale mechanical systems. Each degree of freedom (d.o.f) of the mechanical system is considered as a subsystem. This is a very commonly chosen decentralized structure when the large-scale mechanical systems are in question. Local control is synthesized for each subsystem, neglecting the interconnections among the subsystems. Since the influence of interconnections among subsystems may be too strong, we have introduced nominal programmed control calculated using a centralized model of the system. This programmed control realizes the desired mot ion of the system in an ideal case when no perturbation is acting upon the system. However, this approach is acceptable when the desired mot ion is well known in advance, when parameters of the system are precisely defined and constant (or their variations are defined) and when there are only a few trajectories to be realized (which have to be repeated many times, as is the common case in industrial robotics). However, if these assumptions are not met the synthesis of nominal programmed control using the * Received August 1983; revised 17 April 1984. This paper was recommended for publication by associate editor M. Jamshidi under the direction of editors A. P. Sage and A. H. Levis. t Department of Robotics, "Mihailo Pupin" Institute, Belgrade, Yugoslavia. complete, centralized model is not recommendable. These are the reasons why, in this paper, we shall synthesize completely decentralized control, i.e. we shall avoid the introduction of centralized programmed control. We shall synthesize local controllers and examine the stability of the complete model of the system. However, in our previous papers (Vukobratovi6 and Stoki~, 1980a, b) we have analyzed asymptotic stability of the system around nominal control. Now, when only local controllers are applied, we must directly analyze practical stability of the global system around the nominal trajectory. We shall restrict our considerations to the so-called 'naive decentralized control' which is most often used in practice when mechanical systems are in question. Here we shall, for the first time, consider the practical stability of large-scale mechanical system and estimate ability of the decentralized control to withstand nonlinearities of the model and parameter variation. Our aim is to synthesize simple decentralized control but robust enough to accommodate a large number of desired trajectories and withstand parameter variation. If one finds out that such 'naive decentralized control' cannot stabilize the particular largescale mechanical system, global control may be introduced. Mathematical model and control task definition Let us consider the mechanical large-scale system S consisting of the mechanical part of the system S M and actuators S ~. The system has n degrees of freedom (d.o.f.) each powered by one actuator S i. The model of actuator is given by a linear timeinvariant system Si:~ i = Aix i + biu i q-fiPi, V i i i = {i:i = 1 . n} (1) where x ~ e R"' is the vector of state coordinates of S i, n~ is the order of the model S ~, A i e R"' ~ "' is the system matrix, b ~ e R"' is the input distribution vector, f~ ~ R "' is the load distribution vector, u ~ ~ R is the input to the actuator S ~, P~ e R t is the driving torque (load) acting upon the ith d.o. f. (actuator), which is a function of all state coordinates of the system S, x = ( x 11 , x 21 . . . . ,xn~) 1, x ~ R N, N = ~ n~ and of parameters d i 1 SM:pi =Ji(x, d), ViEI (2) whereJi: R N x R t ~ Rt, and I x 1 vector d of parameters belongs to finite region deD ~_ R t, i.e. D is the set of allowed values of mechanism parameters. It is not required that Pi(0, d) = 0. The adequate smoothness requirements are assumed to be satisfied throughout all the regions of interest. The following control task is imposed. The set of L1 nominal trajectories x°(t), VteTt , l e L = { l : l = 1,2 . L1} is given, where Tl = {t:{t~(0,zt)}, zl are given time periods. We have to synthesize the control which should ensure practical stability of the system S around each nominal trajectory x°(t), so that Vx(O)~X~ and VdED~ implies x(t,x(O)~X~(t)), Vt~TI, Vl~L, where X ~ e R N is the given finite region of allowable initial l | N conditions around the lth nominal trajectory x (0),"Xl(t) _~ R , Vt • T~, is the given finite time-varying region of allowable system states around x°(t) at the moment t e T~, and D~ _= D is the set of