ABSTRACT
The purpose of control of a dynamic system called object or plant is to force the plant to behave su¢ ciently closely to (to follow su¢ ciently precisely, to track su¢ ciently accurately) its desired output behavior over some, usually prespeci…ed, time interval and under real (usually unpredictable and unknown) both external (input) actions and initial conditions [216, pp. 121-127]. The very, the primary, goal of control is to assure that the controlled plant
exhibits a requested kind of output tracking that we will call, for short, tracking in this special sense. However, the term tracking in the wide sense concerns all kinds of the plant real output vector following its desired output vector. Historically considered, tracking studies started as the servomechanism or
servosystem theory in the general sense. Among its pioneers are L. A. MacColl, 1945 [233]; H. Lauer, R. Lesnick and L. E. Matson, 1947 [219]; J. C. West, 1953 [317]; H. Chestnut and R. W. Mayer, 1955 [33]; I. Flügge-Lotz and C. F. Taylor, 1956 [60]; J. C. Lozier, 1956 [231]; and G. S. Brown and D. P. Campbell, 1948 [22]. A. I. Talkin used the term “servo tracking”in 1961 [309]. The name servomechanism or servosystem signi…es the controller that should force the plant output, or forces the controlled plant output, to follow, i.e., to track, its, in general, time-varying desired output rather than to track only a constant desired output. The latter is the purpose of the feedback controller (the controller in the closed-loop control system) called classically regulator . In the control literature tracking has been mainly and largely studied as the
problem of the zero steady state error problem. This means that the control synthesis should assure that the control forces the plant real output (or state) to approach asymptotically the plant desired output (or state) as time t tends to in…nity, respectively. The tracking studies started in this sense, then the control studies began. They have been known under di¤erent names such as studies of servomechanisms/servosystems, or of regulation systems, or of control systems in general comprising the preceding ones, i.e., as studies on the servo-
mechanism/servosystem problem, or studies on the regulation problem, or on the control problem in general incorporating the preceding ones. The problem of the zero steady state error has been commonly treated as a part of stability and stabilization studies, which might be a reason for which the control need for the tracking theory in its own right was not recognized until 1980 [145], [146], [280], [281]. During his …rst visit to Belgrade B. Porter presented the problem of tracking of nonlinear Lurie-Postnikov systems to the author and proposed the common research that resulted in those papers and in [278]. The notion, the sense, and the meaning of tracking herein signify in gen-
eral that the real plant output follows, i.e., tracks, every plant desired output belonging to a family Ykd, k 2 f; 0; 1; :::; n; ::g, - regardless of whether the desired output is constant (in a part of the
control literature this is related to the regulation systems) or time-varying (in another part of the control literature it is associated with the servomechanisms/servosystems), - under the actions of arbitrary external disturbances belonging to a set Dk,
k 2 f; 0; 1; :::; n; ::g, and - under arbitrary (input and output) initial conditions. Therefore, tracking incorporates both the servomechanism/servosystem prob-
lematic and the regulation issues; i.e., it spans the whole control thematic. Subsection 2.1.2 presents the characterization of the disturbance family Dk
and of the desired output family Ykd. Tracking is perfect (ideal) if, and only if, the plant real output behavior is
always equal to the plant desired output behavior. If the initial real output is di¤erent from the initial desired output then tracking is only imperfect . We will consider both perfect and imperfect tracking. The de…nition of any tracking property should clarify the following:
the characterization of the plant behavior we are interested in, whether we are interested in the internal dynamical behavior of the plant, or in the plant output dynamical behavior;
the space in which the demanded closeness is to be achieved, which means that, although originally tracking concerns the output behavior, we can consider the output tracking either via the output space or via the state space;
the de…nition of the distance between the real behavior and the desired behavior of the plant;
the de…nition of the demanded closeness of the real behavior to the desired behavior of the plant;
the nonempty sets of the initial conditions of all plant variables under which the demanded closeness is to be achieved;
the nonempty set D(:) of permitted external disturbances acting on the plant, under which the demanded closeness is to be realized;
the nonempty set Y(:)d of realizable desired plant behaviors that can be demanded;
the time interval over which the demanded closeness is to be guaranteed ; and
the requested quality with which the real behavior is to follow the desired behavior of the plant.
