In (1) t denotes time, t a the initial time t p 0 , y is the output New system and control concepts of the natural trackability and of the natural tracking control are introduced in the framework of time-invariant linear systems described by input-output n-th order vector differential equations. The necessary and sufficient conditions are proved for the system to be naturally trackable and for the natural tracking control synthesis with completely unknown internal dynamics of the system.
I -INTRODUCTIONThe original goal of a control is an appropriate kind of tracking. Although the stability approach to control analysis and synthesis has been governing control theory and engineering, the tracking considerations have been attracting research and design interests at least since 1945 [45]. Asymptotic tracking (in short tracking) has been mainly studied ([1]-[15],[17-29],[31]-[36],[38],[44]-[48],[50]-[52],[54]-[61],[71]-[77]) in a broad sense and from the point of view of robustness ([5]-[13],[31],[32],[34],[36],[46],[66]-[70],[78]-[80]). The related problem of disturbance compensation was considered together with tracking ([ 161, [ 18],[ 321, [ 471, [53], [56 J, [ 591, [ 621, [64], [70],[75],[81]). The problem of optimal tracking was investigated in [ 3 1 , [ 3~1~~~~1 -~~~1~[~~1 , [~~1 , [~~1 .The real system output is to be controlled so as to appropriately track any desired output from a given family, under action of any external disturbance belonging to a specified class and with a requested quality. This aim should be achieved even if the whole internal dynamics of the system is not known. Hence, the question arises whether there exists at all such a control. In order to clarify this we shall introduce the concept of the system natural trackability. The problem of the necessary and sufficient conditions posed in this paper for time-invariant linear systems described by input-output differential equations is also completely solved herein.Another problem of designing a control without using any information about the system internal dynamics originates the concept of the natural tracking control. The necessary and sufficient conditions for the synthesis of a natural tracking control are proved by following the conceptual approach proposed in [261,[291,[301. II -SYSTEM DESCRIPTION A system (plant, process) to be controlled together with sensors and final controller devices (e.g. actuators) is governed by the input-output time-invariant linear vector differential equation (I), vector (function), y:!JWP, y(t) is the output vector at time E%, U is the control vector (function), u:!JbRm, u(t) is the control vector at time t, and d is the disturbance vector (function), d:%%q, d(t) is the disturbance vector at time t. All the matrices have the appropriate dimensions. A class of systems described by(1) to be considered is determined by the next property.
3)The matrices Ab k=O,l ..., n-I may be completed unknown. The matrices A, and 8, k=O,I, ..., m are known, and &tThe matrices E, k=O,I, ...,p may be completely unknown.i,d....
