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...lemma's in section II, we treat target point problems in section m. Although these problems were treated before (see e.g. Meditch [14] th.2.2 and th.2.4,sLudyck [13] th.2.2 and th.2.4, and Kwakernaak =-=[11]-=- th.6.7.), either the solutions there weresincomplete, or the proofs were false. So for completeness sake we give here results for controllability, stabilizability, output and reachability problems, t...

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...e (strong) admissibility of a target path.sI. IntroductionsExactly tracking a target path has attracked a lot of attention both in control theory andseconomics.sAfter the pioneering work of Tinbergen =-=[21]-=-, Brockett and Mesarovic [5] formulated necessarysand sufficient conditions for target path controllability in linear continuous time-invariant systems.sSince then a lot of papers appeared on this sub...

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... Proof:sFollows directly from the equality v (k) = e (k + 1) - A (k) e (k) . osThe second lemma gives necessary and sufficient conditions for solvability of a linear equation.sA proof can be found in =-=[12]-=-, chap. 3.10.sLemma 2:sLet B be an n x m matrix.sThen the equation B u = y is solvable if and only if (iff) rank [B I y] :: rank B. Moreover, thessolution is uniquely detennined iff rank B = m. Then t...

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... time-invariant systems we call attention to two algorithms developed for time-invariant systems which bear a great similarity withsour algorithm.sOne of them is the algorithm developed by Willems in =-=[24]-=-, see also [23] for an extension. toscalculate the controllable L..,-almost output nulling subspace. The difference between this algorithm and the one developed here is the initialization. Our algorit...

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... + p + q) not diagonal. To solve this problem, we have to consider the decoupling problem first.sThe decoupling problem, as defined here, was solved for time-invariant systems by Falb andsWolovich in =-=[8]-=-. A first attempt to generalize this result to time-varying systems was taken bysPorter in [15]. In this paper a sufficient condition for the so called integrator decoupling problem was given. This pr...

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...cked for a period of length k.sSubsequent questions concern the maximum of such numbers k and the reaction time, i.e. thestime needed to reach the target path (also known as policy lead question). In =-=[16,25]-=- this problem was solved for time-invariant systems. Here we derive an algebraic algorithm for timevarying systems. This section is followed by a section in which the decoupling problem iss- 2 - solve...

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...vantage of the last algorithm is that the subspace Siscan be interpreted as:sSi = {x E X I 3 W E Si-1> U E U such that A (I + i - 2) w + B (I + i - 2) U = x andse(t + i-I) w = OJ (see e.g. Schumacher =-=[20]-=-).sFor the derivation of results concerning global TPC is the following lemma imponantsLemma 10:sLet .'E be time-invariant.sIf the lead p is greater than n, then for all i ;:: 2 is IR, C lRi-l .s- 20 ...

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...t.sThe decoupling problem, as defined here, was solved for time-invariant systems by Falb andsWolovich in [8]. A first attempt to generalize this result to time-varying systems was taken bysPorter in =-=[15]-=-. In this paper a sufficient condition for the so called integrator decoupling problem was given. This problem deals with the question under which conditions compensatorssF (k) and G (k) exists such t...

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...oint in time on is more interesting. This question has therefore gained a lot of attention in economics. People who weresengaged in this subject are e.g. Tinbergen [21], Preston and Sieper [17], Aoki =-=[3,4]-=-. Prestonsand Pagan [16] and Wohltmann [26]. The contribution of this paper is among others thatsresults are generalized to time-varying systems.sFollowing Wohltmann [26] especially two questions are ...

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... systems we call attention to two algorithms developed for time-invariant systems which bear a great similarity withsour algorithm.sOne of them is the algorithm developed by Willems in [24], see also =-=[23]-=- for an extension. toscalculate the controllable L..,-almost output nulling subspace. The difference between this algorithm and the one developed here is the initialization. Our algorithm in general s...

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...cessarysand sufficient conditions for target path controllability in linear continuous time-invariant systems.sSince then a lot of papers appeared on this subject [1,3.4,16-19,25].sRecently Wohltmann =-=[27]-=- and Grasse [9] extended the results to continuous time-varying systems.sThis paper is concerned with discrete time-varying systems. After introducing some notationsand establishing some elementary le...

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...oint in time on is more interesting. This question has therefore gained a lot of attention in economics. People who weresengaged in this subject are e.g. Tinbergen [21], Preston and Sieper [17], Aoki =-=[3,4]-=-. Prestonsand Pagan [16] and Wohltmann [26]. The contribution of this paper is among others thatsresults are generalized to time-varying systems.sFollowing Wohltmann [26] especially two questions are ...

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...y too.s"¢:::" By making use again of the structure of the error equation matrix as derived above thisspart is easily proved. 0sWe conclude this section with an illustration of theorem 10.sIn Engwerda =-=[7]-=-, the infinite time quadratic tracking problem was solved under some conditions. As a special case the following problem was treated:sN-lsmin lim L {(y(k)-y*(k)l Q(y(k)-y*(k» + (u(k)-u*(k)lR (u(k)-u*(...

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...ationsand establishing some elementary lemma's in section II, we treat target point problems in section m. Although these problems were treated before (see e.g. Meditch [14] th.2.2 and th.2.4,sLudyck =-=[13]-=- th.2.2 and th.2.4, and Kwakernaak [11] th.6.7.), either the solutions there weresincomplete, or the proofs were false. So for completeness sake we give here results for controllability, stabilizabili...

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...specified point in time on is more interesting. This question has therefore gained a lot of attention in economics. People who weresengaged in this subject are e.g. Tinbergen [21], Preston and Sieper =-=[17]-=-, Aoki [3,4]. Prestonsand Pagan [16] and Wohltmann [26]. The contribution of this paper is among others thatsresults are generalized to time-varying systems.sFollowing Wohltmann [26] especially two qu...

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...hat the closed loop system (D y )(t) ;:: A (t) w (t) tel!. results. HeresA is a diagonal matrix, and D is an operator which is defined below. This result was generalized by Tzafestas and Pimenides in =-=[22]-=-.sDue to the decoupled inputs influencing the outputs succeeding (D y)(1) and which can not bescontrolled by u(t + 1), a solution to the integrator decoupling problem solves in general not thesdecoupl...

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...cked for a period of length k.sSubsequent questions concern the maximum of such numbers k and the reaction time, i.e. thestime needed to reach the target path (also known as policy lead question). In =-=[16,25]-=- this problem was solved for time-invariant systems. Here we derive an algebraic algorithm for timevarying systems. This section is followed by a section in which the decoupling problem iss- 2 - solve...

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...estion has therefore gained a lot of attention in economics. People who weresengaged in this subject are e.g. Tinbergen [21], Preston and Sieper [17], Aoki [3,4]. Prestonsand Pagan [16] and Wohltmann =-=[26]-=-. The contribution of this paper is among others thatsresults are generalized to time-varying systems.sFollowing Wohltmann [26] especially two questions are discussed: how long is the minimumspolicy l...