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...ror prone. u(t) f(x(t), u(t), t) Figure 7. A conceptual block diagram for continuous-time systems. For example, a transfer function has zero-initial-state controllable canonical form (see for example =-=[8]): . x(t-=-) ∫ x(t) Ys ( ) s + b ----------- Us ( ) s 2 = ----------------------------- – a1s – a0 g(x(t), u(t), t) y(t) (2) 16 of 37s· x1 · which can be built as the composite actor shown in Figure 8. x...

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...analysis of the concept of state in the context of ant colony optimization can be found in [4]. A general analysis of the concept of state is given in the classical literature on linear system theory =-=[20]-=-, dynamic programming [2], and optimal control [3]. 3 For a discrete Markov decision process, the following holds by definition: P (x t+1 |x t , u t ) = P (x t+1 |x t , u t ), where x t = (x t , x t-1...

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...form M m m 1 ~ q~ _z _ + 1-5 0 ' K k ' D d ,d 1 k~ 1 q z q1 z _ 0 . (6.3) [We define/,, in the Laplace domain by /„ (s ) = h(s)q(s).] Taking the Laplace transform of equation (6.3) and equating the transfer functions yields the following set of algebraic equations: M-m2=0 M+D-2md = 0 M8+D(32+K8-d2l32-2mk8 = ce D+K-2dk=y/P8 K-k2=0 (0 ( » • ) OVO (iv) (v) (6.4) (6.6) A realization so defined will have the same input-output relationship as the given system. This realization will therefore be zero-state equivalent to the given system. We would also like the realization to be zero-input equivalent [9]. Thus we have constraints imposed by the initial value terms: M-m2=0 (i) D-cP=0 (ii) (6.5) m = d (Hi) and the initial values are chosen so that z0=- dq0 (0 z0=- dq0 (H) Adding equation (6.5) (ii) to our set of equations (6.4) produces a set of six nonlinear algebraic equations in six unknown parameters (M, D, K, m, d, k) and four known parameters (a, y, /3, 8). As we shall see, equation (6.5) (Hi) is consistent with their solution. After a little algebra these equations reduce to M-m2=0 (i) D-d2=0 (ii) K-k2=Q (Hi) (6.7) M+D-2md=0 (iv) M+K-2mk=a/8 (v) D+K-2dk=y/(58 (vi) Eliminating M, D, K fro...

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...e-varying parameters, an output feedback control design is proposed in [5] for linear time-varying systems based on the gradient algorithm. Other output feedback control design include the LQ control =-=[4]-=-, the controllability-grammian-based control [8, 14]. Among the usual approaches to construct the output feedback control of time-invariant 1sor periodic systems, the effective approach is the Lyapuno...

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...s a function of the state, as indeed stated by thesrst of the two properties. The denition of state given here is inspired by and is consistent with the classical denition given by Zadeh and Desoer [2=-=1-=-]. It is nonetheless a proper extension of that denition in the sense that we consider here systems for which the set of admissible inputs varies according to the preceding input-output history, while...

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...(t) such that the system (1) dx dt = Ax+BK(t)Cx, x ∈ Rn, is asymptotically stable. The problem of stabilizing system (1) with the help of a constant matrix K is classical for automatic control theory =-=[2, 3]-=-. From this point of view, Brockett’s problem can be reformulated as follows. To what extent are the possibilities of classical stabilization extended by introducing matrices K(t) that depend on time ...

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...lem. For frequency-response problems an approach as in [17] may be more appropriate; it is not a-priori that poles play the same dominant role as in LTI. In earlier literature on LTV-systems, such as =-=[6]-=-, the system behavior shows characteristic polynomials with the Heaviside-operator as a variable and time functions as constants. Thus, the physical quality 'time' appears in two different ways. To ha...

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... is given by � ∞ L−{f(t)} = F(s) = f(t)e −st dt, (3) where the domain of integration fully includes the origin and any singularities occuring at that time. This form is adopted in many texts, such as =-=[8]-=-–[12]. The definition (3) of the Laplace transform implies the time-derivative rule 0 − L−{f ′ (t)} = sF(s) − f(0 − ), (4) whereby initial conditions existing before t = 0 are brought into the analysi...