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Proof. 68), it follows that it is a monotone nonincreasing function, which is, by assumption, bounded from below. Thus Vfc(x^) must approach a limit as k —>• oo, and W(k) must approach zero as & -> oo. k, A*) -» 0 as & -> oo. If G is bounded or if W(x) is bounded away from zero for all sufficiently large x, then all solutions that remain in G are bounded and approach a closed, bounded set contained in A as k —> oo. If G is unbounded and there exists a sequence {xn} such that xn e Gas||x n || -> ooandW(x n ) -> Oasn -> oo, then it is possible to have an unbounded solution under the conditions of the theorem.

2 PID control for discrete-time systems This section gives a brief account of the simplest case of the so-called proportional-integralderivative (PID) control for a single-input, single-output discrete-time plant which is also chosen to have a particularly simple structure. 11) is the discrete equivalent of the integral of the error (em :— r — ym). Actually, it is the sum of all control errors up to instant m, scaled by the gain fc/. 1. 3. Plant P and controller C in standard unity feedback configuration.

Brief Review of Control and Stability Theory where B e E n x n , x e E", and f (•) is a diagonal function that belongs to the class 5" = S x S x • • • x S. 81) is as follows. 32. SI) is globally asymptotically stable for all f(-) e Sn if the matrix B is Hurwitz diagonally stable. Observe that the theorem is an absolute stability type of result (also called a robust stability result), since it establishes the stability of an entire class of systems. 32 is defined as follows. Let p be a vector with all components positive and P be a diagonal matrix with components of the diagonal equal to that of the vector p.

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