# Banach Lattices and Positive Operators (Grundlehren Der by H. H. Schaefer

By H. H. Schaefer

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Extra resources for Banach Lattices and Positive Operators (Grundlehren Der Mathematischen Wissenschaften Series, Vol 215)

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This facts allows us to rephrase the general control problem (posed above as problem 8) as Problem 9 Given the full plant behavior Pfull : a. Describe a set of speciﬁcations on the controlled plant, namely, the desired properties of the manifest controlled behavior K (the desired controlled behavior). b. , N ⊆ K ⊆ (Pfull )w . In a sense the controller implementability theorem characterizes the limits of performance of the given full plant behavior Pfull : it exactly tells which controlled system behaviors can be obtained.

9s3 Stabilization and pole placement by regular full interconnection This section deals with the synthesis problems of stabilization and pole placement by regular full interconnection. We will give algorithms to compute, for a given plant behavior, controllers that achieve pole placement and stabilization. Both problems require the computation of a unimodular embedding. We will ﬁrst discuss the problem of pole placement by regular full interconnection. This problem is deﬁned as follows. Let P ∈ Lq be a given plant behavior.

It is well known that the space of all linear diﬀerential systems Lq is closed under addition. Suppose that d d B1 , B2 ∈ Lq , where B1 and B2 have kernel representations R1 ( dt )w = 0 and R2 ( dt )w = 0, respectively. The problem to ﬁnd a kernel representation of B1 + B2 was solved in [65] (see also [11]): Proposition 10 : Let [S1 S2 ] be a MLA of col(R1 , R2 ). Then the polynomial matrix S1 R1 = −S2 R2 yields a kernel representation of B1 + B2 In the following theorem we now give a new condition for regular implementability.