By C.T. Leondes (Eds.)

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**Extra info for Discrete-Time-Control System Analysis and Design**

**Sample text**

Figure 4" The system to generate u. C2 D21 B21 0 Note that K stabilizes G iff K stabilizes G . (the two closed-loop systems have identical A-matrices). ~ ........ GLtTLC1U "-......... ~ K ~ Figure 5" Tlle system to generate u,~e~. ~~(,X) - I , A RF C2 B1 -RF1 B2 R-1 ] I ]. D21 0 Now this is the second special problem. So by Theorem 5 the unique optimal controller is kn~,o~t(A) "- [ A + LC2- B2LIC2 + B2F L-B2L1 ] RL1 R ( L 1 C 2 - F) and the minimum cost is min IIRL~II= - II~fll=.

The closed segment joining x and y is the set C({x,y}), and is denoted by [x'y]. The open segment (x'y) joining x and y is given by [x:y] - {x,y}. Let K be a convex set. A convex subset W of K is called an 41 TECHNIQUES FOR REACHABILITY extreme subset if none of its points are included in an open segment joining two points of K, which are not both in W. An extreme subset consisting of one point is an extreme point. A face of K is a convex subset K' of K such that every closed segment in K with a relative interior point in K' has both endpoints in K'.

The space set of at t. all If $ states all reachable states are at t$ reachable at t the system is called reachable at t too. 8 $ Notice that the definition of reachability is characterized by two main facts. The first is that the initial condition is fixed to be the origin of the vector state space. The second is that reachability is a feature of possible evolutions of the system, occurring in the past, relative to the instant of reachability. An important observation, which provides a key argument in the analysis of reachability, is that, if a state z can be reached at t starting from zero at time t , then it can also be reached s r starting from zero at any instant of time, fact it suffices the system x(t)--O, that with and r to steers the apply initial a zero input condition then to concatenate state from zero in set of states reachable at at t t interval to z at t .