### Browsing by Author "Shayman, M.A."

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Item Feedback Control and Clasaification of Generalized Linear Systems.(1987) Shayman, M.A.; Zhou, Z.; ISRWe present a unified theory of control synthesis for generalized linear (i.e. descriptor) systems using constant-ratio proportional and derivative (CRPD) feedback. Our framework includes the theory of static state feedback and output feedback for regular statespace systems as a special case. The main elements of this theory include (1) a covering of the space of all systems, both regular and singular, by a family of open and dense subsets indexed by the unit circle; (2) a group of transformations which may be viewed as symmetries of the cover; (3) an admissible class of feedback transformations on each subset which is specifically adapted to that subset. We obtain a general procedure of control synthesis of CRPD feedback for generalized linear systems which uses the symmetry transformations to systematically reduce each synthesis problem to an ordinary static state feedback (or output feedback) synthesis problem for a corresponding regular system. We apply this approach to obtain natural generalizations of the Disturbance Decoupling Theorem, the Pole Assignment Theorem and the Brunovsky Classification Theorem.Item Generalized Eulerian Numbers and the Topology of the Hessenberg Variety of a Matrix.(1987) Mari, Filippo de; Shayman, M.A.; ISRLet A {AN ELEMENT OF} gl(n,C) and let p be a positive integer. The Hessenberg variety of degree p for A is the subvariety Hess (p, A) of the complete flag manifold consisting of those flags S_1, {SUBSET}... {SUBSET} S_{n-1}in C^n which satisfy the condition AS_i {SUBSET} S_{i+p} for all i. We show that if A has distinct eigenvalues, then Hess (p, A) is smooth and connected. The odd Betti numbers of Hess (p, A) vanish, while the even Betti numbers are given by a natural generalization of the Eulerian numbers.Item Generalized Riccati Equations.(1987) Mari, Filippo de; Shayman, M.A.; ISRThe matrix Riccati equation refers to the quadratic differential equation (RDE)Item Hessenberg Varieties and Generalized Eulerian Numbers for Semisimple Lie Groups: The Classical Cases.(1988) Mari, Filippo de; Shayman, M.A.; ISRCertain subvarieties of flag manifolds arise from the study of Hessenberg and banded forms for matrices. For a matrix A gl(n,C) (or sl(n,C)) and a nonnegative integer p, the p^th Hessenberg variety of A is the subvariety of the (complete) flag manifold consisting of those flags (S_1, . . . ,S_{n-1} ) satisfying the condition AS_i {IS A SUBSET OF, SYMBOL} S_{i+p}, {UPSIDE DOWN A}i. The definition of these varieties extends to an arbitrary connected complex semisimple Lie group G with Lie algebra g using the root-space decomposition. We investigate the topology of these varieties for the classical linear Lie algebras. If A is a regular element, then for p > 1, the pth Hessenberg variety is smooth and connected. The odd Betti numbers vanish, while the even Betti numbers represent (apparently new) generalizations of the classical Eulerian numbers which are determined by the height function on the root system of the Lie algebra. In particular, for g = sl(n, C) they yield a family of symmetric unimodal sequences which link the classical Eulerian numbers (p = 1) to the classical Mahonian numbers (p = n-1 ), while if g = sp(n, C) and p = 1, they are f- Eulerian numbers in the sense of R.P. Stanley.Item Homogeneous Indices, Feedback Invariants and Control Structure Theorem for Generalized Linear Systems.(1987) Shayman, M.A.; ISRWe define a new set of indices for a generalized linear system. These indices, referred to as the homogeneous indices, are a natural generalization of the minimal column indices (Kronecker indices) of an ordinary state-space system. We prove that the homogeneous indices are a complete set of invariants for the action of a natural group of feedback transformations on generalized linear systems. We also show that the homogeneous indices determine exactly which closed loop invariant polynomials can be assigned by feedback, thereby generalizing the Control Structure Theorem of Rosenbrock.Item A New Framework for Supervisory Control of Discrete Event Systems(1995) Shayman, M.A.; Kumar, Ratnesh; ISRWe propose a new framework for supervisory control design for discrete event systems. Some of the features of the proposed approach are: (i) By associating control and observation capabilities and limitations with the plant as well as the supervisor, it models reactive systems, and also treats plant and supervisory processes in a symmetric way. (ii) By introducing a single general interconnection operation, called masked composition, it permits open-loop as well as closed-loop control. (iii) By viewing the uncontrollability of events as corresponding to a projection-type control mask, and considering more general nonprojection-type control as well as observation masks, it treats the controllability and observability of events in a unified way. (iv) It applies to both deterministic and nondeterministic plant models and supervisory design. The sublanguages of a given language that are realizable under control are closed under union. Hence, the supremal realizable sublanguage always exists. In addition, it yields conditions under which existence of a non-deterministic supervisor implies existence of a deterministic supervisor. (v) By encapsulating control and observation masks with process logic to form process objects, and using a single type of interconnection operator to build complex process objects out of simpler component process objects, it provides a foundation for an object-oriented approach to discrete event control.Item Non-blocking Supervisory Control of Nondeterministic Systems via Prioritized, Synchronization(1993) Kumar, Ratnesh; Shayman, M.A.; ISRIn a previous paper [15], we showed that supervisory control of nondeterministic discrete event systems, in the presence of driven events, can be achieved using prioritized synchronous composition as a mechanism of control, and trajectory models as a modeling formalism. The specifications considered in [15] were given by prefix-closed languages. In this paper, we extend the theory of trajectory models and prioritized synchronous composition to include markings so that non-closed specifications and issues such as blocking can be addressed. It is shown that the usual notion of non-blocking, called language model non- blocking, is inadequate in the setting of nondeterministic systems, and a stronger notion, called trajectory model non- blocking, is introduced. Necessary and sufficient conditions for the existence of non-marking and language model non-blocking as well as trajectory model non-blocking supervisors is obtained for nondeterministic systems in the presence of driven events. We also show that our approach is also suitable for modular supervisory control.Item Pole Placement by Dynamic Compensation for Descriptor Systems.(1987) Shayman, M.A.; ISRUsing the recently introduced concept of homogeneous indices, we obtain a generalization to descriptor systems of the Brasch- Pearson Theorem for pole placement by dynamic compensation.Item Supervisory Control of Nondeterministic Systems with Driven Events via Prioritized Synchronization and Trajectory Models(1992) Shayman, M.A.; Kumar, Ratnesh; ISRWe study the supervisory control of nondeterministic discrete event dynamical systems (DEDS's) with driven events in the setting of prioritized synchronization and trajectory models introduced by Heymann. Prioritized synchronization captures the notions of controllable, uncontrollable, and driven events in a natural way, and we use it for constructing supervisory controllers. The trajectory model is used for characterizing the behavior of nondeterministic DEDS's since it is a sufficiently detailed model (in contrast to the less detailed language or failures models), and serves as a language congruence with respect to the operation of prioritized synchronization. We obtain results concerning controllability and observability in this general setting.