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1.
Given two nonlinear input–output systems written in terms of Chen–Fliess functional expansions, i.e., Fliess operators, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic setup is introduced using a new type of grading that has recently appeared in the literature. This grading also facilitates a fully recursive algorithm to compute the antipode of the Hopf algebra of the output feedback group, and thus, the corresponding feedback product can be computed much more efficiently. The final innovation is an improved convergence analysis of the antipode operation, namely, the radius of convergence of the antipode is computed. 相似文献
2.
The primary goal is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input–output systems. In particular, it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of analytic integral operators (Fliess operators) and analytic functions provided that the composite system is well defined and resides in one of these two classes. In each case, the radius of convergence of the interconnection is computed. 相似文献
3.
The changes in the spectrum caused by structured perturbations of pseudo-resolvents and operators on Banach spaces are considered.
In particular, if a point is in the resolvent set of an operator, necessary and sufficient conditions for it to remain in
the resolvent set under structured perturbations are given. The structured perturbations of an operator are specified by an
operator node that has three generating operators and a characteristic function together with an admissible feedback operator.
In addition, the robustness of stability under structured perturbations is analyzed. The results are applied to boundary control
systems and impedance passive systems. 相似文献
4.
The paper extends quadratic optimal control theory to weakly regular linear systems, a rather broad class of infinite-dimensional systems with unbounded control and observation operators. We assume that the system is stable (in a sense to be defined) and that the associated Popov function is bounded from below. We study the properties of the optimally controlled system, of the optimal cost operatorX, and the various Riccati equations which are satisfied byX. We introduce the concept of an optimal state feedback operator, which is an observation operator for the open-loop system, and which produces the optimal feedback system when its output is connected to the input of the system. We show that if the spectral factors of the Popov function are regular, then a (unique) optimal state feedback operator exists, and we give its formula in terms ofX. Most of the formulas are quite reminiscent of the classical formulas from the finite-dimensional theory. However, an unexpected factor appears both in the formula of the optimal state feedback operator as well as in the main Riccati equation. We apply our theory to an extensive example.Part of the results reported here were obtained while the second author was visiting FUNDP Namur, under the Belgian Program on Inter-University Poles of Attraction initiated by the Belgian state, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors. 相似文献
5.
Regular linear systems with feedback 总被引:5,自引:0,他引:5
We consider a rather general class of infinite-dimensional linear systems, called regular linear systems, for which convenient representations are known to exist both in time and in the frequency domain. We introduce and study the concept of admissible feedback operator for such a system and of well-posedness radius. We show that the closed-loop system obtained from a regular linear system with an admissible feedback operator is again regular and we describe the relationship between the generating operators of the open-loop and closed-loop systems.Part of the results reported here were obtained while the author was visiting FUNDP Namur, under the Belgian Program on Inter-University Poles of Attraction initiated by the Belgian state, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the author. 相似文献
6.
We develop a method for the exact determination of frequency responses for a class of infinite dimensional systems. In particular, we consider distributed systems in which a spatial independent variable belongs to a finite interval, and in which the inputs and outputs are spatially distributed over the same interval. We show that an explicit formula for the Hilbert–Schmidt norm of the operator-valued frequency response can be obtained whenever the underlying operators are represented by a forced two point boundary value state-space realizations (TPBVSR). This formula involves finite dimensional computations with matrices whose dimension is at most four times larger than the order of the underlying differential operator. Two examples are provided to illustrate the procedure. 相似文献
7.
Fliess operators as input–output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory. In the classical analysis of these operators, certain growth conditions on the coefficients in their series representations insure uniform and absolute convergence, provided every input is uniformly bounded by some fixed upperbound. In some emerging applications, however, it is more natural to consider other classes of inputs. In this paper, Lp function spaces are considered. In particular, it is shown that the classic growth conditions also provide sufficient conditions for convergence and continuity when the admissible inputs are from a ball in Lp[t0,t0+T], where T is bounded and p1. In addition, stronger global growth conditions are given that apply even for the case where T is unbounded. When the coefficients of a Fliess operator have a state space representation, it is shown that the state space model will always locally realize the corresponding input–output map on Lp[t0,t0+T] for sufficiently small T>0. If certain well-posedness conditions are satisfied then the state space model will globally realized the input–output mapping for unbounded T when the coefficients satisfy the global growth condition. 相似文献
8.
Takao Nambu 《Mathematics of Control, Signals, and Systems (MCSS)》1994,7(1):76-93
We study an infinite-dimensional operator equation XL–BX=C in a separable Hilbert space. The equation arises in the stabilization study of general linear parabolic systems, where the operatorsL, B, and C are coefficient operators describing a feedback control system. The solution to the stabilization naturally leads to an approximation problem of the operator equation. In this paper we propose a concrete algorithm for the approximation with the prescribed convergence rate when the closed operatorL is self-adjoint or more generally a spectral operator with compact resolvent.The original version of the paper was written while the author was with the Department of Mathematics, Faculty of Engineering, Kumamoto University, Kumamoto 860, Japan. 相似文献
9.
In this article, we give an explicit solution to the nonstandard H∞ problem which includes the optimal robust disturbance attenuation problem and the two-disc problem. The solution depends on an operator which is analogous to the Sarason operator for the standard H∞ problem, it allows to quantify the optimal performance of feedback systems and provides an explicit formula for the optimal controller. The nonstandard H∞ problem is perhaps the simplest control problem which deals with significant plant uncertainty. Surprisingly, relatively little work has been done on it despite the large number of papers in the H∞ control literature. 相似文献
10.
11.
Ruth F. Curtain 《Mathematics of Control, Signals, and Systems (MCSS)》2001,14(4):299-337
We consider the question of the existence of solutions to certain linear operator inequalities (Lur'e equations) for strongly
stable, weakly regular linear systems with generating operators A, B, C, 0. These operator inequalities are related to the spectral factorization of an associated Popov function and to singular
optimal control problems with a nonnegative definite quadratic cost functional. We split our problem into two subproblems:
the existence of spectral factors of the nonnegative Popov function and the existence of a certain extended output map. Sufficient
conditions for the solvability of the first problem are known and for the case that A has compact resolvent and its eigenvectors form a Riesz basis for the state space, we give an explicit solution to the second
problem in terms of A, B, C and the spectral factor. The applicability of these results is demonstrated by various heat equation examples satisfying
a positive-real condition. If (A, B) is approximately controllable, we obtain an alternative criterion for the existence of an extended output operator which
is applicable to retarded systems. The above results have been used to design adaptive observers for positive-real infinite-dimensional
systems.
Date received: July 25, 1997. Date revised: February 10, 2001. 相似文献
12.
The nonstandard two-disk problem plays a fundamental role in robust feedback optimization. Here, it is shown via Banach space duality theory that its solutions satisfy an extremal identity, and may be viewed as a dual extremal kernel of a particular L/sup 1/-optimization problem. A novel operator theoretic framework to characterize explicitly its solutions is developed, in particular, the two-disk optimization is shown to be equal to the induced norm of a specific operator defined on a projective tensor product space involving a non-Hilbert version of a vector valued H/sup 2/ space. Moreover, this operator is shown to be a combination of multiplication and Toeplitz operators. Under certain conditions, existence of maximal vectors is established leading to an explicit formula for the optimal controller. An "infinite matrix" representation with respect to a canonical basis is derived, together with an algorithm to compute it. The norm of the relevant operator is approximated by special finite dimensional optimizations whose solutions lead to solving semi-definite programming problems involving the computation of a matrix projective tensor norm. 相似文献
13.
14.
15.
Ekaterina Shemyakova 《Programming and Computer Software》2013,39(4):212-219
The paper describes package LPDO, which is designed for work with linear partial differential operators with symbolic coefficients in the computer algebra system MAPLE. In addition to basic procedures (operator creation, determination, modification, and various simplifications of their coefficients, as well as algebraic operations on them), it implements generating systems of gauge invariants for separate operators and operator pairs, the Laplace transformation method (not related to the integral Laplace method), procedures returning necessary and sufficient conditions for factoring third-order operators in the plane into compositions of operators of one or another form in terms of invariants, and several procedures related to the Darboux transformations. 相似文献
16.
《Journal of Symbolic Computation》2006,41(12):1295-1308
An algorithm is presented that computes explicit generators for the ring of differential operators on an orbifold, the quotient of a complex vector space by a finite group action. The algorithm also describes the relations among these generators. The algorithm presented in this paper is based on Schwarz’s study of a map carrying invariant operators to operators on the orbifold and on an algorithm to compute rings of invariants using Gröbner bases due to Derksen [Derksen, Harm, 1999. Computation of invariants for reductive groups. Adv. Math. 141 (2), 366–384]. It is also possible to avoid using Derksen’s algorithm, instead relying on the Reynolds operator and the Molien series. 相似文献
17.
Ekaterina Shemyakova 《Programming and Computer Software》2016,42(2):112-119
Darboux transformations of type I are invertible Darboux transformations with explicit short formulas for inverse transformations. These transformations are invariant with respect to gauge transformations, and, for gauge transformations acting on third-order hyperbolic operators of two variables, a general-form system of generating differential invariants is known. In the paper, first-order Darboux transformations of type I for this class of operators are considered. The corresponding operator orbits are directed graphs with at most three edges originating from each vertex. In the paper, an algorithm for constructing such orbits is suggested. We have derived criteria for existence of first-order Darboux transformations of type I in terms of the generating invariants, formulas for transforming invariants, and the so-called “triangle rule” property of orbits. The corresponding implementation in the LPDO package is described. The orbits are constructed in two different forms, one of which outputs the graph in the format of the well-known built-in Maple package Graph Theory. 相似文献
18.
Andrea Neufeld Dipl. Inform. Guido Moerkotte Dr.rer.nat. Peter C. Loekemann Dr.ing. 《The VLDB Journal The International Journal on Very Large Data Bases》1993,2(2):173-213
To address the problem of generating test data for a set of general consistency constraints, we propose a new two-step approach: First the interdepen-dencies between consistency constraints are explored and a generator formula is derived on their basis. During its creation, the user may exert control. In essence, the generator formula contains information to restrict the search for consistent test databases. In the second step, the test database is generated. Here, two different approaches are proposed. The first adapts an already published approach to generating finite models by enhancing it with requirements imposed by test data generation. The second, a new approach, operationalizes the generator formula by translating it into a sequence of operators, and then executes it to construct the test database. For this purpose, we introduce two powerful operators: the generation operator and the test-and-repair operator. This approach also allows for enhancing the generation operators with heuristics for generating facts in a goal-directed fashion. It avoids the generation of test data that may contradict the consistency constraints, and limits the search space for the test data. This article concludes with a careful evaluation and comparison of the performance of the two approaches and their variants by describing a number of benchmarks and their results. 相似文献
19.
The standard (four-block) H/sup /spl infin// control problem for systems with multiple input-output delays in the feedback loop is studied. The central idea is to see the multiple delay operator as a special series connection of elementary delay operators, called the adobe delay operators. The adobe delay case is solved and thereby the general case is solved as a nested set of solutions to adobe delay problems. 相似文献
20.
Xin-Zhuang Dong 《International journal of systems science》2013,44(12):2101-2110
This paper investigates the problem of state feedback robust admissible control for singular delta operator systems with norm-bounded coefficient matrix uncertainties. A necessary and sufficient condition is derived to ensure the admissibility of the closed-loop singular delta operator system for all allowable uncertainties. Then, by specifying the structure of some matrix variables, the existence condition and explicit expression of a robust admissible controller are obtained in terms of strict linear matrix inequalities. Moreover, from the relation between singular discrete systems and singular delta operator systems, the corresponding results are also presented for uncertain singular discrete systems. Finally, some numerical examples are provided to demonstrate the effectiveness of the results in this paper. 相似文献