Finite spectrum assignment of multi-input systems with non-commensurate delays

We present a new framework for finite spectrum assignment for multi-input systems with non-commensurate delays using an algebraic approach over multidimensional polynomial matrices. By focusing on the solvability of a Bezout equation over multidimensional polynomial matrices, we derive a necessary and sufficient condition for finite spectrum assignability under which a finite number of spectra can be assigned by a control law using a ring of entire functions, i.e. Laplace transforms of all exponential time functions with compact support. Furthermore, using a solution to the Bezout equation, we present a design method for a controller that achieves finite spectrum assignment.     

Coprime Factorizations of Multivariate Rational Matrices

Coprime factorization is a well-known issue in one-dimensional systems theory, having many applications in realization theory, balancing, controller synthesis, etc. Generalization to systems in more than one independent variable is a delicate matter: First, several nonequivalent coprimeness notions for multivariate polynomial matrices have been discussed in the literature: zero, minor, and factor coprimeness. Here we adopt a generalized version of factor primeness that appears to be most suitable for multidimensional systems: a matrix is prime iff it is a minimal annihilator. After reformulating the sheer concept of a factorization, it is shown that every rational matrix possesses left and right coprime factorizations that can be found by means of computer algebraic methods. Several properties of coprime factorizations are given in terms of certain determinantal ideals. Date received: September 10, 1998. Date revised: February 25, 1999.     

Finite spectrum assignment of linear systems with a class of non-commensurate delays

This paper is concerned with finite spectrum assignment of linear systems with non-commensurate delays. It is proved that if a feedforward pass between states variables does not have multi non-commensurate delays in parallel but only one or no delay in series, then the system is finite spectrum assignable even though the feedback pass has multi non-commensurate delays in parallel.     

Sufficient conditions for instability of delay differential systems

The paper gives sufficient conditions for instability of linear systems of both the neutral and retarded types with commensurate and non-commensurate time delays. The conditions are given in terms of time delays and coefficients of the characteristic quasi-polynomials of these systems. Illustrative examples are presented.     

Finite spectrum assignment of systems with general delays

This article presents a new framework for dealing with pure and distributed delays simultaneously and impartially as general delays in algebraic finite spectrum assignment. The framework based on an important property common to general delays gives a sufficient condition for finite spectrum assignability of scalar systems with several general delays and a design method of a controller achieving finite spectrum assignment. Especially in algebraic control theory for delay systems, pure and distributed delays have never been dealt with simultaneously and impartially. The framework is significantly important also for multidimensional system theory because its potential applicability to the discussion on systems with general delays, i.e. the possibility that especially distributed delay operators are used as variables for describing multidimensional systems, is clarified for the first time.     

Delay-dependent robust stability of linear systems with non-commensurate time varying delays

In this article we investigate the robust stability of uncertain systems with non-commensurate time varying delays. Two types of perturbations are considered. First, we consider time varying and unknown but norm bounded uncertainties. Next systems with polytopic-type uncertainties are studied. Based on Finsler's lemma we propose a new method for achieving less restrictive sufficient conditions for delay-dependent robust stability. The results are presented in terms of linear matrix inequalities. Three examples are given for illustration and comparison with previous results.     

Linearisation via input–output injection of time delay systems

This paper deals with the problem of linearisation of systems with constant commensurable delays by input–output injection using algebraic control tools based on the theory of non-commutative rings. Solutions for the problem of linearisation free of delays, and with delays of an observable nonlinear time-delay systems are presented based on the analysis of the input–output equation. These results are achieved by means of constructive algorithms that use the nth derivative of the output expressed in terms of the state-space variables instead of the explicit computation of the input–output representation of the system. Necessary and sufficient conditions are established in both cases by means of an invertible change of coordinates.     

Generalization of Tychonov''s Theorem with applications to adaptive control of SISO delay systems

This paper deals with single-input single-output delay systems with unknown transfer function parameters, including unknown, non-commensurate, real delays. A generalization of Tychonov's Theorem is obtained. This generalization is used to show the stability of a smooth adaptive controller when applied to a large class of delay systems. This paper generalizes the results in [1,2].     

一类时滞线性系统的变结构控制

研究一类状态变量具有时滞的线性系统的变结构控制问题,指出传统的切换函数在时滞 系统中应当是一个切换泛函,给出了在系统谱能控条件下具有稳定滑动模态的切换泛函的设 计方法.并利用趋近律设计了相应的变结构控制器.得到了一般时滞系统能够变换为所需要 的时滞系统的条件.     

Robust stability of uncertain time-delay systems

This paper presents several sufficient conditions, delay-independent or delay-dependent, that guarantee the robust stability of uncertain time-delay systems subjected to parametric perturbations. Both single and composite uncertain systems with one delayed state are considered and each of these results, expressed by a succinct scalar inequality, permits the assessment of the transient behaviour of uncertain systems. The robustness of the time-delay system with respect to delay uncertainty is also discussed. Furthermore, these results are extended to the perturbed systems with multiple non-commensurate time delays. It is shown that the parameter perturbations may destabilize the system; hence the nominal system should be sufficiently stable to ensure robust stability.     

Robust stability of uncertain singular time-delay systems via LFT approach

The robust stability problem of continuous-time singular systems with multiple state delays and bounded parametric uncertainties is considered. Both commensurate and non-commensurate delays are investigated. On the basis of the linear fractional transformations (LFTs) framework and μ-analysis, a systematic approach is derived to convert the robustness problem to a robust nonsingularity problem. Some explicit conditions are proposed to guarantee the uncertain singular time-delay system being regular, impulse-free and stable independent of delay. Two illustrative examples are given to show the feasibility of the proposed technique.     

A several complex variables approach to feedback stabilization of linear neutral delay-differential systems

In this paper we consider the problem of uniform asymptotic stabilization of neutral delay differential systems by a suitable choice of linear feedback law, where the controller is static and at timet ₀ has full knowledge ofx(t ₀) ⁿ as well asx(t ₀ – d) for finitely many delays present in the system. Using complex analysis in several variables in a Banach algebra context we present a solution to this problem in the form of a sufficient condition for the existence of such a stabilizing feedback gain. This condition is a weak form of (algebraic) reachability together with a condition, which we call resolvability, on the solution of an associated algebraic Riccati equation. In the case of commensurable delays our results apply to neutral systems having aD-operator which is stable in the sense of Cruz and Hale. In the non-commensurate case, our results hold for systems with aD-operator satisfying a stronger condition on the spectrum of the evolution equation, which we call formal stability. A graphical criterion, in the spirit of the multi-dimensional Nyquist theory, is presented for formal stability ofD. We find these results especially interesting in light of recent work [39] which shows that, for a special class of systems, formal stability is necessary for stabilization by a feedback gain of the type considered here. This, in fact, gives a counterexample [39] to the well known condition of Pandolfi [40].Included among our results is an extension to the neutral case of the result in [11] which states that if the pointwise Kronecker indices are constant then the system is feedback equivalent to a delay-free system. As corollaries to this result we obtain some existing results [33] on canonical forms for time-delay systems, and, in addition, exhibit a large class of systems which are resolvable.Research partially supported by NASA under Grant NSG-2265 and NSF under Grant ENG-79-09459.Research partially supported by the NSF under Grants ECS-8017184 an INT-7902976 at Washington University and ECS-8214262 at Cornell University.     

Stability of quaternionic linear systems

The main goal of this paper is to characterize stability and bounded-input-bounded-output (BIBO)-stability of quaternionic dynamical systems. After defining the quaternion skew-field, algebraic properties of quaternionic polynomials such as divisibility and coprimeness are investigated. Having established these results, the Smith and the Smith-McMillan forms of quaternionic matrices are introduced and studied. Finally, all the tools that were developed are used to analyze stability of quaternionic linear systems in a behavioral framework.     

Cross-Term Forwarding for Systems With Time Delay

This paper presents a Lyapunov-based control design method for nonlinear systems with time delay. The key step is construction of a composite Lyapunov functional $V$ for a stable cascade. This functional consists of Lyapunov functionals for the subsystems and a cross-term defined by an infinite integral. The “$L_{g}V$-type” control law is then employed to achieve global asymptotic stability. The design procedure can be applied recursively to stabilize complex feedforward structures. When applied to a linear cascade, it gives a formula for the control law even if some parts of the system can only be stabilized through channels with multiple non-commensurate or distributed delays.      

New results on stability and L∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays

This article addresses the problems of stability and L_∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays for the first time. First, we consider the stability problem of a class of positive linear differential‐algebraic equations with unbounded time‐varying delays. A new method, which is based on the upper bounding of the state vector by a decreasing function, is presented to analyze the stability of the system. Then, by investigating the monotonicity of state trajectory, the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is characterized. It is shown that the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is also independent of the delays and fully determined by the system matrices. Two numerical examples are given to illustrate the obtained results.     

Differential–Algebraic Equations and Dynamic Systems on Manifolds

The authors consider current problems of the modern theory of dynamic systems on manifolds, which are actively developing. A brief review of such trends in the theory of dynamic systems is given. The results of the algebra of dual numbers, quaternionic algebras, biquaternions (dual quaternions), and their application to the analysis of infinitesimal neighborhoods and infinitesimal deformations of manifolds (schemes) are presented. The theory of differential–algebraic equations over the field of real numbers and their dynamics, as well as elements of trajectory optimization of respective dynamic systems, are outlined. On the basis of connection in bundles, the theory of differential–algebraic equations is extended to algebraic manifolds and schemes over arbitrary fields and schemes, respectively.     

Robust absolute stability of Lurie interval control systems

This paper considers robust absolute stability of Lurie control systems. Particular attention is given to the systems with parameters having uncertain, but bounded values. Such so‐called Lurie interval control systems have wide applications in practice. In this paper, a number of sufficient and necessary conditions are derived by using the theories of Hurwitz matrix, M matrix and partial variable absolute stability. Moreover, several algebraic sufficient and necessary conditions are provided for the robust absolute stability of Lurie interval control systems. These algebraic conditions are easy to be verified and convenient to be used in applications. Three mathematical examples and a practical engineering problem are presented to show the applicability of theoretical results. Numerical simulation results are also given to verify the analytical predictions. Copyright © 2007 John Wiley & Sons, Ltd.     

Robust controllability of linear interval systems with multiple control delays

The robust controllability problem for linear interval systems with multiple control delays is studied in this article. The rank preservation problem is converted to the nonsingularity analysis problem of the minors of the matrix in discussion. Based on some essential properties of matrix measures, a new sufficient algebraically elegant criterion for the robust controllability of linear interval systems with multiple control delays is established. A numerical example is given to illustrate the application of the proposed sufficient algebraic criterion.