Algebraic Criteria Research Articles

Controllability of discrete-time inhomogeneous bilinear systems with real spectrum: Algebraically verifiable criteria

One of the drawbacks of the controllability theory for nonlinear systems is that most existing controllability criteria are not algebraically verifiable, which makes them difficult to apply especially if the system dimension is high. Thus, it is a significant task to seek algebraically verifiable controllability criteria for nonlinear systems. In this paper, we study controllability of discrete-time inhomogeneous bilinear systems. In the classical results on controllability of such systems, a necessary condition is that the linear part has to be controllable. However, we will show that this condition is in fact not necessary for controllability. Specifically, we first define the spectrum for discrete-time inhomogeneous bilinear systems and reveal that the spectrum is a fundamental property which is very useful in investigating the controllability problems. We then present controllability criteria for the systems with real spectrum, which are algebraically verifiable. Furthermore, we also provide algorithms for the controllable systems to compute the exact or approximated control inputs to achieve the transition between any given pair of states. The presented controllability criteria and algorithms work for the systems in any finite dimension and are easy to implement. More importantly, through our controllability criteria, we reveal that controllability of the linear part is not necessary for discrete-time inhomogeneous bilinear systems to be controllable. Examples are given to illustrate the presented algebraic controllability criteria.

Subspace foliations and collapse of closed flat manifolds

AbstractWe study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov–Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular.

Invariant output feedback stabilisability: the scalar case

In this paper we prove that stabilisability is a static output feedback (SOF) invariant, for scalar and multivariable systems. Then we examine scalar stabilisability, from an invariant viewpoint. We prove that the signature of Hermite’s Bezoutian is constant within certain intervals that we call critical, and we give a very simple algebraic stabilisability criterion, consisted of a finite number of stability checks, one for each critical interval. We establish the validity of this criterion with Routh, Hurwitz and Lyapunov methods. We prove that the winding number of the Nyquist plot around points of the real axis, is constant within critical intervals. We correlate the stabilisability within critical intervals with the stability of corresponding critical polynomials defined at their limits. Finally, we discuss why our findings constitute a decisive step towards the analytical solution for the persisting problem of static output feedback stabilizability of linear time invariant multivariable systems.

Stability and Stabilization of Takagi–Sugeno Fuzzy Second-Fractional-Order Linear Networks via Nonreduced-Order Approach

The extensively studied fractional-order dynamical networks only contain a uniform fractional derivative, which restricts the scope of fractional-order investigation. One type of Takagi–Sugeno fuzzy second-fractional-order linear networks is established in this article. First, by defining a novel Lyapunov functional, a sufficient criterion is offered to ensure global asymptotic stability of the addressed networks with time delay. Second, a fuzzy state-feedback control scheme is designed for the Takagi–Sugeno fuzzy second-fractional-order delayed linear networks to achieve global stabilization. Third, considering that only a few scholars have worked on adaptive control of fractional-order networks, an adaptive control method is also designed to realize the global stabilization of the Takagi–Sugeno fuzzy second-fractional-order linear networks, which does not contain the sign function and chattering problem can be averted. All results, given as algebraic criteria, are proved directly from the Takagi–Sugeno fuzzy second-fractional-order networks without utilizing the reduced-order approach. Three simulation examples are conducted to show the validity of the theoretical results at last.

Free Cyclic Actions on Surfaces and the Borsuk—Ulam Theorem

Let M and N be topological spaces, let G be a group, and let τ: G × M → M be a proper free action of G. In this paper, we define a Borsuk—Ulam-type property for homotopy classes of maps from M to N with respect to the pair (G, τ) that generalises the classical antipodal Borsuk—Ulam theorem of maps from the n-sphere $${\mathbb{S}^n}$$ to ℝn. In the cases where M is a finite pathwise-connected CW-complex, G is a finite, non-trivial Abelian group, τ is a proper free cellular action, and N is either ℝ2 or a compact surface without boundary different from $${\mathbb{S}^2}$$ and ℝℙ2, we give an algebraic criterion involving braid groups to decide whether a free homotopy class β ∈ [M, N] has the Borsuk—Ulam property. As an application of this criterion, we consider the case where M is a compact surface without boundary equipped with a free action τ of the finite cyclic group ℤn. In terms of the orientability of the orbit space Mτ of M by the action τ, the value of n modulo 4 and a certain algebraic condition involving the first homology group of Mτ, we are able to determine if the single homotopy class of maps from M to ℝ2 possesses the Borsuk—Ulam property with respect to (ℤn, τ). Finally, we give some examples of surfaces on which the symmetric group acts, and for these cases, we obtain some partial results regarding the Borsuk—Ulam property for maps whose target is ℝ2.

Partial impulse observability of linear descriptor systems

A research paper in this journal vol. 61, no. 3, pp. 427-434, 2012, by M. Darouach, provides a functional observer design for linear descriptor systems under the partial impulse observability condition. The observer design is correct, but there was a flaw in the algebraic criterion characterizing partial impulse observability. In the present paper, we derive a novel characterization of partial impulse observability in terms of a simple rank condition involving the system coefficient matrices and an alternative characterization in terms of the Wong sequences.

Refined frequency estimates for stability domains of synchronization systems

n this paper we examine stability of Lur’e-type systems arising as a feedback superpositions of infinite-dimensional linear blocks, described by integrodifferential Volterra equations, and periodic nonlinearities. Such systems have multiple equilibria, so traditional methods of stability investigation, defined for systems with single equilibrium are no good here. In the paper traditional Popov method of a priori integral indices is combined with two special techniques: Leonov’s nonlocal reduction method and the Bakaev-Guzh procedure. As a result new frequency–algebraic stability criteria are established, yielding tightened estimates of stability domains in the space of the system’s parameters.

Bipartite containment control of fractional multi-agent systems with input delay on switching signed directed networks

The bipartite containment control of nonlinear fractional multi-agent systems (FMASs) on fixed and switching signed directed networks is addressed in this article. Delayed control protocols are put forward to provide several simple algebraic criteria by means of using signed graph theory, fractional Razumikhin technique and common Lyapunov function approach. The presented method can well overcome the difficulty resulting from fractional calculus, time delay and switchings. Two numerical examples are presented for further clarifying the theoretical conclusions.

Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two, Webb Counting Bases, Ph.D. thesis, University of Waterloo, 2004 introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy--Binet formula. This paper studies counting on linear matroid problems extending Webb's work. We first introduce “Pfaffian parities” as an extension of Pfaffian pairs to the linear matroid parity problem, which is a common generalization of the linear matroid intersection problem and the matching problem. We show that a large number of efficiently countable discrete structures are interpretable as special cases of Pfaffian pairs and parities. Our study then turns to algorithmic aspects. We observe that the fastest randomized algorithms for the linear matroid intersection and parity problems by Harvey SIAM J. Comput., 39 (2009), pp. 679--702 and Cheung, Lau, and Leung ACM Trans. Algorithms, 10 (2014), pp. 1--26 can be derandomized for Pfaffian pairs and parities. We further present polynomial-time algorithms to count the number of minimum-weight solutions on weighted Pfaffian pairs and parities. Our algorithms make use of Frank's weight-splitting lemma for the weighted matroid intersection problem and the algebraic optimality criterion of the weighted linear matroid parity problem given by Iwata and Kobayashi SIAM J. Comput., 51 (2002) pp. STOC17-238--STOC17-280.

Algebra criteria for global exponential stability of multiple time-varying delay Cohen–Grossberg neural networks

This paper aims at establishing global exponential stability criteria for multiple time-varying delay Cohen–Grossberg neural networks (CGNNs). The considered network models cannot be expressed as the vector-matrix form, which yields that many methods in literature are unavailable. By constructing novel Lyapunov–Krasovskii functionals, two novel algebraic criteria guaranteeing global exponential stability of CGNNs under consideration are given. A pair of numerical examples are used to explain the effectiveness of the obtained algebra criteria relative to the previously stability conditions. It is worth emphasizing that the approach applied in this paper is applicable to CGNNs that may or may not be represented in vector-matrix form.

Stabilization and lag synchronization of proportional delayed impulsive complex-valued inertial neural networks

This paper deals with global exponential stabilization (GES) and exponential lag synchronization (ELS) for impulsive complex-valued inertial neural networks (ICVINNs) with proportional delays, without employing the standard order-reduction transformation and the general separation approach of the real and imaginary part of the ICVINNs. Several algebraic criteria involving multi-parameters for examining the GES and ELS of the ICVINNs are built, via establishing an appropriate Lyapunov functional, and utilizing inequality techniques, which can be achieved by the novel design of the adaptive controller. Numerical simulations are shown to sustain the availability and efficiency of the proposed criteria.

Lagrange Stability of Fuzzy Memristive Neural Networks on Time Scales With Discrete Time Varying and Infinite Distributed Delays

The existing results of Lagrange stability for neural networks with distributed time delays are scale-free, which introduces conservativeness naturally. A class of Takagi–Sugeno fuzzy memristive neural networks (FMNNs) on time scales with discrete time-varying and infinite distributed delays is brought in this article. First, a new scale-limited Halanay inequality is demonstrated by timescale theory. Next, on the basis of inequality techniques on time scales, some new scale-limited algebraic criteria and linear matrix inequality criteria of Lagrange stability are obtained by comparison strategy and generalized Halanay inequality. All scale-limited sufficient criteria of Lagrange stability for FMNNs not only apply to continuous-time FMNNs and their discrete-time analogs, but also could deal with the arbitrary combination of them. Finally, two numerical simulations are given to verify the validity of the obtained theoretical results.

Passive-active fault detection of Boolean control networks

This paper investigates the fault detectability of Boolean control networks. To deal with different circumstances, passive fault detectability and four types of active fault detectability are proposed for the first time. For passive fault detectability, in line with observed input-output data, several discriminant conditions are put forward. For active fault detectability, by resorting to an auxiliary system, the essence of active fault detectabilty is revealed via reachability. Subsequently, corresponding to different types of active fault detectability, algebraic criteria that facilitate verification are provided. At the same time, the relationships of four types of active fault detectability are discussed as well. In addition, an approach for calculating fault detection time is given by combining passive fault detectability with active fault detectability. Finally, the effectiveness of the results obtained is elaborated by several examples.

Time-dependent and Caputo derivative order-dependent quasi-uniform synchronization on fuzzy neural networks with proportional and distributed delays

This paper focuses on studying the quasi-uniform (Q-U) synchronization on fractional-order fuzzy neural networks (FOFNNs) with the proportional and distributed delays. The impacts of the Caputo derivative order, coefficients of network system and control gain constants on the synchronization performance are taken into account. Two novel time-dependent and Caputo derivative order-dependent algebraic criteria on the Q-U synchronization of FOFNNs are established by using Cauchy–Schwartz inequality, Minkowski inequality, Cp inequality and Hölder inequality, which reveal the internal influences of the time and fractional derivative order on Q-U synchronization with the derivative order interval (0,2). The simulation examples further confirm the effectiveness and practicability of the algebraic criteria in terms of the MATLAB toolbox.

State estimation of quaternion-valued inertial neural networks: non-reduced order method.

This paper concentrates on the problem of state estimation for quaternion-valued inertial neural networks (QVINNs) with nonidentical time-varying delay. Without reducing the original second order system into two first order systems, a non-reduced order method is developed to investigate the addressed QVINNs, which is different from the majority of existing references. By constructing a new Lyapunov functional with tuning parameters, some easily checked algebraic criteria are established to ascertain the asymptotic stability of error-state system with the desired performance. Moreover, an effective algorithm is provided to design the estimator parameters. Finally, a numerical example is given out to illustrate the feasibility of the designed state estimator.

Synchronization in Finite Time of Fuzzy Neural Networks with Hybrid Delays and Uncertain Nonlinear Perturbations

This paper deals with the finite-time synchronization problem of a class of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations. By applying the famous finite-time stability theory, combining differential inequality techniques, and the analysis approach, several new algebraic sufficient criteria are obtained to realize finite-time synchronization between the drive system and the response system by designing a state feedback controller and an adaptive controller. Taking discrete delays, distributed delays, and uncertain nonlinear perturbations into account in fuzzy cellular neural networks makes the neural system more general than most existing cellular neural networks. Two different novel types of controllers designed to achieve finite-time synchronization can not only effectively overcome the influence of time delays and perturbations but also change their form according to the change of system state or perturbation to achieve a better control effect. Meanwhile, some algebraic sufficient criteria obtained in this paper can be proved by the parameters of the system itself, and the complex calculation of matrix inequality is avoided. Finally, the validity of our proposed results is confirmed by several examples and simulations. Furthermore, a secure communication problem is presented to further illustrate the fact of the obtained results.

Event-triggered impulsive synchronization of coupled delayed memristive neural networks under dynamic and static conditions

This paper aims to address the exponential synchronization issue of coupled delayed memristive neural networks (CDMNNs) via event-triggered impulsive control (ETIC). To do this, a fresh controller based on the hybrid state feedback and impulsive effects is devised to efficiently lessen the communication load and economic cost. Then, the event-triggered conditions for two types of strategies including static event-triggered impulsive control (SETIC) and dynamic event-triggered impulsive control (DETIC) are presented. Furthermore, some algebraic criteria are derived to synchronize the CDMNNs by utilizing the Lyapunov method, the interval matrix method and impulsive system theory. Moreover, it can be guaranteed that the Zeno phenomenon does not emerge. Ultimately, the availability and feasibility of the propounded scheme are substantiated through an illustrated instance.

Bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks under signed digraph via adaptive strategy

In this work, an adaptive controller, formulated as linear feedback controls plus nonlinear parts, is synthesized to achieve bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks (FMNNs), in which follower FMNNs are linearly coupled under a signed digraph. The salient features of this research lie in two aspects: (1) the assumption on time-varying delays is very weak, since it neither requires boundedness of delays nor restricts the differentiation of time delay functions; (2) the adaptive controller contains no delay term, and it is feasible for both bounded and unbounded activation functions. As the preparatory work for stability analysis of the controlled synchronization error system, the ready-made results on delayed incommensurate fractional-order linear positive systems, especially stability condition and comparison principle, are perfected by relaxing premises of time-varying delays. Besides, a group of differential inclusion inequalities are established as a powerful aid in scaling the bipartite synchronization error system. More relevantly, an algebraic synchronization criterion, formulated in terms of coupling strength, inner coupling matrix, Laplacian matrix and FMNN system parameters, is proved with the benefit of vector Lyapunov function and positive system theory. With the controller utilized, linear feedback controls can be exerted merely on partial neurons of some selected FMNNs, which is exemplified by numerical simulations.

Conormal Spaces and Whitney Stratifications

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps.

Theoretical analysis of the flow stability of HTGR supercritical steam generators using dimensionless numbers

High Temperature Gas-cooled Reactors (HTGR) have the advantages of inherent safety and high temperature. Due to the high temperature, adoption of supercritical steam generator will further increase the power generation efficiency of HTGR. Flow instability may happen in supercritical steam generators due to acute changes in physical properties near the critical region. A theoretical model is established, verified and adopted to analyze the flow stability of HTGR supercritical steam generators. Secondary side supercritical water is divided into three regions. Partial differential governing equations describing supercritical systems are simplified to linear ordinary differential equations using integral and small disturbance linearization. The variables of differential equations are inlet velocity, the length of region 1 and the total length of region 1 and region 2. Routh stability criterion is adopted to determine supercritical system stability. Algebraic criterion is obtained from the coefficient matrix using Routh stability criterion. Similar to subcritical system, supercritical system stability is determined by algebraic criterion expressed by six dimensionless numbers. They are the inlet resistance coefficient, outlet resistance coefficient, sub-pseudo-critical number (similar to subcooling number in subcritical system), true trans-pseudo-critical number (similar to phase change number in subcritical system), friction number and Froude number. The actual supercritical steam generators of HTGR belong to the system with high inlet resistance coefficients and high friction numbers. Supercritical steam generators of HTGR (with high inlet resistance coefficients) can operate stably and have enough safety margin at full power level and partial power levels. Supercritical steam generators (with high inlet resistance coefficients and high friction numbers) become stable when system pressure, the inverse of Froude number (or angle inclination) and tube inner diameter increase. They become unstable when friction number (or friction factor) and tube length increase. The reasonable application range of the figure expressed by true trans-pseudo-critical number (NTPC) and sub-pseudo-critical number (NSUBPC) is discussed when analyzing parameters’ effects. The conclusions about parameters’ effect in supercritical systems are the same as that in subcritical systems. Previous contradictory conclusions about some parameters’ effects (friction number, tube length and tube inner diameter) are explained. The main reason is that the effect of friction numbers on supercritical systems is different at different inlet resistance coefficients.