Search

This paper concerns the stabilization of linear time-invariant (LTI) systems subject to uncertain, possibly time-varying delays. The fundamental issue under investigation, referred to as the delay margin problem, addresses the question: What is the largest range of delay such that there exists a single LTI feedback controller capable of stabilizing all the plants for delays within that range? Drawing upon analytic interpolation and rational approximation techniques, we derive fundamental bounds on the delay margin, within which the delay plant is guaranteed to be stabilizable by a certain LTI output feedback controller. Our contribution is threefold. First, for single-input single-output (SISO) systems with an arbitrary number of plant unstable poles and nonminimum phase zeros, we provide an explicit, computationally efficient bound on the delay margin, which requires computing only the largest real eigenvalue of a constant matrix. Second, for multi-input multi-output (MIMO) systems, we show that estimates on the variation ranges of multiple delays can be obtained by solving LMI problems, and further, by finding bounds on the radius of delay variations. Third, we show that these bounds and estimates can be extended to systems subject to time-varying delays. When specialized to more specific cases, e.g., to plants with one unstable pole but possibly multiple nonminimum phase zeros, our results give rise to analytical expressions exhibiting explicit dependence of the bounds and estimates on the pole and zeros, thus demonstrating how fundamentally unstable poles and nonminimum phase zeros may limit the range of delays over which a plant can be stabilized by a LTI controller. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the use of relaxed logarithmic barrier functions in the context of linear model predictive control. We present results that allow to guarantee asymptotic stability of the corresponding closed-loop system, and discuss further properties like performance and constraint satisfaction in dependence of the underlying relaxation. The proposed stabilizing MPC schemes are not necessarily based on an explicit terminal set or state constraint and allow to characterize the stabilizing control input sequence as the minimizer of a globally defined, continuously differentiable, and strongly convex function. The results are illustrated by means of a numerical example. [ABSTRACT FROM AUTHOR]

This technical note proposes a robust iterative learning control (ILC) strategy to regulate iteratively-operated, finite-duration nonrepetitive systems characterized by iteration-varying uncertainties in initial states, external disturbances, plant model matrices and desired reference trajectories. Our convergence analysis exploits results on input-to-state stability of discrete parameterized systems. For a class of multiple-input, multiple-output discrete-time linear systems, with all iteration-varying uncertainties bounded, we give one condition that ensures boundedness of all system trajectories and an additional, second condition that ensures convergence of tracking error. Notably, we do not require the standard ILC contraction mapping requirement to hold at each iteration. Moreover, we show that it is possible to achieve perfect output tracking if the iteration-varying uncertainties all converge with increasing iteration. [ABSTRACT FROM PUBLISHER]

We develop a predictor-feedback control design for multi-input nonlinear systems with distinct input delays, of arbitrary length, in each individual input channel. Due to the fact that different input signals reach the plant at different time instants, the key design challenge, which we resolve, is the construction of the predictors of the plant's state over distinct prediction horizons such that the corresponding input delays are compensated. Global asymptotic stability of the closed-loop system is established by utilizing arguments based on Lyapunov functionals or estimates on solutions. We specialize our methodology to linear systems for which the predictor-feedback control laws are available explicitly and for which global exponential stability is achievable. A detailed example is provided dealing with the stabilization of the nonholonomic unicycle, subject to two different input delays affecting the speed and turning rate, for the illustration of our methodology. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a self-triggered formulation of model predictive control for continuous-time nonlinear input-affine networked control systems. Our control method specifies not only when to execute control tasks but also provides a way to discretize the optimal control trajectory into several control samples, so that the reduction of communication load will be obtained. Stability analysis under the sample-and-hold implementation is also given, which guarantees that the state converges to a terminal region where the system can be stabilized by a local state feedback controller. Some simulation examples validate our proposed framework. [ABSTRACT FROM PUBLISHER]

This article investigates the stability of positive switched linear systems. We start from motivating examples and focus on the case when each switched subsystem is marginally stable (in the sense that all the eigenvalues of the subsystem matrix are in the closed left-half plane with those on the imaginary axis simple) instead of asymptotically stable. A weak excitation condition is first proposed such that the considered positive switched linear system is exponentially stable. An extension to the case without dwell time assumption is also presented. Then, we study the influence of time-varying delay on the stability of the considered positive switched linear system. We show that the proposed weak excitation condition for the delay-free case is also sufficient for the asymptotic stability of the positive switched linear system under unbounded time-varying delay. In addition, it is shown that the convergence rate is exponential if there exists an upper bound for the delay, irrespective of the magnitude of this bound. The motivating examples are revisited to illustrate the theoretical results. [ABSTRACT FROM PUBLISHER]

This note is concerned with global stabilization of multiple integrators by delayed and bounded controllers. Three types of new nonlinear control laws are proposed based on some special canonical forms and explicit conditions are proposed to guarantee the global stability of the closed-loop systems. Differently from the existing canonical forms, the new special canonical forms used in this note contain not only time delay in the input but also time delays in its state, which are essential in the recursive design since they lead to natural cancellation. Moreover, some free parameters are introduced into these controllers. These advantages can help to improve the transient performance of the closed-loop system significantly. Numerical simulations for the fourth order integrators illustrate that the proposed new controllers can indeed outperform the existing one. [ABSTRACT FROM AUTHOR]

This technical note develops linear matrix inequality (LMI) conditions to test whether an uncertain linear system is exponentially stable with a given decay rate $\alpha$. These new $\alpha$- exponential stability tests are derived for an uncertain system described by an interconnection of a nominal linear time-invariant system and a “troublesome” perturbation. The perturbation can contain uncertain parameters, time delays, or nonlinearities. This technical note presents two key contributions. First, $\alpha$- exponential stability of the uncertain LTI system is shown to be equivalent to (internal) linear stability of a related scaled system. This enables derivation of $\alpha$- exponential stability tests from linear stability tests using integral quadratic constraints (IQCs). This connection requires IQCs to be constructed for a scaled perturbation operator. The second contribution is a list of IQCs derived for the scaled perturbation using the detailed structure of the original perturbation. Finally, connections between the proposed approach and related work are discussed. [ABSTRACT FROM PUBLISHER]

Discrete-time sliding mode controllers with an implicit discretization of the signum function are considered. With a proper choice of the equivalent part of the control, the resulting controller is shown to be Lyapunov stable with finite-time convergence of the sliding variable to 0. The convergence of the control input, as the sampling period goes to 0, to the continuous-time one is shown. The robustness with respect to matching perturbations is also investigated. The discretization performance in terms of the error order is studied for different discretizations of the equivalent part of the input. Numerical and experimental results illustrate and support the analysis. [ABSTRACT FROM AUTHOR]

Classical frequency domain results of robust regulation are extended by requiring only a right or a left coprime factorization of a plant, but not both. The famous internal model principle is generalized first, which leads to a necessary and sufficient solvability condition of the robust regulation problem and to a parametrization of all robustly regulating controllers. In addition, a procedure for constructing robustly regulating controllers is proposed. [ABSTRACT FROM AUTHOR]

Prediction-based transformation is applied to control-affine systems with distributed input delays. Transformed system state is calculated as a prediction of the system's future response to the past input with future input set to zero. Stabilization of the new system leads to Lyapunov-Krasovskii proven stabilization of the original one. Conditions on the original system are: smooth linearly bounded open-loop vector field and smooth uniformly bounded input vectors. About the transformed system which turns out to be affine in the undelayed input but with input vectors dependent on the input history and system state, we assume existence of a linearly bounded stabilizing feedback and quadratically bounded control-Lyapunov function. If all assumptions hold globally, then achieved exponential stability is global, otherwise local. Analytical and numerical control design examples are provided. [ABSTRACT FROM PUBLISHER]

The technical note provides a computation-oriented necessary and sufficient condition for the global exponential stability of linear impulsive systems, whose impulsions are assumed to occur quasi-periodically. Based on the set-theoretic conditions for robust stability of uncertain linear systems, the existence of polyhedral Lyapunov functions is proved to be necessary and sufficient for global exponential stability of quasi-periodic linear impulsive systems. A constructive method is developed for testing the stability of the system and for computing set-induced polyhedral Lyapunov functions. The method leads to an algorithm whose complexity is similar to the standard algorithm related to discrete-time parametric uncertain systems with the state matrix belonging to a convex polytopic set. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the l\infty gain characterizations of linear switched systems (LSS) and present various relevant results on their exact computation and optimization. Depending on the role of the switching sequence, we study two broad cases: first, when the switching sequence attempts to maximize, and second, when it attempts to minimize the l\infty gain. The first, named as worst-case throughout the paper, can be related to robustness of the system to uncontrolled switching; the second relates to situations when the switching can be part to the overall decision making. Although, in general, the exact computation of l\infty gains is difficult, we provide specific classes, the input-output switching systems, for which it is shown that linear programming can be used to obtain the worst-case l\infty gain. This is a sufficiently rich class of systems as any stable LSS can be approximated by one. Certain applications to robust control design are provided where we show that a switched compensation independently of the plant has no advantage over a linear time invariant (LTI) compensation, and further, if the plant is strictly causal, even a switched compensation which has a matched switching with the plant does not provide a better performance over an LTI compensation. Also, we present a new necessary and sufficient condition to check the stability of LSS in form of a model matching problem. On the other hand, if one is interested in minimizing the l\infty gain over the switching sequences, we show that, for finite impulse response (FIR) switching systems the minimizing switching sequence can be chosen to be periodic. For input-only or output-only switching an exact, readily computable, characterization of the minimal l\infty gain is provided, and it is shown that the minimizing switching sequence is constant, which, as also shown, is not true for input-output switching. [ABSTRACT FROM AUTHOR]

This technical note extends the decoupling method for singularly perturbed systems, Chang transformation, from linear slowly time-varying systems to linear time-varying systems with semi-proper (self-commutative) fast subsystems through the constructions of contraction mappings when negative real parts of the parallel differential-eigenvalues are used to guarantee the exponential stability of the perturbed systems. MacLaurin series forms and iterative solutions in an interval of the singular perturbation parameter for the differential Riccati equation and the differential Sylvester equation are provided, which are key steps for the Chang transformation construction. [ABSTRACT FROM AUTHOR]

This note presents a design technique for the delay-based controller called proportional integral retarded (PIR), which solves the regulation problem of a general class of stable second-order LTI systems. Using spectral analysis, the technique yields a tuning strategy for the PIR by placing a triple real dominant root for the closed-loop system. This result ultimately guarantees a desired exponential decay rate \sigmad while achieving the PIR tuning as an explicit function of \sigmad and system parameters. [ABSTRACT FROM AUTHOR]

We prove that the piecewise linear best-response dynamical systems of strategic interactions are asymptotically convergent to their set of equilibria on any weighted undirected graph. We study various features of these dynamical systems, including the uniqueness and abundance properties of the set of equilibria and the emergence of unstable equilibria. We also introduce the novel notions of social equivalence and social dominance on directed graphs, and demonstrate some of their interesting implications, including their correspondence to consensus and chromatic number of partite graphs. Examples illustrate our results. [ABSTRACT FROM AUTHOR]

We consider the problem of global external stochastic stabilization for linear plants with saturating actuators, driven by a stochastic external disturbance, and having random Gaussian-distributed initial conditions. The objective is to control such plants by a possibly nonlinear static state feedback law that achieves global asymptotic stability in the absence of disturbances, while guaranteeing a bounded variance of the state vector for all time in the presence of disturbances and Gaussian distributed initial conditions. Results for continuous-time open-loop critically stable plants as well as for a chain of integrators have been obtained before. The goal of this technical note is to extend this result to critically unstable plants. We view our contribution in this technical note as a critical step in solving the LQG control problem for linear systems subject to saturated inputs which is a research problem with a long history in our field. [ABSTRACT FROM AUTHOR]

We consider discrete-time linear time-invariant (LTI) systems with state feedback, where the actuation signal is subject to switching propagation delays, due to, e.g., routing in a multi-hop communication network. We show how to model these systems as pure switching linear systems and provide an algorithm for robust stability analysis. Next, we turn to the design problem and provide an algorithm that computes in a finite number of steps the look-ahead knowledge of the delays (i.e., routing) necessary to achieve controllability and stabilizability. This generalizes well known controllability criteria for LTI systems. [ABSTRACT FROM AUTHOR]

We analyze the problem of stability of a continuous time linear switching system (LSS) versus the stability of its Euler discretization. In case of matrices with real spectrum, we obtain a lower bound for the Euler step size to decide stability. This leads to a method for computing the Lyapunov exponent with a given accuracy and with a guaranteed computational cost. Our approach is based on the analysis of Chebyshev systems of exponents. [ABSTRACT FROM AUTHOR]

We prove stability of time-varying systems with delays, using linear Lyapunov functionals and positive systems, and we provide robustness of the stability with respect to multiplicative uncertainty in the vector fields. We allow cases where the delay may be unknown, and where the vector fields defining the systems are not necessarily bounded. We illustrate our work using a chain of integrators and other examples. [ABSTRACT FROM AUTHOR]

Lyapunov–Krasovskii approach is applied to parameter-robustness and delay-robustness analysis of the feedback suggested by Manitius and Olbrot for a linear time-invariant system with distributed input delay. A functional is designed based on Artstein's system reduction technique. It depends on the norms of the reduction-transformed plant state and original actuator state. The functional is used to prove that the feedback is stabilizing when there is a slight mismatch in the system matrices and delay values between the plant and controller. [ABSTRACT FROM PUBLISHER]

In this paper, a novel framework to address the problem of parametric estimation for continuous-time linear time-invariant dynamic systems is dealt with. The proposed methodology entails the design of suitable kernels of non-anticipative linear integral operators thus obtaining estimators showing, in the ideal case, “non-asymptotic” (i.e., “finite-time”) convergence. The analysis of the properties of the kernels guaranteeing such a convergence behaviour is addressed and a novel class of admissible kernel functions is introduced. The operators induced by the proposed kernels admit implementable (i.e., finite-dimensional and internally stable) state-space realizations. Extensive numerical results are reported to show the effectiveness of the proposed methodology. Comparisons with some existing continuous-time estimators are addressed as well and insights on the possible bias affecting the estimates are provided. [ABSTRACT FROM PUBLISHER]

Global delay independent stability is analyzed for nonlinear time-delay systems by applying homogeneity theory. It is shown that finite-time stability can be encountered in this class of systems under uniformity of the convergence time with respect to delay. Some additional tools for stability analysis of time-delay systems using homogeneity are also presented: in particular, it is shown that if a time-delay system is homogeneous with nonzero degree and it is globally asymptotically stable for some delay, then this property is preserved for any delay value, which is known as the independent of delay stability. [ABSTRACT FROM AUTHOR]

The aim of this technical note is to provide an efficient control design technique for discrete-time positive periodic systems. In particular, stability, positivity and periodic invariance of such systems are studied. Moreover, the concept of periodic invariance with respect to a collection of boxes is introduced and investigated with connection to stability. It is shown how such concept can be used for deriving a stabilizing state-feedback control that maintains the positivity of the closed-loop system and respects states and control signals constraints. In addition, all the proposed results can be efficiently solved in terms of linear programming. [ABSTRACT FROM AUTHOR]

This technical note addresses the stability analysis of linear systems governed by continuous-time difference equations with multiple delays. We propose new sufficient conditions for L2-exponential stability and exponential stability within the Lyapunov-Krasovskii framework. These conditions are delay-independent, and provide constructive exponential decay rate estimates of the system response. It is also shown that these conditions are less conservative than previous results in the literature. [ABSTRACT FROM AUTHOR]

Motivated by control applications relying on shared resources (such as computation time or bandwidth), we analyze the stability and robustness of discrete-time switched linear systems with stochastic commutations. We show that a wide class of shared resources control strategies can be modeled by a stochastic jump linear system involving two stochastic processes. The class of systems we study encompasses Markov chains and independent and identically distributed switching processes. For these systems, we recall existing definitions of stability and robustness, by relying on the input-to-state stability (ISS) property. We show that, for the class of systems under concern, $\delta$-moment stability is equivalent to $\delta$-moment ISS and that they both imply almost sure ISS. Several sufficient conditions are provided to guarantee these properties. Anytime control design for a translational oscillator/rotational actuator (TORA) system is used to illustrate all these concepts. [ABSTRACT FROM PUBLISHER]

In this technical note, cone-copositive piecewise quadratic Lyapunov functions (PWQ-LFs) for the stability analysis of conewise linear systems are proposed. The existence of a PWQ-LF is formulated as the feasibility of a cone-copositive programming problem which is represented in terms of linear matrix inequalities. A constructive procedure for its solution is provided. Examples show the effectiveness of the approach, also in the case of uncertain conewise linear systems. [ABSTRACT FROM AUTHOR]

In this technical note, we consider the following problem: what passivity properties for a nonlinear system can be inferred when its linearization around an equilibrium point is known to be passive. We consider both continuous-time and discrete-time systems with feedthrough terms. Our main results show that when the linearized model is simultaneously strictly passive and strictly input passive, the nonlinear system is passive as well within a neighborhood of the equilibrium point around which the linearization is done. [ABSTRACT FROM AUTHOR]

In this article, we consider a fundamental decentralized optimal control problem, which we call the two-player problem. Two subsystems are interconnected in a nested information pattern, and output feedback controllers must be designed for each subsystem. Several special cases of this architecture have previously been solved, such as the state-feedback case or the case where the dynamics of both systems are decoupled. In this paper, we present a detailed solution to the general case. The structure of the optimal decentralized controller is reminiscent of that of the optimal centralized controller; each player must estimate the state of the system given their available information and apply static control policies to these estimates to compute the optimal controller. The previously solved cases benefit from a separation between estimation and control that allows the associated gains to be computed separately. This feature is not present in general, and some of the gains must be solved for simultaneously. We show that computing the required coupled estimation and control gains amounts to solving a small system of linear equations. [ABSTRACT FROM PUBLISHER]

In this technical note, we present a new class of event triggering mechanisms for event-triggered control systems. This class is characterized by the introduction of an internal dynamic variable, which motivates the proposed name of dynamic event triggering mechanism. The stability of the resulting closed-loop system is proved and the influence of design parameters on the decay rate of the Lyapunov function is discussed. For linear systems, we establish a lower bound on the inter-execution time as a function of the parameters. The influence of these parameters on a quadratic integral performance index is also studied. Some simulation results are provided for illustration of the theoretical claims. [ABSTRACT FROM AUTHOR]

This note addresses Distributed State Estimation (DSE) over sensor networks. Two existing consensus approaches for DSE, i.e., consensus on information (CI) and consensus on measurements (CM), are combined to provide a novel class of hybrid consensus filters (named Hybrid CMCI) which enjoy the complementary benefits of CM and CI. Novel theoretical results, limitedly to linear systems, on the guaranteed stability of the Hybrid CMCI filters under collective observability and network connectivity are proved. Finally, the effectiveness of the proposed class of consensus filters is evaluated on a target tracking case study with both linear and nonlinear sensors. [ABSTRACT FROM AUTHOR]

Prediction and filtering of continuous-time stochastic processes often require a solver of a continuous-time differential Lyapunov equation (CDLE), for example the time update in the Kalman filter. Even though this can be recast into an ordinary differential equation (ODE), where standard solvers can be applied, the dominating approach in Kalman filter applications is to discretize the system and then apply the discrete-time difference Lyapunov equation (DDLE). To avoid problems with stability and poor accuracy, oversampling is often used. This contribution analyzes over-sampling strategies, and proposes a novel low-complexity analytical solution that does not involve oversampling. The results are illustrated on Kalman filtering problems in both linear and nonlinear systems. [ABSTRACT FROM PUBLISHER]

This technical note addresses the \ell\infty-gain (respectively, L\infty -gain) analysis problem for discrete-time (respectively, continuous-time) positive linear systems with unbounded time-varying delays. For the discrete-time case, by virtue of the monotonicity of an auxiliary system and the corresponding delay-free system, it is shown that the \ell\infty -gain of discrete-time positive systems with unbounded delays is insensitive of the magnitude of delays. For the continuous-time case, we first construct a sampled-data positive system and analyze its monotonicity and asymptotic behavior. Then, by comparing the delayed positive system with the proposed sampled-data positive system and the corresponding delay-free system, it is further proved that the L\infty -gain of continuous-time positive systems with unbounded delays is also independent of the delays and fully determined by the system matrices. The application of the results is illustrated by a sampled-data system with unbounded sampling intervals. [ABSTRACT FROM PUBLISHER]

This technical note provides necessary and sufficient conditions to determine that a linear system with more than one matrix in its state-space representation can be decomposed into cascade or separate sub-systems. In order to perform such decomposition, one needs to determine a linear transformation matrix. Furthermore, the given conditions are adapted to a simple but effective condition to derive all possible scalar sub-systems for a given linear system. Numerical examples are provided to demonstrate the applicability of the presented results. [ABSTRACT FROM PUBLISHER]

In this technical note, the problem of switching stabilization for slowly switched linear systems is investigated. In particular, the considered systems can be composed of all unstable subsystems. Based on the invariant subspace theory, the switching signal with mode-dependent average dwell time (MDADT) property is designed to exponentially stabilize the underlying system. Furthermore, sufficient condition of stabilization for switched systems with all stable subsystems under MDADT switching is also given. The correctness and effectiveness of the proposed approaches are illustrated by a numerical example. [ABSTRACT FROM PUBLISHER]

This study proposes a novel multiple model adaptive control (MMAC) algorithm for a class of nonlinear discrete time systems. The controller consists of a linear indirect adaptive controller, a nonlinear indirect adaptive controller based on neural networks, and a switching mechanism. The control input is generated by the switching mechanism, which selects the candidate controller from the two controllers. The assumption of the nonlinear term is relaxed to linear-bounded when a modified adaptive law is introduced. The restraint that the nonlinear term of the plant should be linear with respect to the control input is removed by resorting to the pole-placement control scheme. The proposed control method can address the properties of non-minimum phase and open-loop instability in the linear part of the plant. The proposed MMAC algorithm can guarantee the bounded-input-bounded-output stability of the proposed closed-loop switching system. A simulation example is presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM PUBLISHER]

Recently, we investigated the problem of leader-following rendezvous with connectivity preservation for a double integrator multi-agent system subject to external disturbances by a full information feedback controller. In this technical note, we further study a generalized problem via the position feedback control. Thus, the result of this technical note is more practical than that of the previous paper. [ABSTRACT FROM PUBLISHER]

In this paper, it is proved that a predictor-based feedback controller can effectively yield asymptotic convergence for a class of linear systems subject to input-dependent input delay. This class is characterized by the delay being implicitly related to past values of the input via an integral model. This situation is representative of systems where transport phenomena take place, as is frequent in the process industry. The sufficient conditions obtained for asymptotic stabilization bring a local result and require the magnitude of the feedback gain to be consistent with the initial conditions scale. Arguments of proof for this novel result include general Halanay inequalities for delay differential equations and build on recent advances of backstepping techniques for uncertain or varying delay systems. [ABSTRACT FROM PUBLISHER]

We study the stability of switched systems whose dynamic modes are described by systems of higher-order linear differential equations not necessarily sharing the same state space. Concatenability of trajectories at the switching instants is specified by gluing conditions, i.e. algebraic conditions on the trajectories and their derivatives at the switching instants. We provide sufficient conditions for stability based on LMIs for systems with general gluing conditions. We also analyse the role of positive-realness in providing sufficient polynomial-algebraic conditions for stability of two-modes switched systems with special gluing conditions. [ABSTRACT FROM AUTHOR]

Switched linear hyperbolic partial differential equations are considered in this technical note. They model infinite dimensional systems of conservation laws and balance laws, which are potentially affected by a distributed source or sink term. The dynamics and the boundary conditions are subject to abrupt changes given by a switching signal, modeled as a piecewise constant function and possibly a dwell time. By means of Lyapunov techniques some sufficient conditions are obtained for the exponential stability of the switching system, uniformly for all switching signals. Different cases are considered with or without a dwell time assumption on the switching signals, and on the number of positive characteristic velocities (which may also depend on the switching signal). Some numerical simulations are also given to illustrate some main results, and to motivate this study. [ABSTRACT FROM AUTHOR]

Petersen's lemma on matrix uncertainty is commonly used in the analysis and design for uncertain systems with time-varying norm-bounded uncertain parameter. In this note, Petersen's lemma is extended to the case where the uncertain inequality contains two uncertain matrix pairs. The condition is given such that the Petersen's lemma gives a necessary and sufficient condition. A numerical example is given to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This note considers the work entitled “Robust Stability and Stabilization of Fractional-Order Interval Systems with the Fractional Order $\alpha$: The $0<\alpha<1$ Case” by Lu and Chen, where the authors meant to provide sufficient and necessary conditions for stability of fractional-order linear-time-invariant interval systems. As this note proves, conditions are only sufficient, since necessity relies unsuitably on the certain case. An example is provided. [ABSTRACT FROM AUTHOR]