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This paper investigates the consensus problem for discrete-time leader-following multi-agent systems subject to large time delays. Building upon two assumptions, a novel fully distributed protocol is devised by utilizing a normalized weighting matrix, depending solely on the relative output measurement. It is shown that, for arbitrarily large yet bounded input and communication delays that are constant and exactly known, the consensus problem can be effectively addressed by both the proposed protocol and its truncated version. Assuming further that followers incorporate solely input delays, then the permitted delays can be time-varying and different. The proposed protocols do not rely on global information of the directed communication topology, thus ensuring robustness against alterations in the communication topology. A numerical example is employed to validate the effectiveness of the suggested approach. • A new fully distributed protocol suitable for discrete-time setting is designed. • The MASs are influenced by large time-varying input and communication delays. • The proposed consensus protocol employs relative output measurements. • The protocol is robust to changes in communication structure. [ABSTRACT FROM AUTHOR]

Summary: This paper studies the problem of sensor fault estimation for linear systems in the presence of unknown inputs in both input and output channels. Relying on matrix equation theory, the unknown inputs are first removed from the output channel. Second, an augmented descriptor system is constructed by treating both system state and sensor fault as (pseudo)‐state variables. Third, the augmented descriptor system is transformed into a regular one, followed by the elimination of the unknown input from the state equation. Finally, a reduced‐order prescribed‐time sensor fault estimator is obtained by using periodic delayed observers. This matrix equation based approach is then complemented by an alternative one, which relies on successive transformations of state, input and outputs. This approach sheds light on the adopted solvability conditions throughout the paper and on the role of different directions in the output space in handling unknown inputs, sensor faults and the core state estimation problem. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed methodologies. [ABSTRACT FROM AUTHOR]

In this paper, by extending the controllability and Brunovsky indices for a matrix pair to a matrix quadruple, three kinds of indices, say, the invertibility indices, the left invertibility indices, and the right invertibility indices, for linear time-invariant systems are introduced and studied. Based on properties of these indices, a neat expression for the Morse lists of a linear system is given without transforming it into its Morse normal form, which is a deep and fundamental result in linear systems theory, and takes an important role in many analysis and design problems for linear systems. Furthermore, the Morse normal form with only algebraic equivalence transformation is also investigated and is simplified. [ABSTRACT FROM AUTHOR]

This article studies the data‐driven H∞ controller design of continuous‐time linear periodic systems. By treating the H∞ optimal control problem as a two‐player zero‐sum game problem, the adaptive dynamic programming technique is utilized. Based on the Newton's method, a one‐loop model‐based policy‐iteration (PI) method is introduced. Since the initial matrix for starting the proposed model‐based PI method is not easy to choose, a data‐driven value iteration method without requiring the knowledge of the initial matrix is then proposed. The convergence of the proposed methods is analyzed and the numerical simulation verifies the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper addresses the leader-following consensus problem for general linear multiagent systems under general directed topology, with a focus on the scenarios where only relative output information is available. To solve such problem, most existing observer-based protocols rely on relative observer information among neighboring agents, which is obtained through the communication network and can be susceptible to network attacks. To overcome this limitation, this paper proposes a novel class of distributed observer-based linear time-varying protocols, in which the feedback gain is designed with two time-varying terms, one of which tends towards infinity as time increases, while the other tends towards zero. Compared to existing protocols, the proposed protocols offer significant advantages. Notably, they do not require information exchange through the communication network, making them immune to network attacks. Furthermore, the protocols operate in a fully distributed manner, which makes them resilient to changes in the topology graph. Simulation results demonstrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Summary: This paper is concerned with the disturbance rejection control problem of multiple integrators systems subject to actuator saturation and external disturbance. By using the solutions to a class of parametric Lyapunov equations (PLEs), extended observer based bounded linear output feedback controller is designed to solve the considered problem. By constructing a novel Lyapunov function and using the properties of the PLE, the stability of the closed‐loop system is proved and the domain of attraction is estimated. In order to improve control performance, bounded linear time‐varying output feedback controller is also proposed such that the state of the closed‐loop system tends to zero as the time approaches to the prescribed setting time which is independent of the initial conditions and can be assigned in advance. In addition, some comparisons of the proposed methods to existing results are given. Finally, numerical examples verify the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This article proposes static/dynamic event-triggered and self-triggered discrete gain scheduled control with a designable parametric minimal interevent time (MIET) to achieve semiglobal stabilization of linear systems with input constraints. First, a novel static event-triggered discrete gain scheduled control, which can improve the control performance and simultaneously save the communication resources, is proposed by utilizing the properties of the parametric Lyapunov equation (PLE). Moreover, the static self-triggered mechanism, in which the next control law updates based on the previous triggered states, is also designed to avoid the monitoring of all states. In order to further increase the interevent times (IETs), the corresponding dynamic event-triggered and self-triggered discrete gain scheduled control are designed, respectively. All the proposed algorithms can not only avoid the Zeno phenomenon but also provide a designable parametric MIET. This allows to easily find a tradeoff between the IETs and the control performance by adjusting the only design parameter. In addition, by exploiting the properties of the PLE, the designed algorithms avoid the complicated relationship between the MIET and the system matrices. In some cases, the MIET can totally avoid the relationship with the system itself and be designed as an arbitrarily large bounded constant. Finally, applications to the spacecraft rendezvous system show the effectiveness of the established algorithms. [ABSTRACT FROM AUTHOR]

This paper addresses the stabilization of linear systems with multiple time-varying input delays. In scenarios where neither the exact delays information nor their bound is known, we propose a class of linear time-varying state feedback controllers by using the solution to a parametric Lyapunov equation (PLE). By leveraging the properties of the solution to the PLE and constructing a time-varying Lyapunov–Krasovskii-like functional, we prove that (the zero solution of) the closed-loop system is asymptotically stable. Furthermore, this result is extended to the observer-based output feedback case. The notable characteristic of these controllers is their utilization of linear time-varying gains. Furthermore, they are designed entirely independent of any knowledge of the time delays, resulting in controllers that are exceedingly easy to implement. Finally, a numerical example demonstrates the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

This article studies the practical prescribed time sampled-data control problem for input-constrained linear time invariant continuous time systems. By using the solution to a parametric Lyapunov equation, a bounded linear time-varying (LTV) sampled-data controller is designed to solve such a problem. In addition, stability analysis and performance analysis of the closed-loop sampled-data control system are presented by using the methodology of scalarization. Finally, the developed LTV sampled-data state controller is utilized to the whole physical simulation system of air-bearing testbed and experimental results verify the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This article designs the static and dynamic event-triggered control (ETC) and self-triggered control (STC) algorithms to achieve the semiglobal stabilization of discrete-time systems with input constraints. First, a novel static ETC algorithm based on the discrete-time parametric Lyapunov equation (DPLE) is designed. In order to further increase the interevent times (IETs), the corresponding dynamic ETC is designed. Next, both static and dynamic STC, where the next control law updates depend on the previous triggered states, are proposed to avoid monitoring the measurement errors. The proposed algorithms are not only capable of reducing the number of transmissions significantly but also build a very simple and clear relationship between the only design parameter and the nontrivial IET (NIET). This allows us to change regularly IETs by adjusting the design parameter so that the nontriviality of static and dynamic ETC and STC is guaranteed and a tradeoff between the IETs and the control performance can be easily found. Specifically, by exploring the properties of DPLE, the designed algorithms avoid the complex relationship between the nontrivial condition and the system matrices. Finally, the designed static and dynamic ETC and STC algorithms are applied to the design of the spacecraft rendezvous control system and their effectiveness is verified by simulation results. [ABSTRACT FROM AUTHOR]

This article investigates the global stabilization problem of the circular orbit rendezvous system with actuator saturation and time-delay. By decomposing the linearized relative motion equations into a cascade of neutral stable linear systems, linear state feedback controllers are proposed in the presence of both actuator saturation and/or time-delay. The global stability of the closed-loop system is proved. Optimal feedback gain is also obtained in the delay-free case. Simulation results are given to show the effectiveness of the presented methods. [ABSTRACT FROM AUTHOR]

This paper proposes a pseudo predictor feedback control scheme for stabilisation of linear systems with both input and state delays. It is proved that the stability of the closed-loop system is determined by the stability of the target closed-loop system, along with the stability of an associated integral delay system, which describes the additional dynamics induced by the pseudo predictor. It is shown that in case of commensurate delays, the control law can be directly expressed as a function of the original system matrices, widening the applicability of analysis and design tools. For the synthesis of the feedback gains, a numerical procedure is presented to solve the simultaneous stabilization problem of the target system and the associated integral delay system. Numerical experiments are carried out to illustrate the effectiveness of the methodology. [ABSTRACT FROM AUTHOR]

In this paper, the prescribed-time control problem for a class of uncertain nonlinear systems is investigated. By fully exploiting key properties of a class of parametric Lyapunov equations and constructing a time-varying Lyapunov-like function, a neat linear time-varying high gain observer-based output feedback controller is designed. The uncertain parameters of the systems are reflected in the selection of the initial gain of the time-varying high gain function in the controller. It is proved that the proposed controller can drive the state to zero at the prescribed time and the control is bounded. Numerical examples verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new value iteration (VI) based method is proposed to solve the H∞ control problem of continuous‐time linear systems. The problem is transformed into solving a nonlinear differential equation, and the local stability of its solution is proved. Then a VI based iteration scheme for getting the approximation of the H∞ optimal controller is proposed. Compared with the existing results, the proposed scheme does not need any special initial matrix or extra condition to solve the problem and provides a relatively small disturbance attenuation bound. The related result is also applied to the quadratic guaranteed cost control of linear time‐delay systems. Simulation examples verify the effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

This article studies event‐triggered and self‐triggered gain scheduling control design for semi‐global stabilization of input constrained linear systems. Using the parametric Lyapunov equation (PLE), a event‐triggered gain scheduling control, which can increase the convergence rate of the closed‐loop systems by designing a gain scheduling scheme and save the communication resources by building an event generator, is designed. In order to avoid continuously monitoring the system states, the self‐triggered gain scheduling control algorithms are also designed. The non‐occurrence of the Zeno phenomenon and the boundedness of the parameter in the PLE are proved. The impact of the parameter on the minimal inter‐event time is analyzed, which allows us to find easily a trade‐off between the inter‐event times and the control performance. Another advantage of the designed methods is that the controller is linear. Finally, the proposed control algorithms are applied to stabilize the spacecraft rendezvous control system and the numerical simulation result shows the efficacy of the control algorithms. [ABSTRACT FROM AUTHOR]

Summary: This article studies semiglobal stabilization of input constrained linear systems based on the static and dynamic event‐triggered control (ETC) and self‐triggered control (STC). First, a static ETC based on the parametric Lyapunov equation (PLE) is designed. Then the corresponding dynamic ETC is also proposed. The main advantage of the proposed static and dynamic ETC algorithms is that the minimum interexecution time (MIET) as a decreasing function of the parameter in the PLE is given, which allows us to find easily a trade‐off between the interexecution times (IETs) and the control performance. The fact that the MIET of the dynamic ETC is never less than that of the static ETC is also guaranteed. Next, in order to avoid continuous monitoring of the states, both static and dynamic STC are designed. The influence of the parameters on IETs of the static and dynamic STC is analyzed in detail. The nonoccurrence of the Zeno phenomenon is proved in all the control algorithms. Finally, the proposed algorithms are applied to the design of spacecraft rendezvous. Numerical simulations on the original nonlinear model of the spacecraft rendezvous control system show the control system effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we study the consensus problem for high-order discrete-time multiagent systems with state, input, and communication delays, where the input and communication delays can be arbitrarily large yet exactly known. Moreover, the communication delays are different for different agents. Nested predictor-based state feedback protocols and full-order/reduced-order observer-based output feedback protocols are established to solve the problem. It is shown that both the input and communication delays can be compensated completely. Furthermore, linear matrix inequalities-based approaches are provided to design state feedback gains and observer gains. A numerical example is worked out to illustrate the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

In this note, the invariant factors assignment and the minimal polynomial assignment problems for multiple-input linear systems are studied. It is found that for any invariant factors with the geometry multiplicity of each eigenvalue less than or equal to the number of the inputs, the invariant factors assignment problems are solvable if and only if the controllability indices set of the linear system satisfies certain condition. For the minimal polynomial assignment problem, a necessary and sufficient condition on the solvability is proposed. Three numerical examples are provided to illustrate the proposed results. [ABSTRACT FROM AUTHOR]

Summary: We study in this paper the linear quadratic optimal control (linear quadratic regulation, LQR for short) for discrete‐time complex‐valued linear systems, which have several potential applications in control theory. Firstly, an iterative algorithm was proposed to solve the discrete‐time bimatrix Riccati equation associated with the LQR problem. It is shown that the proposed algorithm converges to the unique positive definite solution (bimatrix) to the bimatrix Riccati equation with appropriate initial conditions. With the help of this iterative algorithm, the LQR problem for the antilinear system, which is a special case of complex‐valued linear system, was carefully examined and three different Riccati equations–based approaches were provided, namely, bimatrix Riccati equation, anti‐Riccati equation, and normal Riccati equation. The established approach is then used to solve the LQR problem for a discrete‐time time‐delay system with one‐step state delay, and a numerical example was used to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper the prescribed-time output feedback stabilization of linear systems is addressed using both a functional observer, which includes the reduced-order observer as a special case, and a dual observer. The approach is built upon smooth periodic delayed feedback which is a special case of linear periodic time-varying feedback. In contrast to existing periodic control approaches, the proposed feedback achieves prescribed-time convergence of the state to zero rather than asymptotic convergence. At the same time its synthesis requires solving algebraic equations rather than periodic differential equations. Compared with full-order observer based output feedback, the dimensions of the dynamic equations in the reduced-order observer and dual observer based output feedback are respectively n − p and n − m , where n , m and p are respectively the number of state variables, inputs and outputs, which indicates that the established controllers are less expensive than the full-order observer based controllers in implementations. A numerical example is given to verify the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

The consensus problem for multiagent systems (MASs) described by discrete-time linear systems with multiple input delays is investigated. Under two reasonable assumptions, a truncated pseudo-predictor feedback (TPPF) approach is established to solve the consensus problem. The proposed TPPF protocols allow arbitrarily large yet bounded delays and are easy to implement in practice since they are finite dimensional and only use the relative current state information of neighboring agents. Moreover, the proposed protocols can also achieve semi-global consensus of MASs if the actuators are subject to saturations. A numerical example is given to illustrate the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates the global stabilization problem of discrete-time multiple integrators with bounded and delayed feedback. In the absence of input delay, two classes of nonlinear feedback laws are proposed. The first one consists of parallel connections of saturation functions and the other one consists of nested saturation functions. Some free parameters and the so-called state-dependent saturation functions are introduced into these two types of control laws which can help to improve the transient performance of the closed-loop system. In the presence of input delay, with the aid of a special canonical form, two types of nonlinear control laws are also proposed to achieve global stability of the closed-loop system. Two numerical examples are given to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper studies a class of complex-valued linear systems whose state evolution dependents on both the state vector and its conjugate. The complex-valued linear system comes from linear dynamical quantum control theory and is also encountered when a normal linear system is controlled by feedback containing both the state vector and its conjugate that can provide more design freedom. By introducing the concept of bimatrix and its properties, the considered system is transformed into an equivalent real-representation system and a non-equivalent complex-lifting system, which are normal linear systems. Some analysis and design problems including solutions, controllability, observability, stability, eigenvalue assignment, stabilisation, linear quadratic regulation, and state observer design are then investigated. Criterion, conditions, and algorithms are provided in terms of the coefficient bimatrices of the original system. The developed approaches are also utilised to investigate the so-called antilinear system which is a special case of the considered complex-valued linear system. The existing results on this system have been improved and some new results are established. [ABSTRACT FROM AUTHOR]

Summary: This paper is concerned with the stabilization of linear systems with both state and distinct input delays. Nested predictor feedback controllers are designed to predict the future states such that the distinct input delays that can be arbitrarily large yet bounded are compensated completely. It is shown that the compensated closed‐loop system possesses the same characteristic equation as the closed‐loop system without distinct input delays. Both continuous‐time and discrete‐time time‐delay systems are studied in this paper. Moreover, the safe implementation problem for the continuous‐time nested predictor feedback controller is solved via adding input filters. Three numerical examples show the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

This study is concerned with the safe implementation of delayed output feedback (DOF) for linear systems with multiple input delays. It is observed that the numerical implementation of the distributed delay terms involved in the DOF controller may lead to instability of the closed‐loop system. To solve this problem, a generalised low‐pass filter (LPF) is introduced into the control loop. It is shown that the stability of the closed‐loop system can be maintained if the numerical integration is sufficiently precise. An augmented system is then constructed to design the LPF by applying the traditional pole assignment algorithm for linear systems without delay. Two illustrative examples are worked out to show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

The problem of global stabilization for the multiple oscillator systems by bounded and delayed controls is considered. Based on a special canonical form of the considered system, a nonlinear control law consisting of nested saturation functions is proposed to achieve global stabilization. The new special canonical form used in this brief contains not only time delay in the input but also time delays in its state, which leads to natural cancellation and helps to not only provide explicit conditions guaranteeing the global stability of the closed-loop systems but also improves the transient performance of the closed-loop system significantly. The proposed feedback law is also adopted to achieve restricted tracking. A numerical simulation is given to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper considers the consensus problem of linear multi-agent systems with large input and communication delays. Provided that only the relative output among agents is available and the delays are constant and exactly known, a novel reduced-order observer-based finite dimensional protocol is proposed to solve the considered problem. If it is further assumed that the systems only contain input delays, then the delays are allowed to be time-varying and unknown. It should be noted that the delays under consideration are allowed to be arbitrarily large yet bounded and the proposed protocols do not need the a priori global information of the communication topology and thus are robust to changes in communication topology. Finally, simulation results on spacecraft formation flying system verify the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this study, the authors study the input delay compensation problem for discrete‐time linear systems with both state and input delays. Under the assumption that the original time‐delay system without input delay can be stabilised by state feedback, a nested predictor feedback controller is established to predict the future states such that the arbitrarily large yet exactly known input delay in the original system is completely compensated. Consequently, it is shown that the closed‐loop system consisting of the original time‐delay system and the nested prediction feedback controller is asymptotically stable. Under an additional assumption, an explicit nested predictor feedback controller without involving any nested summations is also established. Finally, two numerical examples are carried out to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

This paper studies the problems of stabilization of discrete-time linear systems with a single input delay. By developing the methodology of pseudo-predictor feedback, which uses the (artificial) closed-loop system dynamics to predict the future state, memoryless state feedback control laws are constructed to solve the problem. Necessary and sufficient conditions are obtained to guarantee the stability of the closed-loop system in terms of the stability of a class-difference equations. It is also shown that the proposed controller achieves semi-global stabilization of the system if its actuator is subject to either magnitude saturation or energy constraints under the condition that the open-loop system is only polynomially unstable. Numerical examples have been worked out to illustrate the effectiveness of the proposed approaches. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The main results of the paper are generalizations of the Razumikhin and of the Krasovskii classical stability theorems for stability analysis of time-varying time-delay systems. The condition of negativity of the time-derivative of Razumikhin functions and Krasovskii functionals is weakened. This is achieved by using the notion and properties of uniformly stable functions. We also show how to apply the results to the stability analysis of linear time-varying time-delay systems of retarded type. Both the system matrices and time-delays are allowed to be time-varying. Some constructive sufficient stability conditions are obtained and their effectiveness is demonstrated by some examples. [ABSTRACT FROM AUTHOR]

The normal form, which is also known as Byrnes–Isidori normal form, of a linear system is very important in control theory, as it is a fundamental tool for solving a couple of important problems. It is known that the normal form exists only if the system has a relative degree and a nonsingular coupling matrix, which is not always satisfied. Though both the relative degree and the non-singularity of the coupling matrix are invariant under state and input transformations, they can be changed by output transformation. This paper establishes novel necessary and sufficient conditions for the existence of an output transformation such that the transformed system has a well defined relative degree and nonsingular coupling matrix, namely, the existence of a normal form for linear multivariable systems. The conditions are expressed by two rank identities in terms of the coefficient matrices of the linear system. When these two rank conditions are satisfied, an algorithm for constructing the output transformation matrix is also provided. The output tracking problem is revisited with the help of the obtained results. Examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper provides a new approach referred to as pseudo-predictor feedback (PPF) for stabilisation of linear systems with multiple input delays. Differently from the traditional predictor feedback which is from the model reduction appoint of view, the proposed PPF utilises the idea of prediction by generalising the corresponding results for linear systems with a single input delay to the case of multiple input delays. Since the PPF will generally lead to distributed controllers, a truncated pseudo-predictor feedback (TPPF) approach is established instead, which gives finite dimensional controllers. It is shown that the TPPF can compensate arbitrarily large yet bounded delays as long as the open-loop system is only polynomially unstable. The proposed TPPF approach is then used to solve the consensus problems for multi-agent systems characterised by linear systems with multiple input delays. Numerical examples show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper considers truncated predictor feedback (TPF) stabilization of periodic linear systems with time-varying input delays with its applications to the elliptical spacecraft rendezvous. By carefully investigating the solutions and properties to a class of parametric Lyapunov differential equations, it is proved that TPF controllers can stabilize the periodic linear time-delay systems with arbitrarily large and bounded delays as long as the characteristic multiplier set of the open-loop system is within the closed unit circle. Both single input delay and multiple input delays are addressed. Moreover, the control matrices of the systems are not necessarily periodic. The developed TPF approach is then applied to the elliptical spacecraft rendezvous problem, and numerical simulations validate the effectiveness of the established approaches. [ABSTRACT FROM AUTHOR]

We study in this paper the three-axis magnetic attitude control of small spacecraft by considering the actuator saturation. By noticing that the linearized dynamics is a neutrally stable linear periodic system, a general problem of global stabilization of linear periodic systems with bounded feedback is formulated and solved by both state feedback and observer based output feedback. Explicit saturated linear feedback control laws are established by using solutions to some Lyapunov differential equations associated with the open-loop systems and global stability of the closed-loop system is proved by constructing an explicit Lyapunov function. To apply the theoretical results on the three-axis magnetic attitude control systems, necessary and sufficient conditions are established to guarantee that the linearized open-loop system is neutrally stable and explicit solutions to the associated Lyapunov equations are obtained consequently. The controllers for the three-axis magnetic attitude control systems are then given in closed-form. Simulations show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper studies stabilization of linear systems with both state and input delays. A dynamic input-delay compensator obtained by adding integrators is established to compensate the input delays that can be arbitrarily large. With the input delay compensator, the original stabilization problem reduces to the problem of stabilizing an augmented linear time-delay system without input delay. Three methods are also proposed to design stabilizing controllers for the augmented linear time-delay system. The first method is based on linear matrix inequalities (LMIs) and the second method is based on model reduction. The third method is based on pole placement and is built for the particular case that the original time-delay system has only a pure delayed state vector on its right hand side. For this method, the optimal gain such that the decay rate of the closed-loop system is maximized is also proposed. The effectiveness of the proposed approaches is illustrated by three linear time-delay systems that are open-loop unstable. [ABSTRACT FROM AUTHOR]