Input-to-state Stability Property Research Articles

Second-order consensus with unknown dynamics via cyclic-small-gain method

This study proposes a distributed non-linear consensus protocol for second-order non-linear multi-agent systems with unknown locally Lipschitz dynamics and connected graph. The main analysis is based on a blend of graph-theoretic and non-linear-theoretic tools with the notion of input-to-state stability (ISS) playing a central role. Through the backstepping design, the closed-loop multi-agent system is transformed into a two-cascade interconnected system with proven ISS properties. Correspondingly, the recently developed cyclic-small-gain theorem is then employed to guarantee the asymptotic stability of the closed-loop multi-agent system, which implies consensus.

Quantized feedback stabilization for a class of linear systems with nonlinear disturbances

The quantized feedback stabilization problem for a class of linear system with nonlinear disturbances is addressed, in which the system and controller are connected via a communication channel. In this case, the effect of quantization errors is studied. A practical quantized scheme is designed such that the transmission error decays to zero exponentially. Meanwhile, a sufficient condition in terms of linear matrix inequality for input-to-state stability (ISS) of the system is presented with regard to the transmission error. Therefore, a quantized stabilization of the system is guaranteed based on the ISS property. A simulation example for the single-link flexible joint robot is presented to demonstrate the effectiveness of the result.

MPC Schemes Guaranteeing ISDS and ISS for Nonlinear (Time‐Delay) Systems

New directions in model predictive control (MPC) are introduced. On the one hand, we combine the input‐to‐state dynamical stability (ISDS) with MPC for single and interconnected systems. On the other hand, we introduce MPC schemes guaranteeing input‐to‐state stability (ISS) of single systems and networks with time delays. In both directions, recent results of the stability analysis from the mentioned areas are applied using Lyapunov function(al)s to show that the corresponding cost function(al) of the MPC scheme is a Lyapunov function(al). For networks, we show that under a small‐gain condition and with an optimal control obtained by an MPC scheme for networks, it has the ISDS property or ISS property, respectively.

Input‐to‐state‐stability‐type comparison principles and input‐to‐state stability for discrete‐time dynamical networks with time delays

SUMMARYThis paper investigates the input‐to‐state stability (ISS) issue for discrete‐time dynamical networks (DDNs) with time delays. Firstly, a general comparison principle for solutions of DDNs is proposed. Then, based on this general comparison principle, three kinds of ISS‐type comparison principles for DDNs are established, including the comparison principle for input‐to‐state ‐stability, ISS, and exponential ISS. The ISS‐type comparison principles are then used to investigate stability properties related to ISS for three kinds (linear, affine, and nonlinear) of DDNs. It shows that the ISS property of a DDN can be derived by comparing it with a linear or lower‐dimension DDN with known ISS property. By using methods such as variation of parameters, uniform M‐matrix, and the ISS‐type comparison principle, conditions of global exponential ISS for time‐varying linear DDNs with time delays are derived. Moreover, the obtained ISS results for DDNs are extended to the hybrid DDNs with time delays. As one application, the synchronization within an error bound in the sense of ISS is achieved for DDNs with coupling time delays and external disturbances. Finally, two examples are given to illustrate the results. Copyright © 2011 John Wiley & Sons, Ltd.

Input-to-state stability for networked control systems via an improved impulsive system approach

This paper presents a novel impulsive system approach to input-to-state stability (ISS) analysis of networked control systems (NCSs) with time-varying sampling intervals and delays. This approach is based upon the new idea that an NCS can be viewed as an interconnected hybrid system composed of an impulsive subsystem and an input delay subsystem. A new type of time-varying discontinuous Lyapunov-Krasovskii functional, which makes full use of the information on the piecewise-constant input and the bounds of the network delays, is introduced to analyze the ISS property of NCSs. Linear matrix inequality based sufficient conditions are derived for ISS of NCSs with respect to external disturbances. When applied to the approximate tracking problem for NCSs, the derived ISS result provides bounds on the steady-state tracking error. Numerical examples are provided to show the efficiency of the proposed approach.

Input-to-State Stability and Averaging of Linear Fast Switching Systems

We consider the averaging method for stability of rapidly switching linear systems with disturbances. We show that the notions of strong and weak averages proposed, with partial strong average defined in this note, play an important role in the context of switched systems. Using these notions of average, we show that exponential input-to-state stability (ISS) of the strong and the partial strong average system with linear gain imply exponential ISS with linear gain of the actual system. Similarly, exponential ISS of the weak average guarantees an appropriate exponential derivative ISS (DISS) property for the actual system. Moreover, using the Lyapunov method, we show that linear ISS gains of the actual system and its average converge to each other as the switching rate is increased.

Uniform stability and ISS of discrete-time impulsive hybrid systems

This paper studies the uniform stability and ISS (input-to-state stability) properties for DIHS (discrete-time impulsive hybrid systems) via comparison approach. By employing the vector-value function, the comparison principle is established for DIHS with external inputs. Then the comparison principle is used to establish the uniform stability and ISS criteria for DIHS, respectively. Moreover, regions in which the uniform stability and ISS properties can be guaranteed are estimated. As applications, the comparison principle and the results of uniform stability and ISS are used to study the robustly globally uniformly exponential stability for uncertain DIHS and exponential ISS of DIHS. It is shown that impulses contribute to stability and ISS properties for a discrete-time system which has no such properties. Two examples with numerical simulations are worked out for illustration.

Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions

This article presents a novel model predictive control (MPC) scheme that achieves input-to-state stabilization of constrained discontinuous nonlinear and hybrid systems. Input-to-state stability (ISS) is guaranteed when an optimal solution of the MPC optimization problem is attained. Special attention is paid to the effect that sub-optimal solutions have on ISS of the closed-loop system. This issue is of interest as firstly, the infimum of MPC optimization problems does not have to be attained and secondly, numerical solvers usually provide only sub-optimal solutions. An explicit relation is established between the deviation of the predictive control law from the optimum and the resulting deterioration of the ISS property of the closed-loop system. By imposing stronger conditions on the sub-optimal solutions, ISS can even be attained in this case.

Lyapunov conditions for input-to-state stability of impulsive systems

This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS, but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS, but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS, and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case, we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network.

A degree of flexibility in Lyapunov inequalities for establishing input-to-state stability of interconnected systems

In this paper, a novel approach to constructing flexible Lyapunov inequalities is developed for establishing Input-to-State Stability (ISS) of interconnection of nonlinear time-varying systems. It aims at a useful tool for using nonlinear small-gain conditions by allowing some flexibility in Lyapunov inequalities each subsystem is to satisfy. In the application of the ISS small-gain “theorem”, achieving a Lyapunov inequality conforming to a nonlinear small-gain “condition” is not a straightforward task. The proposed technique provides us with many Lyapunov inequalities with which a single trade-off condition between subsystems gains can establish the ISS property of the interconnected system. Proofs are based on explicit construction of Lyapunov functions.

Predictive Control of Hybrid Systems: Stability results for sub-optimal solutions

This article presents a novel model predictive control (MPC) scheme that achieves input-to-state stabilization of constrained discontinuous nonlinear and hybrid systems. Input-to-state stability (ISS) is guaranteed when an optimal solution of the MPC optimization problem is attained. Special attention is paid to the effect that sub-optimal solutions have on ISS of the closed-loop system. This issue is of interest as firstly, the infimum of MPC optimization problems does not have to be attained and secondly, numerical solvers usually provide only sub-optimal solutions. An explicit relation is established between the deviation of the predictive control law from the optimum (called the optimality margin) and the resulting deterioration of the ISS property of the closed-loop system (called the ISS margin).

Robust tracking control for a class of MIMO nonlinear systems with measurable output feedback

AbstractThis paper proposes a robust output feedback controller for a class of nonlinear systems to track a desired trajectory. Our main goal is to ensure the global input‐to‐state stability (ISS) property of the tracking error nonlinear dynamics with respect to the unknown structural system uncertainties and external disturbances. Our approach consists of constructing a nonlinear observer to reconstruct the unavailable states, and then designing a discontinuous controller using a back‐stepping like design procedure to ensure the ISS property. The observer design is realized through state transformation and there is only one parameter to be determined. Through solving a Hamilton–Jacoby inequality, the nonlinear control law for the first subsystem specifies a nonlinear switching surface. By virtue of nonlinear control for the first subsystem, the resulting sliding manifold in the sliding phase possesses the desired ISS property and to certain extent the optimality. Associated with the new switching surface, the sliding mode control is applied to the second subsystem to accomplish the tracking task. As a result, the tracking error is bounded and the ISS property of the whole system can be ensured while the internal stability is also achieved. Finally, an example is presented to show the effectiveness of the proposed scheme. Copyright © 2007 John Wiley & Sons, Ltd.

Regional Input-to-State Stability for Nonlinear Model Predictive Control

In this note, regional input-to-state stability (ISS) is introduced and studied in order to analyze the domain of attraction of nonlinear constrained systems with disturbances. ISS is derived by means of a non smooth ISS-Lyapunov function with an upper bound guaranteed only in a sub-region of the domain of attraction. These results are used to study the ISS properties of nonlinear model predictive control (MPC) algorithms

A unified approach to controller design for achieving ISS and related properties

A unified approach to the design of controllers achieving various specified input-to-state stability (ISS) like properties is presented. Both full state and measurement feedback cases are considered. Synthesis procedures based on dynamic programming are given using the recently developed results on controller synthesis to achieve uniform l/sup /spl infin// bound. Our results provide a link between the ISS literature and the nonlinear H/sup /spl infin// design literature.

ISS properties of nonholonomic vehicles

The paper presents the first result on nonholonomic systems enjoying input to state stability (ISS) properties. Although it is known that smooth stabilizability implies ISS, the converse is not generally true. This leaves the possibility of non-smoothly stabilizable systems being ISS with respect to a particular input, after an appropriate feedback transformation. This is shown to be true for the case of the unicycle with a dynamic extension, in a particular topology induced by a metric appropriate for this type of systems. A feedback control law renders the closed-loop system locally ISS in the particular topology. Potential applications include stability and robustness analysis of formations of mobile robots.

Robustness of network flow control against disturbances and time-delay

This paper studies robustness of Kelly's source and link control laws in (J. Oper. Res. Soc. 49 (1998) 237) with respect to disturbances and time-delays. This problem is of practical importance because of unmodelled flows, and propagation and queueing delays, which are ubiquitous in networks. We first show L p -stability, for p ∈[1,∞] , with respect to additive disturbances. We pursue L ∞-stability within the input-to-state stability (ISS) framework of Sontag (IEEE Trans. Automat. Control 34 (1989) 435), which makes explicit the vanishing effect of initial conditions. Next, using this ISS property and a loop transformation, we prove that global asymptotic stability is preserved for sufficiently small time-delays in forward and return channels. For larger delays, we achieve global asymptotic stability by scaling down the control gains as in Paganini et al. (Proceedings of 2001 Conference on Decision and Control, Orlando, FL, December 2001, pp. 185–190)

Input-to-state stability for a class of Lurie systems

We analyze input-to-state stability (ISS) for the feedback interconnection of a linear block and a nonlinear element. This study is of importance for establishing robustness against actuator nonlinearities and disturbances. In the absolute stability framework, we prove ISS from a positive real property of the linear block, by restricting the sector nonlinearity to grow unbounded as its argument tends to infinity. When this growth condition is violated, examples show that the ISS property is lost. The result is used to give a simple proof of boundedness for negative resistance oscillators, such as the van der Pol oscillator. In a separate application, we relax the minimum phase assumption of an earlier boundedness result for systems with nonlinearities that grow faster than linear.

Formulas relating [formula omitted] stability estimates of discrete-time and sampled-data nonlinear systems

We provide an explicit KL stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the KL estimate for the corresponding discrete-time system and a K function describing inter-sample growth. It is quite obvious that a uniform inter-sample growth condition, plus an ISS property for the exact discrete-time model of a closed-loop system, implies uniform ISS of the sampled-data nonlinear system. Our results serve to quantify these facts by means of comparison functions. Our results can be used as an alternative to prove and extend results in [1] or extend some results in [4] to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool for some other problems which we indicate.

Input-to-state stability for discrete-time nonlinear systems

In this paper the input-to-state stability (ISS) property is studied for discrete-time nonlinear systems. We show that many ISS results for continuons-time nonlinear systems in earlier papers (Sontag, 1989; Sontag, 1990; Sontag and Wang, 1996; Jiang et al ., 1994; Coron et al ., 1995) can be extended to the discrete-time case. More precisely, we provide a Lyapunov-like sufficient condition for ISS, and we show the equivalence between the ISS property and various other properties. Utilizing the notion of ISS, we present a small gain theorem for nonlinear discrete time systems. ISS stabilizability is discussed and connections with the continuous-time case are made. As in the continuous time case, where the notion ISS found wide applications, we expect that this notion will provide a useful tool in areas related to stability for nonlinear discrete time systems as well.

ISS property for time-varying systems and application to partial-static feedback stabilization and asymptotic tracking

A concept of input-to-state stability (ISS) for time-varying control systems is introduced that constitutes extension of the well-known notion concerning the autonomous case. We use this concept to derive sufficient conditions for global stabilization of triangular systems by means of a time-varying smooth feedback and to achieve asymptotic tracking of unbounded signals.