# Online preprints

Note: Preprints of all of my published journal papers and book chapters are provided below (some conference and unpublished journal papers are also provided). If you would like to obtain a preprint of a conference paper and/or an unpublished journal paper that is not provided below, please send me your request via an email.

F. Mazenc and D. Nesic, "Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem" , Math. Contr. Sign. Syst., vol 19, No. 2, (2007), pp. 151-182.

Abstract: The classical Matrosov theorem concludes uniform asymptotic stability of time varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semidefinite derivative along solutions) and another auxiliary function with derivative that is strictly non-zero where the derivative of the Lyapunov function is zero [M1]. Recently, several generalizations of the classical Matrosov theorem that use a finite number of Lyapunov like functions have been reported in the literature. None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) that is a very useful analysis and controller design tool for nonlinear systems. We provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy inequalities that have a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function based controller redesign. We illustrate our results by constructing a strong Lyapunov function for a simple Euler-Lagrange system controlled by an adaptive controller.

D. Dacic and D. Nesic, "Quadratic stabilization of linear networked control systems via simultaneous controller and protocol synthesis", Automatica, vol. 43, No. 7, July, (2007), pp. 1145-1155.

Abstract: We develop necessary and sufficient conditions for quadratic stabilizability of linear networked control systems by dynamic output feedback and communication protocols. These conditions are used to develop a computationally tractable design for simultaneous synthesis of controllers and protocols in terms of matrix inequalities. The obtained protocols do not require knowledge of controller and plant states but only of the discrepancies between current and the most recently transmitted values of nodes' signals, and are implementable on control area networks. We demonstrate on a batch reactor example that our design guarantees quadratic stability with a significantly smaller network bandwidth than previously available designs.

D. Carnevale, A.R.Teel and D.Nesic, "Further results on stability of networked control systems: a Lyapunov approach", IEEE Trans. Auomat. Contr., Volume 52, Issue 5, (2007), 892 - 897.

Abstract: Simple Lyapunov proofs are given for an improved (relative to previous results that have appeared in the literature) bound on the maximum allowable transfer interval to guarantee global asymptotic or exponential stability in networked control systems and also for semiglobal practical asymptotic stability with respect to the length of the maximum allowable transfer interval. We apply our results to emulation of nonlinear controllers in sampled-data systems.

D. Liberzon and D. Nesic, "Input-to-state stabilization of linear systems with quantized state measurements", IEEE Trans. Automat. Contr., Volume 52, Issue 5, (2007), 767 - 781.

Abstract: We consider the problem of achieving input-to-state stability (ISS) with respect to external disturbances for control systems with linear dynamics and quantized state measurements. Quantizers considered in this paper take finitely many values and have an adjustable .zoom. parameter. Building on an approach applied previously to systems with no disturbances, we develop a control methodology that counteracts an unknown disturbance by switching repeatedly between .zooming out. and .zooming in.. Two specific control strategies that yield ISS are presented. The first one is implemented in continuous time and analyzed with the help of a Lyapunov function, similarly to earlier work. The second strategy incorporates time sampling, and its analysis is novel in that it is completely trajectory-based and utilizes a cascade structure of the closed-loop hybrid system. We discover that in the presence of disturbances, time-sampling implementation requires an additional modification which has not been considered in previous work.

D. Dacic, D. Nesic and P.V. Kokotovic, "Path following for nonlinear systems with unstable zero dynamics", IEEE Trans. Automat. Contr., Volume 52, Issue 3, (2007) 481 - 487.

Abstract: In the path-following problem formulated in this paper, it is required that the error between the system output and the desired geometric path eventually be less than any prespecified constant. If in a nonlinear MIMO system the output derivatives do not enter into its zero dynamics, a condition relating path geometry and stabilizability of the zero dynamics is given under which a solution to this problem exists. The solution is obtained by combining input-to-state stability and hybrid system methodologies.

D. Nesic and A.R. Teel, "Stabilization of sampled-data nonlinear systems via backstepping on their Euler approximate model", Automatica, 2006.

Abstract: Two integrator backstepping designs are presented for digitally controlled continuous-time plants in special form. The controller designs are based on the Euler approximate discrete-time model of the plant and the obtained control algorithms are novel. The two control laws yield, respectively, semiglobal-practical stabilization and global asymptotic stabilization of the Euler model. Both designs achieve semiglobal-practical stabilization (in the sampling period that is regarded as a design parameter) of the closed loop sampled-data system. A simulation example illustrates that the obtained controllers may sometimes be superior to backstepping controllers based on the continuous-time plant model that are implemented digitally.

D. Nesic and L. Grune, "A receding horizon control approach to sampled-data implementation of continuous-time controllers", Syst. Contr. Lett., vol. 55 (2006), 660-672.

Abstract: We propose a novel way for sampled-data implementation (with the zero order hold assumption) of continuous-time controllers for general nonlinear systems. We assume that a continuous-time controller has been designed so that the continuous-time closed-loop satis¯es all performance requirements. Then, we use this control law indirectly to compute numerically a sampled-data controller. Our approach exploits a model predictive control (MPC) strategy that minimizes the mismatch between the solutions of the sampled-data model and the continuous-time closed-loop model. We propose a control law and present conditions under which stability and sub-optimality of the closed loop can be proved. We only consider the case of unconstrained MPC. We show that the recent results in [6] can be directly used for analysis of stability of our closed-loop system.

Y. Tan, D. Nesic and I. M. Y. Mareels, "On non-local stability properties of extremum seeking controllers", Automatica, vol. 42 (2006), 889-903.

Abstract: In this paper, we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted. This non-local stability result is proved by showing semi-global practical stability of the closed-loop system with respect to the design parameters. We show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction. Our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking. Simulation examples illustrate our results.

L.Zaccarian, D.Nesic and A.R.Teel, "Antiwindup designs for time delay linear systems", Syst. Contr. Lett., vol. 54, No. 12 (2005), 1205-1217.

Abstract: In this paper we address and solve the problem of anti-windup augmentation for linear systems with input and output delay. In particular, we give a formal definition of an optimal L2 gain based anti- windup design problem in the global, local, robust and nominal cases. For each of these cases we show that a specific anti-windup compensation structure (which is a generalization of the approach in [29]) is capable of solving the anti-windup problem whenever this is solvable. The effectiveness of the proposed scheme is shown on a simple example taken from the literature, in which the plant is a marginally stable linear system.

S. Huang, M.R. James, D. Nesic and P.M. Dower, "A Unifying Framework for Nonlinear Controller Design to Achieve ISS-Like Properties", IEEE Trans. Automat. Contr., vol. 50, No. 11 (2005), 1681-1697.

Abstract: A unified approach to the design of controllers achieving various specified input- to-state (ISS) like stability properties is presented. A synthesis procedure based on dynamic programming is given. Both full state and measurement feedback cases are considered. Our results make an important connection between the ISS literature and nonlinear H infinity design methods. We make use of recently developed results on controller synthesis to achieve uniform l8 bound [10].

S. Huang, M.R. James, D. Nesic and P.M. Dower, "Analysis of input to state stability for discrete-time nonlinear systems via dynamic programming", Automatica, vol. 41, No. 12 (2005), 2055-2065.

Abstract: The Input-to-state stability (ISS) property for systems with disturbances has received considerable attention in the last ten years, with many applications and characterisations reported in the literature. The main purpose of this paper is to present novel analysis results for ISS that utilise dynamic programming techniques to characterise minimal ISS gains and transient bounds. These characterisations naturally lead to computable necessary and sufficient conditions for ISS. Our results make a connection between ISS and optimisation problems in nonlinear dissipative systems theory (including L2-gain analysis and nonlinear H infinity theory). As such, the results presented address an obvious gap in the literature.

D. Nesic and L. Gruene, "Lyapunov based continuous-time controller redesign for sampled-data implementation", Automatica, vol 41., No. 7 (2005), 1143-1156.

Abstract: Given a continuous-time controller and a Lyapunov function that shows global asymptotic stability for the closed loop system, we provide several results for modification of the controller for sampleddata implementation. The main idea behind this approach is to use a particular structure for the redesigned controller and the main technical result is to show that the Fliess series expansions (in the sampling period T) of the Lyapunov difference for the sampled-data system with the redesigned controller have a very special form that is useful for controller redesign. We present results on controller redesign that achieve two different goals. The first goal is making the lower order terms (in T) in the series expansion of the Lyapunov difference with the redesigned controller more negative. These control laws are very similar to those obtained from Lyapunov based redesign of continuous-time systems for robustification of control laws and they often lead to corrections of the well known ”-LgV ” form. The second goal is making the lower order terms (in T) in the Fliess expansions of the Lyapunov difference for the sampled-data system with the redesigned controller behave as close as possible to the lower order terms of the Lyapunov difference along solutions of the ”ideal” sampled response of the continuous-time system with the original controller. In this case, the controller correction is very different from the first case and it contains appropriate ”prediction” terms. The method is very flexible and one may try to achieve other objectives not addressed in this paper or derive similar results under different conditions. Simulation studies verify that redesigned controllers perform better (in an appropriate sense) than the unmodified ones when they are digitally implemented with sufficiently small sampling period T.

M. Tabbara and D. Nesic, "Discussion on "Development and experimental verification of a mobile client-centric networked controlled system" ", Europ. J. Contr., vol. 11 (2005), 249-251.

Abstract: Discussion on: ‘‘Development and Experimental Verification of a Mobile Client-Centric Networked Controlled System’’, European Journal of Control vol. 11 (2005).

L. Grune, D. Nesic and J. Pannek, "Model predictive control for nonlinear sampled-data systems" , in: F. Allgover, L. Biegler and R. Findeisen, eds., in Assessment and Future Directions of Nonlinear Model Predictive Control (NMPC05), Lecture Notes in Control and Information Sciences, Springer Verlag, in print, 2006.

Abstract: The topic of this paper is a new model predictive control (MPC) approach for the sampled–data implementation of continuous–time stabilizing feedback laws. The given continuous–time feedback controller is used to generate a reference trajectory which we track numerically using a sampled-data controller via an MPC strategy. Here our goal is to minimize the mismatch between the reference solution and the trajectory under control. We summarize the necessary theoretical results, discuss several aspects of the numerical implemenation and illustrate the algorithm by an example.

D.S. Laila, D.Nesic and A.Astolfi, "Sampled-data control of nonlinear systems", Advanced topics in control systems theory II, Lecture notes from FAP (Editors: Antonio Loria, Francoise Lamnabhi-Lagarrigue, Elena Panteley), to appear in 2006.

Abstract: This chapter provides some of the main ideas resulting from recent developments in sampled-data control of nonlinear systems. We have tried to bring the basic parts of the new developments within the comfortable grasp of graduate students. Instead of presenting the more general results that are available in the literature, we opted to present their less general versions that are easier to understand and whose proofs are easier to follow. We note that some of the proofs we present have not appeared in the literature in this simplified form. Hence, we believe that this chapter will serve as an important reference for students and researchers that are willing to learn about this area of research.

D. Liberzon and D. Nesic, "Stability analysis of hybrid systems via small gain theorems", in Proceedings of the Ninth International Workshop on Hybrid Systems: Computation and Control, Santa Barbara, CA, Mar 2006, Lecture Notes in Computer Science, J. P. Hespanha and A. Tiwari (Eds.), Springer, Berlin, in print, 2006.

Abstract: We present a general approach to analyzing stability of hy- brid systems, based on input-to-state stability (ISS) and small-gain theo- rems. We demonstrate that the ISS small-gain analysis framework is very naturally applicable in the context of hybrid systems. Novel Lyapunov- based and LaSalle-based small-gain theorems for hybrid systems are pre- sented. An illustrative application of the proposed approach in the con- text of a quantized feedback control problem is treated in detail. The reader does not need to be familiar with ISS or small-gain theorems to be able to follow the paper.

L. Zaccarian, D. Nesic and A. R. Teel, "First order reset elements and the Clegg integrator revisited" , to appear in Proc. Amer. Contr. Conf., Portland, OR, 2005.

Abstract: We revisit a class of reset control systems containing first order reset elements (FORE) and Clegg integrators and propose a new class of models for these systems. The proposed model generalizes the models available in the literature and we illustrate, using the Clegg integrator, that it is more appropriate for describing the behavior of reset systems. Then, we state computable sufficient conditions for L2 stability of the new class of models. Our results are based on LMIs and they exploit quadratic and piecewise quadratic Lyapunov functions. Finally, a result on stabilization of linear minimum phase systems with relative degree one using high gain FOREs is stated. We present two examples to illustrate our results. In particular, we show that for some systems a FORE can achieve lower L2 gain than the underlying linear controller without resets.

D. Nesic, L. Zaccarian and A. R. Teel, "Stability properties of reset systems", to appear in 16th IFAC World Congress, Prague, Czeck Republic, 2005.

Abstract: In this paper we address and solve the problem of anti-windup augmentation for linear systems with input and output delay. In particular, we give a formal definition of an optimal L2 gain based anti- windup design problem in the global, local, robust and nominal cases. For each of these cases we show that a specific anti-windup compensation structure (which is a generalization of the approach in [29]) is capable of solving the anti-windup problem whenever this is solvable. The effectiveness of the proposed scheme is shown on a simple example taken from the literature, in which the plant is a marginally stable linear system.

Y. Tan, D. Nesic and I.M.Y. Mareels, "On non-local stability properties of extremum seeking control", to appear in Proceeding of IFAC World Congress, Prague, Czeck Republic, 2005. (longer version to appear in Automatica)

Abstract: We revisit the extremum seeking scheme whose local stability properties were analyzed in (Krsti´c and Wang, 2000) and propose its simplified version that still achieves extremum seeking. We show under slightly stronger conditions that this simplified scheme achieves extremum seeking from arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted. This non-local convergence result is proved by showing semi-global practical stability of the closed-loop system with respect to the design parameters. Moreover, we show at the same time that reducing the parameters typically slows down the convergence of the extremum seeking controller. Hence, the control designer faces a tradeoff between the size of the domain of attraction and the speed of convergence when tuning the extremum seeking controller. We present a simulation example to illustrate our results.

S. Huang, M.R. James, D. Nesic and P.M. Dower, "Analysis of input to state stability for discrete-time nonlinear systems via dynamic programming" , to appear in Automatica, 2005.

Abstract: The Input-to-state stability (ISS) property for systems with disturbances has received considerable attention in the last ten years, with many applications and characterisations reported in the literature. The main purpose of this paper is to present novel analysis results for ISS that utilise dynamic programming techniques to characterise minimal ISS gains and transient bounds. These characterisations naturally lead to computable necessary and su±cient conditions for ISS. Our results make a connection between ISS and optimisation problems in nonlinear dissipative systems theory (including L2-gain analysis and nonlinear H infinity theory). As such, the results presented address an obvious gap in the literature.

S. Huang, M. James, D. Nesic and P. M. Dower, "Measurement feedback controller design to achieve input to state stability" , in Proc. 43rd Conf. Decis. Contr., Bahamas, pp. 2613-2618, 2004. (expanded version to appear in IEEE Trans. Automat. Contr.)

Abstract: An approach for design of measurement feedback controllers achieving input-to-state (ISS) stability properties is presented. A synthesis procedure based on dynamic programming is given. We make use of recently developed results on controller synthesis to achieve uniform l infinity bound [6]. Our results make an important connection between the ISS literature and nonlinear H infinity design methods.

G. Liu, D. Nesic and I. M. Y. Mareels, "Modelling and stabilization of a spherical inverted pendulum",to appear in 16th IFAC World Congress, Prague, Czeck Republic, 2005.

Abstract: We design a nonlinear control law for a four degree of freedom spherical inverted pendulum based on the forwarding technique. We first explore the forwarding structure of the spherical inverted pendulum model and then find a control law to stabilize the angle variables. Next, we develop a nested saturating controller for the whole system. The control law is evaluated through simulations.

L.Zaccarian, D.Nesic and A.R.Teel, "L2 antiwindup for linear dead-time systems", to appear in Syst. Contr. Lett., 2005.

Abstract: In this paper we address and solve the problem of anti-windup augmentation for linear systems with input and output delay. In particular, we give a formal definition of an optimal L2 gain based anti- windup design problem in the global, local, robust and nominal cases. For each of these cases we show that a specific anti-windup compensation structure (which is a generalization of the approach in [29]) is capable of solving the anti-windup problem whenever this is solvable. The effectiveness of the proposed scheme is shown on a simple example taken from the literature, in which the plant is a marginally stable linear system.

D. Nesic and A.R. Teel, "A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models", IEEE Trans. Automat. Contr., vol. 49 (2004), pp. 1103-1034.

Abstract: A unified framework for design of stabilizing controllers for sampled-data differential inclusions via their approximate discrete-tim models is presented. Both fixed and fast sampling are considered. In each case, sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete-time plant models also stabilizes the exact discrete-time plant model for sufficiently small integration and/or sampling periods. Previous results in the literature are extended to cover: (i) continuous-time plants modelled as differential inclusions; (ii) general approximate discrete-time plant models; (iii) dynamical discontinuous controllers modelled as difference inclusions; (iv) stability with respect to closed arbitrary (not necessarily compact) sets.

D. Nesic and A. Loria, "On uniform asymptotic stability of time-varying parameterized discrete-time cascades", IEEE Trans. Automt. Contr., vol. 49 (2004), pp. 875-887.

Abstract: Recently, a framework for controller design of sampled- data nonlinear systems via their approximate discrete-time models has been proposed in the literature. In this paper, we develop novel tools that can be used within this framework and that are useful for tracking problems. In particular, results for stability analysis of parameterized time-varying discrete-time cascaded systems are given. This class of models arises naturally when one uses an approximate discrete-time model to design a stabilizing or tracking controller for a sampled-data plant. While some of our results parallel their continuous-time counterparts, the stability properties that are considered, the conditions that are imposed, and the the proof techniques that are used, are tailored for approximate discrete-time systems and are technically different from those in the continuous-time context. A result on constructing strict Lyapunov functions from nonstrict ones that is of independent interest, is also presented. We illustrate the utility of our results in the case study of the tracking control of a mobile robot. This application is fairly illustrative of the technical differences and obstacles encountered in the analysis of discrete-time parameterized systems.

D.Nesic and A.R.Teel, "Matrosov theorem for parameterized families of discrete-time systems", Automatica, vol. 40 (2004), pp. 1025-1034.

Abstract: A version of Matrosov's theorem for parameterized discrete-time time-varying systems is presented. The theorem is a discrete-time version of the continuous-time result in [2]. Our result facilitates controller design for sampled-data nonlinear systems via their approximate discrete-time models. An application of the theorem to establishing uniform asymptotic stability of systems controlled by model reference adaptive controllers designed via approximate discrete-time plant models is presented.

A.R.Teel, L.Moreau and D.Nesic, "Input-to-state set stability of pulse width modulated controllers with disturbances", Systems and Control Letters, vol. 51 (2004), pp. 23-32.

Abstract: New results on set stability and input-to-state stability in pulse-width modulated (PWM) control systems with disturbances are presented. The results are based on a recent generalization of two time scale stability theory to differential equations with disturbances. In particular, averaging theory for systems with disturbances is used to establish the results. The non smooth nature of PWM systems is accommodated by working with upper semicontinuous set-valued maps, locally Lipschitz inflations of these maps, and locally Lipschitz parameterizations of locally Lipschitz set-valued maps.

D.Nesic and A.R.Teel, "Input output stability properties of networked control systems", IEEE Trans. Automat. Contr., vol. 49 (2004), pp. 1650-1667.

Abstract: Results on input-output Lp stability of networked control systems (NCS) are presented for a large class of network scheduling protocols. It is shown that static protocols and a recently considered dynamical protocol called Try-Once-Discard (TOD) belong to this class. Our results provide a unifying framework for generating new scheduling protocols that preserve Lp stability properties of the system if a design parameter is chosen sufficiently small. The most general version of our results can be used to treat NCS with data packet dropouts. The model of NCS and, in particular, of the scheduling protocol that we use appears to be novel and we believe that it will be useful in further study of these systems. The proof technique we use is based on the small gain theorem and it lends itself to an easy interpretation. We prove that our results are guaranteed to be better than existing results in the literature and we illustrate this via an example of a batch reactor.

M.Arcak and D.Nesic, "A framework for nonlinear sampled-data observer design via approximate discrete time models an emulation", Automatica, vol. 40 (2004), pp. 1931-1938.

Abstract: We present results on observer design for sampled-data nonlinear systems using two approaches: (i) the observer is designed via an approximate discrete-time model of the plant; (ii) the observer is designed based on the continuous-time plant model and then discretized for sampled-data implementation (emulation). Since exact discrete-time models are often unavailable for nonlinear sampled-data systems, it is suitable to employ approximate discrete-time models for observer design. We investigate under what conditions, and in what sense, such an approximate design achieves convergence for the unknown exact discrete-time model. We first present examples which show that designs via approximate discrete-time models may indeed lead to instability when implemented on the exact model. We then present conditions for approximate designs that guarantee robustness for the exact discrete-time model. We finally characterize convergence properties for emulation designs where the discrete-time observer is derived from a continuous-design via an approximate discretization.

D.Nesic and A.R.Teel, "Input to state stability of networked control systems", Automatica, vol. 40 (2004), pp. 2121-2128.

Abstract: A new class of Lyapunov uniformly globally asymptotically stable (UGAS) protocols in networked control systems (NCS) is considered. It is shown that if the controller is designed without taking into account the network so that it yields input-to- state stability (ISS) with respect to external disturbances (not necessarily with respect to the error that will come from the network implementation), then the same controller will achieve semi-global practical ISS for the NCS when implemented via the network with a Lyapunov UGAS protocol. Moreover, the ISS gain is preserved. The adjustable parameter with respect to which semi-global practical ISS is achieved is the maximal allowable transfer interval (MATI) between transmission times.

F. Mazenc and D.Nesic, "Strong Lyapunov functions for systems satisfying the conditions of La Salle", IEEE Trans. Automat. Contr., vol. 49 (2004), pp. 1026-1030.

Abstract: We present a construction of a (strong) Lyapunov function whose derivative is negative definite along the solutions of the system using another (weak) Lyapunov function whose derivative along the solutions of the system is negative semi-definite. The construction can be carried out if a Lie algebraic condition that involves the (weak) Lyapunov function and the system vector field is satisfied. Our main result extends to general nonlinear systems the strong Lyapunov function construction presented in \cite{FauPo} that was valid only for homogeneous systems.

A. Loria and D.Nesic, "On uniform boundedness of parameterized discrete-time cascades with decaying inputs: applications to cascades", Syst. Contr. Lett., vol. 49 (2003), pp. 163-174.

Abstract: A framework for controller design of sampled-data nonlinear systems via their approximate discrete-time models has been established recently. Within this framework naturally arises the need to investigate stability properties of parameterized discrete-time systems. Further results that guarantee appropriate stability of the parameterized family of discrete-time systems that is used within this framework have been also established for systems with cascaded structure. A fundamental condition that is required in this framework is uniform boundedness of solutions of the cascade. However, this is difficult to check in general. In this paper we provide a range of sufficient conditions for uniform boundedness that are easier to check. These results further contribute to the toolbox for controller design of sampled-data nonlinear systems via their approximate discrete-time models.

A.R. Teel, L. Moreau and D. Nesic, "A unification of time-scale methods for systems with disturbances", IEEE Trans. Automat. Contr, vol.48 (2003), pp. 1526-1544.

Abstract: This paper develops a unified framework for studying robustness of the input-to-state stability (ISS) property and presents new results on robustness of ISS to slowly varying parameters, to rapidly varying signals, and to generalized singular perturbations. The common feature in these problems is a time-scale separation between slow and fast variables which permits the definition of a boundary layer system like in classical singular perturbation theory. To address various robustness problems simultaneously, the asymptotic behavior of the boundary layer is allowed to be complex and it generates an average for the derivative of the slow state variables. The main results establish that if the boundary layer and averaged systems are ISS then the ISS bounds also hold for the actual system with an offset that converges to zero with the parameter that characterizes the separation of time-scales. The generality of the framework is illustrated by making connection to various classical two time-scale problems and suggesting extensions.

L. Zaccarian, A.R. Teel and D. Nesic, "On finite gain Lp stability of nonlinear sampled-data systems", Systems and Contol Letters, vol. 49 (2003), pp. 201-212.

Abstract: It is shown that uniform global exponential stability of the input-free discrete-time model of a globally Lipschitz sampled-data time-varying nonlinear system with inputs implies finite gain Lp stability of the sampled-data system for all p. This result generalizes results on Lp stability of sampled-data linear systems and it is an important tool for analysis of robustness of sampled-data nonlinear systems with inputs.

L. Grune and D. Nesic, "Optimization based stabilization of sampled-data nonlinear systems via their approximate discrete-time models", SIAM J. Opt. Contr., vol. 42 (2003), pp. 98-122.

Abstract: We present results on numerical regulator design for sampled-data nonlinear plants via their approximate discrete-time plant models. The regulator design is based on an approximate discrete-time plant model and is carried out either via an infinite horizon optimization problem or via a finite horizon with terminal cost optimization problem. In both cases we discuss situations when the sampling period $T$ and the integration period $h$ used in obtaining the approximate discrete-time plant model are the same or they are independent of each other. We show that using this approach practical and/or semiglobal stability of the exact discrete-time model is achieved under appropriate conditions.

D.S.Laila and D.Nesic, "Lyapunov based small-gain theorem for parameterized discrete-time interconnected ISS systems", IEEE Trans. Automat. Contr., vol. 48 (2003), pp. 1783-1788.

Abstract: Input-to-state stability (ISS) of a feedback interconnection of two discrete-time ISS systems satisfying an appropriate small gain condition is investigated via the Lyapunov method. In particular, an ISS Lyapunov function for the overall system is constructed from the ISS Lyapunov functions of the two subsystems. We consider parameterized families of discrete-time systems that naturally arise when an approximate discrete-time model is used in controller design for a sampled-data system.

D.S. Laila, D. Nesic and A.R. Teel, "Open and closed loop dissipation inequalities under sampling and controller emulation", Europ. J. Contr., vol. 18 (2002), pp. 109-125. [invited]

Abstract: We present a general and unified framework for the design of nonlinear digital controllers using the emulation method for nonlinear systems with disturbances. It is shown that if a (dynamic) continuous-time controller, which is designed so that the continuous-time closed-loop system satisfies a certain dissipation inequality, is appropriately discretized and implemented using sample and zero-order-hold, then the discrete-time model of the closed-loop sampled-data system satisfies a similar dissipation inequality in a semiglobal practical sense (sampling period is the parameter that we can adjust). We consider two different forms of dissipation inequalities for the discrete-time model: the weak'' form and the strong'' form. The results are also applicable for open-loop systems.

D. Nesic and D.S. Laila, "A note on input-to-state stabilization for nonlinear sampled-data systems", IEEE Trans. Automat. Contr., vol. 47 (2002), pp. 1153-1158.

Abstract: We present sufficient conditions that guarantee that a discrete-time controller that input-to-state stabilizes an approximate discrete-time model of a nonlinear sampled-data plant with disturbances would also input-to-state stabilize (in an appropriate sense) the exact discrete-time plant model.

D. Nesic and D. Angeli, "Integral versions of ISS for sampled-data nonlinear systems via their approximate discrete-time models", IEEE Trans. Automat. Contr. Vol. 47 (2002), pp. 2033-2038.

Abstract: Two integral versions of input-to-state stability are considered: integral input to state stability (iISS) and integral input to integral state stability (iIiSS). We present sufficient conditions that guarantee that if a controller achieves semi-global practical iISS (respectively iIiSS) of an approximate discrete-time model of a nonlinear sampled-data system, then the same controller achieves semi-global practical iISS (respectively iIiSS) of the exact discrete-time model by reducing the sampling period. Recent results on numerical methods for systems with measurable disturbances can be used to generate approximate models that we consider. Results are presented for arbitrary dynamic controllers that can be discontinuous in general.

D. Nesic and A.R.Teel, "Input-to-state stability for nonlinear time-varying systems via averaging", Math. Contr. Sign. Syst. (MCSS), vol.14 (2001), pp. 257-280.

Abstract: We introduce two definitions of an averaged system for a time-varying ordinary differential equation with exogenous disturbances (strong average'' and weak average''). The class of systems for which the strong average exists is shown to be strictly smaller than the class of systems for which the weak average exists. It is shown that input-to-state stability (ISS) of the strong average of a system implies uniform semi-global practical ISS of the actual system. This result generalizes the result of [18] which states that global asymptotic stability of the averaged system implies uniform semi-global practical stability of the actual system. On the other hand, we illustrate by an example that ISS of the weak average of a system does not necessarily imply uniform semi-global practical ISS of the actual system. However, ISS of the weak average of a system does imply a weaker semi-global practical "ISS like" property for the actual system when the disturbances w are absolutely continuous and disturbance and its derivative are bounded. ISS of the weak average of a system is shown to be useful in a stability analysis of time-varying cascaded systems.

D. Nesic and A.R. Teel, "Changing supply functions in input-to-state stable systems: the discrete-time case", IEEE Trans. Automat. Contr., vol. 46, No. 6 (2001), pp. 960-962.

Abstract:We characterize possible supply rates for input-to-state stable discrete-time systems and provide results that allow some freedom in modifying the supply rates. In particular, we show that the results reported in [7] for continuous-time systems are achievable for discrete-time systems.

D. Angeli and D. Nesic, "Power characterizations of input-to-state stability and integral input-to-state stability", IEEE Trans. Automat. Contr, vol.46, No. 8 (2001), pp. 1298-1303.

Abstract: New notions of external stability for nonlinear systems are introduced, making use of average powers as signal norms and comparison functions as in the Input-to-State Stability (ISS) framework. Several new characterizations of ISS and integral ISS are presented in terms of the new notions. An example is discussed to illustrate differences and similarities of the newly introduced properties.

D. Nesic, I. M. Y. Mareels, T. Glad and M. Jirstrand, "Software for control system analysis and design: symbol manipulation", Encyclopedia of Electrical and Electronics Engineering, J. Webster (Ed.), J. Wiley, 2001, available online http://www.interscience.wiley.com:83/eeee/.

Abstract: Some results on the use of computer algebra in control systems analysis and design are presented. The results make use of the quantifier elimination and the Grobner basis algorithms to address controllability, observability, stabilization and identifiability of a large class of nonlinear polynomial systems.

D. Nesic and A.R.Teel, "Sampled-data control of nonlinear systems: an overview of recent results", Perspectives on Robust Control, R.S.O.Moheimani (Ed.), Springer-Verlag: New York, pp. 221-239, 2001.

Abstract: Some recent results on design of controllers for nonlinear sampled-data systems are surveyed.

D. Nesic and P.M. Dower, "A note on input to state stability and averaging of systems with inputs", IEEE Trans. Automat. Contr., vol. 46, No. 11 (2001), pp. 1760-1765.

Abstract: Two different definitions of an average for time-varying systems with inputs and a small parameter that were recently introduced in the literature are considered: "strong" and "weak" averages. It is shown that if the strong average is input-to-state stable (ISS), then the solutions of the actual system satisfy an integral bound in a semi- global practical sense. The integral bound that we prove can be viewed as a generalization of the notion of finite-gain L2 stability, that was recently introduced in the literature. A similar result is proved for weak averages but the class of inputs for which the integral bound holds is smaller (Lipschitz inputs) than in the case of strong averages (measurable inputs).

D. Angeli and D. Nesic, "A trajectory based approach for stability robustness of systems with inputs", Math. Contr. Sign. Syst. (MCSS),vol. 15 (2001) pp. 336-355.

Abstract: We show, for two different definitions of semiglobal practical external stability, that the stability property holds on semi-infinite time intervals if and only if it holds on arbitrarily long but finite time intervals. These results have immediate applications in analysis of stability properties of highly oscillatory systems with inputs using averaging or for systems with inputs that are slowly varying. Results are stated for general flows and the stability is given with respect to arbitrary (not necessarily compact) sets.

D.S. Laila and D. Nesic, "Changing supply rates for input-output to state stable discrete-time nonlinear systems with applications", Automatica, vol. 39 (2001), pp. 821-835.

Abstract: We present results on changing supply rates for input-output to state stable (IOSS) discrete-time nonlinear systems. Our results can be used to combine two Lyapunov functions, none of which can be used to verify that the system has a certain property, into a new composite Lyapunov function from which the property of interest can be concluded. The results are stated for parameterized family of discrete-time systems that naturally arise when an approximate discrete-time model is used to design a controller for a sampled-data system. We present several applications of our results: (i) a LaSalle criterion for input to state stability (ISS) of discrete-time systems; (ii) constructing ISS Lyapunov functions for time-varying discrete-time cascaded systems; (iii) testing ISS of discrete-time systems using positive semidefinite Lyapunov functions; (iv) observer-based input to state stabilization of discrete-time systems. Our results are exploited in a case study of a two link manipulator and some simulation results that illustrate advantages of our approach are presented.

A. R. Teel and D. Nesic, "Averaging with disturbances and closeness of solutions", Sys. Contr. Lett, vol. 40, No. 5 (2000), pp. 317-323.

Abstract: We establish that, under appropriate conditions, the solutions of a time-varying system with disturbances converge uniformly on compact time intervals to the solutions of the system's average as the rate of change of time increases to infinity. The notions of "average" used for systems with disturbances are the "strong" and "weak" averages introduced in [5].

D. Nesic, "Output feedback stabilization of a class of Wiener systems", IEEE Trans. Automat. Contr., vol. 45, No. 9 (2000), pp. 1727-1731.

Abstract: A globally stabilizing output feedback controller is designed for a class of continuous-time Wiener systems. The Wiener systems we consider consist of a linear dynamical block and an output polynomial nonlinearity connected in series. The (hybrid) controller consists of three modes of operation which are periodically applied to the system. The controller achieves a dead-beat response of the closed-loop system.

D. Nesic and I. M. Y. Mareels, "Controllability of structured polynomial systems", IEEE Trans. Automat. Contr., vol. 44, No. 4 (1999), pp. 761-765.

Abstract: Two algorithms, based on the Grobner basis method, which facilitate the controllability analysis for a class of polynomial systems are presented. The authors combine these algorithms with some recent results on output dead-beat controllability in order to obtain sufficient, as well as necessary, conditions for complete and state dead-beat controllability for a surprisingly large class of polynomial systems. Our results are generically applicable to the class of polynomial systems in strict feedback form.

D. Nesic, E. Skafidas, I. M. Y. Mareels and R. J. Evans, "Minimum phase properties of input non-affine nonlinear systems", IEEE Trans. Automat. Contr., vol. 44, No. 4 (1999), pp. 868-872.

Abstract: For input nonaffine nonlinear control systems, the minimum phase property of the system in general depends on the control law. Switching or discontinuous controllers may offer advantages in this context. In particular, there may not exist a continuous control law which would keep the output identically equal to zero and for which the zero output constrained dynamics are locally stable, whereas a discontinuous controller which achieves this exists. For single-input/single-output input nonaffine nonlinear systems we give sufficient conditions for existence and present a method for the design of discontinuous switching controllers which yield locally stable zero dynamics.

D. Nesic, "Controllability for a class of simple Wiener-Hammerstein systems", Sys. Contr. Lett., vol. 36, No. 1 (1999), pp. 51-59.

Abstract: Controllability for a class of simple Wiener{Hammerstein systems is considered. Necessary and sucient conditions for dead-beat and complete controllability for these systems are presented. The controllability tests consist of two easy-to-check tests for the subsystems.

D. Nesic and I. M. Y. Mareels, "Stabilizability and stability for implicit and explicit polynomial systems: a symbolic computation approach", Europ. J. Contr., vol. 5 (1999), pp. 32-43.

S. Diop, D. Nesic and I. M. Y. Mareels, "Discussion on: Stability of implicit and explicit polynomial systems: symbolic computation approaches", Europ. J. Contr., vol. 5 (1999), pp. 44-45.

Abstract: Stabilizability and stability for a large class of discrete-time polynomial systems can be decided using symbolic computation packages for quanti¯er elimination in the first order theory of real closed fields. A large class of constraints on states of the system and control inputs can be treated in the same way. Stability of a system can be checked by constructing a Lyapunov function, which is assumed to belong to a class of polynomial positive definite functions. Moreover, we show that stability/stabilizability is possible to decide in a rather unexpected way, namely directly from the espilon-delta definition.

D. Nesic and G. Bastin, "Stabilizability and dead-beat controllers for two classes of Wiener-Hammerstein systems", IEEE Trans. Automat. Contr., vol. 44, No. 11 (1999), pp. 2068-2072.

Abstract: Two classes of block oriented models of the Wiener–Hammerstein type are considered. We prove that a generic condition is sufficient for a null controllable discrete-time system of this form to have a stabilizing minimum-time dead-beat controller. When the condition is violated, we show how to design a nonminimum time stabilizing (dynamic) dead-beat controller. The result is used to obtain stabilizability conditions for these systems.

D. Nesic, A. R. Teel and E. D. Sontag, "Formulas relating KL-stability estimates of discrete-time and sampled-data nonlinear systems", Sys. Contr. Lett., vol. 38, No.1 (1999), pp. 49-60.

Abstract: We provide an explicit ${\cal K}{\cal L}$ stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the ${\cal K}{\cal L}$ estimate for the corresponding discrete-time system and a ${\cal K}$ 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 \cite{aeyels} or extend some results in \cite{chen} to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool for some other problems which we indicate.

D. Nesic, "Controllability for a class of parallelly connected polynomial systems", Math. Contr. Sign. Syst., vol. 12 (1999), pp. 270-294.

Abstract: Null controllability for a class of parallelly connected discrete-time polynomial systems is considered. We prove for this class of systems that a necessary and sufficient condition for null controllability of the parallel connection is that all its subsystems are null controllable. Consequently, the control- lability test splits into a number of easy-to-check tests for the subsystems. The test for complete controllability is also presented and it is subtly different from the null controllability test. A similar statement is then given for complete controllability of a class of parallelly connected continuous-time polynomial systems. The result is somewhat unexpected when compared to the classical linear systems result. We identify the phenomenon which shows the difference between the linear and nonlinear cases.

D. Nesic, A. R. Teel and P. V. Kokotovic, "Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations", Sys. Contr. Lett., vol. 38, No. 4-5 (1999), pp. 259-270.

Abstract: Given a parameterized (by sampling period T) family of approximate discrete-time models of a sampled nonlinear plant and given a family of controllers stabilizing the family of plant models for all T sufficiently small, we present conditions which guarantee that the same family of controllers semi-globally-practically stabilizes the exact discrete-time model of the plant for sufficiently small sampling periods. When the family of controls is locally bounded, uniformly in the sampling period, the inter-sample behavior can also be uniformly bounded so that the (time-varying) sampled-data model of the plant is uniformly semi-globally-practically stabilized. The result justifies controller design for sampled-data nonlinear systems based on the approximate discrete-time model of the system when sampling is sufficiently fast and the conditions we propose are satisfied. Our analysis is applicable to a wide range of time-discretization schemes and general plant models.

D. Nesic, A note on observability tests for general polynomial and simple Wiener-Hammerstein systems'', Sys. Contr. Lett., vol. 35 (1998), pp. 219-227.

Abstract: We propose an algorithm, based on symbolic computation packages, for testing observability conditions of general polynomial systems, which were formulated in Sontag, SIAM J. Control Optim. 17 (1979) 139{151. Computational complexity of the observability test can be reduced and the test simplified for classes of polynomial systems. We illustrate this by considering the class of simple Wiener{Hammerstein systems, which consist of a series of two linear dynamic blocks between which a static nonlinearity is \sandwiched". We consider the case when the nonlinearity is a monomial N()=()q; q>1. Simple necessary and sufficient conditions for observability are given and they resemble, but are subtly different from, the well known result on observability for the series connection of two linear systems.

D. Nesic and E. D. Sontag, Input-to-state stability of linear systems with positive measurements'', Sys. Contr. Lett., vol. 35 (1998), pp. 245-255.

Abstract: This paper considers the problem of stabilization of linear systems for which only the magnitudes of outputs are measured. It is shown that, if a system is controllable and observable, then one can find a stabilizing controller, which is robust with respect to observation noise (in the ISS sense).

D. Nesic and I. M. Y. Mareels, Dead beat control of simple Hammerstein systems'', IEEE Trans. Automat. Contr., vol. 43, No. 8 (1998), 1184-1189.

Abstract: Dead-beat controllers for simple Hammerstein systems are investigated. Several designs for nonminimum-time state dead-beat controllers are given for certain classes of simple Hammerstein systems. A general minimum-time state dead-beat controller is presented for a class of simple Hammerstein systems. A design for a family of minimum-time control laws is provided. This enables, to a certain extent, shaping of transient response via choosing an appropriate control law. Finally, the authors design an output dead-beat controller for a class of Hammerstein systems that are not necessarily state dead-beat controllable.

D. Nesic and I. M. Y. Mareels, Dead beat controllability of polynomial systems: symbolic computation approaches'', IEEE Trans. Automat. Contr., vol. 43, No. 2 (1998), 162-175.

Abstract:State and output dead beat controllability tests for a very large class of polynomial systems with rational coefficients may be based on the QEPCAD symbolic computation program. The method is unified for a very large class of systems and can handle one or two sided control constraints. Families of minimum time state/output dead beat controllers are obtained. The computational complexity of the test is prohibitive for general polynomial systems but by constraining the structure of the system, we may beat the curse of complexity. A computationally less expensive algebraic test for output dead beat controllability for a class of odd polynomial systems is presented. Necessary and sufficient conditions are given. They are still very difficult to check. Therefore, a number of easier-to-check sufficient conditions are also provided. The latter are based on the Grobner basis method and QEPCAD. It is shown on a subclass of odd polynomial systems how it is possible to further reduce the computational complexity by exploiting the structure of the system.

D. Nesic, I. M. Y. Mareels, G. Bastin and R. Mahony, Output dead beat control for a class of planar polynomial systems'', SIAM J. Contr. Optimiz., vol. 36, No. 1 (1998), 253-272.

Abstract: Output dead beat control for a class of non linear discrete time systems, which are described by a single input-output polynomial difference equation, is considered. The class of systems considered is restricted to systems with a two dimensional state space description. It is assumed that the highest degree with which the present input appears in the equation is odd. Necessary and sufficient conditions for the existence of output dead beat control and for the stability of the zero output constrained dynamics are presented. We also design a minimum time output dead beat control algorithm (feedback controller) which yields stable zero dynamics, whenever this is feasible. A number of interesting phenomena are discussed and illustrated by examples.

D. Nesic, "A note on controllability of generalized Hammerstein systems", Sys. Contr. Lett., vol. 29 (1997), 223-231.

Abstract: Necessary and sufficient conditions for dead-beat and complete controllability for a class of generalised Hammerstein systems are presented. Since the system's structure is very close to linear, only linear algebra is used for the controllability test. The test is very simple and easy to use. A closed-loop minimum-time dead-beat controller can be designed for a subclass of generalised Hammerstein systems using the Grobner basis method.

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