Showing papers in "International Journal of Robust and Nonlinear Control in 2005"
TL;DR: In this article, the disturbance attenuation and rejection problem is investigated for a class of MIMO nonlinear systems in the disturbance-observer-based control (DOBC) framework.
Abstract: In this paper the disturbance attenuation and rejection problem is investigated for a class of MIMO nonlinear systems in the disturbance-observer-based control (DOBC) framework. The unknown external disturbances are supposed to be generated by an exogenous system, where some classic assumptions on disturbances can be removed. Two kinds of nonlinear dynamics in the plants are considered, respectively, which correspond to the known and unknown functions. Design schemes are presented for both the full-order and reduced-order disturbance observers via LMI-based algorithms. For the plants with known nonlinearity, it is shown that the full-order observer can be constructed by augmenting the estimation of disturbances into the full-state estimation, and the reduced-order ones can be designed by using of the separation principle. For the uncertain nonlinearity, the problem can be reduced to a robust observer design problem. By integrating the disturbance observers with conventional control laws, the disturbances can be rejected and the desired dynamic performances can be guaranteed. If the disturbance also has perturbations, it is shown that the proposed approaches are infeasible and further research is required in the future. Finally, simulations for a flight control system is given to demonstrate the effectiveness of the results. Copyright © 2005 John Wiley & Sons, Ltd.
677 citations
TL;DR: In this paper, an augmented Lyapunov functional is proposed to investigate the asymptotic stability of neutral systems and two methods with or without decoupling the LyAPunov matrices and system matrices are developed and shown to be equivalent to each other.
Abstract: In this paper, an augmented Lyapunov functional is proposed to investigate the asymptotic stability of neutral systems Two methods with or without decoupling the Lyapunov matrices and system matrices are developed and shown to be equivalent to each other The resulting delay-dependent stability criteria are less conservative than the existing ones owing to the augmented Lyapunov functional and the introduction of free-weighting matrices The delay-independent criteria are obtained as an easy corollary Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods Copyright © 2005 John Wiley & Sons, Ltd
260 citations
TL;DR: In this article, a stochastic model for chemical reactions is presented, which represents the population of various species involved in a chemical reaction as the continuous state of a polynomial Stochastic Hybrid System (pSHS).
Abstract: A stochastic model for chemical reactions is presented, which represents the population of various species involved in a chemical reaction as the continuous state of a polynomial stochastic hybrid system (pSHS). pSHSs correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. We show that for pSHSs, the dynamics of the statistical moments of its continuous states, evolves according to infinite-dimensional linear ordinary differential equations (ODEs), which can be approximated by finite-dimensional nonlinear ODEs with arbitrary precision. Based on this result, a procedure to build this types of approximation is provided.
This procedure is used to construct approximate stochastic models for a variety of chemical reactions that have appeared in literature. These reactions include a simple bimolecular reaction, for which one can solve the Master equation; a decaying–dimerizing reaction set which exhibits two distinct time scales; a reaction for which the chemical rate equations have a continuum of equilibrium points; and the bistable Schogl reaction. The accuracy of the approximate models is investigated by comparing with Monte Carlo simulations or the solution to the Master equation, when available. Copyright © 2005 John Wiley & Sons, Ltd.
174 citations
TL;DR: Some key themes that emerged during a Caltech/AFOSR Workshop on ‘Principles and Applications of Control in Quantum Systems’ (PRACQSYS) are described, which concluded that the number of experimentally accessible quantum control systems is steadily growing, and that quantum control motivates extensive study of model classes that have previously received scant consideration.
Abstract: We describe in this article some key themes that emerged during a Caltech/AFOSR Workshop on ‘Principles and Applications of Control in Quantum Systems’ (PRACQSYS), held 21–24 August 2004 at the California Institute of Technology. This workshop brought together engineers, physicists and applied mathematicians to construct an overview of new challenges that arise when applying constitutive methods of control theory to nanoscale systems whose behaviour is manifestly quantum. Its primary conclusions were that the number of experimentally accessible quantum control systems is steadily growing (with a variety of motivating applications), that appropriate formal perspectives enable straightforward application of the essential ideas of classical control to quantum systems, and that quantum control motivates extensive study of model classes that have previously received scant consideration. Copyright © 2005 John Wiley & Sons, Ltd.
171 citations
TL;DR: This article motivates the stochastic modelling of genetic networks and demonstrates the approach using several examples, and discusses the mathematics of molecular noise models including the chemical master equation, the chemical Langevin equation, and the reaction rate equation.
Abstract: SUMMARY Gene regulatory networks are dynamic and stochastic in nature, and exhibit exquisite feedback and feedforward control loops that regulate their biological function at different levels. Modelling of such networks poses new challenges due, in part, to the small number of molecules involved and the stochastic nature of their interactions. In this article, we motivate the stochastic modelling of genetic networks and demonstrate the approach using several examples. We discuss the mathematics of molecular noise models including the chemical master equation, the chemical Langevin equation, and the reaction rate equation. We then discuss numerical simulation approaches using the stochastic simulation algorithm (SSA) and its variants. Finally, we present some recent advances for dealing with stochastic stiffness, which is the key challenge in efficiently simulating stochastic chemical kinetics. Copyright # 2005 John Wiley & Sons, Ltd.
153 citations
TL;DR: In this article, a relaxation approach based on homogeneous polynomial forms is proposed to solve the double nonconvex optimization problem of estimating the domain of attraction (DA) of equilibria.
Abstract: SUMMARY The problem of estimating the domain of attraction (DA) of equilibria is considered for odd polynomial systems. Specifically, the computation of the optimal quadratic Lyapunov function (OQLF), i.e. the quadratic Lyapunov function (QLF) which maximizes the volume of the largest estimate of the DA (LEDA), is addressed. In order to tackle this double non-convex optimization problem, a relaxation approach based on homogeneous polynomial forms is proposed. The first contribution of the paper shows that a lower bound of the LEDA for a fixed QLF can be obtained via linear matrix inequalities (LMIs) based procedures. Also, condition for checking tightness of the lower bound are provided. The second contribution is a strategy for selecting a good starting point for the OQLF search, which is based on the volume maximization of the region where the time derivative of the QLFs is negative and is given in terms of LMIs. Several application examples are presented to illustrate the numerical behaviour of the proposed approach. Copyright # 2005 John Wiley & Sons, Ltd.
110 citations
TL;DR: In this article, an integral sliding mode technique robustifying the optimal controller for linear stochastic systems with input and observation delays is presented, which is based on integral sliding modes compensation of disturbances.
Abstract: SUMMARY This paper presents an integral sliding mode technique robustifying the optimal controller for linear stochastic systems with input and observation delays, which is based on integral sliding mode compensation of disturbances The general principles of the integral sliding mode compensator design are modified to yield the basic control algorithm oriented to time-delay systems, which is then applied to robustify the optimal controller As a result, two integral sliding mode control compensators are designed to suppress disturbances in state and observation equations, respectively, from the initial time moment Moreover, it is shown that if certain matching conditions hold, the designed compensator in the state equation can simultaneously suppress observation disturbances, as well as the designed compensator in the observation equation can simultaneously suppress state disturbances The obtained robust control algorithm is verified by simulations in the illustrative example, where the compensator in the observation equation provides simultaneous suppression of state and observation disturbances Copyright # 2005 John Wiley & Sons, Ltd
104 citations
TL;DR: In this article, an improved delay-dependent asymptotic stability criterion is developed without using bounding techniques on the related cross product terms, which significantly improves the allowed upper bounds of the delay size over some existing ones in the literature.
Abstract: This paper deals with the problem of robust stability analysis for uncertain neutral systems. In terms of a linear matrix inequality (LMI), an improved delay-dependent asymptotic stability criterion is developed without using bounding techniques on the related cross product terms. Based on this, a new delay-dependent LMI condition for robust stability is obtained. Numerical examples are provided to show that the proposed results significantly improve the allowed upper bounds of the delay size over some existing ones in the literature. Copyright © 2005 John Wiley & Sons, Ltd.
98 citations
87 citations
TL;DR: In this paper, the robust integrated control/diagnosis approach using ℋ∞-optimization techniques to the nonlinear longitudinal dynamics of a Boeing 747-100/200 aircraft is presented.
Abstract: In this paper, an application of the robust integrated control/diagnosis approach using ℋ∞-optimization techniques to the nonlinear longitudinal dynamics of a Boeing 747-100/200 aircraft is presented. The integrated approach allows to address directly the trade-off between the conflicting controller and fault diagnosis objectives. The integrated design formulation (interconnection and weight selection) is defined using five LTI plants obtained through out the Up-and-Away flight envelope. Linear and nonlinear closed-loop time simulations are carried out under a realistic turbulence and noise environment. A comparison drawn with the non-integrated design of a controller and a diagnosis filter with the same objectives shows that the integrated case results in similar diagnosis characteristics but improved fault tolerant performance and ease of design. Copyright © 2005 John Wiley & Sons, Ltd.
73 citations
TL;DR: For a family of uncertain nonlinear systems dominated by a triangular system that satisfies linear growth condition with an output dependent growth rate, the authors showed that global robust stabilization can be achieved by smooth output feedback.
Abstract: For a family of uncertain nonlinear systems dominated by a triangular system that satisfies linear growth condition with an output dependent growth rate, we prove that global robust stabilization can be achieved by smooth output feedback. This conclusion has incorporated and generalized the recent output feedback stabilization results, for instance, the work (IEEE Trans. Automat. Control 2002; 47:2068–2073) where the same conclusion was already shown to be true for planar systems, and the work (Proceedings of the 42nd IEEE, CDC, 2003; 1544–1549) where the growth rate is required to be a polynomial function of the system output. There are two key ingredients in the present contribution. One of them is the introduction of a rescaling transformation with a dynamic factor that is tuned on-line through a Riccati-like differential equation, which turns out to be extremely effective in dealing with the system uncertainty. The other one is the development of a recursive observer design algorithm making it possible to assign the robust observer gains in a step-by-step fashion. Both a smooth state feedback controller and a robust observer are explicitly constructed for the rescaled system using only the knowledge of the bounding system. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this article, a robust continuous sliding mode controller for output regulation of a class of minimum-phase nonlinear systems is proposed, where the servocompensator is designed as a conditional one that provides servocensation only inside the boundary layer.
Abstract: SUMMARY We consider the design of a robust continuous sliding mode controller for the output regulation of a class of minimum-phase nonlinear systems. Previous work has shown how to do this by incorporating a linear servocompensator in the sliding mode design, but the transient performance is degraded when compared to ideal sliding mode control. Extending previous ideas from the design of ‘conditional integrators’ for the case of asymptotically constant references and disturbances, we design the servocompensator as a conditional one that provides servocompensation only inside the boundary layer; achieving asymptotic output regulation, but with improved transient performance. We give both regional as well as semi-global results for error convergence, and show that the controller can be tuned to recover the performance of an ideal sliding mode control. Copyright # 2005 John Wiley & Sons, Ltd.
TL;DR: In this article, the robust stabilization and robust H1 control for uncertain discrete-time stochastic bilinear systems with Markovian switching were studied. But the problem of robust stabilization was not addressed.
Abstract: SUMMARY This paper is concerned with the problems of robust stochastic stabilization and robust H1 control for uncertain discrete-time stochastic bilinear systems with Markovian switching. The parameter uncertainties are time-varying norm-bounded. For the robust stochastic stabilization problem, the purpose is the design of a state feedback controller which ensures the robust stochastic stability of the closed-loop system irrespective of all admissible parameter uncertainties; while for the robust H1 control problem, in addition to the robust stochastic stability requirement, a prescribed level of disturbance attenuation is required to be achieved. Sufficient conditions for the solvability of these problems are obtained in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, explicit expressions of the desired state feedback controllers are also given. An illustrative example is provided to show the effectiveness of the proposed approach. Copyright # 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, the authors provide an introduction to modelling, identification, and control of a spherical particle trapped in an optical tweezer, and apply a recursive least squares method to a trapped 9.6-µm diameter polystyrene bead to study the possibility of obtaining faster calibrations of characteristic frequency.
Abstract: We provide an introduction to modelling, identification, and control of a spherical particle trapped in an optical tweezer. The main purpose is to analyse the properties of an optical tweezer from a control systems point of view. By representing the non-inertial dynamics of a trapped particle using a stochastic differential equation, we discuss probability distributions and compute first mean exit times. Within the linear trapping region, experimentally measured mean passage times for a 9.6-µm diameter polystyrene bead show close agreement with theoretical calculations. We apply a recursive least squares method to a trapped 9.6-µm diameter polystyrene bead to study the possibility of obtaining faster calibrations of characteristic frequency. We also compare the performance of proportional control, LQG control, and nonlinear control to reduce fluctuations in particle position due to thermal noise. Assuming a cubic trapping force, we use computer simulations to demonstrate that the nonlinear controller can reduce position variance by a factor of 65 for a 1-µm diameter polystyrene bead under typical conditions. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, the authors considered a linear, discrete-time system with uncertain, real-time-varying parameters, and proposed a new sufficient condition for the stability of this class of systems, in terms of a feasibility problem involving linear matrix inequalities (LMIs), under the hypothesis that a bound on the rate of variation of the parameters is known.
Abstract: In this paper we consider a linear, discrete-time system depending multi-affinely on uncertain, real time-varying parameters. A new sufficient condition for the stability of this class of systems, in terms of a feasibility problem involving linear matrix inequalities (LMIs), is obtained under the hypothesis that a bound on the rate of variation of the parameters is known. This condition, obtained by the aid of parameter dependent Lyapunov functions, obviously turns out to be less restrictive than that one obtained via the classical quadratic stability (QS) approach, which guarantees stability in presence of arbitrary time-varying parameters. An important point is that the methodology proposed in this paper may result in being less conservative than the classical QS approach even in the absence of an explicit bound on the parameters rate of variation. Concerning the synthesis context, the design of a gain scheduled compensator based on the above approach is also proposed. It is shown that, if a suitable LMI problem is feasible, the solution of such problem allows to design an output feedback gain scheduled dynamic compensator in a controller-observer form stabilizing the class of systems which is dealt with. The stability conditions are then extended to take into account L2 performance requirements. Some numerical examples are carried out to show the effectiveness and to investigate the computational burden required by the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, a novel methodology for fast interrogation of material that exploits the transient part of the cantilever motion is developed, which effectively addresses the perceived fundamental limitation on bandwidth due to high quality factors.
Abstract: SUMMARY In the existing dynamic-mode operation of atomic force microscopes (AFMs) steady-state signals like amplitude and phase are used for detection and imaging of material. Due to the high quality factor of the cantilever probe the corresponding methods are inherently slow. In this paper, a novel methodology for fast interrogation of material that exploits the transient part of the cantilever motion is developed. This method effectively addresses the perceived fundamental limitation on bandwidth due to high quality factors. It is particularly suited for the detection of small time scale tip–sample interactions. Analysis and experiments show that the method results in significant increase in bandwidth and resolution as compared to the steady-state-based methods. This article demonstrates the effectiveness of a systems perspective to the field of imaging at the nano-scale and for the first time reports realtime experimental results and scanning applications of the transient method. Copyright # 2005 John Wiley & Sons, Ltd.
TL;DR: In this article, the multivariable circle criterion is used to check the poles of a low-dimensional transfer function matrix and the eigenvalues of a matrix whose dimension is independent of the horizon length.
Abstract: The quadratic programme that must be solved with certain output–feedback model predictive controllers can be expressed as a continuous sector-bounded nonlinearity together with two linear transformations. Thus, the multivariable circle criterion gives a simple test for stability, with or without model mismatch. In particular, it may be applied if the open-loop plant is stable and the actuators are subject to simple saturation constraints. In the case of single horizon model predictive control, it suffices to check for positive realness a transfer function matrix whose dimension corresponds to the number of inputs. For an arbitrary length receding horizon it suffices to check the poles of a low dimension transfer function matrix and the eigenvalues (over an appropriate range of operator values) of a matrix whose dimension is independent of the horizon length. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, an optimal control problem for the control of the atomic force microscope (AFM) is proposed, which includes the design of a sample-profile estimate signal in addition to the set-point regulation and resolution objectives.
Abstract: SUMMARY The atomic force microscope (AFM) is a powerful tool for imaging and manipulating matter at the nanoscale. An optimal control problem is proposed for the control of AFMs which includes the design of a sample-profile estimate signal in addition to the set-point regulation and resolution objectives. A new estimate signal for the sample profile is proposed and it is proved that the transfer function between the profile signal and the estimate signal is unity. The main contribution in comparison to existing designs is that there is no bandwidth limitation on estimation of sample profiles! Experimental results are presented to corroborate these results. Copyright # 2005 John Wiley & Sons, Ltd.
TL;DR: In this article, a geometric framework for the stabilization and control of a general class of electrostatically actuated mechanical systems is presented, where the configuration space of the movable body is assumed to have the structure of a simple mechanical system on a Lie group, and stabilizing static and dynamic feedback control laws are derived in terms of co-ordinate-independent geometric formulas.
Abstract: This paper presents a geometric framework for the stabilization and control of a general class of electrostatically actuated mechanical systems. Microelectromechanical systems (MEMS), such as micromirrors, are one motivating application for this work. There wavelengths of applications of interest lead to positioning requirements on the order of 40–100 nm. Furthermore, electrostatic actuation is poised to be the method of choice for the emerging field of nanoelectromechanical systems (NEMS), and the approach presented should be applicable there as well. The class of devices under study consists of a movable, rigid, grounded electrode, with a variety of allowable rotational and/or translational degrees of freedom, and a set of multiple, fixed, independently addressable, drive electrodes. A key contribution of this paper places general electrostatic forces in a framework suitable for passivity-based control. The configuration space of the movable body is assumed to have the structure of a simple mechanical system on a Lie group, and stabilizing static and dynamic feedback control laws are derived in terms of co-ordinate-independent geometric formulas. To obtain controllers for a specific device it is then necessary only to evaluate these formulas. Appropriate approximations may be applied to make the computations more tractable. The static output feedback controller requires only measurement of the charge and voltage on each drive electrode to provide almost-global stabilization of a desired feasible configuration, but performance is limited by the natural dynamics of the mechanical subsystem. Performance may be improved using dynamic output feedback, but additional information is needed, typically in the form of a model relating electrode capacitances to the system configuration. We demonstrate the controller computations on a representative MEMS, and validate performance using ANSYS simulations. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, the authors describe results and challenges in feedback control of microfluidic systems and present two representative examples: control of liquid droplets by electrically actuated surface tension forces and steering of many particles at once by micro flow control.
Abstract: SUMMARY This paper describes results and challenges in feedback control of microfluidic systems. Results are provided for two representative examples: control of liquid droplets by electrically actuated surface tension forces and steering of many particles at once by micro flow control. Common themes and challenges are outlined based on the authors’ research programs and on the results of the March 2004 NSF workshop on ‘Control and System Integration of Micro- and Nano-Scale Systems’ organized by the author. Copyright # 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, a set of linear matrix inequalities (LMIs) and non-common LMI solutions are used to reduce the conservatism arising from seeking a common LMI solution in the past results.
Abstract: This paper addresses less conservative control design for multiple design specifications. Problems are described by a set of linear matrix inequalities (LMIs) and solved with non-common LMI solutions in order to reduce the conservatism arising from seeking a common LMI solution in the past results. Noticing that completing the square can split two variables in bilinear matrix inequality (BMI) terms into two different LMI ones, we propose an iterative algorithm in which non-positive quadratic terms are successively replaced by their upper bounds. An illustrated example is included. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this article, a conditional integrator is used to provide integral action inside a boundary layer while acting as stable systems outside, leading to improvement in transient response while achieving asymptotic regulation in the presence of unknown constant disturbances or parameter uncertainties.
Abstract: This paper studies the regulation of nonlinear systems using conditional integrators. Previous work introduced the tool of conditional integrators that provide integral action inside a boundary layer while acting as stable systems outside, leading to improvement in transient response while achieving asymptotic regulation in the presence of unknown constant disturbances or parameter uncertainties. The approach, however, is restricted to a sliding mode control framework. This paper extends this tool to a fairly general class of state feedback control laws, with the stipulation that we know a Lyapunov function for the closed-loop system. Asymptotic regulation with improvement in transient response is done by using the Lyapunov redesign technique to implement the state feedback control as a saturated high-gain feedback and introducing a conditional integrator to provide integral action inside a boundary layer. Improvement in the transient response using conditional integrators is demonstrated with an experimental application to the pendubot. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this article, the authors considered the design of an H∞ trade-off dependent controller, that is, a controller such that, for a given Linear Time-Invariant plant, a set of performance trade-offs parameterized by a scalar is satisfied.
Abstract: In this paper, we consider the design of an H∞ trade-off dependent controller, that is, a controller such that, for a given Linear Time-Invariant plant, a set of performance trade-offs parameterized by a scalar θ is satisfied. The controller state space matrices are explicit functions of θ. This new problem is a special case of the design of a parameter dependent controller for a parameter dependent plant, which has many application in Automatic Control. This last design problem can be naturally formulated as a convex but infinite dimensional optimization problem involving parameter dependent Linear Matrix Inequality (LMI) constraints. In this paper, we propose finite dimensional (parameter independent) LMI constraints which are equivalent to the parameter dependent LMI constraints. The parameter dependent controller design is then formulated as a convex finite dimensional LMI optimization problem. The obtained result is then applied to the trade-off dependent controller design. A numerical example emphasizes the strong interest of our finite dimensional optimization problem with respect to the trade-off dependent control application. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, the optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances.
Abstract: In this paper, the optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. As a result, the optimal estimate equation similar to the traditional Kalman–Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman–Bucy filter. The resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is however demonstrated that a finite set of the filtering equations, whose number is specified by the ratio between the current filtering horizon and the delay values, can be obtained in the particular case of equal or commensurable (τ=qh, q is natural) delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays is verified against the best Kalman–Bucy filter available for linear systems without delays and two versions of the extended Kalman–Bucy filter for time delay systems. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, the robustness of linear time-delay systems under multi-perturbation and affine perturbation of coefficient matrices via the concept of strong delay-independent stability radius (shortly, strong stability radius) was studied.
Abstract: In this paper, we study robustness of the strong delay-independent stability of linear time-delay systems under multi-perturbation and affine perturbation of coefficient matrices via the concept of strong delay-independent stability radius (shortly, strong stability radius). We prove that for class of positive time-delay systems, complex and real strong stability radii of positive linear time-delay systems under multi-perturbations (or affine perturbations) coincide and they are computed via simple formulae. Apart from that, we derive solution of a global optimization problem associated with the problem of computing of the strong stability radii of a positive linear time-delay system. An example is given to illustrate the obtained results. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: Standard input/state/output notions, such as existence and uniqueness of solutions, continuous dependence on initial data, observability, controllability, stabilizability, detectability, and minimality to the state/signal setting are extended.
Abstract: This is the first paper in a series of several papers in which we develop a state/signal linear time-invariant systems theory. In this first part we shall present the general state/signal setting in discrete time. Our following papers will deal with conservative and passive state/signal systems in discrete time, the general state/signal setting in continuous time, and conservative and passive state/signal systems in continuous time, respectively. The state/signal theory that we develop differs from the standard input/state/output theory in the sense that we do not distinguish between input signals and output signals, only between the “internal” states x and the “external” signals w. In the development of the general state/signal systems theory we take both the state space X and the signal space W to be Hilbert spaces. In later papers where we discuss conservative and passive systems we assume that the signal space W has an additional Kre1˘n space structure. The definition of a state/signal system has been designed in such a way that to any state/signal system there exists at least one decomposition of the signal space W as the direct sum W = Y ∔ U such that the evolution of the system can be described by the standard input/state/output system of equations with input space U and output space Y. (In a passive state/signal system we may take U and Y to be the positive and negative parts, respectively, of a fundamental decomposition of the Kre1˘n space W.) Thus, to each state/signal system corresponds infinitely many input/state/output systems constructed in the way described above. A state/signal system consists of a state/signal node and the set of trajectories generated by this node. A state/signal node is a triple Σ = (V ; X, W), where V is a subspace with appropriate properties of the product space X × X × W. In this first paper we extend standard input/state/output notions, such as existence and uniqueness of solutions, continuous dependence on initial data, observability, controllability, stabilizability, detectability, and minimality to the state/signal setting. Three classes of representations of state/signal systems are presented (one of which is the class of input/state/output representations), and the families of all the transfer functions of these representations are studied. We also discuss realizations of signal behaviors by state/signal systems, as well as dilations and compressions of these systems. (Duality will be discussed later in connection with passivity and conservativity.)
TL;DR: In this paper, the authors employ semidefinite programming duality theory to develop new alternative linear matrix inequality (LMI) tools for eventually periodic systems, which are then utilized to rederive an important version of the Kalman-Yakubovich-Popov (KYP) Lemma.
Abstract: SUMMARY This paper employs semidefinite programming duality theory to develop new alternative linear matrix inequality (LMI) tools for eventually periodic systems. These tools are then utilized to rederive an important version of the Kalman–Yakubovich–Popov (KYP) Lemma for such systems, and further give new synthesis results. Copyright # 2005 John Wiley & Sons, Ltd. In this paper, we continue our work started in Reference [1] on the control of eventually periodic systems. Such systems are aperiodic for an initial amount of time, and then become periodic afterwards. Eventually periodic dynamics arise in various scenarios. One of these scenarios is when linearizing a system along a trajectory composed of an aperiodic manoeuver and a subsequent periodic orbit. Another is when considering plants with uncertain initial states. It is worth noting that both finite horizon and periodic systems are subclasses of eventually periodic systems. Primarily, this paper serves as a gateway for the use of semidefinite programming duality results in control problems involving eventually periodic systems. In fact, we will show that all analysis and synthesis convex conditions pertaining to the ‘2-induced control of such systems can be provided in terms of finite-dimensional semidefinite programming problems. Then, by appealing to the vast literature on duality, it is possible to develop new alternative tools that would potentially offer new theoretical insight and possibly help provide new results in this area of research. Specifically, in this paper, we will invoke one of the theorems of alternatives of Reference [2], which is itself a special case of the Hahn–Banach separation theorem, to develop new alternative linear matrix inequality (LMI) tools that would be later used to rederive an
TL;DR: In this article, the integral sliding mode (ISM) is used for robustification of a linear time invariant min-max multi-model problem with uncertainties, starting from the beginning of the process, and guarantees that the unmatched part of uncertainties is minimized and not the entire uncertainty.
Abstract: SUMMARY The concept of the integral sliding mode (ISM) is revised and applied for robustification of a linear time invariant min–max multi-model problem with uncertainties. Modified version of ISM ensures the insensitivity of the designed min–max control law with respect to matched uncertainty, starting from the beginning of the process, and guarantees that the unmatched part of uncertainties is minimized and not ’
TL;DR: In this article, the authors focus on the problem of characterizing the structure of the omega-limit set that attracts the trajectories that start inside the large ball and stay inside thereafter.
Abstract: In nonlinear control theory, the equilibrium of a system is semiglobally practically stabilizable if, given two balls centred at the equilibrium, one of arbitrarily large radius and one of arbitrarily small radius, it is possible to design a feedback so that the resulting closed-loop system has the following property: all the trajectories originating in the large ball enter into the small ball and stay inside thereafter.
In this work, given certain classes of nonlinear systems that are semiglobally practically stabilized, we focus on the problem of characterizing the structure of the omega-limit set that attracts the trajectories that start inside the large ball. Copyright © 2005 John Wiley & Sons, Ltd.