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The problem of an adaptive Lyapunov redesign is revisited for a class of systems without a priori knowledge of the function majoring nonlinear uncertainties and disturbances. An adaptive barrier function-based gain for unit control is proposed, ensuring an arbitrary a priori predefined uniform ultimate bound for solutions despite of the presence of uncertainties and disturbances. The usage of positive semi-definite barrier function generates a continuous control signal adjusting the chattering, when the perturbations are decreasing to zero.

This article deals with the robust control design for linear systems with input and state delays, subject to nonvanishing matched disturbances. The controller is synthesized with the Lyapunov redesign technique in two stages: first, an optimal predictor compensates the input delay of the nominal system while a quadratic performance index is minimized. Then, the robust component guarantees that the solutions are ultimately bounded. The theoretical results are experimentally validated through a dehydration process, with the controller digital implementation on an industrial programmable automation controller cDAQ-9132.

Integrated-control solutions will play a significant role in future tokamaks, in which a variety of coupled control problems will need to be solved simultaneously by means of a limited number of actuators. In this article, the problem of simultaneously regulating the global toroidal rotation, $\Omega _\phi $ , and total plasma energy, $W$ , is tackled. These two 0-D variables, $\Omega _\phi $ and $W$ , depend on the ion toroidal rotation and electron temperature profiles, respectively. Both $\Omega _\phi $ and $W$ also depend on the electron density and safety factor profiles. The actuation methods considered in this article are co-current and counter-current neutral beam injection. A nonlinear, robust controller that makes use of Lyapunov redesign techniques is synthesized based on 0-D, control-oriented models of the $\Omega _\phi $ and $W$ dynamics. In addition, an actuator management scheme is designed to handle variations in the control priorities and availability of the neutral beam injectors. The actuator manager solves an optimization problem in real time in order to find the most appropriate course of action when unexpected changes occur. The integrated control architecture is tested for a DIII-D scenario by means of the 1-D code Control-Oriented Transport Simulator (COTSIM), which predicts the time evolution of the electron temperature, electron density, ion toroidal rotation, and safety factor profiles.

This paper investigates the problem of synchronization for nonlinear systems. Following a Lyapunov approach, we first study the global synchronization of nonlinear systems in the canonical control form with both distributed proportional-derivative and proportional-integral-derivative control actions of any order. To do so, we develop a constructive methodology and generate in an iterative way inequality constraints on the coupling matrices that guarantee the solvability of the problem or, in a dual form, provide the nonlinear weights on the coupling links between the agents such that the network synchronizes. The same methodology allows us to include a possible distributed integral action of any order to enhance the rejection of heterogeneous disturbances. The considered approach does not require any dynamic cancellation, thus preserving the original nonlinear dynamics of the agents. The results are then extended to linear and nonlinear systems admitting a canonical control transformation. Numerical simulations validate the theoretical results.

This paper studies stability of stochastic impulsive switched systems. Different from exponential Lyapunov function and average dwell-time in the previous works, general Lyapunov function and fixed dwell-time are implemented in this paper to analyze input-to-state stability and global stability of stochastic impulsive switched systems. Two cases are investigated, that is, the case that the continuous dynamics is stable and the case that the discrete dynamics is stable. To implement multiple Lyapunov functions, there are two subcases considered in this paper: the subcase that the estimates on the derivatives of the multiple Lyapunov functions are the same and the subcase that the growths of the multiple Lyapunov functions by the jumps are the same. For aforementioned different cases, sufficient stability conditions are established for stochastic impulsive switched systems. Finally, the developed theory is illustrated through examples from networked control systems and synchronization problem of 3-D novel chaotic circuit systems.

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and sufficient to ensure input-to-state stability (ISS) with respect to closed sets. Toward this end, we first develop a Lyapunov characterization of $\omega$ ISS via finite-step $\omega$ ISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee $\omega$ ISS of a network of systems. Finally, applications of our results to partial ISS, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given.

This paper investigates the distributed adaptive control problem for synchronization of multi-agent systems where the dynamics of the agents are nonlinear, nonidentical, unknown and subject to external disturbances. In our recent work, we solved this problem for general higher order systems under two types of communication topologies, represented, respectively, by a fixed strongly connected directed graph and by a switching connected undirected graph. A common Lyapunov function technique is employed to establish the results. In this paper, we solve the problem for a more general communication topology which is represented by a switching strongly-connected directed graph. We construct a sequence of different Lyapunov functions and use them to establish our results. Simulation study verifies the effectiveness of our theoretical results.

In wake of frequency stability challenges enhanced by the volatile renewable energy based generation, there is a renewed interest in the methods that can provide stability insights for broad range of operating conditions of the oscillating multi-machine systems, such as power system. With this perspective in sight, this work proposes an analysis of transient stability with emphasis on the Gram–Schmidt and covariant Lyapunov vectors. Numerical accuracy of these Lyapunov vectors is validated by comparison with Floquet exponents obtained from the Jacobian matrix leading to an increased confidence in their potential application. In addition, a study of global stability using Lyapunov exponents and of local stability by Lyapunov vectors is performed. The assumptions necessary for the derivation of energy functions are avoided resulting in a broader stability perspective that is also applicable on the operating point outside the stability manifold. The methods are evaluated for the IEEE 9 bus system.

The Lyapunov equation is widely employed in the engineering field to analyze stability of dynamic systems. In this paper, based on a new evolution formula, a novel finite-time recurrent neural network (termed finite-time Zhang neural network, FTZNN) is proposed and studied for solving a nonstationary Lyapunov equation. In comparison with the original Zhang neural network (ZNN) model for a nonstationary Lyapunov equation, the convergence performance has a remarkable improvement for the proposed FTZNN model and can be accelerated to finite time. Besides, by solving the differential inequality, the time upper bound of the FTZNN model is computed theoretically and analytically. Simulations are conducted and compared to validate the superiority of the FTZNN model to the original ZNN model for solving the nonstationary Lyapunov equation. At last, the FTZNN model is successfully applied to online tracking control of a wheeled mobile manipulator.

We consider an uncertain mechanical system. The uncertainty includes the initial condition and system parameter. The uncertain parameter in the system is (possibly fast) time varying. The only known information about the uncertainty is that it lies in a fuzzy set. The mechanical system is to follow a set of constraints, which may include many engineering applications, even in the presence of uncertainty. For this purpose, we propose a $\beta$ -measure for constraint obedience, which reflects how much this constraint is obeyed. Based on a very general Lyapunov function, a control scheme is proposed to render a twofold performance: guaranteed and optimal. In the guaranteed phase, the $\beta$ -measure is assured to be uniformly bounded and uniformly ultimately bounded, regardless of the actual value of the uncertainty. In the optimal phase, a fuzzy-theoretic-based performance, by which both the “average” $\beta$ -measure and control effort are considered, is minimized. As a result, the control serves the practical engineering purposes: The mechanical system is guaranteed to follow the desired task with the minimum cost. This paper is part of a unique endeavor to cast both the descriptions of the uncertainty and desired performance index into a fuzzy framework.

The problem of designing $C^1$ or $C^\infty$ controllers for a class of stochastic nonlinear systems (SNSs) in lower-triangular form is studied in this note. By using the well-known backstepping method, the concept of homogeneity with monotone degrees (HWMD) and the sign function approach, we construct a $C^1$ state feedback controller recursively. Meanwhile, by employing a polynomial Lyapunov function with sign functions, we prove that the solution to SNSs is globally asymptotically stable in probability. Furthermore, based on the concept of HWMD, it shows that $C^1$ controllers for a class of three-dimensional SNSs can be modified to $C^{\infty }$ controllers under certain conditions. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed controllers.

In this brief, we investigate the consensus problem of nonlinear multiagent systems by means of general Lyapunov functions. That is, by constructing general Lyapunov functions, such as polynomial Lyapunov functions beyond quadratic forms, a sufficient condition is established for achieving global consensus. Compared with the existing consensus criteria deduced by quadratic Lyapunov functions, our consensus criterion is less conservative and can be applicable for more systems. In particular, for polynomial systems and even certain classes of nonpolynomial systems, our consensus criterion can be formulated based on polynomial Lyapunov functions, which can be calculated by using the sum-of-squares (SOS) programming tools such as SOSTOOLS. Finally, two simulation examples are given to illustrate the effectiveness of our proposed theoretical results.

This paper addresses the design of output controllers of chosen order in a given set for ensuring a number of desired properties, such as robust D-stability and robust performance. It is shown that a sufficient condition for determining such controllers can be obtained by solving convex optimization problems with linear matrix inequality constraints through the introduction of a Lyapunov function depending polynomially on both uncertainty and controller. The proposed condition is also necessary for sufficiently large degrees of the polynomials involved under some assumptions.

A general class of uncertain nonlinear systems with dynamic input uncertainties (nonlinear input unmodeled dynamics) is considered. While the proposed approach can, in general, be applied to multi-input systems, single-input systems are considered here for simplicity. It is shown that if a nominal control law is available to globally asymptotically stabilize the nominal system (without the input unmodeled dynamics), then a dynamic scaling based redesign of the control law can be performed to achieve global asymptotic stabilization of the system with the input unmodeled dynamics. The proposed control redesign approach is applicable to a wide class of nonlinear systems including triangular and non-triangular structures as long as a set of structural and inequality conditions on the system dynamics is satisfied and yields a global robust output-feedback stabilizing controller.

This technical note considers the problem of input-to-state stability (ISS) for switched nonlinear time-varying systems. First, a sufficient condition is proposed to verify the ISS of nonlinear time-varying systems by using an improved Lyapunov function. Then, the results obtained are extended to study the ISS of switched nonlinear time-varying systems. Three relaxed conditions are given by using the methods of multiple Lyapunov functions, minimum dwell time, and infinite switchings, respectively. Comparing with the existing results, the conditions obtained have two relaxations, i.e., the derivative of Lyapunov functions of subsystems are allowed to be indefinite, and all subsystems are allowed to be unstable in the case of infinite switchings. All results obtained also be used to study the uniformly asymptotic stability (UAS) of systems. Finally, a numerical example is given to illustrate the theoretical results.

The goal of this paper is to study the exponential stabilization problem for autonomous discrete-time switched linear systems (SLSs), where only the discrete mode can be controlled. Our approach is based on periodic control Lyapunov functions whose value decreases periodically instead of at each time step as in the classical control Lyapunov functions. Using periodic control Lyapunov functions, we develop stabilizability analysis and controller synthesis conditions that are less conservative than existing results in that they apply to a larger class of SLSs. Utilizing recent results on the switched optimal control problems, a constructive way to find periodic control Lyapunov functions is presented.

International audience; We present some results on stability of linear time-varying systems with particular structures. Such systems appear in diverse problems , which include the analysis of adaptive systems, persistently-excited observers and consensus of systems interconnected through time-varying links. The originality of our statements rely in the fact that we provide smooth strict Lyapunov functions hence, our proofs are constructive and direct. Moreover, we establish uniform global exponential stability with explicit stability and decay estimates. For illustration we address a brief but representative case-study of consensus of Lagrangian systems interconnected through unreliable links.

In this technical note, the problem of stability for a class of slowly switched systems is investigated. By developing a novel multiple discontinuous Lyapunov function approach and exploring the feature of mode-dependent dwell time switching, new stability conditions are established for systems with a designed switching strategy where fast switching and slow switching are respectively applied to unstable and stable subsystems. In particular, stability conditions for linear switched systems are also given via choosing multiple discontinuous Lyapunov functions in the quadratic form. Moreover, stability criteria for the systems consisting of stable subsystems are also derived. It is shown that our proposed results cover some existing ones in literature as special cases, and provide tighter bounds on the dwell time. Finally, some simulation results are provided to show the advantages of the theoretic results obtained.

In this paper, a new time-dependent discontinuous Lyapunov functional, namely, free-matrix-based time-dependent discontinuous (FMBTDD) Lyapunov functional is introduced for stability analysis of sampled-data systems. First, a modified free-matrix-based integral inequality (MFMBII) is derived based on the existing free-matrix-based integral inequality [1] and it is applied to develop a stability criterion for sampled-data systems. And then, inspired by MFMBII, FMBTDD term is established that leads to efficient stability conditions. Four numerical examples are given to demonstrate the effectiveness of the proposed methods.

In this paper we study the stability of nonlinear and time-varying switched systems under restricted switching. We approach the problem by decomposing the system dynamics into a nominal-like part and a perturbation-like one. Most stability results for perturbed systems are based on the use of strong Lyapunov functions, i.e. functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state. However, switched systems under restricted switching may not admit strong Lyapunov functions, even when asymptotic stability is uniform over the set of switching signals considered. The main contribution of the current paper consists in providing stability results that are based on the stability of the nominal-like part of the system and require only a weak Lyapunov function. These results may have wider applicability than results based on strong Lyapunov functions. The results provided follow two lines. First, we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal-like and the perturbation-like parts of the system. Second, we provide input-to-state stability (ISS) results for the case when the nominal-like part is switched linear-time-varying. We provide two types of ISS results: Standard ISS that involves the essential supremum norm of the input and a modified ISS that involves a power-type norm. Fil: Mancilla Aguilar, Jose Luis. Instituto Tecnológico de Buenos Aires; Argentina Fil: Haimovich, Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentina Fil: Garcia Galiñanes, Rafael Antonio. Instituto Tecnológico de Buenos Aires; Argentina

Multiple rotational translational actuator (RTAC) systems are nonlinear complicated mechatronic systems introduced to investigate the self-synchronized phenomenon in mechanical engineering, which consist of a series of single-RTAC subsystems connected one by one through elastic springs. This paper studies the globally stabilizing control problem for multiple-RTAC systems by considering that the actuators are constrained in torque amplitudes and velocity signals are unavailable for feedback. More precisely, by carefully analyzing the system energy storage function and respecting the maximum allowable control torques, a novel amplitude-restricted control law is developed without requiring velocity feedback, which can stabilize multiple-RTAC systems from any initial condition theoretically. It is beneficial that the proposed control law does not involve any plant parameters, which consequently makes it insensitive to parameter uncertainties. The stability and convergence characteristics of the designed control system are rigorously supported with Lyapunov techniques. To the best of our knowledge, the proposed controller is the first one that can stabilize a multiple-RTAC system while simultaneously ensuring the practical control amplitude-restricted constraints, by using only output feedback. Both numerical simulation and hardware experiments are included to verify the effectiveness of the proposed control approach.

In this paper, a new control law is proposed for the nonlinear underactuated system. It is proved that the proposed control achieves asymptotic stability of the whole system. A broad class of underactuated systems, termed as commensurate underactuated systems, is identified where numbers of actuated and unactuated degrees of freedom are equal; for which the proposed control can be used as the general stabilizing control. The design methodology is simple and uses the prevalent total energy like Lyapunov function. The control strategy is tested on slosh-container system, a benchmark underactuated system of the class considered herein. We discern some important features of the proposed control law and analyze its effectiveness through physical interpretations. The findings are verified through simulations and experimental results for the slosh-container system.

This paper investigates a problem of stabilization control design. A novel triggering mechanism, called parallel-triggering mechanism, is proposed to handle the stabilization of nonlinear/linear systems. Compared with the existing triggering conditions, through fully considering the relationship between the variation of the Lyapunov function and the variation of system states, we propose the parallel-triggering mechanism, without the requirement that the derivative of the Lyapunov function is negative at all-time instants. Under the new parallel-triggering mechanism, the number of executions of control tasks can be significantly reduced and the similar stability performance can be kept. Particularly, the existing event-triggered conditions can be seen as special cases of the proposed parallel-triggering mechanism. Theoretical analysis and simulation results are given to show the advantages of the proposed mechanism in this paper.

Unlike the traditional analysis and synthesis approach of a switched system that requires a monotonic decrease of Lyapunov function (LF), this paper investigates an ${N}$ -step-ahead LF approach. Such an approach allows a non-monotonic behavior both at the switching instants and during the running time of each subsystem but guarantees an average decrease at every ${N}$ sampling steps. The asymptotic stability criterion is improved as well as the capability of disturbance attenuation. By introducing a series of auxiliary variables, the future knowledge of states and exogenous noises can be properly used to derive sufficient conditions for the existence of a robust ${H}_{\infty }$ controller in the form of a set of numerical testable conditions. Note that ${N}$ has direct impact on the number of inequality constraints. The essential difficulty is to construct an exponential damping law of the decreasing points of LF, i.e., to find the joint point between the switching interval and the predictive horizon. Moreover, the relationship between ${N}$ -step time difference of LF and switching rate, i.e., the average dwell time constraint, is thoroughly discussed. An ecology system is employed to demonstrate practical potentials of the presented design framework.

In this technical note, an optimization-based synthesis of a multi-objective reset controller for linear plants is presented. The result allows designing via convex tools with a line-search a multi-objective reset controller optimizing both the exponential converging rate and the ${\mathcal L}_{2}$ gain. A novel peculiarity of our scheme is that the underlying linear flow dynamics is not necessarily stabilizing. This last property is the consequence of taking into account where the flowing trajectories lay and lead to improved performance, especially in terms of decay rate. Some simulations illustrate the usefulness of our techniques.

This technical note deals with (local) asymptotic stabilization of uncertain nonlinear systems by output feedback control. Our study provides a new design technique from a vector control Lyapunov function perspective. A comparison principle is developed and as a major consequence, an effective estimation method of the region of attraction (ROA) is established. Application to a class of minimum-phase strict-feedback nonlinear systems gives rise to a constructive stabilization design as well as an estimate of the resulting ROA.

This paper investigates the problem of the decentralized state feedback controller design for a class of nonlinear large-scale systems described by uncertain affine fuzzy models with unknown interconnections. First, based on the structural information encoded in the fuzzy rules, the affine fuzzy large-scale system is represented by multiple operating-regime-based models. Then, a decentralized state feedback controller is designed, and a cyclic-small-gain condition is introduced to address the unknown interconnections, such that the resulting closed-loop system is asymptotically stable with the disturbance attenuation. By constructing a piecewise Lyapunov function combined with S-procedure and adding slack matrix variables, a convex decentralized controller design condition is derived in the formulation of linear matrix inequalities, where the system matrix is separated from Lyapunov matrix, such that the controller parameterization is independent of the Lyapunov matrix. In contrast with the existing results, the derived stabilizability condition loosens restrictions on the decentralized controller design, and can be applied to handle the nonlinear large-scale systems without satisfying the matching condition. Finally, an example is given to illustrate the effectiveness and superiority of the new results.

In this technical note, implicit iterative algorithms with some tunable parameters are developed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. A significant feature of the proposed algorithms is that the iterative sequences are updated by using not only the information in the last step, but also the information in the current step and the previous steps. Also the convergence rate of the proposed algorithms can be significantly improved by choosing appropriate parameters in the algorithms.

Robust controller design problem is investigated for a class of fractional order nonlinear systems with time varying delays. Firstly, a reduced-order observer is designed. Then, an output feedback controller is designed. Both the designed observer and controller are independent of time delays. By choosing appropriate Lyapunov functions, we prove the designed controller can render the fractional order system asymptotically stable. A simulation example is given to verify the effectiveness of the proposed approach.

Using the Lyapunov function method, this paper investigates the design of state feedback stabilization controllers for fractional order nonlinear systems in triangular form, and presents a number of new results. First, some new properties of Caputo fractional derivative are presented, and a sufficient condition of asymptotical stability for fractional order nonlinear systems is obtained based on the new properties. Then, by introducing appropriate transformations of coordinates, the problem of controller design is converted into the problem of finding some parameters, which can be certainly obtained by solving the Lyapunov equation and relevant matrix inequalities. Finally, based on the Lyapunov function method, state feedback stabilization controllers making the closed-loop system asymptotically stable are explicitly constructed. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.

An integral-type multiple Lyapunov functions (MLFs) approach for switched nonlinear systems is set up for the first time, which gives a more general condition for analyzing the behavior of switched nonlinear systems since the Branicky's nonincreasing condition is no longer assumed and the generalized MLFs condition is a special case of the condition provided. Meanwhile, based on the integral-type MLFs approach, global stabilization for a class of switched nonlinear systems in $p$ - normal form is achieved by constructing state-feedback controllers of subsystems and a proper switching law, where the solvability of the problem under study for individual subsystems is not assumed. Two examples are also provided to demonstrate the effectiveness of the proposed design method.

In this paper, we present adaptive neural network tracking control of a robotic manipulator with input deadzone and output constraint. A barrier Lyapunov function is employed to deal with the output constraints. Adaptive neural networks are used to approximate the deadzone function and the unknown model of the robotic manipulator. Both full state feedback control and output feedback control are considered in this paper. For the output feedback control, the high gain observer is used to estimate unmeasurable states. With the proposed control, the output constraints are not violated, and all the signals of the closed loop system are semi-globally uniformly bounded. The performance of the proposed control is illustrated through simulations.

This paper addresses integral input-to-state stability (iISS) for stochastic nonlinear dynamical systems and develops Lyapunov-type criteria. The developments offer an iISS small-gain framework for verifying robustness of interconnected systems with respect to disturbances. The framework not only includes input-to-state stability (ISS) as a special case, but also highlights fundamental differences between the stochastic and deterministic systems. In particular, dealing with nonlinearities in interconnection gives rise to a unique issue of degradation in the presence of stochastic noises. This paper addresses this issue by coping judiciously with the effect of the gradient of Lyapunov functions of subsystems through diffusion fields in composing a Lyapunov function of the entire system. This paper employs two types of robustness notions, and demonstrates that the influence of the stochastic degradation on one notion is considerably different from the influence on the other notion in dealing with interconnected systems.

This technical note presents stability analysis and stabilization for a general class of switched systems characterized by nonlinear functions. The proposed approach is composed of approximating the switched nonlinear system with a switched affine system that has a mixture of controlled and autonomous switching behavior. Utilizing a joint polyhedral partitioning approach, a stabilizing switching law based on quadratic Lyapunov functions and with considering the autonomous switching between polyhedral regions is proposed. To ensure the decrease of the overall Lyapunov function, two approaches are proposed: 1) guarantee continuity of the Lyapunov function over boundaries of polyhedral regions and 2) relax the continuity requirement by using additional matrix inequalities. The second approach is less conservative but with more variables and matrix inequalities than in the first method. With fixing one scalar variable, the stabilization conditions will have the form of linear matrix inequalities (LMIs). Further, the sufficient conditions for stabilizing the original switched nonlinear system using the proposed switching schemes are presented. Finally, through two examples, the performance of the proposed stabilization methods is demonstrated.

This technical note introduces a novel control law that dynamically modifies the reference of a pre-compensated nonlinear system to ensure the satisfaction of constraints. This is done by translating the state space constraints into an upper bound on the value of the Lyapunov function and by manipulating the velocity of the applied reference to enforce this bound. The main advantage of this approach is that it systematically provides a closed-form control law able to enforce constraints. The theory is introduced for general nonlinear systems subject to constraints and is then specialized to the large and highly relevant class of nonlinear systems admitting a Lyapunov function lower-bounded by a quadratic form and subject to linear constraints. Numerical simulations show the effectiveness of the proposed method.

A new class of Lyapunov functions is proposed for analysis of incremental stability for nonlinear systems. This class of Lyapunov functions allows to establish input-dependent incremental stability criteria. Two substantially different situations are considered: when incremental stability is guaranteed by the inputs of sufficiently small amplitude and when, similar to the excited van der Pol oscillator, the stability is induced by sufficiently large inputs.

While nonlinear scalings of Lyapunov functions are also Lyapunov functions, we provide examples that the same statement does not necessarily hold for Input-to-State Stable (ISS) Lyapunov functions or for integral ISS (iISS) Lyapunov functions. We provide sufficient conditions under which a nonlinear scaling of an ISS or iISS Lyapunov function is also an ISS or iISS Lyapunov function. We also introduce a generalization of the iISS Lyapunov function, which we term a dissipative iISS-Lyapunov function, that allows for a larger class of nonlinear scalings so that nonlinear scalings of dissipative iISS-Lyapunov functions are again dissipative iISS-Lyapunov functions.

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite-time convergence implies nonuniqueness of system solutions in reverse time, such systems possess non-Lipschitzian dynamics. Sufficient conditions for finite-time stability have been developed in the literature using continuous Lyapunov functions. In this technical note, we develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finite-time stability and finite-time stabilization. Finite-time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that satisfies a differential inequality involving fractional powers. This Lyapunov function can clearly be seen to be the solution to a partial differential equation that corresponds to a steady-state form of the Hamilton-Jacobi-Bellman equation, and hence, guaranteeing both finite-time stability and optimality.

This paper studies the tracking control problem for an uncertain ${n}$ -link robot with full-state constraints. The rigid robotic manipulator is described as a multiinput and multioutput system. Adaptive neural network (NN) control for the robotic system with full-state constraints is designed. In the control design, the adaptive NNs are adopted to handle system uncertainties and disturbances. The Moore–Penrose inverse term is employed in order to prevent the violation of the full-state constraints. A barrier Lyapunov function is used to guarantee the uniform ultimate boundedness of the closed-loop system. The control performance of the closed-loop system is guaranteed by appropriately choosing the design parameters. Simulation studies are performed to illustrate the effectiveness of the proposed control.

In this technical note, the stability and failure tolerant control are investigated for linear time-invariant (LTI) control systems with occasional controller failures. At first, a collection of non-convex sufficient conditions ensuring the closed loop to be globally asymptotically stable is presented, which are simple to use but difficult to consider for control purpose. To overcome this, a time-scheduled Lyapunov function approach is considered and alternative convex conditions are obtained. Based on this result, a time-dependent failure-tolerant controller is designed. The approach is finally illustrated on a networked control system.

This paper addresses the problem of multi-agent coordination and control under multiple objectives, and presents a set-theoretic formulation amenable to Lyapunov-based analysis and control design. A novel class of Lyapunov-like barrier functions is introduced and used to encode multiple, non-trivial control objectives, such as collision avoidance, proximity maintenance and convergence to desired destinations. The construction is based on recentered barrier functions and on maximum approximation functions. Thus, a single Lyapunov-like function is used to encode the constrained set of each agent, yielding simple, gradient-based control solutions. The derived control strategies are distributed, i.e., based on information locally available to each agent, which is dictated by sensing and communication limitations. Furthermore, the proposed coordination protocol dictates semi-cooperative conflict resolution among agents, which can be also thought as prioritization, as well as conflict resolution with respect to an agent (the leader) which is not actively participating in collision avoidance, except when necessary. The considered scenario is pertinent to surveillance tasks and involves nonholonomic vehicles. The efficacy of the approach is demonstrated through simulation results.

Analysis of transient stability of strongly nonlinear post-fault dynamics is one of the most computationally challenging parts of Dynamic Security Assessment. This paper proposes a novel approach for assessment of transient stability of the system. The approach generalizes the idea of energy methods, and extends the concept of energy function to a more general Lyapunov Functions Family (LFF) constructed via Semi-Definite-Programming techniques. Unlike the traditional energy function and its variations, the constructed Lyapunov functions are proven to be decreasing only in a finite neighborhood of the equilibrium point. However, we show that they can still certify stability of a broader set of initial conditions in comparison to the traditional energy function in the closest-UEP method. Moreover, the certificates of stability can be constructed via a sequence of convex optimization problems that are tractable even for large scale systems. We also propose specific algorithms for adaptation of the Lyapunov functions to specific initial conditions and demonstrate the effectiveness of the approach on a number of IEEE test cases.

This paper presents a novel fuzzy adaptive controller for controlling a class of dead-zone output nonlinear systems with time delays. A new approximate model is first designed to describe a special dead-zone phenomenon encountered by the output mechanism of nonlinear systems, and the proposed smooth model can be conveniently fused with available adaptive fuzzy control techniques. In addition, the coupling effect that the dead-zone output and the time-delayed states coexist in a common coupling function makes the tracking control design more complicated. To further address this difficulty, a compensation method using mean-value theorem with Lyapunov–Krasovskii function is presented in this paper. By using the proposed output dead-zone model, and based on Lyapunov synthesis, a new optimized algorithm is developed to guarantee the prescribed convergence of tracking error and the boundedness of all the signals in the closed-loop systems. Simulations have been implemented to verify the performance of the proposed fuzzy adaptive controller.

For continuous-time nonlinear systems with inputs, we introduce the notion of an asymptotic ratio input-to-state stability (ISS) Lyapunov function. The derivative of such a function along solutions is upper-bounded by the difference of two terms whose ratio is asymptotically smaller than 1 for large states. This asymptotic ratio condition is sometimes more convenient to check than standard ISS Lyapunov function conditions. We show that the existence of an asymptotic ratio ISS Lyapunov function is equivalent to ISS. A related notion of ISS with nonuniform convergence rate is also explored.

The modular multilevel converter (MMC) has become one of the most promising candidates for medium-/high-power applications, specifically for high-voltage direct current transmission systems. This paper proposes a control strategy for the MMC based on the sum-of-squares decomposition method. A bilinear mathematical discrete model of the MMC is derived in the rotating reference frame by including the dynamics of ac-side currents, circulating currents, dc-side current, and stored energy of the converter. The coordinates of the developed model are transformed to the origin, and the steady-state operating point of the MMC is calculated based on the desired reference values of the states. A quadratic Lyapunov function is defined and by converting the Lyapunov difference inequality to an SOS problem, a stabilizing controller is designed. The designed controller not only guarantees the Lyapunov stability of the states, but also controls the external and internal dynamics of the MMC with only one controller, instead of using several proportional-integral controllers. To improve the region of stability of the MMC, an optimization problem is formulated and solved, so the controller can stabilize the MMC from an initial operating point far from the steady-state operating point. The performance of the proposed strategy for a 20-level MMC is evaluated based on the time-domain simulation studies and is compared with the existing control methods.

In this paper, two event-based robust sampled-data model predictive control (MPC) strategies are proposed based on the non-monotonic Lyapunov function approach for continuous-time systems with disturbances. Each event-triggering mechanism consists of the event-based MPC law and the triggering conditions. We show that although the proposed event-triggering conditions are only checked at the sampling instants and the control law is piecewise constant, the feasibility of the event-based sampled-data MPC algorithm and the stability of the closed-loop system are guaranteed in continuous time. Besides, the implementation issue is discussed, and we show that the proposed triggering conditions can be checked rapidly without obviously increasing the computational burden. Finally, an application to a nonholonomic robot system is provided to illustrate the effectiveness of the proposed results.

This paper considers the second-order sliding (2-sliding) mode control design problem for nonlinear systems with uncertainties bounded by positive functions. Two different 2-sliding mode control algorithms are proposed, including a discontinuous one and a quasi-continuous one. The discontinuous algorithm is built based upon the Lyapunov method, and the global finite-time Lyapunov stability rather than the global finite-time convergence is established for the resulting closed-loop system by using the modified adding a power integrator technique. On this basis, a quasi-continuous 2-sliding mode algorithm is further developed, and the chattering problem can be considerably reduced. Simulation results show that the control strategies proposed in this paper are effective.

In this paper, we present the vibration control design for a Euler–Bernoulli beam with the boundary output constraint. To prevent the constraint violation, a novel barrier Lyapunov function is employed for the control design and stability analysis. This paper represents an important step in extending the barrier Lyapunov function theory to distributed parameter systems. Model-based barrier control is proposed to suppress the vibration of a flexible Euler–Bernoulli beam under the boundary output constraint. Then, adaptive control is designed to handle the system parametric uncertainties. The vibration suppression is well achieved without violation of the constraint. Numerical simulations are provided to illustrate the performance of the control system.

In this paper, boundary control laws are developed to stabilize the transverse vibration for a nonlinear vertically moving string system. The control system is considered with varying length, varying speed, and the constrained boundary output. Based on the integral-barrier Lyapunov function, the exponential stability is proved with the proposed control without consideration of the disturbance. When the external boundary disturbance is taken into account, the disturbance observer is designed to eliminate its effect. The vibration is regulated and the boundary output always remains in the constrained space by appropriately choosing the control parameters. The control design and the stability analysis are based on the original infinite-dimensional dynamic equations. Extensive numerical examples illustrate the performance of the control system.

This work is concerned with the design and effects of the synchronization gains on the synchronization problem for a class of networked distributed parameter systems. The networked systems, assumed to be described by the same evolution equation in a Hilbert space, differ in their initial conditions. To address the synchronization problem, a coupling term containing the pairwise state differences of all the networked systems weighted by the synchronization gains is included in the controller of each networked system. By considering the aggregate closed-loop systems, an optimization scheme for the synchronization gains is proposed by minimizing an appropriate measure of synchronization. The integrated control and synchronization design is subsequently cast as an optimal control problem, the solution of which is found via the solution of parameterized operator Lyapunov equations. An alternative to the optimization of the synchronization gains is also proposed in which the adaptation of synchronization gains is derived from Lyapunov redesign methods. Both choices of the proposed synchronization controllers aim at achieving both the control and the synchronization objectives. An extensive numerical study examines the various aspects of the optimization and adaptation of the gains on the control and synchronization of networked 1D parabolic differential equations.