Lyapunov-like Function Research Articles

Global Leader-Following Consensus of Double-Integrator Multiagent Systems by Fully Distributed Bounded Linear Protocols

This article is concerned with the global leader-following consensus problem of the input-constrained double-integrator multiagent systems. By utilizing a Lyapunov-like function consisting of a positive semidefinite term and an integral term, a class of bounded static linear protocols are proposed to ensure that the global leader-following consensus problem for homogeneous multiagent systems is solved in a fully distributed manner for all undirected communication graphs. As a further result, the global consensus problem for heterogeneous multiagent systems under directed communication graphs is also considered and two different distributed observer-based bounded linear protocols are proposed to solve such a problem also in a fully distributed manner. The most significant advantages of this article are that the system under consideration is more general, the designed protocols are linear, and the global consensus in a fully distributed manner can be guaranteed. A numerical example shows the effectiveness of the proposed approaches.

Multi-UAV cooperative target encirclement within an annular virtual tube

Continuously encircling and monitoring an individual static target is an important application scenario for a multi-UAV system. In this paper, to address the problem of multi-UAV encircling the target arbitrarily, evenly and densely in 3D space, a distributed vector field control method with an annular virtual tube is proposed. Some Lyapunov-like functions are designed elaborately, and a formal analysis based on the invariant set theorem is made to show that all UAVs can complete the encirclement task successfully. Simulations and experiments are carried out to show the effectiveness of the proposed method.

Practical Control for Multicopters to Avoid Non-Cooperative Moving Obstacles

Unmanned Aerial Vehicles (UAVs) are now becoming increasingly accessible to amateur and commercial users alike. The main task for UAVs is to keep a prescribed separation with obstacles in the air. In this paper, a collision-avoidance control method for non-cooperative moving obstacles is proposed for a multicopter with the altitude hold mode by using a Lyapunov-like barrier function. Lyapunov-like functions are designed elaborately, based on which formal analysis and proofs of the proposed control are made to show that the collision-avoidance control problem can be solved if the moving obstacle is slower than the multicopter. The result can be extended to some cases of multiple obstacles. What is more, by the proposed control, a multicopter can keep away from obstacles as soon as possible, once obstacles enter into the safety area of the multicopter accidentally, and converge to the waypoint. Simulations and experiments are given to show the effectiveness of the proposed method by showing the distance between UAV and waypoint, obstacles respectively.

Quantized Interval Type-2 Fuzzy Control for Persistent Dwell-Time Switched Nonlinear Systems With Singular Perturbations.

This article investigates the problem of quantized fuzzy control for discrete-time switched nonlinear singularly perturbed systems, where the singularly perturbed parameter (SPP) is employed to represent the degree of separation between the fast and slow states. Taking a full account of features in such switched nonlinear systems, the persistent dwell-time switching rule, the technique of singular perturbation and the interval type-2 Takagi-Sugeno fuzzy model are introduced. Then, by means of constructing SPP-dependent multiple Lyapunov-like functions, some sufficient conditions with the ability to ensure the stability and an expected H∞ performance of the closed-loop system are deduced. Afterward, through solving a convex optimization problem, the gains of the controller are obtained. Finally, the correctness of the proposed method and the effectiveness of the designed controller are demonstrated by an explained example.

Practical Distributed Control for VTOL UAVs to Pass a Virtual Tube

Unmanned Aerial Vehicles (UAVs) are now becoming increasingly accessible to amateur and commercial users alike. An air traffic management (ATM) system is needed to help ensure that this newest entrant into the skies does not collide with others. In an ATM, airspace can be composed of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">airways, intersections</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nodes.</i> In this paper, for simplicity, distributed coordinating the motions of Vertical TakeOff and Landing (VTOL) UAVs to pass an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">airway</i> is focused. This is formulated as a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tube passing problem</i> , which includes passing a virtual tube, inter-agent collision avoidance and keeping within the virtual tube. Lyapunov-like functions are designed elaborately, and formal analysis based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">invariant set theorem</i> is made to show that all UAVs can pass the virtual tube without getting trapped, avoid collision and keep within the virtual tube. What is more, by the proposed distributed control, a VTOL UAV can keep away from another VTOL UAV or return back to the virtual tube as soon as possible, once it enters into the safety area of another or has a collision with the virtual tube during it is passing the virtual tube. Simulations and experiments are carried out to show the effectiveness of the proposed method and the comparison with other methods.

Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay

The work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regularization method utilizes the parallelism between Lyapunov stability and input-to-state stability, which is well established in the deterministic case, but less known for stochastic differential equations. In its practical implementation, the method is based on seeking an auxiliary equation, which is used to regularize the equation to be studied. In the final step, estimation of the norm of an integral operator or verification of the property of positivity of solutions is performed. In the latter case, one applies the theory of positive invertible matrices. This report contains a systematic presentation of how the regularization method can be applied to stability analysis of linear stochastic delay equations with random coefficients and random initial conditions. Several stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained. A number of verifiable sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Itô equations are offered as well.

Semi-time-dependent stabilization for a class of continuous-time impulsive switched linear systems

In this paper, the stability and stabilization problems of a class of impulsive switched linear systems with mode-dependent persistent dwell-time (MPDT) switching are investigated in continuous-time domain. The proposed switching law is more general, which not only covers the widely probed dwell-time (DT) and average dwell-time (ADT) as special cases but also extends the persistent dwell time (PDT) in the literature. By invoking novel modal partition time-dependent multiple Lyapunov-like functions, the stability criteria for the systems under MPDT switching in nonlinear settings are established as a first attempt, upon which the desired mode-dependent and semi-time-dependent (STD) stabilizing controllers for continuous-time impulsive switched linear systems are designed. It is demonstrated that the control performance is improved as both the length of admissible MPDT and the number of time partition segment are added. Finally, a numerical example is provided to illustrate the potential of the obtained theoretical results.

Design of multi-module adaptive fuzzy power tracking control for nonlinear four-point reactor core model under multiple transient conditions

In this work, a multi-module adaptive fuzzy controller is designed for multipoint reactor core model of VVER-1000 to minimize the axial power variations among the corresponding nodes under efficient load tracking abilities and disturbance rejection. The validated four-point reactor model is syndicated using diffusion strategy and consideration of thermal hydraulics, reactivity feedbacks, poison concentration and three groups of delayed neutron precursors. The control approach followed here is parametric adaptive control which uses the online identification of a process model as its performance monitor. The adaptive design makes adjustments among the various modules according to the changing dynamic properties of the nonlinear reactor process. The multi-module structure is based on the absolute error level and develops online switching of control modes with anticipation of optimized gains for all operating conditions. The selection of fuzzy parameters regarding severity level of control output is based on desired performance measures and actuator design capacity limitations under multiple transient conditions. A Lyapunov-like function is established for the stability criteria of the four-point reactor control model. Simulation results show the effectual and robust load following power control of model under large load rejection, parametric perturbations and external reactivity variations with axial power deviations as low as 0.004% in comparison with PID configured controller.

Hybrid variables-dependent event-triggered model predictive control subject to polytopic uncertainties

This paper focuses on the model predictive control (MPC) issue where the measurable state is released under an event-triggered mechanism (ETM) to implement MPC over an infinite horizon. A new dynamic ETM (DETM) is devised to conserve network resources, which contains an additive internal dynamic variable (IDV), a multiplicative adaptively adjusting variable, a time-varying weighting matrix and several flexible scalars. The MPC problem is formulated as a ‘min–max’ optimisation problem (OP), where a hard constraint and robust positive invariant set on the predictive state/IDV are considered simultaneously. By resorting to a Lyapunov-like function that depends on the IDV, we put forward an auxiliary OP with matrix-inequality-based constraints. By the feasible solutions of such an auxiliary OP, the feedback gain matrix is designed which ensures the asymptotic stability of the closed-loop system. Two examples are presented to demonstrate the validity of the devised DETM and the DETM-based MPC algorithm. The study verifies that the devised DETM has advantages over an existing counterpart in conserving network resources while achieving the desired performance.

Stochastic input-to-state stability for nonlinear impulsive switched systems with random impulses coupled by multiple jump maps

This article focuses on a class of nonlinear random impulsive and switched systems for stochastic input-to-state stability. The impulse is divided into two types: single jump map and multiple jump maps. When the impulse is chosen as single jump map, it means that the impulse between the two adjacent switching times is single as well as impulse and switching occur simultaneously, in which the impulsive moments are fixed. When the impulse is chosen as multiple jump maps, in this model, the impulse is not only limited by multiple maps, but also the impulse instants are assumed to be random. Therefore, this model is an extension of the traditional impulsive and switched system. In view of admissible edge-dependent average dwell time switching signal and the impulse dynamic satisfying Poisson’s process, sufficient conditions for the stochastic input-to-state stability are introduced into nonlinear impulsive switched system with random impulses. Three significant distinctions from the existing results rest with that (1) a new multiple Lyapunov functions are first created; the novelty contains discontinuity at the discrete points including switching moments and the impulse jump instants of Lyapunov-like functional; (2) the coefficient of derivative of a Lyapunov-like functions is permitted to be either positive or negative, which is less conservative; (3) the adopted admissible edge-dependent average dwell time switching scheme delves into less conservative stochastic input-to-state stability condition than mode-dependent average dwell time switching, nonlinear impulsive switched system with various jump maps. Numerical examples are simulated to manifest the validity of the main results.

A finite‐time adaptive Taylor series tracking control of electrically‐driven wheeled mobile robots

This research seeks to address a new integrated kinematic/dynamic adaptive Taylor series-based control design for the robust tracking of electrically-driven differential drive wheeled mobile robots (WMRs). This control design includes two loops, namely the outer loop (a kinematic control law) and the inner loop (a dynamic controller). Being capable of compensating for far initial conditions from a desired trajectory, a new kinematic control law is designed to make the posture tracking error converge to zero asymptotically as well as to generate a desired trajectory for a dynamic controller. The key role of the dynamic controller is to compensate for lumped uncertainties. To do this, the proposed chattering-free dynamic controller guarantees that the defined sliding surface which is a function of tracking error and its time derivative will be converged to zero within a finite time. The exact stability analysis of inner closed-loop system is developed via two Lyapunov-like positive definite functions to ensure not only the boundedness of all signals but also the finite-time convergence of sliding surface to zero. The proposed control algorithm is validated by means of various simulations, including comparisons with well-designed kinematic and integrated kinematic/dynamic control literature.

Approximately bisimilar symbolic model for switched systems with unstable subsystems

This paper deals with approximate bisimulation for the switched nonlinear system with mode-dependent dwell time. A criterion for incremental stability is presented for this switched nonlinear system by constructing incremental Lyapunov-like functions. Then for the case that all the subsystems are linear, a more solvable criterion is provided in terms of linear matrix inequalities. A symbolic model which is approximately bisimilar to the original switched nonlinear system is developed by using the grid-based approach, and the bisimilar precision is also given. Numerical examples are provided to show the application of the proposed results.

Adaptive modeling of nonnegative environmental systems based on projectional Differential Neural Networks observer

A new design of a non-parametric adaptive approximate model based on Differential Neural Networks (DNNs) applied for a class of non-negative environmental systems with an uncertain mathematical model is the primary outcome of this study. The approximate model uses an extended state formulation that gathers the dynamics of the DNN and a state projector (pDNN). Implementing a non-differentiable projection operator ensures the positiveness of the identifier states. The extended form allows producing continuous dynamics for the projected model. The design of the learning laws for the weight adjustment of the continuous projected DNN considered the application of a controlled Lyapunov-like function. The stability analysis based on the proposed Lyapunov-like function leads to the characterization of the ultimate boundedness property for the identification error. Applying the Attractive Ellipsoid Method (AEM) yields to analyze the convergence quality of the designed approximate model. The solution to the specific optimization problem using the AEM with matrix inequalities constraints allows us to find the parameters of the considered DNN that minimizes the ultimate bound. The evaluation of two numerical examples confirmed the ability of the proposed pDNN to approximate the positive model in the presence of bounded noises and perturbations in the measured data. The first example corresponds to a catalytic ozonation system that can be used to decompose toxic and recalcitrant contaminants. The second one describes the bacteria growth in aerobic batch regime biodegrading simple organic matter mixture.

Logic-based distributed switching control for agents in power-chained form with multiple unknown control directions

This work studies logic-based distributed switching control for nonlinear agents in power-chained form, where logic-based (switching) control arises from the online estimation of the control directions assumed to be unknown for all agents. Compared to the state-of-the-art logic-based mechanisms, the challenge of power-chained dynamics is that in general asymptotic tracking cannot be obtained, even for a single agent. To address this challenge, a new logic-based mechanism is proposed, which is orchestrated by a dynamic boundary function. The boundary function is decreasing in-between switching instants and monotonically increasing at the switching instants, depending on the jumps of an appropriately designed Lyapunov-like function. To remove chattering (i.e. two or more switching instants occurring consecutively with zero dwell time), a dynamic threshold is proposed, based on selecting the maximum values of the Lyapunov-like function before and after switching.

Existence theory and generalized Mittag-Leffler stability for a nonlinear Caputo-Hadamard FIVP via the Lyapunov method

&lt;abstract&gt;&lt;p&gt;This paper discusses the existence, uniqueness and stability of solutions for a nonlinear fractional differential system consisting of a nonlinear Caputo-Hadamard fractional initial value problem (FIVP). By using some properties of the modified Laplace transform, we derive an equivalent Hadamard integral equation with respect to one-parametric and two-parametric Mittag-Leffer functions. The Banach contraction principle is used to give the existence of the corresponding solution and its uniqueness. Then, based on a Lyapunov-like function and a $ \mathcal{K} $-class function, the generalized Mittag-Leffler stability is discussed to solve a nonlinear Caputo-Hadamard FIVP. The findings are validated by giving an example.&lt;/p&gt;&lt;/abstract&gt;

Practical stability for fractional impulsive control systems with noninstantaneous impulses on networks

This paper investigates practical stability for a class of fractional-order impulsive control coupled systems with noninstantaneous impulses on networks. Using graph theory and Lyapunov method, new criteria for practical stability, uniform practical stability as well as practical asymptotic stability are established. In this paper, we extend graph theory to fractional-order system via piecewise Lyapunov-like functions in each vertex system to construct global Lyapunov-like functions. Our results are generalization of some known results of practical stability in the literature and provide a new method of impulsive control law for impulsive control systems with noninstantaneous impulses. Examples are given to illustrate the effectiveness of our results

Smooth Output-to-State Stability for multistable systems on compact manifolds

Output-to-State Stability (OSS) is a notion of detectability for nonlinear systems that is formulated in the ISS framework. We generalize the notion of OSS for systems which possess a decomposable invariant set and evolve on compact manifolds. Building upon a recent extension of the ISS theory for this very class of [systems [D. Angeli and D. Efimov, IEEE Trans. Autom. Control 60 (2015) 3242–3256.], the paper provides equivalent characterizations of the OSS property in terms of asymptotic estimates of the state trajectories and, in particular, in terms of existence of smooth Lyapunov-like functions.

Practical prescribed-time sampled-data control of triangular nonlinear systems satisfying a linear growth condition by linear time-varying feedback

In this paper, the practical prescribed-time sampled-data control problem of a class of triangular nonlinear systems satisfying a linear growth condition is studied. With the help of some properties of a class of parametric Lyapunov equations (PLEs) and by constructing a time-varying Lyapunov-like function, a linear time-varying feedback controller is designed to make the states of the closed-loop system converge to an arbitrarily small neighbourhood of the origin in any prescribed time, while the control siginals are guaranteed to be bounded. A numerical example validates the effectiveness of the proposed method.

On Stochastic ISS of Time-Varying Switched Systems With Generic Lévy Switching Signals

Switched systems in which switching among subsystems occurs at random time instants find numerous applications in engineering. Stability analysis of such systems, however, is quite challenging. This letter investigates the stochastic input-to-state stability of this class of switched systems. The random switching instants are modeled by a non-decreasing, positive, and real-valued L&#x00E9;vy process, which, at every time instant, selects the active subsystem from a family of deterministic systems. No assumption on the stability of subsystems is presumed; they can be stable or unstable. Stochastic properties of the switching signal are coupled with a family of Lyapunov-like functions to obtain a sufficient condition for stochastic input-to-state stability.

Hybrid Control of Flowrate in Microextrusion-Based Direct-Write Additive Manufacturing

Flowrate control has been a continuous challenge in Direct-Write Additive Manufacturing (DW AM) due to capacitive energy storage in the system and the lack of suitable flowrate sensors at the microscale, resulting in poor dimensional control of material addition. While a pressure sensor can be used for pressure-based feedback control, the system will become marginally stable if the material loses contact with the pressure sensor, and thus feedback is lost. In this letter, we design and implement a pressure-based hybrid controller with a state-dependent switching strategy that eliminates the stability problem due to a loss of pressure feedback, and thus enables the real time, stable control of flowrate in DW AM. The stability of the hybrid controller is assessed using phase portraits and Multiple Lyapunov-like Functions. The hybrid dynamical model and the hybrid controller are experimentally implemented and validated. Stability results indicate that the hybrid controller resolved the marginal stability issue, and is stable in the sense of Lyapunov. Moreover, the hybrid control scheme implementation in the case study demonstrated that the shape fidelity of the parts is appreciably enhanced compared to the open loop control case.