Lyapunov Inequality Research Articles

New stability theory of model predictive control: modified stage cost approach

This paper presents a promising new approach to establish the stability of finite receding horizon control with a terminal cost. Departing from the traditional approaches of using the property of the terminal cost or relaxed Lyapunov inequalities, this paper establishes its stability based on the property of a modified stage cost. First, we rotate the stage cost with the terminal cost. Then a one-step optimisation problem is defined based on this augmented stage cost. It is shown that a slightly modified Model Predictive Control (MPC) algorithm is stable if the value function of the augmented one-step cost (OSVF) is a Control Lyapunov Function (CLF). Stability for MPC algorithms with zero terminal cost or even negative terminal cost can be unified with this new approach. Combining it with the existing MPC stability theories, we are able to significantly relax the stability requirement on MPC and extend the stabilising MPC design space to the region that no existing MPC stability theories can cover. The proposed stage cost-based approach will help to further reduce the gap between stability theory and practical applications of MPC and other optimisation-based control methods.

Finite-time hybrid impulsive formation tracking control of multi-agent systems via aperiodic intermittent communication

This article studies the problem of formation tracking control in multi-agent systems, achieved in finite time, under challenging conditions such as strong nonlinearity, aperiodic intermittent communication, and time-delay effects, all within a hybrid impulsive framework. The impulses are categorized as either stabilizing control impulses or disruptive impulses. Furthermore, by integrating Lyapunov-based stability theory, graph theory, and the linear matrix inequality (LMI) method, new stability criteria are established. These criteria ensure finite-time intermittent formation tracking while considering weak Lyapunov inequality conditions, intermittent communication rates, and time-varying gain strengths. Additionally, the approach manages an indefinite number of impulsive moments and adjusts the control domain’s width based on the average impulsive interval and state-dependent control width. Numerical simulations are provided to validate the applicability and effectiveness of the proposed formation tracking control protocols.

Observer-based quantized control for networked interconnected PDE systems with actuator failures

This paper proposes an observer-based quantized controller for parabolic partial differential equation systems interconnected by a nonlinear coupling protocol. First, a Markov jump model is introduced to describe various randomly occurring actuator failures, and an observer-based pointwise controller is designed under the averaged measurement scheme. Furthermore, taking into account the limitation of network communication resources, a quantization method is adopted to relieve bandwidth pressure. In addition, stability conditions of the closed-loop system with ℋ∞ disturbance attenuation performance are derived by utilizing appropriate Lyapunov functional and inequality techniques. Ultimately, the proposed method is applied to the Fisher equation to verify its feasibility and effectiveness.

New synchronization results for fractional neural networks with parameter uncertainty and reaction–diffusion terms

New synchronizations of fractional neural networks (FNNs) with uncertain parameters and reaction diffusion terms are investigated in this paper. Firstly, considering a system with mixed boundary conditions, a mixed boundary controller is designed. Sufficient criteria for the global exponential synchronization of the designed FNNs are given through Lyapunov function, inequality techniques and Razumikhin theorem. Furthermore, the exponential synchronization of the system can be explored under Neumann boundary conditions. An example is provided to verify the theoretical part and present the synchronization results for different fractional orders and diffusion coefficients.

Bipartite quasi synchronization of stochastic delayed cooperation–competition networks with gain-waving intermittent control

Given the cooperation and competition bi-directional mechanisms between subsystems, this paper aims at addressing bipartite quasi synchronization (BQS) of stochastic delayed cooperation–competition networks (SDCNs). We propose a gain-waving intermittent control policy, that includes more of the existing works, to assist the BQS, in an effort to cope with external interferences and the possibility of actuator damage caused by rapid switching. Intermittent gain is to be quantized legitimately via the postulate of waving average gain. Within this context, novel Lyapunov inequalities concerning bounded waving gain and unbounded one are advanced to ensure the BQS of SDCNs in conjunction with graph theory, since the present differential inequalities are incapable of being inoperative. Then we develop two kinds of BQS criteria and the bounds for the BQS error are estimated where the scenarios of bounded waving gain and unbounded one are involved separately. Besides, corresponding criteria about complete bipartite synchronization are presented and several design methods of gain-waving intermittent control are introduced by specific examples. Finally, the validity of the theoretical results is examined by an application on Chua’s circuit with numerical simulations being shown.

Mean-square exponential stabilization of memristive neural networks: Dealing with replay attacks and communication interruptions

This paper investigates the mean-square exponential stabilization (MSES) of memristive neural networks (MNNs) under replay attacks and communication interruptions. The research will revolve around the following two questions: Firstly, facing replay attacks and communication interruptions, how to design an appropriate controller? Secondly, how to ensure the MSES of MNNs under higher replay attack rate and communication interruption rate? To address these challenges, a novel intermittent sampled-data control scheme is established by considering the mathematical characteristics of replay attacks and communication interruptions. Furthermore, interval-dependent Lyapunov functions are constructed based on the Lyapunov stability theory and inequalities techniques, and two sufficient criteria for MSES of MNNs under the mentioned risks are derived. Thus, the control gain is obtained by solving a series of linear matrix inequalities. In addition, two algorithms are designed to investigate the maximum allowable replay attack rate and communication interruption rate for the system, respectively. Finally, two numerical examples are given to verify the effectiveness of the proposed scheme.

Lowering carbon emissions from a zinc oxide rotary kiln using event-scheduling observer-based economic model predictive controller

Background:The leading carbon emitting unit of the zinc smelting industry is the zinc oxide rotary kiln, which preserves a trade-off between carbothermic reduction and the zinc recovery rate. To scale up a substantial zinc restoration rate during the hydrometallurgical process, a substantial amount of coke is combusted, which leads to significant greenhouse gas emissions. It is imperative to mitigate the emission of carbon as reduced as possible. Since the kiln operates in three different regions — gas, solid, and wall — several highly non-uniform paths are typically linked to one another to yield complex looping processes. In industrial settings, solid maldistribution and gas refluxing pose prevalent challenges in the gas–solid flow mechanism. Consequently, control over processes and observation of the internal states/concomitant parameters become critical for ensuring safe and reliable operation. Therefore, it is imperative to devise effective control layouts to optimize the functioning of the kiln system. Method:(i) The usual form of model-predictive control rule is paired with an economic stage cost (using the CasADi optimization method) to offer optimal consumption of the coal and the rotation speed, thereby effectively alleviating the adverse effects of the unmeasured perturbations, actuator faults, or multi-sensor failures. (ii) With the aid of accessible state sensors, a cubature Kalman filter infused with a singular-value-decomposition strategy reliably predicts uncertainties/faults of the actuators or the unmeasured states of the kiln. (iii) cubature Kalman filter and event-triggered scheduling are utilized together, to consume less communication resources. Significant findings:A variety of indicators, such as zinc recovery rate (1.35% improvement), carbon emissivity (11.54% improvement), residence time (6% shorter), accuracy of the controllers/observers were well assessed and compared with standard form of the non-linear model predictive control law to figure out the efficiency and acceptability of the economic model predictive controller, demonstrating on an industrial rotary kiln. An in-depth study of the proposed control method satisfying Lyapunov inequality has been addressed.

On the hyper-Lyapunov matrix inclusions

By adding a scalar parameter to the classical Lyapunov matrix inequality, the underlying structure turns to be richer: Partial order of stability sets is introduced. This is then used to improve estimates of trajectories associated with differential inclusions.Technically, the spectrum of all matrices, satisfying a given Lyapunov inequality, lies within a special disk in the right-half plane: Under inversion such a disk is mapped onto itself.As a by-product, it is shown that these disks are a natural tool to understanding the Matrix-Sign-Function iteration scheme, used in matrix computations.Hyper-Lyapunov inclusions are formulated through Matrix-Quadratic-Form Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within sub-unit disks.

Tensor Product Alternatives for Nonlinear Field-Oriented Control of Induction Machines

The paper presents a nonlinear field-oriented control technique based on the tensor product representation of the nonlinear induction machine model and the solvability of linear matrix inequalities. The nonlinear model has 32 quasi linear parameter-varying equivalent variants, and it is shown that only half of the models result in feasible controller. Two control goals are realized: torque control and speed control. The controller is a nonlinear state feedback controller completed by integral action. A new block diagram is investigated for speed control. The controller gains are designed by the solution of linear matrix inequalities to solve the Lyapunov inequality to obtain a stable and fast response and constraints on the control signal. The presented methods are verified and compared by simulations.

Event-triggered passivity and synchronization of multiple derivative coupled reaction–diffusion neural networks

The paper studies the event-triggered passivity and synchronization problems for multiple derivative coupled reaction–diffusion neural networks. Two types of multiple derivative coupled reaction–diffusion neural networks, with and without time-varying delay, are considered. According to the characteristics of multiple derivative coupled reaction–diffusion neural networks, an event-triggered condition is constructed. By imposing appropriate Lyapunov functionals and inequality techniques, several passivity conditions are established for these two networks under the designed event-triggered controllers. Then, synchronization criteria of these two networks are derived based on the relationship between output-strict passivity and synchronization. Moreover, the non-occurrence of the Zeno behavior in the proposed control is proved. Numerical example is illustrated to verify the effectiveness of the proposed criteria. Finally, theoretical results and the associated image encryption application are validated through numerical simulations and statistical analyses.

Exponential Synchronization of Complex Dynamic Networks with Time Delay and Uncertainty via Adaptive Event-Triggered Control

In this paper, exponential synchronization problem of uncertain complex dynamic networks with time delay is studied via adaptive event-triggered control. Considering the influence of external environment, a new dynamic event-triggered mechanism is proposed, in order to reduce the transmission signal among nodes and reduce the consumption of communication resources. Moreover, in the proposed control mechanism, the controller is adaptive, that is, it only works when the triggering conditions are satisfied. Then, according to the designed adaptive event-triggered control strategy, the sufficient conditions for exponential synchronization are obtained by using Lyapunov functions and inequality technique. In addition, it is proved that the system can avoid Zeno behavior. At last, using two examples to verify the feasibility of the results.

Exponential Output Synchronization of Complex Networks With Output Coupling via Intermittent Event-Triggered Control

This paper considers the exponential output synchronization of complex networks with output coupling (CNOC) based on the intermittent event-triggered control (IE-TC) strategy. Unlike existing intermittent control strategies, the proposed IE-TC strategy is event-triggered instead of time-triggered during the control intervals. Moreover, the average control rate is adopted for intermittent control, which makes the results less conservative. A new Lyapunov function is proposed to simplify the proof, which does not require the traditional mathematical induction method for intermittent control. Based on the Lyapunov method and inequality technique, sufficient conditions for achieving exponential output synchronization of CNOC for undirected graph and digraph topologies are given, respectively. Moreover, the Zeno phenomenon can be excluded. Finally, an example with simulations is provided to verify the feasibility of the theoretical results.

LQR controller design for affine LPV systems using reinforcement learning

In this paper, a data-driven sub-optimal state feedback is designed for a continuous time linear parameter varying (LPV) system using reinforcement learning. Time-varying parameters lie in a poly top and the system matrix has an affine representation for the parameters. Two novels, on-policy and off-policy algorithms, are proposed using available data from vertex systems of polytop to minimise a performance index and admit a common Lyapunov function (CLF). A convex optimisation problem is derived for each iteration based on Lyapunov inequality. Algorithms yield stabilising feedback gain in each iteration and convergence to a common lyapunov function. We demonstrate the efficacy of the proposed method by simulation of two case studies.

Prescribed-time cluster practical consensus for nonlinear multi-agent systems based on event-triggered mechanism.

This paper investigates the prescribed-time event-triggered cluster practical consensus problem for a class of nonlinear multi-agent systems with external disturbances. To begin, to reach the prescribed-time cluster practical consensus, a new time-varying function is introduced and a novel distributed continuous algorithm is designed. Based on the Lyapunov stability theory and inequality techniques, some sufficient conditions are given, ensuring the prescribed-time cluster practical consensus. Moreover, to avoid different clusters' final states overlapping, a virtual leader is considered for each cluster. In this case, an event-triggered distributed protocol is further established and some related conditions are given for achieving prescribed-time cluster practical consensus. Additionally, it is proven that the Zeno behavior can be avioded by choosing parameters appropriately. Finally, some numerical examples are presented to show the effectiveness of the theoretical results.

Automated tight Lyapunov analysis for first-order methods

AbstractWe present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider (i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components, (ii) first-order methods that can be written as a linear system on state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and (iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions. We present a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality within a predefined class of Lyapunov inequalities, which amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality gap convergence in the Chambolle–Pock method when the linear operator is the identity mapping.

Synthesis of ship systems optimal regulators based on matrix inequalities

The output and state regulators synthesis for ship control systems based on linear matrix inequalities using the CVX software package for solving optimization problems by the method of semi-definite programming is considered. Estimates parameters for regulators providing asymptotically stable systems regimes on Lyapunov inequalities solutions sets are obtained. An algorithm for controlling the ship course using a robust regulator in state feedback, followed by the use of CVX technologies to implement the minimum energy consumption control algorithm, is presented. The control vector estimation is reduced to the minimization of the L2 norm and it is performed in two stages, namely, the best mode is first determined, and then the mode under constraints is defined. The use of Lyapunov inequalities with the calculations implementation in CVX format on proprietary solvers has allowed us to obtain compact program texts and expand the set of restrictions introduced. In the algorithm for solving the boundary value problem of controlling a discrete model of an object, in contrast to existing optimization methods, a mode of transfer from the initial state to the final one with the right boundary, not necessarily equal to zero, is implemented. For a class of observable and controllable time–invariant systems, using the Krylov’s function contained in the MATLAB gallery of functions, L2‑norms to be minimized are formed. Examples of regulator calculations with a given degree of stability and optimal systems, confirming proposed solutions correctness, are given.

Adaptive Output Synchronization of Coupled Fractional-Order Memristive Reaction-Diffusion Neural Networks

This article discusses the adaptive output synchronization problem of coupled fractional-order memristive reaction-diffusion neural networks (CFOMRDNNs) with multiple output couplings or multiple output derivative couplings. Firstly, by using Lyapunov functional and inequality techniques, an adaptive output synchronization criterion for CFOMRDNNs with multiple output couplings is proposed. Then, an adaptive controller is designed for ensuring the output synchronization of CFOMRDNNs with multiple output derivative couplings. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.

Observer-based aperiodically intermittent pinning synchronization of complex-valued dynamical networks with time-varying delay

This paper addresses the impact of time delay and unavailability of system states on synchronization in complex networks. Specifically, it focuses on observer-based synchronization of complex-valued dynamical networks (CDNs) with time-varying delays, encompassing both complex-valued node dynamics and outer coupling forms. Two aperiodically intermittent control schemes, with and without pins, are proposed to reduce control costs by minimizing control frequency and the number of controllers. The proposed schemes establish sufficient criteria for guaranteeing synchronization in CDNs with time-varying delays, utilizing a combination of Lyapunov function and inequality technology in the complex field. Numerical examples are provided to illustrate the effectiveness of the proposed theoretical results.

Almost sure stability criteria for linear Markovian switching systems

We present some stability criteria mainly for continuous-time Markovian switching systems, which can provide us with an insight into the mechanism that drives the subsystems to interact with each other so as to achieve stability. On the one hand, directly computing the Lyapunov exponent through the state-transition matrix of the system under consideration allows us to capture the interaction of subsystems by using the method developed in discrete-time setting. On the other hand, we can make sense of how the subsystems are orchestrated by the statistical property of switching signal through a family of coupled Lyapunov inequalities, which naturally covers the stability in the mean-square sense as a special case. Three examples are included to illustrate the theoretical results.

On a class of Lyapunov's inequality involving $ \lambda $-Hilfer Hadamard fractional derivative

<abstract><p>In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.</p></abstract>