Continuous-time Systems Research Articles

Distributed Nash Equilibrium Seeking for Aggregative Games With Quantization Constraints

The problem of seeking Nash equilibrium (NE) based on aggregative games under quantization constraints is full of challenges. Although the NE seeking algorithm in continuous-time systems has been studied, this problem in discrete-time systems still needs to be solved urgently. To address this problem, three distributed algorithms are first proposed under three quantization cases, adaptive, random, and time-varying quantizations, based on doubly stochastic communication topology networks. Then, the actions of players would eventually converge to NE under the conditions of vanishing step size and strong monotonicity are proved. Moreover, the convergence rate of the three quantization cases are analyzed, respectively. Finally, numerical experiments are implemented on plug-in hybrid electric vehicles (PHEVs) to validate the effectiveness of the proposed distributed algorithms. Comparing the convergence rates of the three proposed algorithms, the convergence effect of the adaptive quantization is better than that of the other two quantization cases.

Output Bipartite Consensus for Second-Order Heterogeneous Uncertain Agents With State-Dependent Cooperation-Competition Interactions

For state-dependent cooperation-competition networks, the output bipartite consensus problem of heterogeneous uncertain agents is studied in this paper. In our framework, all the agents are described by second-order continuous-time nonlinear systems with different intrinsic dynamics, and the agents' uncertainties are characterized by unknown parameters in the intrinsic term. Then, the edge evolution rules with hysteresis coefficients are proposed via the classification strategy. To solve this problem, the distributed Lyapunov-based redesign method with the potential function term is first applied to such second-order heterogeneous uncertain multi-agent system. Through the three-step correlation design, the explicit expressions of the distributed controllers and the unknown parameter estimators are obtained, and it is shown that the state-dependent cooperation-competition network is always connected and maintains structural balance for all time if an initial structurally balanced and connected network topology is provided. On this basis, it follows from the total Lyapunov function that output bipartite consensus can be achieved asymptotically for the heterogeneous uncertain multi-agent system. Finally, a numerical simulation is provided to validate the structural balance of state-dependent networks and output bipartite consensus of heterogeneous uncertain agents.

Learning dynamics on invariant measures using PDE-constrained optimization.

We extend the methodology in Yang et al. [SIAM J. Appl. Dyn. Syst. 22, 269-310 (2023)] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.

Event-Triggered Optimal Adaptive Control of Partially Unknown Linear Continuous-Time Systems With State Delay

Event-Triggered Optimal Adaptive Control of Partially Unknown Linear Continuous-Time Systems With State Delay

Inaccuracy Matters: Accounting for Solution Accuracy in Event-Triggered Nonlinear Model Predictive Control

We consider the effect of using approximate system predictions in event-triggered control schemes. These approximations often result from using numerical transcription methods for solving continuous-time optimal control problems. Mesh refinement can guarantee upper bounds on the error in the differential equations that model the system dynamics. We employ the accuracy guarantees of a mesh refinement scheme to show that the proposed event-triggering scheme, which compares the measured system with approximate state predictions, can be used with a guaranteed strictly positive inter-update time. Furthermore, if knowledge of the employed transcription scheme or the approximation errors are available, then better online estimates of inter-update times can be obtained. We also detail a method of tightening constraints on the approximate system trajectory to guarantee constraint satisfaction of the continuous-time system. This is the first work to incorporate prediction accuracy in triggering metrics to guarantee reliable lower bounds for inter-update times and perform solution-dependent constraint tightening.

Optimal Adaptive Control of Uncertain Nonlinear Continuous-Time Systems With Input and State Delays.

In this article, an actor-critic neural network (NN)-based online optimal adaptive regulation of a class of nonlinear continuous-time systems with known state and input delays and uncertain system dynamics is introduced. The temporal difference error (TDE), which is dependent upon state and input delays, is derived using actual and estimated value function and via integral reinforcement learning. The NN weights of the critic are tuned at every sampling instant as a function of the instantaneous integral TDE. A novel identifier, which is introduced to estimate the control coefficient matrices, is utilized to obtain the estimated control policy. The boundedness of the state vector, critic NN weights, identification error, and NN identifier weights are shown through the Lyapunov analysis. Simulation results are provided to illustrate the effectiveness of the proposed approach.

Design of Photovoltaic Power Generation Servo System Based on Discrete Adaptive Network Dynamic Surface Control Technology

In recent years, under the development of the dual carbon goal, the energy crisis has become increasingly serious, and China has also experienced serious power rationing. However, the research on dynamic surface control technology in solar tracking systems in nonlinear control systems is mostly based on continuous-time systems, while adaptive dynamic surface control based on discrete-time nonlinear control systems can describe an actual control system more accurately in the production process. It can effectively suppress interference with extremely high stability and safety. To solve the problem of low efficiency in photovoltaic power generation, this research first built a photovoltaic power generation servo system model based on the parameter of uncertainty. Then, a discrete adaptive neural network dynamic surface (DANNDS) controller was designed to solve the problems in the design of the traditional backstepping method. Finally, based on the designed method of a dynamic surface controller, a discrete adaptive neural network quantization controller (DANNQC) for the photovoltaic power generation servo system was designed by introducing external disturbance. The control parameters and their studied ranges were as follows: The reference signals were or1=sin(0.1t) and or2=cos(0.1t). The parameters of the virtual control law and the final control law were m11=0.01, m22=0.01, m12=0.02, m13=0.02, and m23=0.02. The time constant of the low-pass filter was ζ12=ζ13=ζ22=ζ23=0.005. The parameters of the parameter regulation law were ρ12=ρ13=ρ22=ρ23=0.0005 and a12=a13=20, a22=a23=22. The research results show that the MTE, RMSTE, and 2NTE scores of the height angle servo motor of the DANNDS control method were 0.0026, 7.0279 × 10−4, and 0.3552, respectively. The scores for each index of the azimuth servo motor were 0.0028, 8.9237 × 10−4, and 0.4511, respectively. The height angle tracking error for the DANNQC control method was [−0.02,0.022]. The azimuth tracking error was [−0.03,0.03]. In summary, the photovoltaic power generation servo system based on the DANNQC has a better control performance. By controlling the height angle and azimuth angles, it can better track the position of the sun and adjust the position of the photovoltaic panel in real time. The sun’s rays illuminate the photovoltaic panel at an appropriate angle to achieve maximum power generation efficiency, which is of great practical significance for the development of solar technology.

Continuous-time approach of sampled-data systems

This paper studies the continuous-time behavior of sampled-data control systems. In these systems, a discrete-time control law is applied to a continuous-time system model using sampling and zero-order hold. Traditionally, sampled-data systems are modeled in discrete-time, and thus, the intersample dynamics cannot be examined. This paper shows that the classical methods (such as the d2c function of MATLAB) for obtaining continuous-time models can be inaccurate between the sampling instants, even when Shannon’s sampling theorem is satisfied. In this paper, a new method is also presented which can be used to characterize the continuous-time dynamics of sampled-data control systems. The new continuous-time system model including the intersample behavior is derived analytically from the discrete-time model, and the results are verified by simulations.

Robust model predictive control for constrained linear system based on a sliding mode disturbance observer

For perturbed continuous-time systems, this paper proposes a robust model predictive control (RMPC) strategy for the regulation problem, exploiting a sliding mode disturbance observer. The main advantage is that it effectively enables the RMPC to be designed based on a model with reduced uncertainties. The proposed sliding mode observer (SMO) is finite-time convergent allowing the estimation error of the additive disturbance to be explicitly bounded by a predictable and decreasing limit. Due to the compensation of the estimated disturbance, the uncertainty that the RMPC has to handle is reduced from the original disturbance to the estimation error of the disturbance. This ensures all the admissible state trajectories are limited within a shrinking neighborhood of the origin and the steady-state error is therefore reduced. Simulation results show the effectiveness of the proposed method.

An innovative approach based on optimization for the determination of initial conditions of continuous-time chaotic system as a random number generator

Security has been one of the important problems in the processing, storage and transmission of information. The transfer of information to virtual environments with technological developments has made it necessary to examine the concept of security with different dimensions. Data security can be achieved to a great extent by using randomly generated unpredictable bit sequences in encryption. In this study, approximately 1,000,000 random bits were generated for use in computer science. The Mode2(x) method and the unpredictable nature of chaotic systems are used for random number generation. With this hybrid structure used in the study, it is aimed to gain a different perspective to future studies. However, it is known that the sensitivity of chaotic systems to initial conditions is very high. The most important contribution of the study to the literature is that the initial conditions were determined using optimization algorithms. In this study, Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) were used to calculate the initial conditions. To the authors' knowledge, this is the first study to use optimization to determine initial conditions for continuous-time chaotic systems. NIST SP 800-22 Statistical Test Suite was used to analyze the reliability of the generated random bits. It was seen that the proposed method passed all statistical tests successfully (100 %). Histogram and floating frequency analysis were used for the analysis of random numbers obtained from the generated bits. Histogram plots showed uniform distribution. It is seen that there is a homogeneous distribution in the sliding frequency analysis.

Decentralized event-triggered resilient control for a class of networked control systems under denial-of-service attacks

This paper studies the problem of resilient dynamic output-feedback control schemes for continuous-time networked control systems (NCSs). A decentralized hybrid strategy is utilized while network imperfections, external disturbances, and noise are considered. Moreover, during data transmission, a practical denial-of-service (DoS) jamming attack, which periodically disturbs the network channels, is considered. To preserve the network resources, a decentralized event-triggered mechanism is employed to transmit only the sampled signals that are required. It is assumed that outputs and control inputs of each subsystem are transmitted to the corresponding decentralized controllers and actuators, respectively, over the different individual channels based on independent triggering mechanisms. At first, the NCS is modeled as a decentralized hybrid system with exogenous disturbance and noise. Then, sufficient conditions that guarantee the [Formula: see text]-stability of the NCS in the presence of external disturbances and noise, which is resilient to periodic DoS attacks, are provided in terms of the linear matrix inequalities. Finally, sufficient conditions are derived to deal with the time-varying delays. It has been demonstrated that the proposed technique can be effectively applied to a well-known continuous stirred tank reactor (CSTR) as a benchmark example.

Partial stability of nonlinear dissipative feedback systems

Partially stable systems involve dynamical systems whose motion lie in a subspace of the state space resulting in system stability with respect to part of the system's states. In this paper, we develop partial stability theorems for nonlinear continuous-time and discrete-time dissipative feedback systems. Specifically, by invoking additional structural constraints on the forward loop and feedback loop system storage functions, we develop feedback interconnection partial stability results for dissipative nonlinear dynamical systems. Our results provide extensions of the positivity and small gain theorems for guaranteeing partial stability of feedback interconnected systems.

Predictive-based control of complex dynamic networks

This paper addresses the problem of designing a robust controller for a class of complex dynamical networks (CDNs). This class of CDNs is characterized by a scale-free typical structure with Rossler and Lorenz-type nodes. Direct control of every node in a CDN with a large number of nodes might be unnecessary. Based on the concept of pinning control of continuous-time chaotic systems and the matrix measure theory, some simple stability criteria are derived to guarantee the asymptotic stability of the system states such that all the system states of the closed-loop system converge towards the desired unstable equilibrium points. Consequently, the proposed controller developed here for scale-free networks can be employed to handle a broader class of CDNs with different types of nodes. Numerical simulations are provided to show the effectiveness of the proposed control method.

A flow of quantum genetic Lotka-Volterra algebras on [formula omitted

In the present paper, we define a quantum analog of Lotka-Volterra algebras using a coalgebra scheme. This new framework provides a fresh perspective for the treatment of generic algebras. Additionally, a flow of quantum analogs of Lotka-Volterra genetic algebras is investigated. It's worth mentioning that such types of algebras are first introduced in this work. We observe that a flow of algebras is a particular type of continuous-time dynamical system, with states that are algebras and a structural constant matrix that depends on time and satisfies certain analogs of the Kolmogorov-Chapman equations. Using quantum quadratic operators, it is constructed a flow of quantum Lotka-Volterra algebras for the given multiplication. Furthermore, time-dependent behavior properties of these flow algebras are examined. The algebraic properties of the introduced flows are also studied, such as finding idempotents and examining an algebra generated by a pair of idempotents. It is shown that the later one is associative, while the flow is not associative. Additionally, derivations of the flow of the algebras are described.

Space manipulator optimal impedance control using integral reinforcement learning

This paper examines the optimal impedance control problem for large-scale space manipulator operational tasks with unknown contact dynamics and partial measurements. More specifically, by quantifying the interaction performance using a discounted value function, the optimal impedance control problem is tactfully transformed into a linear quadratic tracking problem. By resorting to the historical inputs and outputs, an improved state reconstruction method is presented, which obviates the velocity measurement. Unlike the existing state reconstruction method for continuous-time systems, the estimation bias caused by probing noise is completely eliminated under the improved state reconstruction method. Based on this, a novel model-free value iteration integral reinforcement learning algorithm is developed to approximate optimal impedance parameters. Compared with the earlier integral reinforcement learning algorithms, the proposed algorithm not only averts any prior contact dynamics knowledge and full-state measurements, but also eliminates the heavy dependence on the specific initial stabilization control. In addition, the implementation and convergence of the proposed algorithm are discussed successively. Finally, numerical simulations verify the effectiveness of the theoretical results.

Input-output finite-time stabilization of periodic piecewise systems with multiple disturbances

In this work, the problems of input-output finite-time stability and disturbance rejection for continuous-time periodic piecewise systems with linear fractional uncertainty, matched and mismatched disturbances are investigated. In detail, the matched disturbances that are ensuing from the exogenous systems can be tackled by modeling a periodic piecewise disturbance observer (PPDO), which effectively estimates the disturbance with high precision. The mismatched part can be sorted out by implementing H∞ control protocol and meanwhile the state feedback is quantized in accordance with the logarithmic quantizer. On the whole, by combining the quantized state-feedback control law with the output of the PPDO, the anti-disturbance control protocol is developed. Moreover, by constructing a Lyapunov function with periodic piecewise positive definite matrices, a collection of adequate criteria affirming the system’s input-output finite-time stability are procured in the context of linear matrix inequalities (LMIs). Subsequently, the time-varying periodic piecewise gain values of the crafted disturbance observer and developed controller are acquired by working on the LMIs. Conclusively, the simulation results are provided, including the 2-degree of freedom vibration system, to verify the potential of the developed control strategy.

Online Reinforcement Learning Control of Nonlinear Dynamic Systems: A State-action Value Function Based Solution

In this paper, we present an online reinforcement learning-based solution to the optimal control problem of continuous-time nonlinear input-affine systems. The proposed approach contains a concurrent identifier that estimates time derivatives of states of the system in some arbitrary points. The identifier is utilized to simulate a so-called Bellman error in some unvisited points. The simulated errors together with errors obtained along the trajectory of the system are used to estimate the state-action value function, which is then employed to derive the estimated optimal controller. The designed approach does not explicitly require the input dynamics, which is hard to segregate it from the drift dynamics in optimal regulation problems. In addition, the simulated Bellman errors relax the confining persistence of excitation condition, which is needed for convergence in deterministic systems. A Lyapunov-based analysis was conducted to derive convergence conditions. Simulation studies demonstrated the effectiveness of the developed control scheme.

Formal Verification of Unknown Discrete- and Continuous-Time Systems: A Data-Driven Approach

This paper is concerned with a formal verification scheme for both discrete- and continuous-time deterministic systems with unknown mathematical models. The main target is to verify the safety of unknown systems based on the construction of barrier certificates via a set of data collected from trajectories of systems while providing an a-priori guaranteed confidence on the safety. In our proposed framework, we first cast the original safety problem as a robust convex program (RCP). Solving the proposed RCP is not tractable in general since the unknown model appears in one of the constraints. Instead, we collect finite numbers of data from trajectories of the system and provide a scenario convex program (SCP) corresponding to the original RCP. We then establish a probabilistic closeness between the optimal value of SCP and that of RCP, and as a result, we formally quantify the safety guarantee of unknown systems based on the number of data points and a required level of confidence. We propose our framework in both discrete-time and continuous-time settings. We illustrate the effectiveness of our proposed results by first applying them to an unknown continuous-time room temperature system. We verify that the temperature of the room maintains in a comfort zone with some desirable confidence by collecting data from trajectories of the system. To show the applicability of our techniques to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">higher dimensional</i> systems with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> dynamics, we then apply our results to a continuous-time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> jet engine compressor and a discrete-time DC motor.

Distributed State Estimation With Deep Neural Networks for Uncertain Nonlinear Systems Under Event-Triggered Communication

This work explores the distributed state estimation problem for an uncertain, nonlinear, and continuous-time system. Given a sensor network, each agent is assigned a deep neural network (DNN) which is used to approximate the system's dynamics. Each agent updates the weights of their DNN through a multiple timescale approach, i.e., the outer layer weights are updated online with a Lyapunov-based gradient descent update law, and the inner layer weights are updated concurrently using a supervised learning strategy. To promote the efficient use of network resources, the distributed observer uses event-triggered communication. A nonsmooth Lyapunov analysis demonstrates that the distributed event-triggered observer achieves uniformly ultimately bounded state reconstruction. A simulation example of a 5-agent sensor network estimating the state of a two-link robotic manipulator tracking a desired trajectory is provided to validate the result and showcase the performance improvements afforded by DNNs.

Relaxed Actor-Critic With Convergence Guarantees for Continuous-Time Optimal Control of Nonlinear Systems

This paper presents the Relaxed Continuous-Time Actor-critic (RCTAC) algorithm, a method for finding the nearly optimal policy for nonlinear continuous-time (CT) systems with known dynamics and infinite horizon, such as the path-tracking control of vehicles. RCTAC has several advantages over existing adaptive dynamic programming algorithms for CT systems. It does not require the ``admissibility" of the initialized policy or the input-affine nature of controlled systems for convergence. Instead, given any initial policy, RCTAC can converge to an admissible, and subsequently nearly optimal policy for a general nonlinear system with a saturated controller. RCTAC consists of two phases: a warm-up phase and a generalized policy iteration phase. The warm-up phase minimizes the square of the Hamiltonian to achieve admissibility, while the generalized policy iteration phase relaxes the update termination conditions for faster convergence. The convergence and optimality of the algorithm are proven through Lyapunov analysis, and its effectiveness is demonstrated through simulations and real-world path-tracking tasks.