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This dissertation investigates the challenges of improving the robustness and fairness of deep neural networks through the lens of optimal control theory. Deep neural networks, despite their extensive application across various engineering fields,are vulnerable to imperceptible perturbations and often exhibit biased performance towards underrepresented demographic populations. The first part of this dissertation presents a novel self-healing framework designed to improve the robustness of deep neural networks against unforeseen perturbations.This framework is realized by a novel closed-loop control approach grounded in optimal control theory, which adaptively generates control signals to identify and correct potential errors in the state trajectory of perturbed input data during inference. The second part of this dissertation introduces a PID control framework that generalizes the closed-loop method with additional integral and derivative controllers.We derive an analytical solution for fast online inference, making our control framework applicable to large-scale models.The third part of this dissertation addresses the fairness issue in machine learning within dynamic environments, where undesired model biases against minority users could lead to significant user churn, thereby diminishing the training data for model tuning in subsequent time steps.This negative feedback loop can further exacerbate demographic disparity.To address this, we introduce the concept of asymptotic fairness to maintain consistent model performance across all demographic groups and propose an optimal control solution to achieve this goal.

The security and economy of multienergy systems (MESs) are directly threatened by the potential cyberattacks. It is of great importance to investigate the effects of cyberattacks, e.g., denial of service (DoS) attacks, on distributed energy management algorithms. To this end, this article focuses on exploring how the frequency and the time of duration of DoS attacks influence the behavior of Newton-Raphson-based distributed energy management (NRBDEM) algorithm for MES and in which condition the optimal operations can still be obtained. First, a switched NRBDEM algorithm is presented, which is composed of the normal operation mode and the attack mode. In the attack mode, the attackers are able to change the communication structure at will and make it unconnected to destroy the convergence of the switched NRBDEM algorithm. Then, by making use of automatons to generate the hybrid time domain, the switched NRBDEM algorithm is further modeled and formulated as a hybrid dynamical system, which provides a mathematical model for the subsequent convergence analysis. Therein, the generated hybrid time domain satisfies the average dwell-time constraint and time-ratio constraint to limit the persistent attacks. Furthermore, we analyze the restrained conditions for persistent attacks, under which the optimality and convergence of the switched NRBDEM algorithm can be guaranteed still. Finally, simulation results demonstrate the effectiveness of the proposed method.

The simultaneous parameter and state estimation for a multi-input multi-output (MIMO) state space system from a set of measurement data is taken into account in this paper. Firstly, in line with the number of the system outputs, the considered MIMO system is transformed to some subsystems, which lessens the dimensions and the number of the parameters to be estimated. Secondly, by designing the moving data window that contains the latest batch of collected data, we develop a moving data window-based partially-coupled average extended stochastic gradient algorithm for parameter estimation. Thirdly, once the parameter estimates are obtained, a new state filter is designed to produce the estimates of the unmeasurable states by means of the Kalman filtering principle. Then we propose a combined state filtering and moving data window-based partially-coupled average extended stochastic gradient (CSF-MDW-PC-A-ESG) algorithm to produce the estimates of the parameters and states simultaneously. To reveal the superiority of the CSF-MDW-PC-A-ESG algorithm, a combined state filtering and partially-coupled average extended stochastic gradient (CSF-PC-A-ESG) algorithm is given to make a comparison. Finally, the effectiveness and superiority of the proposed CSF-MDW-PC-A-ESG algorithm are proved in a simulation example. The results from the illustrative example show that the CSF-MDW-PC-A-ESG algorithm is effective to produce the estimates of the parameters and states and that the CSF-MDW-PC-A-ESG algorithm has the higher efficient data utilization, the more accurate parameter estimation capability and the better model fitting ability than the CSF-PC-A-ESG algorithm.

In this article, we apply a high-order control to a dynamical system with uncertainty. There are two characteristics. First, the uncertain part, which is time-varying but bounded, is described in a fuzzy aspect. Specifically, the uncertainty lies within a fuzzy set and the bound is regarded as a fuzzy number. Second, the systems are uniformly bounded and uniformly ultimately bounded with a deterministic controller based on the Lyapunov theory. To obtain better system performance and lower control input, we apply the noncooperative game theory to optimize the parameters by establishing a Nash game. Then, the D-operation is proposed for the uncertainty related to fuzzy numbers. Finally, we perform the numerical simulations of the steering-by-wire system for verification.

This article studies fault-tolerant resilient control (FTRC) problems for uncertain Takagi-Sugeno fuzzy systems when subjected to additive actuator faults and/or malicious injections on control input signals. The effects of faults and malicious injections are modeled by unknown bounded signals. The signals are produced by any finite-L₂-gain dynamical system and a Lipschitz and derivable function with respect to states, so that the considered fault model contains some reported ones as special cases. By employing the available state and input data, a function, which is equivalent to a fictitious dynamical system comprising the information about compensation errors for unknown actuator faults, is presented. Then, based on the virtual system, a novel actuator failure compensator (AFC) with the structure of dynamic feedbacks is proposed, so that the compensation capability is improved via cooperative interaction designs between the virtual dynamical systems and closed-loop systems. Furthermore, through the equivalence class and Lyapunov theories, it is proved that the presented robust dynamic AFC-based fuzzy controller ensures the asymptotic convergence of system states to zero. Different from the existing FTRCs, good transient performance is guaranteed, even in the presence of unforeseen actuator faults. Two illustrative examples verify the effectiveness of the presented method.

We approach the problem of stabilizing a dynamical system while optimizing a cost and satisfying safety and control constraints. For affine control systems and quadratic costs, it has been shown that Control Barrier Functions (CBF) guaranteeing safety and Control Lyapunov Functions (CLF) enforcing convergence can be used to reduce the optimal control problem to a sequence of Quadratic Programs. In this paper, we propose High Order CBFs (HOCBF), which can accommodate systems of arbitrary relative degree. We construct a set of controls that renders the intersection of a set of sets forward invariant for the system, which implies the satisfaction of the original safety constraint. Our notion of HOCBF is more general than the recently proposed exponential CBF. We formulate optimal control problems with constraints given by HOCBF and CLF, and propose two methods - the penalty and the parameterization methods - to address the feasibility problem. Finally, we show how our methodology can be extended for safe navigation in unknown environments with long-term feasibility. We illustrate the proposed framework on a cruise control system and on a robot control problem.

In this paper, we investigate the distributed state estimation problem for a nonlinear dynamical system. Relying upon the assumption of the existence of a block-triangular ``uniform observability" canonical form, local high-gain observers are used to stabilize the estimation error for the observable portion of the state and exchange of information between neighbors is used to achieve consensus in the estimation of the state of the unobservable part. The paper addresses first the case of an autonomous system, and then the case of a controlled dynamical system under the assumption that the exogenous input is accessible by each node in the network. Based on the designed state estimator, each distributed sensor node can achieve asymptotic estimation of the full state, with an arbitrarily fixed decay rate of the observation error.

In this article, we propose a learning approach to analyze dynamic systems with an asymmetric information structure. Instead of adopting a game-theoretic setting, we investigate an online quadratic optimization problem driven by system noises with unknown statistics. Due to information asymmetry, it is infeasible to use the classic Kalman filter nor optimal control strategies for such systems. It is necessary and beneficial to develop an admissible approach that learns the probability statistics as time goes forward. Motivated by the online convex optimization (OCO) theory, we introduce the notion of regret, which is defined as the cumulative performance loss difference between the optimal offline-known statistics cost and the optimal online-unknown statistics cost. By utilizing dynamic programming and linear minimum mean square biased estimate (LMMSUE), we propose a new type of online state-feedback control policy and characterize the behavior of regret in a finite-time regime. The regret is shown to be sublinear and bounded by O(lnT) . Moreover, we address an online optimization problem with output-feedback control policy and propose a heuristic online control policy.

In this article, an output regulation problem is considered for nonlinear multiagent systems with unity relative degree, in which nodes are coupled by dynamic edges. The inputs of the edge dynamic systems are determined by the error outputs of the node dynamic systems. Similarly, the neighboring inputs of the node dynamic systems are formed by the outputs of the edge dynamic systems and can influence node outputs. By introducing some coordinate transformations, we can transform the output regulation problem into a robust stabilization problem for an augmented system. Then, using the relative outputs of neighboring agents, we design a distributed output-feedback control law and suitable dynamic couplings between the nodes. Finally, it is shown that the global stabilization of the augmented system can be achieved using the proposed controller. An example is presented to demonstrate the effectiveness of our control strategy.

Guided policy search algorithms have been proven to work with incredible accuracy not only for controlling complicated dynamical systems, but also in learning optimal policies from exploration of various unseen instances. This paper deals with a trajectory optimization problem for an unknown dynamical system subject to measurement noise using expectation maximization and extends it to learning (optimal) policies which have less stochasticity in trajectories because of the higher exploitation efficiency. Theoretical and empirical evidence of learned optimal policies for the new approach is depicted in comparison to some well known baselines which are evaluated on an autonomous system with widely used performance metrics.

We consider the problem of learning a realization for a linear time-invariant (LTI) dynamical system from input/output data. Given a single input/output trajectory, we provide finite time analysis for learning the system's Markov parameters, from which a balanced realization is estimated using the classical Ho-Kalman algorithm. By proving a robustness result for the Ho-Kalman algorithm and combining it with the sample complexity results for Markov parameters, we show how much data is needed to approximate the balanced realization of the system up to a desired accuracy with high probability.

This paper considers a dynamic optimization problem (DOP) of 1, 3-propanediol fermentation process (1,3-PFP). Our main contributions are as follows. Firstly, the DOP of 1, 3-PFP is modeled as an optimal control problem of switched dynamical systems. Unlike the existing switched dynamical system optimal control problem, the state-dependent switching method is applied to design the switching rule. Then, in order to obtain the numerical solution, by introducing a discrete-valued function and using a relaxation technique, this problem is transformed into a nonlinear parameter optimization problem (NPOP). Although the gradient-based algorithm is very efficient for solving NPOPs, the existing algorithm is always trapped in a local minimum for such problems with multiple local minima. Next, in order to overcome this challenge, a gradient-based random search algorithm (GRSA) is proposed based on an improved gradient-based algorithm (IGA) and a novel random search algorithm (NRSA), which cannot usually be trapped in a local minimum. The convergence results are also established, and show that the GRSA is globally convergent. Finally, a DOP of 1, 3-PFP is provided to illustrate the effectiveness of the GRSA proposed by this paper.

In this article, we study the application of Rough Set theory to the representation of uncertainty and partial knowledge in Dynamical Systems. Our approach draws from the abstract notion of an observable pattern, and for this purpose we first propose an abstract knowledge representation formalism that encompasses the main classes of discrete Dynamical Systems. Drawing on the proposed representational formalism, we define appropriate notions of rough approximations and reducts, show how these can be applied for uncertainty representation, and discuss their theoretical properties and characterizations.

This article establishes an approach, based on distributed optimization, for solving continuous-time Lyapunov equations (CTLE) over multiagent networks. Each agent in the network knows partial information of the CTLE and has a dynamical system to estimate exact or least-squares solutions. The aim of agents is to find a solution to CTLE by sharing information with connected agents over a network. This article develops distributed algorithms with an exponential rate of convergence for CTLE via the convex optimization design. Finally, this article presents numerical simulations to show the efficacy of the main results.

Recent decades have witnessed a trend that control-theoretical techniques are widely leveraged in various areas, e.g., design and analysis of computational models. Computational methods can be modeled as a controller and searching the equilibrium point of a dynamical system is identical to solving an algebraic equation. Thus, absorbing mature technologies in control theory and integrating it with neural dynamics models can lead to new achievements. This work makes progress along this direction by applying control-theoretical techniques to construct new recurrent neural dynamics for manipulating a perturbed nonstationary quadratic program (QP) with time-varying parameters considered. Specifically, to break the limitations of existing continuous-time models in handling nonstationary problems, a discrete recurrent neural dynamics model is proposed to robustly deal with noise. This work shows how iterative computational methods for solving nonstationary QP can be revisited, designed, and analyzed in a control framework. A modified Newton iteration model and an improved gradient-based neural dynamics are established by referring to the superior structural technology of the presented recurrent neural dynamics, where the chief breakthrough is their excellent convergence and robustness over the traditional models. Numerical experiments are conducted to show the eminence of the proposed models in solving perturbed nonstationary QP.

Aviation and aerospace system are typical Phased-Mission Systems (PMSs) featured with varying configuration, phased load condition and cross phase failure correlation. The coupling effect of Functional Dependency (FDEP) and Physical Dependency (PDEP) has a unique influence on the failure behavior of PMS. In this article, the coupling effect is analyzed, and the degradation of components is modeled with the positive drift wiener process, in which the drift coefficient is related to environmental conditions and operation stress. Finally, failure behavior and system reliability are simulated. An avionics controller is studied as a case, with the degradation time distribution model and simulation algorithm, the coupling effect and dynamical system reliability can be achieved. Results show that this modeling method can describe the coupling effects of FDEP and PDEP and their influence on the failure behavior and reliability of the PMS system.

In this article, a new intelligent hybrid controller is proposed. The controller is based on the combination of the orthogonal endocrine neural network (OENN) and orthogonal endocrine ANFIS (OEANFIS). The orthogonal part of the controller consists of Chebyshev orthogonal functions, which are used because of their recursive property, computational simplicity, and accuracy in nonlinear approximations. Artificial endocrine influence on the controller is achieved by introducing excitatory and inhibitory glands to the OENN part of the structure, in the form of postsynaptic potentials. These potentials provide a network with the capability of additional self-regulation in the presence of external disturbances. The intelligent structure is trained using a developed learning algorithm, which consists of both offline and online learning procedures: online learning for fitting OENN substructure and offline learning for adjusting OEANFIS parameters. The learning process is expanded by introducing the learning rate adaptation algorithm, which bases its calculations on the sign of the error difference. Finally, the proposed intelligent controller was experimentally tested for control of a nonlinear multiple-input-multiple-output two rotor aerodynamical system. During the test phase, an additional four related intelligent control logics and default PID-based controllers were used, and tracking performance comparisons were performed. The proposed controller showed notably better online results in comparison to other control algorithms. The major deficiencies of the structure are complexity and noticeably large training computation time, but these drawbacks can be neglected if tracking performances of a dynamical system are of the highest importance.

Path integral policy improvement (PI^2) is known to be an efficient reinforcement learning algorithm, particularly, if the target system is a high-dimensional dynamical system. However, PI^2 , and its existing extensions, have adjustable parameters, on which the efficiency depends significantly. This article proposes an extension of PI^2 that adjusts all of the critical parameters automatically. Motion acquisition tasks for three different types of simulated legged robots were performed to test the efficacy of the proposed algorithm. The results show that the proposed method cannot only eliminate the burden on the user to set the parameters appropriately but also improve the optimization performance significantly. For one of the acquired motions, a real robot experiment was conducted to show the validity of the motion.

In this article, we propose a new method to ob-tain feedback controllers of an unknown dynamical systemdirectly from noisy input/state data. The key ingredient ofour design is a new matrix S-lemma that will be proven inthis article. We provide both strict and nonstrict versionsof this S-lemma, which are of interest in their own right.Thereafter, we will apply these results to data-driven con-trol. In particular, we will derive nonconservative designmethods for quadratic stabilization, H2 and H∞ control, allin terms of data-based linear matrix inequalities. In contrastto previous work, the dimensions of our decision variablesare independent of the time horizon of the experiment. Ourapproach, thus, enables control design from large datasets.

Though the government has ruled ``Say No to Plastic'', plastic usage is increasing day by day. It is observed since last three decades that each process of plastic creates pollution affecting degradation of environmental resources like air, soil and water. The proposed mathematical model contains these three pollutions with its issues having a dynamical system of nonlinear differential equations. Feasible region contains three equilibrium points pollution-free, water-pollution free and optimum issue point. Basic reproduction number and center manifold theory is used for analyzing the local stability of a model. Castillo-Chavez theory with detailed computation has been used to check global dynamical behavior of equilibrium points and also gives the existence of backward bifurcation of plastic usage. To optimize the pollution, optimal control has been applied to the system. Graphical analysis is carried out.

In this paper, it is addressed by mathematical analysis how network-oriented modeling relates to the dynamical systems perspective on mental processes. It has been mathematically proven that any dynamical system can be modeled as a temporal-causal network model and that any adaptive dynamical system (of any order) can be modeled by a self-modeling network (of the same order).

This article studies resilient control problems for partially unknown nonlinear systems subjected to malicious injections on the control input signals. The injection model is assumed to be Lipschitz continuous and derivable regarding an unknown bounded signal, and the signal is produced from an unknown finite $L_{2}$ -gain dynamical system. First, based on neural network identifier and adaptive dynamic programming techniques, a novel controller with two fictitious dynamical systems, as co-workers of the closed-loop systems resisting attacks, is proposed. Furthermore, a cooperative interaction framework between the virtual dynamical systems and the closed-loop systems is developed, and through optimal control theory and Lyapunov function methods, it is proved that, the robust resilient controller designed in the framework ensures the attacked system states are uniformly ultimately bounded. Contrary to the presented approach, the impact of attacks is not considered in the existing results, then the stability for partially unknown nonlinear systems might not be guaranteed. Two illustrative examples validate the presented method.

We propose an analytical construction of observable functions in the extended dynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method for approximating the spectral properties of the Koopman operator. The choice of observable functions is fundamental for the application of EDMD to nonlinear problems arising in systems and control. Existing methods either start from a set of dictionary functions and look for the subset that best fits the underlying nonlinear dynamics or they rely on machine learning algorithms to “learn” observable functions. Conversely, in this letter, we start from the dynamical system model and lift it through the Lie derivatives, rendering it into a polynomial form. This proposed transformation into a polynomial form is exact, and it provides an adequate set of observable functions. The strength of the proposed approach is its applicability to a broader class of nonlinear dynamical systems, particularly those with nonpolynomial functions and compositions thereof. Moreover, it retains the physical interpretability of the underlying dynamical system and can be readily integrated into existing numerical libraries. The proposed approach is illustrated with an application to electric power systems. The modeled system consists of a single generator connected to an infinite bus, where nonlinear terms include sine and cosine functions. The results demonstrate the effectiveness of the proposed procedure in off-attractor nonlinear dynamics for estimation and prediction; the observable functions obtained from the proposed construction outperform methods that use dictionary functions comprising monomials or radial basis functions.

In this article, we study the leader-following practical attitude consensus problem of a group of multiple uncertain rigid spacecraft systems over jointly connected networks by a distributed event-triggered control law. We first establish a lemma that allows the problem to be converted to a distributed practical stabilization problem of a well-defined uncertain dynamical system. Then, we combine the adaptive distributed observer technique and the adaptive control technique to design an event-triggered adaptive control law and an event-triggered mechanism to solve our problem. The effectiveness of our design is illustrated by a numerical example.

In this work, we study the generalized Nash equilibrium (GNE, see Definition 1 ) seeking problem for monotone generalized noncooperative games with set constraints and shared affine inequality constraints. A novel projected gradient-based regularized penalized dynamical system is proposed to solve this issue. The idea is to use a differentiable penalty function with a time-varying penalty parameter to deal with the inequality constraints. A time-varying regularization term is used to deal with the ill-poseness caused by the monotonicity assumption and the time-varying penalty term. The proposed dynamical system extends the regularized dynamical system in the literature to the projected gradient-based regularized penalized dynamical system, which can be used to solve generalized noncooperative games with set constraints and coupled constraints. Furthermore, we propose a distributed algorithm by using leader-following consensus, where the players have access to neighboring information only. For both cases, the asymptotic convergence to the least-norm variational equilibrium of the game is proven. Numerical examples show the effectiveness and efficiency of the proposed algorithms.

Rectified linear unit (ReLU) networks have become widely used in machine learning and automated inference using neural networks. Various forms of hardware accelerators based on ReLU networks have also been under development. In this brief, the stability problem in analog ReLU networks is addressed. Using the Lyapunov stability theory, it is shown that the origin of an unforced, analog ReLU dynamical system is globally asymptotically stable if the induced Euclidean norm of its connectivity matrix is less than one. An example is given to demonstrate that this upper bound is the best that can be achieved. In particular, the stability result holds for the case of a nonsymmetric connectivity matrix as may occur in some mathematical models of neurobiology.

A new algorithm for the estimation of the robustness of a dynamical system’s equilibrium is presented. Unlike standard approaches, the algorithm does not aim to identify the entire basin of attraction of the solution. Instead, it iteratively estimates the so-called local integrity measure, that is, the radius of the largest hypersphere entirely included in the basin of attraction of a solution and centred in the solution. The procedure completely overlooks intermingled and fractal regions of the basin of attraction, enabling it to provide a significant engineering quantity in a very short time. The algorithm is tested on four different mechanical systems of increasing dimension, from 2 to 8. For each system, the variation of the integrity measure with respect to a system parameter is evaluated, proving the engineering relevance of the results provided. Despite some limitations, the algorithm proved to be a viable alternative to more complex and computationally demanding methods, making it a potentially appealing tool for industrial applications.

We have investigated the role of competitive mode for the generation of chaotic behavior in a financial dynamical system. Such type of events are very important in the light of stock market crush or volatile behavior. The competitive mode is a good approach other than the fixed point analysis. The character of mode frequencies and the attractor is analyzed numerically . Also, using an analytical method and Lagrange optimization, we were able to calculate the ultimate bound of the chaotic financial system. The method we have presented is simpler and more accurate than other methods that implicitly calculate the final boundary. The estimation of the ultimate bound can be used to study chaos synchronization. Numerical simulations illustrate the analytical results.

This paper discusses finite/fixed-time (FNT/FXT) synchronization of multi-layer networks based on energy consumption estimation. Firstly, a novel FXT stability theorem of dynamical system is presented. Secondly, sufficient criteria for FNT/FXT synchronization in multi-layer networks are given, and the estimation of the energy consumption are also gained. Especially, the range of the number of layers of network is gained based on the least convergence time and the estimation of the least energy consumption. The results show that shorter synchronization time and less energy consumption does not mean less network layers. Finally, the validity of the theoretical method is verified by numerical simulation.

Differential operators based on convolution have been recognized as powerful mathematical operators able to depict and capture chaotic behaviors, especially those that are not able to be depicted using classical differential and integral operators. While these differential operators have being applied with great success in many fields of science, especially in the case of dynamical system, we have to confess that they were not able depict some chaotic behaviors, especially those with additionally similar patterns. To solve this issue new class of differential and integral operators were proposed and applied in few problems. In this paper, we aim to depict chaotic behavior using the newly defined differential and integral operators with fractional order and fractal dimension. Additionally we introduced a new chaotic operators with strange attractors. Several simulations have been conducted and illustrations of the results are provided to show the efficiency of the models.

This article addresses the problem of extremum seeking of a continuous-time dynamical system with a single input and a single output. A Super-Twisting-based optimization algorithm is proposed to compute the input that leads to the extremum value of an unknown convex objective function. Our optimization algorithm requires the input–output gradient of the system’s response at steady state, which we compute throughout a Super-Twisting-based differentiator. Feasibility of the proposed extremum seeking strategy is demonstrated by two simulation examples. The first one is an example of interest in academy. The second one is a novel biohydrogen production process.

An adaptive neural networks (NN) backstepping optimized tracking learning control approach is presented for nonlinear strict-feedback systems with state constraints. In the control design, neural networks are used to learn the unknown nonlinear dynamics, and a NN state identifier is proposed. By constructing Barrier Lyapunov functions and optimal Barrier type cost functions for identifier error dynamic systems and the subsystems of the identifier systems, and under the actor-critic architecture, the virtual and actual optimal controllers are constructed by the backstepping recursive control design algorithm. The proposed adaptive NN optimal control scheme can guarantee that the closed-loop system is uniformly ultimately bounded (UUB) and the states of the controlled system are ensured not to transgress their constrained sets. Moreover, the proposed optimized controller can guarantee that the system output can track the given reference signal. The effectiveness of the proposed control method is verified by a numerical example.

This article proposes a neural networks (NNs)-based tracking control approach for a class of uncertain high-order self-restructuring nonaffine dynamic systems. Unlike most existing NN-based works that normally ignore the precondition on the functionality and reliability of NN unit and thus can only ensure semiglobal stability, the proposed method explicitly addresses the issue of reliable in-loop operation of NN approximation-based control unit, resulting in a safeguarded NN-based control solution capable of ensuring globally stable tracking. Furthermore, the control method proposed guarantees exponentially globally stable tracking for systems with self-restructuring nonlinearities and uncertainties, distinguishing itself from those that only yield uniformly ultimately bounded (UUB) regulation/tracking results for nonlinear systems with fixed structures. All of these features are achieved by the proposed strategy consisting of two cooperative control units: 1) safeguard control and 2) NN-based control. The role of the safeguard control is to force the states (starting from any initial condition) to enter a stable region, so that the NN-based control can be activated trustworthily and safely. It is such cooperation of the two units that not only ensures the tracking error entering the stable region first within a prespecified finite time but also guarantees the tracking error converging to zero exponentially thereafter, resulting in global zero-error tracking. Both the theoretical analysis and numerical simulation authenticate the effectiveness of the proposed method.

The attackers–target–defenders problem is formulated and formalized as a positional zero-sum game with incomplete information. A method for estimating the target survival probability given some estimates of attackers’ dynamic capabilities is proposed, and recurrence expressions for its calculation are obtained. The corresponding numerical target survival probability estimations with optimal behavior of the players are made. The defenders in the game are assigned three different roles: the role of a false target, the target tracking role, and the attacker intercepting role. The survival probability considered as a function of the attackers–target–defenders problem parameters is analyzed. The obtained probability estimates can be used as heuristic estimates for the performance of the corresponding dynamical system. The obtained optimal strategies can be considered as heuristics for the corresponding attackers–target–defenders differential game. The optimal strategies contain orders for the players indicating their current roles. Using the Python language, we developed a software module that allows one to calculate the target survival probability based on known problem parameters.

In this article, the formation mission design problem for commercial aircraft is studied. Given the departure times and the departure and arrival locations of several commercial flights, the relevant weather forecast, and the expected fuel savings during formation flight, the problem consists in establishing how to organize them in formation or solo flights and in finding the trajectories that minimize the overall direct operating cost of the flights. Each aircraft can fly solo or in different positions inside a formation. Therefore, the mission is modeled as a switched dynamical system, in which the discrete state describes the combination of flight modes of the individual aircraft and logical constraints in disjunctive form establish the switching logic among the discrete states of the system. The formation mission design problem has been formulated as an optimal control problem of a switched dynamical system and solved using an embedding approach, which allows switching decision among discrete states to be modeled without relying on binary variables. The resulting problem is a classical optimal control problem which has been solved using a knotting pseudospectral method. Several numerical experiments have been conducted to demonstrate the effectiveness of this approach. The obtained results show that formation flight has great potential to reduce fuel consumption and emissions.

We consider an optimal control problem for a dynamical system whose motion is described by a linear differential equation with the Caputo fractional derivative of order $$\alpha\in(0,1)$$ . The time interval of the control process is fixed and finite. The control actions are subject to geometric constraints. The aim of the control is to minimize a given terminal-integral performance index. In order to construct a solution, we develop the following approach. First, from the considered problem, we turn to an auxiliary optimal control problem for a first-order linear system with lumped delays, which approximates the original system. After that, the auxiliary problem is reduced to an optimal control problem for an ordinary differential system. Based on this, we propose a closed-loop scheme of optimal control of the original system that uses the approximating system as a guide. In this scheme, the control in the approximating system is formed with the help of an optimal positional control strategy from the reduced problem. The effectiveness of the developed approach is illustrated by a problem in which the performance index is the norm of the terminal state of the system.

Combinatorial optimization problems belong to the non-deterministic polynomial time (NP)-hard complexity class, and their computational requirements scale exponentially with problem size. They can be mapped into the problem of finding the ground state of an Ising model, which describes a physical system with converging dynamics. Various platforms, including optical, electronic and quantum approaches, have been explored to accelerate the ground-state search, but improvements in energy efficiencies and computational abilities are still required. Here we report an Ising solver based on a network of electrically coupled phase-transition nano-oscillators (PTNOs) that form a continuous-time dynamical system (CTDS). The bi-stable phases of the injection-locked PTNOs act as artificial Ising spins and the stable points of the CTDS act as the ground-state solution of the problem. We experimentally show that a prototype with eight PTNOs can solve an NP-hard MaxCut problem with high probability of success (96% for 600 annealing cycles). We also show via numerical simulations that our Ising Hamiltonian solver can solve MaxCut problems of 100 nodes with energy efficiency of 1.3 × 107 solutions per second per watt, offering advantages over other approaches including memristor-based Hopfield networks, quantum annealers and photonic Ising solvers. An Ising solver that is based on a network of electrically coupled phase-transition nano-oscillators, which provides a continuous-time dynamical system, can be used to efficiently solve a non-deterministic polynomial time (NP)-hard MaxCut problem.

In this paper, we investigate the parameter identification problem in dynamical systems through a deep learning approach. Focusing mainly on second-order, linear time-invariant dynamical systems, the topic of damping factor identification is studied. By utilizing a six-layer deep neural network with different recurrent cells, namely GRUs, LSTMs or BiLSTMs; and by feeding input-output sequence pairs captured from a dynamical system simulator, we search for an effective deep recurrent architecture in order to resolve damping factor identification problem. Our study results show that, although previously not utilized for this task in the literature, bidirectional gated recurrent cells (BiLSTMs) provide better parameter identification results when compared to unidirectional gated recurrent memory cells such as GRUs and LSTM. Thus, indicating that an input-output sequence pair of finite length, collected from a dynamical system and when observed anachronistically, may carry information in both time directions for prediction of a dynamical systems parameter.

This paper proposes a novel approach to analyze the observability of a stochastic linear or non-linear dynamical system. Unlike the traditional observability analysis approach, our approach is fully derivative-free, resulting in a low complexity and an easy implementation. Furthermore, it can not only analyze the observability of each system state, but also it can assess the effect of the observation and the system noise on the system observability, which is not considered by the traditional approach. Specifically, using the generalized polynomial chaos expansion, an observability-coefficient matrix is first calculated, which enables us to judge whether the system is observable or not. Then, the puny and brawny observability indexes are assessed to quantify the degree of observability. The equivalence between the conventional method and the proposed method for a continuous-time linear time-invariant system is proved in the appendix. The effectiveness of the proposed approach is mathematically proved and its good performance is demonstrated in several test cases.

In this paper, we propose a general decentralised probabilistic control framework for a class of complex stochastic systems with switching modes. Probabilistic state space models are exploited to characterise the subsystems’ dynamical behaviours constituting a complex dynamical system, thus providing a complete description of the subsystems components. To address the variations in the operational modes of the subsystems, the Mixture Density Network (MDN) is applied here to identify the subsystems modes and provides estimates for the system dynamic distributions. Besides, to harmonise the actions between the subsystems, the probabilistic message passing methodology is utilised to provide communication between neighbouring subsystems. Based on the MDN model and the neighbours subsystems information via message passing, the general solution of the fully probabilistic decentralised randomised controller which minimises the Kullback-Leibler divergence (KLD) between the actual and its ideal distributions is then obtained. Moreover, a numerical example is presented to illustrate the effectiveness and the usefulness of our novel proposed framework.

In this paper, a method based on alternating direction method of multipliers (ADMM) algorithm is proposed to solve the energy management problem for fuel cell vehicles (FCVs). Taking the minimum hydrogen consumption as the objective function, a new fuel cell dynamical system and a whole vehicle model that conform to the property of convex function are constructed, and an augmented Lagrangian equation with its scaled dual form is established, which satisfies the standard normal form of ADMM algorithm. For the first time, the ADMM algorithm is applied to energy management system of FCVs. In the meantime, a cyclic constraint inspection strategy is introduced to have a further precise control of the power of the fuel cell system so that it can meet the dynamic change constraints of the power. Compared with dynamic programming (DP) algorithm off-line calculation results, the simulation results of the proposed method indicate that the calculation of this algorithm is approximately 20–200 times faster than the former in 9 standard driving cycles. And under the two energy management modes by power maintenance and consumption, the deviation results in terms of the hydrogen consumption derived from the proposed method and DP were 2.37% and 1.06% respectively.

The bond graph (BG) method is an appropriate tool to model hybrid systems. The automated extraction of state equations from BG is the main drawback of this method. In this paper, a new method has b...

Differing from the traditional wavelet neural network, a special type of discrete-time multiscale wavelet neural network (MWNN) using mesh grid is presented and investigated to solve the problem of identification of the autonomous nonlinear dynamical system. Inspired by the multiscale perception of biological neurons and the concept of continuous wavelet theory, multiscale and mesh grid proposed in this paper can be regarded as scale transformation and time translation in the mechanism of MWNN. For the convenience of digital processor realization, discrete-time expressions of weights updating and errors iteration are inferred by the Taylor expansion. In order to ensure the convergence of performance of this discrete-time model, the relation between the constant C in the equation of error iteration and sampling interval has been discovered by applying Z transform theory. The tracking error of autonomous nonlinear dynamical system will converge to the neighborhood of zero, which has been testified by discrete-time Lyapunov stability theory. For comparative purposes, discrete-time MWNN, Raised-Cosine Radial Basis Function Neural Network (RCRBFNN) and Gaussian Radial Basis Function Neural Network (GRBFNN) are used for solving the problem of autonomous nonlinear dynamical system identification. The Lorenz system and clinical electrocardiogram (ECG) dynamical system are applied to test the efficacy and superiority of the proposed discrete-time MWNN, in comparison with GRBFNN and RCRBFNN.

Dynamical behaviors and generalized synchronization play an important role in chaotic application. In this paper, bi-stability and the Hamilton energy are discussed in a one-parameter chaotic system in complex field. Then, based on the Lyapunov stability theory, a scheme of complex generalized synchronization with regard to a complex vector map (CGS) is designed for complex dynamical system with complex parameters. Finally, CGS of coexisting chaotic attractors as well as two strictly different complex chaotic systems is achieved.

How to make full use of the evolution information of chaotic systems for time-series prediction is a difficult issue in dynamical system modeling. In this article, we propose a maximum information exploitation broad learning system (MIE-BLS) for extreme information utilization of large-scale chaotic time-series modeling. An improved leaky integrator dynamical reservoir is introduced in order to capture the linear information of chaotic systems effectively. It can not only capture the information of the current state but also achieve the compromise with historical states in the dynamical system. Furthermore, the feature is mapped to the enhancement layer by nonlinear random mapping to exploit nonlinear information. The cascading mechanism promotes the information propagation and achieves feature reactivation in dynamical modeling. Discussions about maximum information exploration and the comparisons with ResNet, DenseNet, and HighwayNet are presented in this article. Simulation results on four large-scale data sets illustrate that MIE-BLS could achieve better performance of information exploration in large-scale dynamical system modeling.

Many control systems incorporate input from actuators. The actuators can be represented as force or displacement input. When an actuator is modeled as a displacement input, there are many practical applications for which the displacement is not smooth and its time derivative, i.e. the velocity, is not continuous everywhere. Numerical study for such displacement input becomes tedious since the state variables experience a finite jump at the time of acceleration singularity. Although the acceleration singularity can be avoided if the actuator dynamics is modeled by a force input, such an approach increases the order of the overall system. To preserve the simplicity of the displacement input of the actuators, we propose transforms of state variables so that only the velocity of the displacement input appears explicitly in the equations of the dynamical system. In short, we introduce transforms that reduce the order of time derivatives of the input. This approach simplifies the numerical solutions for non-smooth displacement input. Moreover, when the transform is applied to a parametrically excited pendulum, the stabilizing effect of the parametric excitation is shown explicitly.

In this article, an observer-based adaptive hybrid attitude controller is presented that overcomes the topological obstructions associated with the global attitude stabilization. This hybrid controller consists of two major parts: an adaptive trimodal hybrid controller to eliminate $unwinding$ phenomenon, and a centrally synergistic potential function. In the adaptive trimodal hybrid controller, the first logic variable selects, which quaternion representation of the desired attitude should be tracked. Furthermore, an adaptive technique is introduced to collaborate with the second logic variable to adapt the hysteresis width. The third logic variable controls the mismatch between the auxiliary dynamical system and the actual attitude. Moreover, an auxiliary dynamical system and a nonlinear angular velocity observer are presented whose outputs are utilized in the proposed feedback control law to attain the necessary damping. Furthermore, a new centrally synergistic potential function is presented to cooperate with the trimodal hybrid controller to overcome the topological obstruction associated with the vector bundle structure. The robust global asymptotic stability of the consequent closed-loop system is guaranteed through a Lyapunov analysis. A comparative analysis in simulations illustrates the effectiveness and superiority of the proposed control scheme.

Discrete-time stochastic systems are an essential modelling tool for many engineering systems. We consider stochastic control systems that are evolving over continuous spaces. For this class of models, methods for the formal verification and synthesis of control strategies are computationally hard and generally rely on the use of approximate abstractions. Building on approximate abstractions, we compute control strategies with lower- and upper-bounds for satisfying unbounded temporal logic specifications. Firstly, robust dynamic programming mappings over the abstract system are introduced to solve the control synthesis and verification problem. These mappings yield a control strategy and a unique lower bound on the satisfaction probability for temporal logic specifications that is robust to the incurred approximation errors. Secondly, upper-bounds on the satisfaction probability are quantified, and properties of the mappings are analysed and discussed. Finally, we show the implications of these results for linear stochastic dynamic systems with a continuous state space. This abstraction-based synthesis framework is shown to be able to handle infinite-horizon properties. Approximation errors expressed as deviations in the outputs of the models and as deviations in the probabilistic transitions are allowed and are quantified using approximate stochastic simulation relations.