Dynamical Control Systems Research Articles

Decomposition and equivalence of general nonlinear dynamical control systems

AbstractProjection and equivalence concepts have been widely studied in the literature. In the present paper two nonlinear systems with the same number of inputs, but not the same number of state variables, are considered. Under mild assumptions, necessary and sufficient conditions are given for the existence of a submersion such that the higher dimensional system projects locally onto the other one. The solution to this problem has relevant applications, for instance in robotics. It includes as a special case the equivalence of nonlinear systems with no particular structure.

On regularity conditions in higher-order state-constrained control problems

This work is aimed at the study of optimal control problems with state constraints of higher order. Such problems are due to multi-level dynamical control systems of triangle-type which are often encountered in various engineering applications, notably in Robotics. In the framework of the proposed investigation, a notion of higher-order regularity with respect to the state constraints (termed as k-regularity) is formulated. Under such a condition, a strengthened variant of Pontryagin’s maximum principle is derived in which the non-triviality condition is pointwise and thereby allows for the informative maximum condition. An example of a 3-order control problem is considered, which demonstrates the essentiality of k-regularity for pointwise non-triviality and also shows how the presented necessary optimality conditions can be applied.

Analytical modelling and design of linear controlled dynamic systems

A generalized canonical mathematical model of multidimensional controlled mining transportation complexes and other technical systems in form by A.M. Letov is considered. A principle of symmetry and an algebraic concept are in the basis of the developed analytical methods of design. A principle of symmetry is realized on a set of indexes of roots of a characteristic equation of the system and on a set of indexes of phase coordinates of the mathematical model. The problem of quality of dynamic processes in time is reduced to an algebraic problem of distribution of roots of a corresponding characteristic equation in a complex plane. An analogy in a procedure of transformation of a characteristic determinant into a polynomial and a structure of elementary symmetrical polynomials of roots is established. A new formulation of an analytical representation of change of phase coordinates in time in a form of ordered determinants with respect to indexes of roots and indexes of phase coordinates is obtained based on residue theorem. An illustration of the developed analytical method of design is performed on a special case of a well-known controlled technical system of the fourth order.

Contract composition for dynamical control systems: Definition and verification using linear programming

Designing large-scale control systems to satisfy complex specifications is hard in practice, as most formal methods are limited to systems of modest size. Contract theory has been proposed as a modular alternative, in which specifications are defined by assumptions on the input to a component and guarantees on its output. However, current contract-based methods for control systems either prescribe guarantees on the state of the system, going against the spirit of contract theory, or are not supported by efficient computational tools. In this paper, we present a contract-based modular framework for discrete-time dynamical control systems. We extend the definition of contracts by allowing the assumption on the input at a time k to depend on outputs up to time k−1, which is essential when considering feedback systems. We also define contract composition for arbitrary interconnection topologies, and prove that this notion supports modular design, analysis and verification. This is done using graph theory methods, and specifically using the notions of topological ordering and backward-reachable nodes. Lastly, we present an algorithm for verifying vertical contracts, which are claims of the form “the conjunction of given component-level contracts implies given contract on the integrated system”. These algorithms are based on linear programming, and scale linearly with the number of components in the interconnected network. A numerical example is provided to demonstrate the scalability of the presented approach, as well as the modularity achieved by using it.

Spike sorting in the presence of stimulation artifacts: a dynamical control systems approach

Objective. Bi-directional electronic neural interfaces, capable of both electrical recording and stimulation, communicate with the nervous system to permit precise calibration of electrical inputs by capturing the evoked neural responses. However, one significant challenge is that stimulation artifacts often mask the actual neural signals. To address this issue, we introduce a novel approach that employs dynamical control systems to detect and decipher electrically evoked neural activity despite the presence of electrical artifacts. Approach. Our proposed method leverages the unique spatiotemporal patterns of neural activity and electrical artifacts to distinguish and identify individual neural spikes. We designed distinctive dynamical models for both the stimulation artifact and each neuron observed during spontaneous neural activity. We can estimate which neurons were active by analyzing the recorded voltage responses across multiple electrodes post-stimulation. This technique also allows us to exclude signals from electrodes heavily affected by stimulation artifacts, such as the stimulating electrode itself, yet still accurately differentiate between evoked spikes and electrical artifacts. Main results. We applied our method to high-density multi-electrode recordings from the primate retina in an ex vivo setup, using a grid of 512 electrodes. Through repeated electrical stimulations at varying amplitudes, we were able to construct activation curves for each neuron. The curves obtained with our method closely resembled those derived from manual spike sorting. Additionally, the stimulation thresholds we estimated strongly agreed with those determined through manual analysis, demonstrating high reliability ( R2=0.951 for human 1 and R2=0.944 for human 2). Significance. Our method can effectively separate evoked neural spikes from stimulation artifacts by exploiting the distinct spatiotemporal propagation patterns captured by a dense, large-scale multi-electrode array. This technique holds promise for future applications in real-time closed-loop stimulation systems and for managing multi-channel stimulation strategies.

Hybrid quantum-classical control problems

<p>The notion of hybrid quantum-classical control system was introduced as a control dynamical system which combined classical and quantum degrees of freedom. Classical and quantum objects were combined within a geometrical description of both types of systems. We also considered the notion of hybrid quantum-classical controllability by means of the usual definitions of geometric control theory, and we discussed how the different concepts associated to quantum controllability are lost in the hybrid context because of the nonlinearity of the dynamics. We also considered several examples of physically relevant problems, such as the spin-boson model or the notion of hybrid spline.</p>

On the boundedness of solutions of some fuzzy dynamical control systems

<abstract> <p>The asymptotic behavior of solutions of fuzzy control systems is a component of the study of fuzzy control theory. The study of stability for T-S (Takagi-Sugeno) fuzzy systems, which process qualitative data through linguistic expressions, is the subject of this paper. Asymptotic stability is conservative in many real-world applications due to measurement noise and other disruptions. The ultimate limit, which indicates that the mistakes stay in a specific area close to the origin after a long enough amount of time, is a crucial characteristic that is frequently defined for such systems. We are interested with the problem of the state feedback controller for T-S fuzzy models with uncertainties where the global exponential ultimate boundedness of solutions is studied for certain fuzzy control systems. We use common quadratic Lyapunov function and parallel distributed compensation controller techniques to study the asymptotic behavior of the solutions of fuzzy control system in presence of perturbations. An example demonstrating the validity of the main result is discussed.</p> </abstract>

ATTAINABLE SETS OF INTEGRAL CONSTRAINED SEIR CONTROL SYSTEM WITH NONLINEAR INCIDENCE

In this survey, we consider the dynamics of a contagious disease spread by employing a nonlinear dynamical control system of differential equations. It considers treatment and vaccination as key control parameters to discern their influence on disease control. The study, approximate the attainable sets of a given control system and presents visual results, while also discussing potential biological applications of their findings.

Neural Network Trajectory Tracking Control on Electromagnetic Suspension Systems

A new adaptive-like neural control strategy for motion reference trajectory tracking for a nonlinear electromagnetic suspension dynamic system is introduced. Artificial neural networks, differential flatness and sliding modes are strategically integrated in the presented adaptive neural network control design approach. The robustness and efficiency of the magnetic suspension control system on desired smooth position reference profile tracking can be improved in this fashion. A single levitation control parameter is tuned on-line from a neural adaptive perspective by using information of the reference trajectory tracking error signal only. The sliding mode discontinuous control action is approximated by a neural network-based adaptive continuous control function. Control design is firstly developed from theoretical modelling of the nonlinear physical system. Next, dependency on theoretical modelling of the nonlinear dynamic system is substantially reduced by integrating B-spline neural networks and sliding modes in the electromagnetic levitation control technique. On-line accurate estimation of uncertainty, unmeasured external disturbances and uncertain nonlinearities are conveniently evaded. The effective performance of the robust trajectory tracking levitation control approach is depicted for multiple simulation operating scenarios. The capability of active disturbance suppression is furthermore evidenced. The presented B-spline neural network trajectory tracking control design approach based on sliding modes and differential flatness can be extended to other controllable complex uncertain nonlinear dynamic systems where internal and external disturbances represent a relevant issue. Computer simulations and analytical results demonstrate the effective performance of the new adaptive neural control method.

From Zeroing Dynamics to Zeroing-Gradient Dynamics for Solving Tracking Control Problem of Robot Manipulator Dynamic System with Linear Output or Nonlinear Output

With the vigorous development of mechanical intelligence in industrial manufacturing, tracking control dynamic systems have been widely applied in many aspects of industry. In this paper, we present one theorem to discuss the validity condition of a ZD model with order-n for solving the tracking control problem of a nonlinear problem by utilizing a Lie derivative. Moreover, we also give the unified formula of the ZD model with order-n and rigorously prove it mathematically. In addition, we present three other theorems to give the global exponential convergence property of the ZD controller u(t), and the steady-state tracking error bound of the ZGD controller u(t), and the radius bound where the steady-state tracking error converges exponentially. Finally, simulations are conducted to demonstrate the validity and parameter influences of the ZD model and ZGD model for solving the tracking control problem with a single linear or nonlinear output of the single-link manipulator with flexible joints.

Delayed reset control design for uncertain fractional-order systems with actuator faults via dynamic output feedback scheme

In this work, the stabilization of uncertain fractional-order dynamical control system with actuator faults and external disturbances will be addressed. Specifically, a fractional-order reset control system consisting of two sub-systems namely flow and jump subsystems will be designed. Based on the triggering conditions of reset law, a particular subsystem of the control system will be invoked. Meanwhile, the uncertainty and time-delay will be incorporated in the reset control system. Then the output of the reset control system, that is the control input of the system is designed to be fault-tolerant due to the presence of actuator faults. Further, the fractional-order dependent linear matrix inequalities (LMIs) will be derived by an appropriate choice of a Lyapunov function and other matrix inequalities. Thus, by solving the obtained LMIs, the gain values can be obtained guaranteeing the stabilization of the closed loop system with prescribed disturbance attenuation level. Later, the theoretical results are authenticated by providing numerical simulation results.

A New Approach to Nonlinear State Observation for Affine Control Dynamical Systems

In this work, a Nonlinear Higher Order Extended State Observer (NHOESO) is presented to replace the Linear Extended State Observer (LESO) used in Conventional Active Disturbance Rejection Control (C-ADRC) solutions. In the NHOESO, the standard LESO is completed with a two-term smooth nonlinear function with saturation-like characteristics. The proposed novel NHOESO enables precise observation of the generalized disturbances with higher-order derivatives. The stability of the NHOESO is examined with the aid of the Lyapunov method. A simulation of an uncertain nonlinear Single-Input–Single-Output (SISO) system with time-varying external disturbances confirms that the proposed NHOESO copes well with the generalized disturbance, which is not true for other ESOs.

Robust missile autopilot design based on dynamic surface control

Since the dynamical system and control system of the missile are typically nonlinear, an effective acceleration tracking autopilot is designed using the dynamic surface control (DSC) technique in order to make the missile control system more robust despite the uncertainty of the dynamical parameters and the presence of disturbances. Firstly, the nonlinear mathematical model of the tail-controlled missile is decomposed into slow acceleration dynamics and fast pitch rate dynamics based on the naturally existing time scale separation. Secondly, the controller based on DSC is designed after obtaining the linear dynamics characteristics of the slow and fast subsystems. An extended state observer is used to detect the uncertainty of the system state variables and aerodynamic parameters to achieve the compensation of the control law. The closed-loop stability of the controller is derived and rigorously analyzed. Finally, the effectiveness and robustness of the design is verified by Monte Carlo simulation considering different initial conditions and parameter uptake. Simulation results illustrate that the missile autopilot based DSC controller achieves better performance and robustness than the other two well-known autopilots. The method proposed in this paper is applied to the design of a missile autopilot, and the results show that the acceleration tracking autopilot based on the DSC controller can ensure accurate tracking of the required commands and has better performance.

Nonlinear dynamical systems to study epileptic seizures and extract average amount of mutual information from encephalographs – Part I

In this study, we apply the non-linear dynamical systems theory for the assessment built on recurrence-quantification analysis technique for characterizing—differentiating non-linear electro encephalograph (EEG) signals dynamics. The technique offers convenient quantifiable data plus information over normal, tumultuous, or probability and statistical stochastic properties of inherent systems dynamics theory. The R.Q.A-established processes as the quantifiable mathematical features of non-linear electroencephalograph signals dynamics. Average amount of mutual information (AAMI) applied to compute highly applicable feature-manifestation sub-sets out of R.Q.A-built centered-features. The chosen features were then fed into the computer using artificial intelligence based neural net-works for clustering the data of encephalograph-signals to identify ictic(i.e.,ictal), inter ictal, followed by state of controls. The study is implemented by validating R.Q.A with a data base for various issues of categorization. Results showed that the combination of five selected features created on AAMI attained the precision of100% and proves dominance of R.Q.A. Nonlinear dynamical control systems theory and analysis techniques centered on R.Q.A can be used as an appropriate methodology for distinguishing the non-linear systems dynamics of encephalograph signals data also epileptic seizures tracing.

Model-Free Undetectable Attacks on Linear Systems Using LWE-Based Encryption

We show that the homomorphic property, a desired property in encrypted control, can lead to failure in the cyber defense of a dynamical control system from undetectable attacks, even though individual signal sequences remain unknown to the attacker. We consider an encryption method based on the Learning with Errors (LWE) problem and demonstrate how model-free undetectable attacks on linear systems over integers can be computed from sampled inputs and outputs that are encrypted. Previous work has shown that computing such attacks is possible on nonencrypted systems. Applying this earlier work to our scenario, with minor modifications, typically amplifies the error in encrypted messages unless a short vector problem is solved. Given that an attacker obtains a short vector, we derive the probability that the attack is detected and show how it explicitly depends on the encryption parameters. Finally, we simulate an attack obtained by our method on an encrypted linear system over integers and conduct an analysis of the probability that the attack will be detected.

Controllability of damped dynamical systems modelled by Hilfer fractional derivatives

In this article, we investigate the controllability of damped dynamical system modelled by Hilfer derivative of fractional order and . The Mittag-Leffler matrix function of two parameters has been used to represent the solution of Hilfer fractional problem of damped dynamical system. With the use of the Schawder fixed point theorem and the contraction mapping theorem, the existence and uniqueness of the linear and non-linear system of Hilfer dynamical control system are determined. More specifically, fractional calculus are used to provide sufficient criteria for the controllability results. Finally, examples are given to show how the key results might be justified.

Guest Editorial: Learning, optimisation and control of cyber‐physical systems

Guest Editorial: Learning, optimisation and control of cyber‐physical systems

Contraction criteria for incremental stability of differential systems with discontinuous right-hand sides

Incremental stability analysis, which plays a crucial role in dynamical systems, especially nonlinear systems, has attracted more and more concern for its applications in real world control systems nowadays. This paper presents a constructive approach to derive sufficient conditions for incremental exponential stability of the Filippov solutions of a class of differential systems with discontinuous right-hand sides, by introducing a sequence of continuous dynamical systems which is uniformly contracting and approximating the Filippov systems in terms of the evolution map graphs. Afterwards, several applications of the derived theoretical results are explored. Some specific classes of control dynamical systems with discontinuous right-hand sides are studied and relative detailed conditions are presented to show the power of the present approach to investigate the stability of switched dynamical systems, Hopfield neural network with discontinuous activations and sliding mode control.

Generating Nesterov’s accelerated gradient algorithm by using optimal control theory for optimization

Nesterov’s accelerated gradient algorithm is derived from first principles. The first principles are founded on the recently-developed optimal control theory for optimization. This theory frames a generic optimization problem as an optimal control problem. The necessary conditions for optimal control produce a controllable dynamical system. Stabilizing this system by dissipating a simple, quadratic “control” Lyapunov function generates an equation that produces Nesterov’s algorithm. In this context, the proposed theory solves the long-standing mystery surrounding the algorithm.

Finite-Data Error Bounds for Koopman-Based Prediction and Control

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein–Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.