Trajectories Of Control System Research Articles

Adaptive control of an amplified piezoelectric-driven micropositioning stage based on notch filter structure inspiration and neural network online tuning

Abstract Piezoelectric-driven flexure micro/nanopositioning stages often exhibit a low-damping resonance mode, which can easily excite mechanical resonance during high-speed movement, and significantly impact the control system's stability, control bandwidth, and trajectory tracking accuracy. To mitigate the reliance on the precise modeling of stage dynamics inherent in current resonant controllers, an adaptive control method based on a back propagation (BP) neural network was designed to suppress resonance in real-time. First, a piezoelectric-driven flexure micro/nanopositioning stage system was constructed. Next, a feedback controller similar to a notch filter was designed, with bilinear transformation applied based on the system's inherent parameters to determine the initial values. Finally, the designed adaptive control method was tested through trajectory tracking experiments using a triangular wave signal. The experimental results showed that, when tracking the triangular wave signal, the maximum tracking error was reduced by 72.66% compared to proportional-integral (PI) control alone and by 68.66% compared to proportional integral control combined with a traditional notch filter. The tracking results demonstrate a significant improvement in the stage's stability and tracking accuracy.

On Almost Periodic Trajectories of Control Systems with Feedback in the Form of Sweeping Processes

On Almost Periodic Trajectories of Control Systems with Feedback in the Form of Sweeping Processes

Linear obstacles in linear systems, and ways to avoid them

We describe the cohomology ring of the space of trajectories of linear control systems avoiding linear obstacles.

Trajectories of Affine Control Systems and Geodesics of a Spacetime with a Causal Killing Vector Field

We study the geodesic connectedness of a globally hyperbolic spacetime (M,g) admitting a complete smooth Cauchy hypersurface S and endowed with a complete causal Killing vector field K. The main assumptions are that the kernel distribution D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal D$\\end{document} of the one-form induced by K on S is non-integrable and that the gradient of g(K,K) is orthogonal to D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {D}$\\end{document}. We approximate the metric g by metrics gε smoothly depending on a real parameter ε and admitting K as a timelike Killing vector field. A known existence result for geodesics of such type of metrics provides a sequence of approximating solutions, joining two given points, of the geodesic equations of (M,g) and whose Lorentzian energy turns out to be bounded thanks to an argument involving trajectories of some affine control systems related with D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {D}$\\end{document}.

Approximations of the set of trajectories and integral funnel of the non-linear control systems with Lp norm constraints on the control functions

Abstract In this paper, approximations of the set of trajectories and integral funnel of the control system described by non-linear ordinary differential equation with integral constraint on the control functions are considered. The set of admissible control functions is replaced by a set, consisting of a finite number of piecewise-constant control functions. It is shown that the set of trajectories generated by a finite number of piecewise-constant control functions is an internal approximation of the set of trajectories. Further, each trajectory generated by a piecewise-constant control function is substituted by appropriate Euler’s broken line and it is proved that the set consisting of a finite number of Euler’s broken lines is an approximation of the set of trajectories of given control system. An approximation of the system’s integral funnel by a set consisting of a finite number of points is obtained.

Viability and invariance of systems on metric spaces

We consider a generalized control system on a metric space and investigate necessary and sufficient conditions for viability and invariance of proper subsets, describing state constraints. Viability means that for every initial condition in the set of constraints we can find trajectories of control system starting at this condition and satisfying state constraints forever. Invariance means that every trajectory of control system starting in the set of constraints never violates them. As examples of application we consider controlled continuity equations on the metric space of Borel probability measures having compact support, endowed with the Wasserstein distance, and controlled morphological systems on the space of nonempty compacts subsets of the Euclidean space endowed with the Hausdorff metric. We also provide sufficient conditions for the existence and uniqueness of contingent solutions to the Hamilton–Jacobi–Bellman equation on proper metric spaces.

On the properties of the set of trajectories of nonlinear control systems with integral constraints on the control functions

MSC: 93C10, 93B03, 93B35 DOI: 10.21538/0134-4889-2022-28-3-274-284 The control systems described by nonlinear differential equations and integral constraints on the control functions are studied. Admissible control functions are chosen from a closed ball of the space $L_p,$ $p\in (1,\infty]$, with radius $r$ and centered at the origin. It is proved that the set of trajectories

Introducing the Effects of Road Geometry Into Microscopic Traffic Models for Automated Vehicles

Road geometry (e.g. slope and curvature) has significant impacts on driving behaviours of low-level automated vehicles (AVs), but it has been largely ignored in microscopic traffic models. To capture these effects, this study proposes a generic approach to extend any (free-flow or car-following) microscopic models characterized by acceleration functions. To this end, three model extensions are developed, each of which can use different submodels for comparison. Their effectiveness is demonstrated with a microscopic free-flow model that represents the AVs’ control logic. Finally, all possible combinations of the microscopic model and the model extensions are calibrated and cross-validated against data sets, which contain empirical trajectories of commercial adaptive cruise control (ACC) systems on different test tracks. Results suggest that two submodels, i.e., the nonlinear vehicle dynamics (NVD) and the radius difference method (RDM), can extend the microscopic model to effectively capture the effects of road slope and road curvature, respectively, on automated driving. Specifically, the NVD is the dominant factor contributing to increasing (by 34.9% on average) model accuracy. In addition, when simulating reckless turning behaviours (i.e. the vehicle turns at high speeds), the inclusion of the developed RDM is significant for model performance, and the models extended with both the NVD and the RDM can achieve the largest accuracy gains (39.6%).

Design of Picking Robot Manipulator Control System Based on Fuzzy Compensation RBF Neural Network

Industrial manipulator occupies a very important position in industrial production. The tracking control of its control system and joint trajectory has always been a research hotspot. But the manipulator is a multi input multi output system, which has the characteristics of nonlinearity and strong coupling. Radial basis function (RBF) neural network has high nonlinear mapping ability. In this paper, the structure characteristics, learning algorithm and application of RBF neural network in manipulator control are analyzed. In this paper, the nonlinear approximation property of RBF neural network is theoretically verified. This paper analyzes the basic structure of picking manipulator system in detail. At the same time, the Lagrange Euler method is used to deduce the dynamic equation of the two degree of freedom series manipulator, and the inertia characteristics, Coriolis force and centripetal force characteristics, heavy torque characteristics are analyzed. The nonlinear system model of manipulator based on S-function is established in MATLAB, and the dynamic model is transformed into the form of second-order differential equation to facilitate the introduction of the designed algorithm.

Control Homotopy of Trajectories

The main purpose of this paper is to establish the machinery for doing homotopy of (regular) trajectories of control systems. In a mildly different setting than our earlier work in Colonius et al. (J Differ Equ. 2005; 216:324–53), we require this time two trajectories of a (conic) control system to be homotopic by means of their control parameters and simply call them control homotopic. More precisely, let p be a fixed inial point of the state space manifold and let ep denote the end-point mapping that associates to a given control the terminal point of the corresponding trajectory. Then, we say two trajectories α and β are control homotopic if their corresponding controls u and v belong to the same path component of the fiber (ep)− 1(m) for m = ep(u) = ep(v). Due to this point of view, we constrain in the present work our attention to the study of the set $\mathcal {U}$ of addmissible control as an open subset of a certain Banach space. Control homotopy may hence be viewed as an equivalence relation on $\mathcal {U}$ for which the equivalence classes are the path components of the sets (ep)− 1(m), where m belongs to the (regular) accessible set from p. This interpretation also motivates to deal with notions such as control homotopically trivial andcontrol homotopy chain.

Control and numerical analysis for cancer chaotic system

This article investigates the problem of control of chaotic dynamics of tumor cells, immune system cells, and healthy tissue cells in a three-dimensional cancer model by adaptive control technique. Adaptive control law is derived such that the trajectory of controlled system asymptotically approaches equilibrium point with estimated parameter converges to stabilizing values. A nonlinear control law is designed which change the original chaotic system into a controlled one linear system. In addition, we present and analyze numerical solution of the cancer dynamical system with the help of a discretization technique. Achieved solutions show a comparable results with Runge–Kutta methods. The reliability and accuracy of the proposed technique is presented by comparing numerical results. The used technique has displayed a brilliant prospective in dealing with the numerical solutions of nonlinear dynamical systems.

Trajectory Tracking Control of Autonomous Vehicle With Random Network Delay

Random network delay will introduce uncertainty into trajectory tracking model of the autonomous vehicle, which seriously deteriorates the vehicle's control system stability and trajectory tracking accuracy. In this paper, considering steering angle oscillation caused by random network delay, trajectory tracking system robustness and stability is analyzed and a linear uncertain time-delay system is established. Comprehensively considering control system accuracy, robustness, and computational efficiency in the rolling optimization of Model Predictive Control (MPC), Adaptive Model Predictive Control for Uncertain model (UM-AMPC) algorithm is proposed to predict control variables for the next sampling time and alleviate the target angle discontinuity. This is achieved by operating target angle signal and augmented state variables, which are received by the lower nodes during the period from the current sampling time to network delay upper bound. The hardware-in-the-loop simulation results show that the proposed algorithm can effectively guarantee system stability and tracking accuracy of the autonomous vehicle under random network delay.

Cascade Linearization of Invariant Control Systems

Let Pfaffian system ω define an intrinsically nonlinear control system on manifold M that is invariant under the free, regular action of a Lie group G. The problem of identifying and constructing static feedback linearizable G-quotients of ω was solved in De Dona et al. (2016). Building on these results, the present paper proves that the trajectories of ω can often be expressed as the composition of the trajectories of a static feedback linearizable quotient control system, ω/G, on quotient manifold M/G, and those of a separate control system, γ G , evolving on a principal G-bundle over a jet space. Furthermore, we point out that ω may not only have a static feedback linearizable quotient, ω/G but additionally, γ G itself may possess a static feedback linearizable reduction as well. This enables one to express the trajectories of an intrinsically nonlinear control system as the composition of the trajectories of static feedback linearizable control systems, thereby providing a geometric criterion for the explicit integrability of intrinsically nonlinear systems. Moreover, special integrability properties arise when G is solvable. Examples are presented in which the above phenomena are explicitly demonstrated. An important aspect of the examples is that they gather evidence for the conjecture that our sufficient conditions for explicit integrability are also necessary.

Parameter optimization of the program trajectory of the cellular concrete autoclaving control system

Manufacturing of cellular-concrete products that are widely used in modern construction involves high energy consumption attributable mostly to the process of autoclaving. Insufficient elaboration of issues of mathematical description of autoclaving that would be usable in engineering practice and, consequently, the lack of adequate computational models aimed for practicing technologists, make it necessary to use heuristic approaches to parameter determination for the program trajectories of autoclave automatic control systems (ACS) in manufacturing. Therefore, design and development of autoclaving computational models aimed at parameter optimization of the program trajectory is a very relevant issue. The authors have developed mathematical models of cellular concrete autoclave curing and computational algorithms enabling determination of energy consumption under the variation in dynamic parameters of the automatic system's driver unit. Using the methods of conducting computational experiments under the discrete parameter change of the program trajectory, under the conditions of limitation on the temperature derivative of semi-finished concrete and autoclaving process duration, we have established that completion of the known program trajectory as a polygonal line with two variable slope sections of the third section forming unit with a variable slope, enables reduction of energy consumption up to 8%. The research conducted is specifically focused on the use of the obtained results for optimization of cellular concrete autoclaving ACSs.

Adaptive-Robust Stabilization of the Furuta's Pendulum Via Attractive Ellipsoid Method

This paper focuses on the issue of adaptive-robust stabilization of the Furuta's pendulum around unstable equilibrium where the dynamical model is unknown. The control scheme lies at the lack of the dynamical model as well as external disturbances. The stabilization analysis is based on the attractive ellipsoid method (AEM) for a class of uncertain nonlinear systems having “quasi-Lipschitz” nonlinearities. Even more, a modification of the AEM concept that permits to use online information obtained during the process is suggested here. This adjustment (or adaptation) is made only in some fixed sample times, so that the corresponding gain matrix of the robust controller is given on time interval too. Furthermore, under a specific “regularized persistent excitation condition,” the proposed method guarantees that the controlled system trajectories remain inside an ellipsoid of a minimal size (the minimal size is refereed to as the minimal trace of the corresponding inverse ellipsoidal matrix). Finally, the adaptive process describes a region of attraction (ROA) of the considered system under adaptive-robust nonlinear control law.

Process control of ore crushing using block-oriented predictive model

The problem of developing a system of predictive ore crushing control is considered. A method of forming forecasting control of ore crushing, which is based on static nonlinearities inverting input-output of block-oriented model and approximation of trajectories of control systems of orthonormal Laguerre functions, allowing reduce the problem of determining the sequence of actions to control the problem of quadratic programming. By means of simulation it is found that the proposed system provides higher quality control of transients and calculated load on the control unit as compared to non-linear predictive control. Using the proposed system will improve the efficiency of ore dressing in mining enterprises through the formation and stabilization of the required granulate characteristics of crush ore, which will reduce energy consumption in the next stages of processing, and as a result, reduce the cost of the final product.

Synthesis of a Switching Control Approach Based on the Input-Output Feedback Linearization

In this work, we explore the input-output feedback linearization in an attempt to suggest a switching rule that is liable to find out the appropriate diffeomorphism that can improve the transient behavior and let the switching systems stable. Unlike the approach based on one diffeomorphism, our approach may guarantee the behavior of the controlled system trajectories as it rests on multi-diffeomorphisms. Finally, the study of a numerical example is presented in order to show the effectiveness of the proposed approach.

An ISMC Approach to Robust Stabilization of Uncertain Stochastic Time-Delay Systems

Integral sliding mode control (ISMC) of uncertain stochastic systems with time delay is investigated in this paper. Aiming to remove a very restrictive assumption required in most existing ISMC schemes for stochastic systems, a new ISMC approach is proposed. It is shown that the closed-loop control system trajectories can be kept on the integral sliding surface almost surely since the initial time and the stochastic stability of the sliding motion can be guaranteed in terms of linear matrix inequalities. MATLAB simulation results on a numerical example and a practical stream water quality preserving system are provided to show the effectiveness and advantage of the proposed approaches.

Pontryagin Maximum Principle for Control Systems on Infinite Dimensional Manifolds

We discuss a mathematical framework for analysis of optimal control problems on infinite-dimensional manifolds. Such problems arise in study of dynamic optimization for partial differential equations with some symmetry. We develop nonsmooth analysis methods and Lagrangian charts techniques which can be used for study of global variations of optimal trajectories of such control systems and derivation of a maximum principle for them.

A new design of robust H∞ sliding mode control for uncertain stochastic T-S fuzzy time-delay systems.

In this paper, a novel dynamic sliding mode control scheme is proposed for a class of uncertain stochastic nonlinear time-delay systems represented by Takagi-Sugeno fuzzy models. The key advantage of the proposed scheme is that two very restrictive assumptions in most existing sliding mode control approaches for stochastic fuzzy systems have been removed. It is shown that the closed-loop control system trajectories can be driven onto the sliding surface in finite time almost certainly. It is also shown that the stochastic stability of the resulting sliding motion can be guaranteed in terms of linear matrix inequalities; moreover, the sliding-mode controller can be obtained simultaneously. Simulation results illustrating the advantages and effectiveness of the proposed approaches are also provided.