Control Of Nonlinear Dynamical Systems Research Articles

A new result on PD controller design for second order nonlinear uncertain systems

Under what condition a proportional-derivative (PD) controller can stabilize globally second order nonlinear uncertain systems? It is an important and attractive question. This paper will broaden the range of a function space F which is used to describe the uncertainty of nonlinear systems in literature, such that the corresponding result still holds for nonlinear dynamical control systems with more uncertainty. The method is based on some related results in the research about Markus-Yamabe Conjecture.

Nonlinear Dynamics

The modeling and control of nonlinear dynamic systems is challenging in mathematics and engineering [...]

Orbital decompositions and integrable pseudosymmetries of control systems

Local decompositions of nonlinear dynamical control systems are investigated. We use orbital equivalences to transform systems into decomposed forms. In the language of infinite jets, every control system is interpreted as an infinite dimensional manifold. Each orbital decomposition of the system defines an invariant distribution on this manifold. A local basis of this distribution is an integrable pseudosymmetry of the system. An algorithm for constructing orbital decompositions is given and demonstrated by the example of the Kapitsa pendulum system. It is also shown how to use orbital decompositions to plan trajectories.

A Note on the Accurate Computation of Structural Properties for Dynamic Control Systems

In analysing the observability, identifiability, and controllability of smooth (possibly non-linear) dynamic control systems, the authors identified numerous ways to improve both the accuracy and efficiency of their computations. We summarize a few techniques that help to pinpoint exactly what kind of linear dependencies exist between parametric output sensitivities. These relations, in turn, can be traced back with the help of computer algebra software to an exact re-parametrizations of the original model. The suggested techniques to increase the accuracy of results will be presented on the basis of a few examples.

Neuro-Evolution of Continuous-Time Dynamic Process Controllers

Artificial neural networks are means which are, among several other approaches, effectively usable for modelling and control of non-linear dynamic systems. In case of modelling systems input and output signals are a-priori known, supervised learning methods can be used. But in case of controller design of dynamic systems the required (optimal) controller output is a-priori unknown, supervised learning cannot be used. In such case we only can define some criterion function, which represents the required control performance of the closed-loop system. We present a neuro-evolution design for control of a continuous-time controller of non-linear dynamic systems. The controller is represented by an MLP-type artificial neural network. The learning algorithm of the neural network is based on an evolutionary approach with genetic algorithm. An integral-type performance index representing control quality, which is based on closed-loop simulation, is minimised. The results are demonstrated on selected experiments with controller reference value changes as well as with noisy system outputs.

Improved model-free adaptive predictive control method for direct data-driven control of a wastewater treatment process with high performance

In this paper, an improved model-free adaptive predictive control (IMFAPC) method with controller parameters sensitivity discrimination and optimization is proposed, to solve the problem that using conventional methods is difficult to effectively control the nitrate concentration and dissolved oxygen concentration of wastewater treatment process (WWTP). First, aiming at the nonstationary dynamic characteristics of WWTP and the limitation of control accuracy of conventional MFAPC method in the nonlinear dynamic system control, a novel improved MFAPC method is proposed considering the effect of the WWTP output tracking error variation on the control of nitrate concentration and dissolved oxygen concentration. Meanwhile, in order to obtain the optimum control performance, the sensitivity discrimination and optimization for the multi-parameter set of the proposed IMFAPC method are performed by combining the Monte Carlo experiment based multi-parameter sensitivity analysis and the hybrid intelligent optimization. Then, the stability of the proposed IMFAPC scheme is analyzed and proved by the BIBO method. Finally, various data experiments are carried out to verify the proposed method. The experimental results demonstrate that the proposed method can control the nitrate nitrogen and dissolved oxygen concentration rapidly and accurately under different operating conditions.

Modeling, simulation and experimental validation of flow rate of electro-hydraulic hitch control valve of agricultural tractor

Advances in non-linear control theory have made it possible to develop controllers for non-linear dynamic systems in their nature. The low-cost proportional control valves of the hydraulic control system are examples of such a system where there is no linearity due to the structure of the valves and the flaws in the spool. However, the non-linear analysis and regulation of these hydraulic systems cannot be done without a proper valve model. This paper presents a methodology for the development of an efficient unified model of a three-point hitch (TPH) electro-hydraulic proportional control valve control system for agricultural tractors by means of a parameter estimation technique. Modeling and simulation of the proportional control valve was performed using MATLAB Simulink software. Parameter estimation methodology was used to optimize the effective orifice opening of the solenoid valve to meet the flow characteristics available in the manufacturer's technical data sheet for the proportional control valve model parameters. Such unified Simulink models are useful for simulation, practical capability testing, and non-linear control design. Modeling and simulation of electro-hydraulic hitch (EHH) control valve were made and simulation results were compared with actual experimental results. The simulation results of the TPH lifting and lowering time were found to vary 4 and 12.5 %, respectively with the experimental results. This type of parameterized valve models facilitates their implementation in dynamic simulation models of complex hydraulic systems.

A new fuzzy fractal control approach of non-linear dynamic systems: The case of controlling the COVID-19 pandemics

This article is presenting a first attempt on a proposed fuzzy fractal control method for efficiently controlling nonlinear dynamic systems. The main goal is to combine the main advantages of fractal theoretical concepts and fuzzy logic theory for achieving efficient control of nonlinear dynamic systems. The concept coming from Fractal theory, known as the fractal dimension, can be utilized to measure the complexity of the dynamic behavior of a non-linear plant. On the other hand, fuzzy logic theory can be used to represent and capture the expert knowledge in controlling a plant. In addition, fuzzy logic enables to manage the uncertainty involved in the decision-making process for achieving efficient control of a non-linear plant. We illustrate the proposed fuzzy fractal control method with the current worldwide situation that requires achieving an efficient control of the COVID-19 pandemics.

Output feedback fuzzy control of nonlinear dynamic systems: Event‐triggered case

AbstractThe event‐triggered output feedback controller design problem is investigated in this article for a class of discrete‐time Takagi–Sugeno fuzzy systems. We focus on constructing a closed‐loop system by considering the event generator communication mechanism, interval time‐varying delays, and packet dropouts. Under this newly established system, a less conservative stability condition is obtained using the Wirtinger‐based inequality and reciprocally convex approach, which guarantees that the corresponding augmented system is asymptotically stable in the mean‐squared sense with a prespecified – performance. The solution of the designed output feedback controller parameters can then be obtained by variable substitution method and by conversion of the controller parameters. Finally, two examples are presented to demonstrate the effectiveness of the proposed controller design technique.

Analysis of the Zhang Neural Network and its Application for the control of Nonlinear Dynamical Systems

This research article presents a development of the Zhang Recurrent Neural Network (Z-RNN) model to estimate the on-line solution to linear time-varying matrix equations. This Z-RNN model is further used to develop a control strategy for nonlinear systems by estimating the solution to a Lyapunov equation on-line, without approximations and linearization. The proposed concept is validated using a 4th order Inverted Pendulum on a Cart model and it has been compared with the Linear Quadratic Regulator. The proposed technique offers a significantly lower cost and better performance. The issues with stability of the Z-RNN and sampling rate is also discussed in detail, along with the choice of parameters involved.

Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

SummaryIn this article, we consider extremum seeking problems for a general class of nonlinear dynamic control systems. The main result of the article is a broad family of control laws which optimize the steady‐state performance of the system. We prove practical asymptotic stability of the optimal steady‐state and, moreover, propose sufficient conditions for the asymptotic stability in the sense of Lyapunov. The results generalize and extend existing results which are based on Lie bracket approximations. In particular, our approach does not rely on singular perturbation theory, as commonly used in extremum seeking of nonlinear dynamic systems.

Uniqueness Solution Of Abstract Fractional Order Nonlinear Dynamical Control Problems

The aim of this paper is to investigate the Uniqueness solution of Cauchy Problem represented for fractional order nonlinear dynamical control system involving certain control input and their approach of investigated depended on commutative composite semigroup and some certain conditions in certain space.

Direct Adaptive Controllers for a Multi Input Multi Output Process

Control of nonlinear dynamic systems is of great practical importance, since virtually all real world systems belong to this class. The prevalent approach is to use a model of the process linearized about the steady state operating point to design linear controllers. The Multi Input Multi Output (MIMO) process considered in this paper has varying system dynamics. This paper focuses on the performance evaluation of multi-loop Adaptive MIT(AMIT) controller and Adaptive PI(API) controller with vanishing adaptation technique for a laboratory interacting coupled tank MIMO process.

Robust Under-Approximations and Application to Reachability of Non-Linear Control Systems With Disturbances

We describe a set-based approach, relying on mean-value extensions, for computing guaranteed under-approximations of ranges (or images) of continuously differentiable functions f from Rm to Rn, including what we call robust ranges, i.e., ranges of functions under adversarial uncertainties. Our method is capable of computing efficiently, at a low computational cost, full n-dimensional subsets of the image of f. As an application, we show how to compute under-approximations of robust reachable sets of non-linear controlled dynamical systems under timevarying uncertainties, which is central to many verification problems in control theory.

Data-Driven Model Predictive Control using Interpolated Koopman Generators

In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via Dynamic Mode Decomposition, a quantization approach was recently proposed in [Peitz \& Klus, Automatica 106, 2019]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls. We show that when using the Koopman generator, this relaxation --- realized by linear interpolation between two operators --- does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.

CONTROL AND ESTIMATION FOR A CLASS OF IMPULSIVE DYNAMICAL SYSTEMS

The nonlinear dynamical control system with uncertainty in initial states and parameters is studied. It is assumed that the dynamic system has a special structure in which the system nonlinearity is due to the presence of quadratic forms in system velocities. The case of combined controls is studied here when both classical measurable control functions and the controls generated by vector measures are allowed. We present several theoretical schemes and the estimating algorithms allowing to find the upper bounds for reachable sets of the studied control system. The research develops the techniques of the ellipsoidal calculus and of the theory of evolution equations for set-valued states of dynamical systems having in their description the uncertainty of set-membership kind. Numerical results of system modeling based on the proposed methods are included.

Event-triggered sliding mode control of nonlinear dynamic systems

In this work, the problem of sliding mode observer design is investigated for a class of repeated scalar nonlinear systems via dynamic event-triggered approach. By constructing an event-triggered observer and a sliding mode surface, the system dynamics is obtained and an event-triggered controller is designed to ensure that the closed-loop system trajectories are restricted to the pre-specified sliding region. Sufficient conditions are provided to guarantee that the system dynamics is asymptotically stable with a desired H∞ performance level. Two illustrative examples are provided to demonstrate the effectiveness and potential of the proposed design method.

Theory and implementation of the multiple valued iterative dynamics algorithms for a class of singularly disturbed nonlinear control dynamical systems

We discuss the visualization algorithms and their justification theorems of the steady state sets of a class of nonlinear disturbed control dynamical systems. We focus upon the systems that include sudden large disturbance that is significant enough to cause a bifurcation (sudden change of the qualitative behavior) of the dynamics. We call such a disturbance, the singular disturbance. Under the singular disturbance, the traditional models of control theory generally fail to produce the desired point-convergence. Instead, the steady state sets often exhibit a fractal structure. The multiple valued iterative dynamics modeling proved to be a useful tool establishing the theoretical framework for this case. The main objective of this paper is to establish the theoretical foundation of this framework. First, we present our model and state the basic theorems that justify the algorithmic aspect of the model. Next, we prove the justification theorems. We use the iteration of the predecessor operators and the successor operators as the main tool. Afterwards, we discuss the usage of the justification theorems. We review the known visualization algorithms, with a due emphases on the usage of our main theorems.

U-neural network-enhanced control of nonlinear dynamic systems

Neural networks have been widely used as an approximation for nonlinear dynamic plants in control system design, but they are almost never used as a proper dynamic inverter, particularly with dynamic models available in advance. This study presents a universally feasible U-neural network (U-NN) structure to facilitate the designing control of all dynamic systems modelled with linear/nonlinear polynomial/state space equations. With the presented U-NN, this study proposes a procedure for a model independent control system design, U-control framework/platform. The procedure, against a traditional model based on control system design and a model free/data driven-based design (such as PID control, iterative learning control, and model free control), removes the boundaries of the linear/nonlinear and polynomial/state space model sets, where the model structures are universally treated within the new framework. Furthermore, this study analyses the U-control properties and gives a step-by-step implementation procedure. Several bench test examples demonstrate the effectiveness of the design procedure, as well as the routines in applications.

Pattern-based NN control for uncertain pure-feedback nonlinear systems

Implementing human-like learning and control for nonlinear dynamical systems operating in different control situations is an important and challenging issue. This paper presents a pattern-based neural network (NN) control strategy for nonlinear pure-feedback systems via deterministic learning (DL). Firstly, an appropriately designed adaptive neural dynamic surface controller is proposed to achieve the finite time tracking control. By analyzing the recurrent property of NN input signals, a partial persistent excitation (PE) condition for radial basis function (RBF) network is established, the implicit desired control dynamics under different control situations are accurately identified via DL in the case that the dimension of NN input is reduced. And a set of pattern-based experienced actual and virtual controllers is constructed using the learned knowledge. Secondly, to classify different control situations, when the system is operating in different control situations but controlled by current normal experienced controller, the dynamics of each subsystem are accurately identified via DL, n sets of dynamical estimators are constructed using the learned knowledge. Thirdly, in the recognition phase, n sets of residuals are achieved by comparing each set of estimators with the monitored system, sudden change in the control situation is rapidly recognized based on the principle of the earliest occurrence of the minimum residual. Finally, in the control phase, according to the recognition result, the correct experienced actual and virtual controllers will be selected to control the plant, guaranteed stability and superior control performance are achieved without any further re-adaptation online. Simulation studies are given to verify the proposed scheme can not only acquire and memorize knowledge like humans, but also reuse the learned knowledge to achieve rapid recognition and control of current control situation.