Unstable Nonlinear System Research Articles

A novel control approach for discrete-time 2D Roesser systems under input saturation and intrinsic nonlinearity: A region of stability investigation

This paper presents the control system design for the nonlinear discrete-time Roesser systems under saturated control input. A deficiency in the existing literature has been addressed in this study for the control of two-dimensional (2D) systems. A generalized boundary condition analysis is presented in the form of tractable invariant sets for the derivation of region of stability to design a suitable state feedback control. A mechanism for the evaluation of controller parameters is presented by considering the generalized structure of 2D Lyapunov function. The work has been extended when external disturbances are present in the 2D systems, and a robust design condition by accounting the perturbations as well as the input saturation has been established. In comparison to the existing works, a control system design for 2D Roesser systems under input saturation and intrinsic nonlinearity by application of a generalized Lyapunov function has been revealed. Furthermore, a tractable 2D region of stability for the local control of the 2D systems is achieved, rather than the conventional global controllers. Furthermore, a method has also been studied for obtainment of the controller gain matrix through convex optimization regarding the complete structure of the 2D Lyapunov function. Simulation results are provided for the stabilization of an unstable nonlinear 2D system under input saturation.

One-Degree Aerial Device: Control and Experimental Development

The ball and beam experimental platform is an unstable nonlinear system widely used as a benchmark control setup for testing different controller approaches, especially for beginners on automatic control to improve their control knowledge skills. In this paper, we innovate it by governing the angular position of the beam with a twin-rotor system. Our experiment consists of a beam that rotates through a pivot, in which two propellers are attached to the ends of this beam. Hence, we have a recent one-degree aerial device, and instead of using a ball, we employ a mass moving on the beam, presenting friction on position to its movements on the beam. Then, the control objective is to regulate the mass position at some predefined zone on the beam, ensuring stability and robustness in front of external perturbations and unmodeled uncertainties. To do so, we define a classical PI controller. To assess closed-loop robustness, a mass was introduced to one propeller to induce perturbation, thereby simulating modeling variations or disturbances. The experimental results prove the goodness of our experimental platform for drone applications.

Simulation of Rarefied Gas Flow in a Channel Applying Artificial Neuron Network

The purpose of the present paper is to confirm numerically applying our ANN algorithm for transitional gas flow between continuum and free-molecular flow. The main advantage of this approach is substantial computing time economy compared to usual DSMC procedure. The learning of ANN is based on well known rarefied gas flow examples, analytic solutions in limit cases, calculations by lattice Boltzmann method (LBM), and on quite few Monte Carlo simulations. Our previous results obtained for large Knudsen numbers have been applied near the free-molecular regime. In this region the flow has demonstrated the properties of unstable nonlinear dynamic system. The flow of nitrogen in a microchannel with aspect ratio L/h = 20 and by pressure ratio P between 1.84 and 4.14 is simulated by ANN. This example has shown a good coincidence of ANN and DSMC calculations even in the case when only a small part of original DSMC computation has been applied for learning ANN. Thus, computing time is reduced up to 10 times in this case.

No excuses to avoid observing unstable nonlinear systems: an LMI-based discontinuous solution

No excuses to avoid observing unstable nonlinear systems: an LMI-based discontinuous solution

Training Sample Pattern Optimization Based on a Swarm Intelligence Algorithm for Tiltrotor Flight Dynamics Model Approximation

Neural networks have been widely used as compensational models for aircraft control designs and as surrogate models for other optimizations. In the case of tiltrotor aircraft, the total number of aircraft states and controls is much greater than that of both traditional fixed-wings and helicopters. This requires, in general, a huge amount of training data for the network to reach a satisfactory approximation precision and makes the network size rise considerably. To solve the practical problem of reducing the size of the approximating network, efforts have to be made in the efficient utilization of the limited amount of training data. This work presents the methodology of optimizing the sample pattern of the training data set by adopting the metaheuristic algorithm of the particle swarm optimizer improved by the fourth-order Runge–Kutta algorithm. A 6-degree-of-freedom nonlinear flight dynamics model of the tiltrotor aircraft is derived, along with its approximation radial basis function neural network. An example case of approximating a highly nonlinear function is studied to illustrate the principle and main parameters of the optimizer, and the approximation performance of the time-domain response of the unstable nonlinear system is revealed by the study of a Van der Pol oscillator. Then, the presented method is applied to the modeled tiltrotor aircraft for its early and late conversion modes. The optimization scheme shows great improvement in both cases, as the function approximation error is reduced significantly.

Off‐policy model‐based end‐to‐end safe reinforcement learning

AbstractSafety and stability considerations play a crucial role in the development of learning based strategies for control design of systems that require high levels of safety. Safe reinforcement learning (RL) based approaches traditionally seek learning of the control laws that are optimal with respect to system performance whilst ensuring system stability and safety. In this article, an off‐policy safe RL based approach is proposed for nonlinear systems affine in control in continuous time. In this novel work, safety and stability are guaranteed during initialization and exploration phases by adjusting the control input with the solution of a quadratic programming problem combining both input to state stable‐control Lyapunov function and robust control barrier function (R‐CBF) conditions. Moreover, the safety of the learned policy is assured by augmenting the cost function with a CBF to maintain safety and optimize performance simultaneously. Novel mathematically rigorous proofs are provided to establish the stability and safety guarantees, offering a sound theoretical foundation for the approach. To demonstrate the effectiveness of the algorithm, two examples are presented: engine surge and stall dynamics, and an unstable nonlinear system.

Finite-Time Height Control of Quadrotor UAVs

The quadrotor Unmanned Aerial Vehicle (UAV) belongs to an open-loop unstable nonlinear system, which also has the characteristics of underdrive, strong coupling and external disturbance. In the height control of quadrotor UAVs, the traditional sliding mode control (SMC) and PID methods cannot quickly and effectively eliminate disturbance effects caused by gust, aerodynamic drag and other factors, which indicates that the quadrotor UAV cannot return to its predetermined trajectory. To this end, this paper proposes a dual closed-loop finite-time height control method for the quadrotor UAV. The proposed method is able to estimate and compensate for the disturbance in the height control and make up for the lack of anti-disturbance ability in the control process. More specifically, a finite-time Extended State Observer (ESO) and a finite-time super-twisting controller are designed for the velocity control system to compensate for the total disturbance and track the rapidly changing expected signal. An integral sliding mode controller is designed for the height control system. Simulation results show that the proposed method can reduce the chattering phenomenon of traditional SMC and improve both control accuracy and convergence speed.

Performance Comparison of PID and LQR Controllers for Control of Non-Linear Magnetic Levitation System

Magnetic Levitation System (MLS) has become a current study in the field of engineering due to its advantages such as low energy consumption and minimum friction. MLSs are nonlinear unstable systems. Due to the complexity of the structure and the difficulty of the controls, many advanced control theories can be applied on these systems and the performance of the controllers can be evaluated. In this article, Proportional-Integral-Derivative (PID) and Linear-Quadratic Regulator (LQR) controller methods are applied on MLS mathematically modeled in MATLAB environment. Controller performances were compared in the results found. The results obtained on the applicability of PID and LQR control methods for MLS were evaluated. In addition, the system performance of the controllers was compared with five parameters. These are rise time, settling time, maximum overshoot, overshoot and steady-state error. LQR controller produced great stability and homogeneous response compared to PID controller.

An Extended Experimental Study on Control of Unstable and Non-Minimum Phase Plants With the Cascaded Form of a Fractional Order Compensator

With the advent and emergence of calculus with fractional order (FO), the generalized form of the integer order (IO) has created significant advancement in the field of control. The real time systems represented by the FO models are more accurate enough developing improved performance in contrast to the classical IO one. In this context, the area is still deficient of methodical design algorithms with the different available structures of the FO compensators to control the various types of any unstable and non-minimum phase (NMP) Linear Time Invariant (LTI) plants. There is dearth of detailed analytical and experimental work on the cascaded form of this FO lead compensator in the available literatures. The cascaded structure of the non-integer order compensator is therefore employed here through classical control theory to exhibit the simplified design method in frequency domain explicitly through numerical studies and MATLAB simulation results. These FO compensators with its non-integer order helps one to add the required exact magnitude and phase to the plant to be compensated at a particular frequency thus leading to the unique solution of the desired compensator parameters. This proposition has been further extended to compensate a highly non-linear unstable NMP Cart-Inverted Pendulum System. Experimental results have been appended in support of the method explained to show the applicability of this study in control system design.

Stroboscopic thermally-driven mechanical motion

Unstable nonlinear systems can produce a large displacement driven by a small thermal initial noise. Such inherently nonlinear phenomena are stimulating in stochastic physics, thermodynamics, and in the future even in quantum physics. In one-dimensional mechanical instabilities, recently made available in optical levitation, the rapidly increasing noise accompanying the unstable motion reduces a displacement signal already in its detection. It limits the signal-to-noise ratio for upcoming experiments, thus constraining the observation of such essential nonlinear phenomena and their further exploitation. An extension to a two-dimensional unstable dynamics helps to separate the desired displacement from the noisy nonlinear driver to two independent variables. It overcomes the limitation upon observability, thus enabling further exploitation. However, the nonlinear driver remains unstable and rapidly gets noisy. It calls for a challenging high-order potential to confine the driver dynamics and rectify the noise. Instead, we propose and analyse a feasible stroboscopically-cooled driver that provides the desired detectable motion with sufficiently high signal-to-noise ratio. Fast and deep cooling, together with a rapid change of the driver stiffness, are required to reach it. However, they have recently become available in levitating optomechanics. Therefore, our analysis finally opens the road to experimental investigation of thermally-driven motion in nonlinear systems, its thermodynamical analysis, and future quantum extensions.

General stabilization of non-autonomous hybrid systems with delays and random noises via delayed feedback control

The feedback control, derived from delayed state observations, was triumphantly imported to H∞ and exponentially stabilize an unstable nonlinear hybrid system with time-invariant delay and random noise, which is adapt to autonomous systems. Spontaneously, given an unstable non-autonomous hybrid system with time-dependent delay and random noise, can one import a feedback control, derived from delayed state and mode observations, to stabilize it? Fortunately, this will be well done by the approach of multiple Lyapunov functionals. Surprisingly, it is revealed that the stabilization can be achieved not only in the forms of H∞ stability and exponential speed but also of polynomial speed and general speed. Furthermore, the interesting result with respect to polynomial growth at most is as well yielded here.

Design of Approximate Explicit Model Predictive Controller Using Parametric Optimization

AbstractThis paper introduces a new technique, called state-parameterized nonlinear programming control (sp-NLPC), for designing a feedback controller that can stabilize intrinsically unstable nonlinear dynamical systems using parametric optimization. Stability-preserving constraints are included in the optimization problem solved offline by the predictive parameterized Pareto genetic algorithm (P3GA), a constrained nonlinear parametric optimization algorithm. The optimal control policy is approximated from P3GA output using radial basis function (RBF) metamodeling. The sp-NLPC technique requires fewer assumptions and is more data-efficient than alternative methods. Two nonlinear systems (single and double inverted pendulums on a cart) are used as benchmarking problems. Performance and computational efficiency are compared to several competing control design techniques. Results show that sp-NLPC outperforms and is more efficient than competing methods. A qualitative investigation on phase plane analysis for the controlled systems is presented to ensure stability. The approximating state-dependent solution for the control input lends itself to applications of control design for control co-design (CCD). Such extensions are discussed as part of future work.

Impulsive Control for Nonlinear Systems Under DoS Attacks: A Dynamic Event-Triggered Method

In this brief, the impulsive control problem is addressed for a class of unstable nonlinear systems under denial-of-service (DoS) attacks. A dynamic event-triggered impulsive control (DETIC) method is proposed to achieve asymptotic stability for the resulting closed-loop systems. A new class of impulsive control systems with dynamic event-triggered impulse instants are formulated, and some easily verifiable conditions are provided. The introduction of internal dynamic variable facilitates the regulation of the control frequency. Here, Zeno behaviors can be easily excluded by the proposed method. Finally, two examples, which include a Chua’s circuit and a numerical example, are given to demonstrate the efficiency of the method.

A simple modelling strategy for integer order and fractional order interval type-2 fuzzy PID controllers with their simulation and real-time implementation

This paper presents a simple approach to the mathematical modelling of the Integer Order (IO) and Fractional Order (FO) Interval Type-2 Fuzzy Proportional Integral Derivative (IT2FPID) controllers. In this approach, the control effort produced by the IT2FPID controller is obtained by combining the individual control efforts generated by the Interval Type-2 Fuzzy Proportional (IT2FP), Interval Type-2 Fuzzy Integral (IT2FI), and Interval Type-2 Fuzzy Derivative (IT2FD) controllers. The analytical structures of each of the IT2FPID components are unveiled using one-dimensional (1D) input space and without using any AND or OR operator. To the best of the authors’ knowledge, such a strategy was never used and is completely new for the modelling of the IT2FPID controllers. Properties, computational issues, and design guidelines of the newly obtained controllers are discussed, and their applicability is delineated using five simulation examples and one experimental case study. A chemical reactor plant, two fractional order plants, a nonlinear plant, and a Twin Rotor Multi Input Multi Output System (TRMS) are considered in the simulation study. The experimental study is conducted on an unstable nonlinear Magnetic Levitation System (MLS). Moreover, to exhibit the usefulness of the newly derived controllers, simulation and real-time results are also compared with the existing results available in the literature. As the proposed controllers are plant model-free, they can be used for other control applications also.

Stabilization of the Magnetic Levitation System

This paper contributes toward research on the control of the magnetic levitation plant, representing a typical nonlinear unstable system that can be controlled by various methods. This paper shows two various approaches to the solution of the controller design based on different closed loop requirements. Starting from a known unstable linear plant model—the first method is based on the two-step procedure. In the first step, the transfer function of the controlled system is modified to get a stable non-oscillatory system. In the next step, the required first-order dynamic is defined and a model-based PI controller is proposed. The closed loop time constant of this first-order model-based approach can then be used as a tuning parameter. The second set of methods is based on a simplified ultra-local linear approximation of the plant dynamics by the double-integrator plus dead-time (DIPDT) model. Similar to the first method, one possible solution is to stabilize the system by a PD controller combined with a low-pass filter. To eliminate the offset, the stabilized system is supplemented by a simple static feedforward, or by a controller proposed by means of an internal model control (IMC). Another possible approach is to apply for the DIPDT model directly a stabilizing PID controller. The considered solutions are compared to the magnetic levitation system, controlled via the MATLAB/Simulink environment. It is shown that, all three controllers, with integral action, yield much slower dynamics than the stabilizing PD control, which gives one motivation to look for alternative ways of steady-state error compensation, guaranteeing faster setpoint step responses.

Comparison of Nonlinear and Linear Controllers for Magnetic Levitation System

Nonlinear system control belongs to advanced control problems important for real plants control design. Various techniques have been developed in this field. In this paper we compare two different approaches to a nonlinear unstable Magnetic levitation system control. The first control design approach further develops our recent results on robust discrete-time pole-placement, based on convex DR-regions. The second studied approach is based on feedback linearization and the simplified development of the corresponding nonlinear control law is provided. Both approaches are compared and evaluated. The efficiency of robust discrete-time pole-placement controller is shown as well as its competitiveness in comparison with nonlinear control for Magnetic levitation system.

‘Androgynous Music’: Pauline Oliveros’s Early Cybernetic Improvisation

In the late 1950s and early 1960s, Pauline Oliveros developed new techniques for improvising with new electronic instruments, which she described using language and metaphors from cybernetics and information theory. First using just a tape recorder, and later using a series of tape recorders connected to an unstable pair of oscillators, Oliveros created what she later called a ‘very unstable nonlinear musicmaking system’ (Oliveros, Pauline. 2016a. “Improvising Composition: How to Listen to the Time Between.” In Negotiated Moments: Improvisation, Sound, and Subjectivity, edited by Gillian H. Siddall, and Ellen Waterman, 75–90. Durham, NC: Duke University Press). By positioning herself as only one member of an instrumental system that facilitated the self-regulating flow of sound as signal—operating as air pressure waves, electrical waves, psychoacoustic phenomena, and even human consciousness—Oliveros began to reconceptualize the role of her body in musical performance. By the early 1970s, Oliveros began to embrace the new kinds of musical performance and musical subjectivities produced by this system as a part of her exploration of gender in music, positing the concept of ‘androgynous music’—defined as music that is simultaneously ‘linear’ and ‘nonlinear’. Oliveros thought of her performance practice as one that could destabilise socially constructed musical identities of composition, performance, and listening, and also integrate different modalities of interacting with musical ‘signal’. This article maps Oliveros’s cybernetic, improvisatory practices with her earliest electronic systems, showing how her performance within these systems laid the groundwork for her later theorisations of technology, gender, and the body.

New strategy to control covid-19 pandemic using lead/lag compensator

COVID-19 is still the main worldwide issue since the outbreak. Many strategies were implemented such as suppression, mitigation, and mathematical-engineering strategies, to control this pandemic. In this work, a lead/lag compensator is proposed to control an unstable Covid-19 nonlinear system after using some required assumptions. The control theory is involved with the unstable pandemic and other existing strategies until the invention of the vaccine is approved. In addition, the Most Valuable Player Algorithm (MVPA) is used to optimize the parameters of the proposed controller and to determine whether it is a lead or lag compensator. Finally, the simulation results are based on the daily reports of two pandemic cities: Hubei (China), and Lazio (Italy) since the outbreak began. It can be concluded that the lead/lag compensator can effectively control the COVID-19 system.

Pareto Optimal Design of a Fuzzy Adaptive Hierarchical Sliding-mode Controller for an X-Z Inverted Pendulum System

ABSTRACT The control problem of a three-degree-of-freedom (3-DOF) X-Z inverted pendulum as an unstable, multi-input-multi-output, under-actuated and high-order nonlinear system is facing many challenges. In this paper, a Fuzzy Adaptive Hierarchical Sliding-Mode Controller (FAHSMC) is optimally designed for the X-Z inverted pendulum system utilizing the Multi-Objective Genetic Algorithm (MOGA). To this end, at first, the state variables of this system are converted to the new state variables via five steps of a forward transformation process. In this transformation, the dynamical equations of the X-Z inverted pendulum system are reorganized to an appropriate form for control implementations. Then, the novel FAHSMC is designed for stabilization and tracking control of the system, and the stability analysis of the controller is proved via the Lyapunov stability theory. After that, the state and control variables of the X-Z inverted pendulum are computed through six steps of a backward transformation process. Ultimately, the MOGA is exerted for Pareto optimal design of the proposed control approach applied on the regarded X-Z inverted pendulum system. The tracking error of the cart position in the X-direction, the tracking error of the cart position in the Z-direction and the angle error of the pendulum are selected as three inconsistent objective functions for multi-objective optimization operations. Numerical results and diagrams illustrate that the proposed approaches can accurately converge the system states to the desirable trajectories in a short time and yield superior Pareto optimal fronts in comparison with the Hierarchical Sliding-Mode Controller (HSMC) and the Adaptive Hierarchical Sliding-Mode Controller (AHSMC).

A Simple Framework for Identifying Dynamical Systems in Closed-Loop

In this paper, we propose a simple framework for closed-loop system identification: the stabilized output error method. Traditional closed-loop system identification methods rely on the linearity of the target system, and they require the identification of noise models or prior knowledge or identifiability of the feedback controllers to obtain unbiased estimates. But in many real-world applications, the noise dynamics and feedback controllers are complex and difficult to identify, and the nonlinearity may not be ignored. The proposed framework introduces a virtual controller that stabilizes the error between model prediction and the output of the target system. This enables us to apply the output error method, which gives unbiased estimates without depending on the noise model and is applicable to a wide range of models, including nonlinear systems, to closed-loop system identification problems. The paper describes the framework and gives the theoretical support and design guidelines for virtual controllers. Through numerical examples, we show the effectiveness of the proposed framework in various situations, which include identification of (a) linear gray box models, (b) systems in the presence of disturbances having realistic complexity, and (c) a nonlinear unstable system in a human-in-the-loop environment.