Open-loop Unstable Systems Research Articles

Tracking Control of X-Z Inverted Pendulum with Block Backstepping

As the extension of traditional linear (or X) inverted pendulum (IP), X-Z IP is a multiple-input multiple-output (MIMO), underactuated, open-loop unstable, and nonlinear system. In the tracking control of the X-Z IP, the equilibrium point changes with the pivot position of the pendulum. This makes linear control theories have difficulties in realization of the tracking control for the pendulum. The underactuated feature of the pendulum makes the feedback linearization unsuitable to simplify the control design. With the present model of the X-Z IP, there is no way to realize the backstepping design. This paper gives a novel state transformation method for the X-Z IP. Through the state transformation, the block backstepping can be easily deployed in the controller design of the X-Z IP. The proposed controller can achieve the tracking control in the vertical plane. Simulation results certify the rightness and effectiveness of the proposed tracking controller.

Detailed modelling and LQG\LTR control of a 2-DOF radial active magnetic bearing for rigid rotor

The paper presents detailed modelling of rotor, electro-magnetic actuators, and accompanying electronics of an in-house-developed active magnetic bearing (AMB) test rig in a unified manner. The work is the extension of our previous work on a 1-degree-of-freedom (DOF) cantilever beam while in here, the challenging problem of levitating and spinning a 2 DOF rotor on a pair of opposing EMs is investigated. Classical three-term PID controllers are designed as a benchmark controllers, in frequency domain to stabilize the 2-DOF open-loop unstable system, and the performance is compared with the modern state-space control strategies, namely Linear Quadratic Gaussian (LQG) and combined LQG-Loop Transfer Recovery (LQG\LTR) compensators. The tuned robust controllers are deployed on a custom-designed hardware using Simulink Real-Time software. Dual axes controllers performed reasonably well when subjected to static and dynamic tests both in simulation and in experimentation. The key time domain performance objectives such as reference tracking, disturbance rejection, and minimal control efforts as well as frequency domain stability margins are accomplished. The validity of the developed model is thus confirmed through successful synthesis and implementation of both PID and LQG\LTR controllers, wherein the robust controller (LQG\LTR) outperformed the benchmark controller in terms of performance, i.e. minimal overshoot, superior disturbance rejection capabilities, and improved loop gain characteristics. The AMB actuator also showed reasonable dynamic behaviour when the levitated rotor was spun at 1000 r/min. The dynamic performance was evaluated through a two-dimensional rotor orbit plot. The study goes beyond theory and underscores practical design considerations and demonstrates real-time controller implementation.

Research and Analysis of Generalized Predictive Identification Algorithm Based On Nonlinear System in Control Field of Hydropower Industry

Due to the uncertainties, constraints, nonlinearity and the correlation among variables in the control of hydropower industry, it is difficult to obtain the accurate mathematical model. In this paper, a generalized predictive identification algorithm based on the nonlinear system in the control field of hydropower industry is presented, and a model of the generalized predictive identification algorithm based on the nonlinear system in the control field of hydropower industry is established AB simulation. The simulation results show that the algorithm is robust, can effectively overcome the system delay, and can be applied to the open-loop unstable non minimum phase system. It has strong applicability in the field of electric industry control and can produce good economic benefits.

Exponential Stabilization of Unstable Bilinear Systems in Finite- and Infinite-Dimensional Spaces

This article discusses the uniform exponential stabilization of bilinear systems with multiplicative control inputs. We first consider the case of finite-dimensional bilinear systems, and we present stabilization results concerning the case of open-loop stable systems using a bang-bang feedback control. For open-loop unstable systems, a normalized quadratic time-varying feedback control is proposed. These results are then applied to infinite-dimensional case. Necessary and sufficient conditions are formulated in terms of unique continuation properties and integral estimates involving the control operator and the state of the uncontrolled system. Applications to finite-dimensional systems and coupled partial differential equations (PDE) are further provided.

Pre-stabilised Predictive Functional Control for Open-loop Unstable Dynamic Systems

Abstract Predictive functional control (PFC) is the simplest model-based algorithm, equipped with the attributes of a fully fledged predictive controller but at the cost and complexity threshold of a standard PID regulator. It has proven benefits in controlling stable SISO dynamic systems, but similarly to its competitor PID, it loses efficacy when a challenging application is introduced. In this paper, we present a modified PFC approach, especially tailored for open-loop unstable processes, using pre-stabilisation to efficiently control the undesirable dynamics at hand. This is essentially a two-stage design scheme with implications for PFC tuning and constraint handling. The proposal, nevertheless, is straightforward and intuitive, and provides improved closed-loop control in the presence of external perturbations against the standard PFC, and significantly better performance overall compared to the common PID algorithm, as demonstrated in a numerical case-study.

Particle Swarm Sliding Mode-Fuzzy PID Control Based on Maglev System

The magnetic levitation system is a typical open-loop unstable system. To weaken the chattering phenomenon of sliding mode control (SMC), a hybrid control strategy of particle swarm sliding mode-fuzzy PID control was designed. This strategy uses particle swarm optimization (PSO) to optimize the SMC using the method of exponential approach law, so as to obtain the best sliding mode surface parameters and exponential approach rate coefficients, thus shortening the time required to approach the stable region when the nonlinear system is far from the stable point; a state supervision controller was designed by analyzing the stable conditions of sliding mode. When the system approaches the stable region, it smoothly transitions to fuzzy PID control, thereby weakening the chattering phenomenon of SMC. To verify the dynamic performance of the designed algorithm, a magnetic levitation ball system was selected as the controlled object. Through simulation and experiment, it was verified that the controller had strong robustness and strong anti-interference performance. It effectively weakened the chattering and had good adaptability to external disturbance and state variable crossing.

Design of an Intelligent NPID Controller Based on Genetic Algorithm for Disturbance Rejection in Single Integrating Process with Time Delay

The integrating process with time delay (IPTD) is a fundamentally unstable open-loop system due to poles at the origin of the transfer function, and designing controllers with satisfactory control performance is very difficult because of the associated time delay, which is a nonlinear element. Therefore, this study focuses on the design of an intelligent proportional-integral-derivative (PID) controller to improve the regulatory response performance to disturbance in an IPTD, and addresses problems related to optimally tuning each parameter of the controller with a real coded genetic algorithm (RCGA). Each gain of the nonlinear PID (NPID) controller consists of a product of the gains of the linear PID controller and a simple nonlinear function. Each of these nonlinear functions changes the gains in the controller to on line by nonlinearly scaling the error signal. A lead-lag compensator or first-order filter is also added to the controller to mitigate noise, which is a disadvantage of ideal derivative action. The parameters in the controller are optimally tuned by minimizing the integral of time-weighted absolute error (ITAE) using a RCGA. The proposed method is compared with three other methods through simulation to verify its effectiveness.

A Distributed Luenberger Observer for Linear State Feedback Systems With Quantized and Rate-Limited Communications

This article addresses the problem of simultaneous distributed state estimation, and control of linear systems with linear state feedback, subjected to process, and measurement noise, under the constraints of quantized, and rate-limited network data transmission. In the set-up adopted, sensors and actuators communicate through a network with a strongly connected topology. Unlike the case of centralized linear systems, for which the separation principle holds, the above practical assumption prevents the separate design of observers, and controller because each of the nodes does not necessarily have access to the control inputs generated at all the other nodes. We derive a linear distributed Luenberger observer, and a set of sufficient conditions that guarantee ultimate boundedness of the estimation error, and system state vectors, with bounds that depend on the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm of the noise signals, and the number of bits used in the transmissions. A numerical example illustrates the performance and effectiveness of the proposed algorithm in controlling a network of open-loop unstable systems.

Tuning optimal PID controllers for open loop unstable first order plus time delay systems by minimizing ITAE criterion

The present paper gives an optimal tuning procedure for design of Proportional Integral Derivative (PID) controllers for open loop unstable First Order plus Time Delay systems. Estimation of the optimal PID controller parameters: kc, τI and τD is carried out by minimizing the time integral performance criteria:ITAE(Integral of Time-weighted Absolute Error) for a setpoint tracking problem. Results of the simulation for linear transfer functions and nonlinear isothermal CSTR reveals the proposed tuning approach provides enhanced performance for the set-point tracking and disturbance rejection with improved time domain specifications. To evaluate the superiority of the proposed method over others, robustness analysis with complimentary sensitivity function is carried out.

Disturbance Rejection for Input-Delay System Using Observer-Predictor-Based Output Feedback Control

This article addresses the control problem of an open-loop unstable linear time-invariant system with an input delay and an unknown disturbance. An observer-predictor-based control method is presented for such a system in which only output information is available. First, the system state is reconstructed and the disturbance is estimated using the equivalent-input-disturbance approach. Next, the future information on the state and the disturbance is predicted to reduce the effect of the input delay. Then, a new predictive control scheme is developed. The closed-loop system is simplified into two subsystems for analysis. Stability conditions are derived for each subsystem separately. Finally, a comparison with previous approaches through simulations shows the superiority of the presented method over others for disturbance rejection.

Hands-on MPC Tuning for Industrial Applications

This paper proposes a practical tuning of closed loops with model based predictive control. The data assumed to be known from the process is the result of the bump test commonly applied in industry and known in engineering as step response data. A simplified context is assumed such that no prior know-how is required from the plant operator. The relevance of this assumption is very realistic in the context of first time users, both for industrial operators and as educational competence of first hand student training. A first order plus dead time is approximated and the controller parameters immediately follow by heuristic rules. Analysis has been performed in simulation on representative dynamics with guidelines for the various types of processes. Three single-input-single-output experimental setups have been used with no expert users available in different locations - both educational and industrial - these setups are representative for practical cases: a variable time delay dominant system, a non-minimum phase system and an open loop unstable system. Furthermore, in a multivariable control context, a train of separation columns has been tested for control in simulation, followed by experimental tests on a laboratory system with similar dynamics, i.e. a sextuple coupled water tank system. The results indicate the proposed methodology is suitable for hands-on tuning of predictive control loops with some limitations on performance and multivariable process control.

Introduction and realization of four fractional-order sliding mode controllers for nonlinear open-loop unstable system: a magnetic levitation study case

The concept of magnetic levitation attracts the attention of researchers and developers on the global platform for its serviceability in both industrial and research applications. Sliding mode controller is a nonlinear and robust control strategy to provide efficient control over variable structure systems. This paper is initiating with nonlinear modelling of magnetic levitation system dynamics. Various sliding mode controllers for controlling the current through electromagnetic coil to levitate the ferromagnetic ball are presented. This paper is mainly focusing on design robust and energy-efficient fractional sliding mode controller for a nonlinear system. Four novel controllers called fractional static sliding mode controller, fractional integral static sliding mode controller, fractional dynamic sliding mode controller and fractional integral dynamic sliding mode controller are proposed. Stability proof of Lyapunov criteria for the proposed controller has been done. Performance of the proposed controllers is analysed in comparison with the existing sliding mode controllers. Experimental validation is done on a benchmark Feedback 33–210 magnetic levitation system.

Set point tracking of Ball and Beam System Using Genetic Algorithm based PI-PD Controller

The ball and beam system is one of the commonly used benchmark control apparatus for evaluating numerous different real systems and control strategies. It is an inherently nonlinear and open-loop unstable system. In this paper, we have suggested an Evolutionary Algorithm (EA) based Proportional Integral-Proportional Derivative (PI-PD) controller for the set point tracking of this well-known ball and beam system. A linearized model of the ball and beam system is deduced and PI-PID control methodology is employed. The popular EA technique such as Genetic algorithm (GA) is used for tuning of the controller. The optimized values of the controller parameters are achieved by solving a fitness function using GA. The transient performance of the proposed GA based PI-PD controller (GA-PI-PD) is evaluated by carrying set point tracking analysis of the ball and beam system through MATLAB/Simulink simulations. Furthermore, the performance of GA-PI-PD controller is investigated using four different performance indices such as Integral of squared value of error (ISE), Integral of time multiplied by squared value of error (ITSE), Integral of absolute value of error (IAE) and Integral of time multiplied by absolute value of error (ITAE). The comparison of transient performance including rising time, settling time and % overshoot is made with SIMC-PID and H-infinity controllers. The comparison reveals that GA-PI-PD controller yielded transient response with small % overshoot and settling time. The superior performance of the GA-PI-PD controller has witnessed that it is highly effective for maintaining good stability and the setpoint tracking of ball and beam system with fast settling time and less overshoot than SIMC-PID and H-infinity controllers.

The design of NMSS fractional-order predictive functional controller for unstable systems with time delay

Open-loop unstable systems are more difficult to control than stable processes. In presence of time delay and uncertainty, the complexity of problem increases. In this paper, non-minimal state space predictive functional control (NMSS-PFC) infrastructure has been generalized for control of unstable systems with time delay. At first, NMSS system representation has been extended for a so-called coprime-factorized equivalent model of the unstable processes. Then, the proposed NMSS-fractional-order PFC (NMSS-FOPFC) has been formulated via a fractional order cost function in terms of the output tracking error vector. In the developed formulation and via the NMSS structure, the constraints on inputs of the system could be easily formulated and employment of fractional order cost function led to improved performance even in case of disturbances, perturbations or uncertainties. Closed loop robust stability analysis was also performed based on small gain theorem and verified via simulations. Simulation examples show that the proposed NMSS-FOPFC results in improved nominal and perturbed responses compared to conventional methods. Comparison has been carried out by considering integral of absolute error (IAE) and integral of squared error (ISE) as well as step response transient and steady state properties and the control effort.

Practical approach of input delay nonlinear systems: Application to spinal cord injury sitting stability

Abstract This paper is concerned with the design of an application-oriented control law for a biomechanical model of people living with spinal cord injury sitting stability. A major difficulty is to stabilize the resulting open-loop unstable nonlinear system with a delayed control input. A solution is proposed based on Pade approximation of delays to obtain a set of design conditions in the form of linear matrix inequalities which can be effectively solved by convex optimization techniques. Numerical simulations are conducted to show the effectiveness of the proposed control law compared to previous works.

Control Algorithms of Magnetic Suspension Systems Based on the Improved Double Exponential Reaching Law of Sliding Mode Control

This paper proposes an improved double power reaching law integral SMC algorithm to overcome the chattering, large overshoot, slow response. This improved algorithm has two advantages. Firstly, the designed control law can reach the approaching equilibrium point quickly when it is away from or close to the sliding surface. The chattering and response speed problems can be resolved. Secondly, the proposed algorithm has a good anti-jamming performance, and can maintain a good dynamic quality under the condition of the uncertain external disturbance. Finally, the proposed algorithm is applied to the open-loop unstable magnetic suspension system. Theoretical analysis and Matlab simulation results show that the improved algorithm has better control performances than the traditional SMC and the power reaching law integral SMC algorithm, such as less chattering, smaller overshoots, and faster response speed.

Nonstandard use of anti-windup loop for systems with input backlash

Control systems with backlash at the input are considered in this paper. The goal of this work is to characterize the attractor of such nonlinear dynamical systems, and to design anti-windup inspired loops such that the system is globally asymptotically stable with respect to this attractor. The anti-windup loops affect the dynamics of the controllers, and allow to increase the performance of the closed-loop systems. Different performance issues are considered throughout the paper such as robustness with respect to uncertainty in the backlash, and L2 gain when external disturbances affect the dynamics. Numerically tractable algorithms with feasibility guarantee are provided, as soon as the linear closed-loop system, obtained by neglecting the backlash effect, is asymptotically stable. The results are illustrated on an academic example and an open-loop unstable aircraft system.

English

In this paper, a new robust approach based on combining the model reference control with H-infinity technique is proposed. The goal of the proposed controller is to obtain an adequate transient response which may not be obtained when only H-infinity technique is used. This is done by adding the model reference block to the standard H-infinity feedback configuration. Then the overall block diagram is formulated by linear fractional transformation (LFT). The Magnetic Levitation which is a highly nonlinear, open loop unstable and uncertain system is used to show the effectiveness of the proposed controller. The results show that the proposed controller is very effective in compensating the system parameters variations with forcing the system output to track the output of the model reference. A variation of ±10% in system parameters is considered.

A Newborn Hybrid Anti-windup Scheme for Fractional Order Proportional Integral Controller

Significant amount of work has been done in the field of fractional order control. However fractional controllers have been applied usually to stable processes regardless of actuator saturation. In this paper, a new concept to remove the actuator saturation as well as stabilizing the open-loop unstable system simultaneously has been introduced based on the fractional order control theory. As a benchmark problem, magnetic levitation system (MLS) Feedback 33-210 is used here. The restructuring of $$PI^\lambda $$ controller dematerializes the requirement of dedicated anti-windup scheme for removing the actuator saturation problem. In addition, to further improve the performance of system, a novel hybrid anti-windup scheme is proposed. The stability analysis of overall system has been carried out by drawing the fractional order root locus in both s-plane and w-plane. Experimental demonstration of proposed scheme on MLS affirm the potential of controller, to reject actuator saturation as well as stabilize the system.

Robust Indirect Adaptive Control for a Class of Nonlinear Systems and Its Application to Shape Memory Alloy Actuators

In this paper, a new robust adaptive control method has been proposed for nonlinear systems with uncertainties. This method combines the advantages of self-tuning control and sliding mode control. A simple parameterization model is first derived based on a linear dynamic model and unmodeled dynamics. Based on a modified sliding surface, the design procedure is based on the indirect adaptive control concept. This controller consists of four parts: 1) the system parameters estimation; 2) the unmodeled dynamics estimation; 3) the weighting polynomials updating; and 4) the control law calculation. The key merits of this controller are as follows: 1) the controller is applicable to non-minimum phase and open-loop unstable systems; 2) the estimation of the unmodeled dynamics is introduced as a feedback compensation control to improve the response; and 3) the strict stability condition is eliminated and a desirable performance is ensured during a wide operation region. Moreover, the control problem of a shape memory alloy actuator system is considered. In the literature, the mechanism model-based controllers have been extensively reported to address this issue. As an alternative, we describe this plant as a gray-box model. The adaptation algorithm and the control law have been implemented through the Beckhoff controller. The experimental results have demonstrated that the proposed controller has wider applicability than some existing methods.