Robust Proportional Integral Derivative Controller Research Articles

Robust PID controller design for positioning systems with hysteresis and time-varying delay

This paper presents Linear Matrix Inequalities (LMIs) conditions for designing Proportional-Integral-Derivative (PID) controllers that guarantee asymptotic stability with exponential convergence rate for positioning systems affected by hysteresis nonlinearity and time-varying delay. The hysteresis is modelled using the Prandtl-Ishlinskii operator, and the effects of both the time-varying delay and hysteresis are encapsulated into a bounded uncertain parameter confined within a known convex polytope. Numerical results support the theoretical developments, confirming that the controller effectively mitigates the effects of disturbances while suppressing both the hysteresis and time-varying delay influences.

Robust fractional-order PID controller assisted by active disturbance rejection control for the first-order plus time-delay systems

Time-delay characteristics of various industrial processes may degrade the stability and dynamic performance of the control systems. Aiming at the problems of the existing methods in dealing with the time delay plant, a modified fractional-order proportional–integral–derivative (FOPID) controller for the first-order plus time-delay (FOPTD) system is developed. Assisted by a modified active disturbance rejection control (ADRC) scheme with increased observer bandwidth, the proposed FOPID controller inherently obtains good robustness to time-delay uncertainties and external disturbances. In addition, taking advantage of the fractional-order operator, the proposed controller can provide larger stability margin over the proportional–integral–derivative (PID) controller. By suitably establishing the relation between ADRC and FOPID controller parameters, the proposed controller can be analytically tuned based on the common design indices. A practical tuning guideline is developed according to frequency-domain characteristic analysis, making the proposed controller more acceptable to industrial application. The performance of the ADRC-based FOPID controller is tested by the control simulation of some typical FOPTD systems and a diesel engine speed regulation system. The efficiency of the ADRC-based FOPID controller is demonstrated by the comparisons with some existing controllers.

A robust PID and RLS controller for TCP/AQM system

Transmission Control Protocol (TCP) controlling congestion by peer-to-peer is challenging to handle communication with numerous concurrent TCP flows and heavy traffic loads. Therefore, TCP requires the active queue management (AQM) to assist in avoiding buffer bloat in intermediate devices. Thus, Some AQM controllers, such as Red, Codel, and Pie, have been proposed to control congestion better. However, these controllers are unable to overcome the drawbacks, including micro-burst streams, unfair competition of flows with different workloads, as well as sluggishly responding to congestion in heavy traffic, which degrade the performance of data transmission. To address these problems, in this paper, we proposed a queueing-delay-based AQM controller for TCP/AQM systems, namely PID-R. PID-R adopts two-level control components, i.e., the Proportional-Integral-Derivative (PID) algorithm and the recursive least squares (RLS) filter, and combines the advantages of the two parts. Meanwhile, we also constructed a new TCP/PID-R fluid model derived from TCP/AQM classical model. According to the model, PID-R parameters are determined to keep the TCP/PID-R system steady. The comprehensive simulation reveals that PID-R outperforms the state-of-the-art AQM controllers.

Robust PID control of a tuned mass damper system and its stability analysis validation by Kharitonov and Edge theorems

ABSTRACT A robust control and stability analysis of a tuned mass damper attached to benchmark the model of mass-spring-damper system via different Kharitonov-related approaches, including the post-Kharitonov Edge Theorem, is presented in this paper. The control of uncertain parameters of the given system is demonstrated via robust Proportional-Integral-Derivative (PID) system tuning techniques in MATLAB. Usually, the Kharitonov theorem is sufficient to check the stability analysis of a controlled interval system. A detailed stability analysis of the given system by forming the Kharitonov rectangular parameter boxes and testing the positivity of Hurwitz determinants of the enclosed segments connecting opposite corners of the box is explained here. Stability checking is validated by the analysis based on the Hurwitz matrices of two Kharitonov polynomials. The multi-linear affine system model in the overall system plant is controlled and analysed via Edge theorem.

Robustness evaluation of the damped outrigger structure

Robustness is defined as the insensitivity of the structure to uncertainties like earthquakes, fire, cyclones, explosions, tsunamis, etc. Robustness analysis of Kalman observer with robust controller for damped outrigger structure is studied to analyze its design performance in the presence of various effects like an earthquake. This is a novel innovative study undertaken in evaluating the robustness of the proposed controller to signify its performance in the field of structural control. The damped outrigger structure is modeled using the finite element approach by finding mode shapes, fundamental natural frequency, and period. The Kalman observer is modeled according to the requirement of the structure with the Riccati equation. The robust proportional–integral–derivative controller is designed according to the input disturbance with Ziegler–Nichols ultimate gain approach. The issue of deterioration in the system performance due to saturation was observed during the analytical investigation of the robust proportional–integral–derivative controller with Kalman observer, which has been addressed by the anti-windup approach. The robustness index of the structure is calculated using sensitivity and complementary sensitivity. The maximum amplitude ratio of the sensitivity for viscous damper-controlled structure is 1.4723 and the value decreases for the other controllers, with a minimum value of 1.0 for the proposed anti-windup robust proportional–integral–derivative controller with Kalman observer. Respectively, the percentage overshoot for the uncontrolled case is 23.4% that values decreasing for other controlled cases, with a minimum of 7.8% for proposed anti-windup robust proportional–integral–derivative controller with Kalman observer. This robustness index and performance indices discriminate the significant robust performance of the proposed robust proportional–integral–derivative controller with Kalman observer-based damped outrigger structure in comparison with other controlled and uncontrolled cases.

Neuromechanical Model-Based Corrective Torque Estimation During Weight Shifting in Lower Limb Amputees

As a consequence of limb loss, unilateral lower limb amputees (LLAs) exert additional flexion torque in the intact limb during postural balance, leading to other secondary complications. A prosthetic device with torque delivery is required to avoid such complications. However, estimating desired torque is an acknowledged challenge in this domain. Motivated by the above, this paper focuses on estimating the corrective torque required from an assistive device to balance the individual while performing weight-shifting exercises. Further, a healthy individual’s inverted pendulum (IP) model is explored to model LLA’s dynamics during weight-shifting. The model and the control strategy are validated using forward and backward body lean angles recorded from healthy and LLA individuals. Finally, a robust Proportional Integral Derivative controller is designed to estimate the corrective torque for maintaining the postural balance during weight-shifting. Along with the IP dynamics, a suitable control law based on Coefficient Diagram Method achieves the experimentally recorded body lean angle. The proposed control scheme is validated through simulation studies to evaluate the tracking performance in two different biomechanically relevant IP postural models. The findings assist in determining the external torque required to maintain the postural balance during gait initiation and compensatory measures for pathologically reduced torque. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Gait initiation is a transition between vertical posture and gait. Since vertical stance is inherently unstable, a postural adjustment occurs before gait initiates, propelling the center of mass forward and towards the first stance leg. The absence of flexor torque in the amputated limb causes a loss of postural stability during weight shifting activities. In the case of powered exoskeletons and prosthetic devices, a prior estimation of corrective torque through a computational approach helps to select a suitable actuator that provides an appropriate joint torque while maintaining the postural balance. As a result, an efficient computational model of gait initiation that accounts for neuromuscular system components before designing assistive devices is necessary. The controllers further provide corrective action to the assistive device to obtain robust and optimal coordination between the prosthetic element and gait. Therefore, this study demonstrates the significance of neuromuscular controller and assists in determining how much corrective torque from an external device is required to maintain postural stability and adaptability during weight shifting exercises. The effectiveness of the neural controller is also tested for real-time data to compensate for the effect of external disturbance and noise, thereby achieving a proper and controlled degree of movement in amputee users. These findings could provide one of the fundamental aspects of gait initiation model paradigms of compensatory measures for pathologically diminished torque and contribute to developing low-cost prosthetic solutions for the benefit of the rehabilitation society in developing countries.

A new MOPSO algorithm for solving mathematical test functions and control engineering problems

In this article, a new multiobjective particle swarm optimization (MOPSO) algorithm is introduced to improve the performance of a sliding mode based robust fuzzy proportional–integral–derivative (PID) controller. In this regard, the non-dominated solution having minimum number of neighbors is considered as the global best position, while the sigma values of the members are employed to determine the personal best position. A modified multiple-crossover operator is combined with the operators of the particle swarm optimization to significantly increase the convergence speed of the algorithm. To limit the size of the archive, a dynamical elimination scheme defined in the Euclidean space is introduced. Besides, iteration-based linear relations are implemented to adaptively compute the inertia weight and learning coefficients. To evaluate the effectiveness of the introduced MOPSO algorithm, the requirements are conducted by means of three benchmark functions with regard to generational distance, spacing, and maximum spread metrics. This analysis demonstrates that the proposed algorithm operates better through comparison with well-known elitist multiobjective evolutionary algorithms. Moreover, the MOPSO algorithm is applied for optimal design of a hybrid robust fuzzy PID controller for a pneumatic system with two bellows. Conflicting objective functions are considered as the normalized values of overshoot and settling time of the displacement between the bellows that should be simultaneously minimized. The feasibility and efficiency of the strategy are assessed in comparison with the conventional controllers.

Robust PID controllers tuning based on the beetle antennae search algorithm

The core components of both traditional and contemporary control systems are the proportional–integral–derivative (PID) control systems, which have established themselves as standards for technical and industrial applications. Therefore, the tuning of the PID controllers is of high importance. Utilizing optimization algorithms to reduce the mean square error of the controller’s output is one approach of tuning PID controllers. In this paper, an appropriately modified metaheuristic optimization algorithm dubbed beetle antennae search (BAS) is employed for robust tuning of PID controllers. The findings of three simulated experiments on stabilizing feedback control systems show that BAS produces comparable or higher performance than three other well-known optimization algorithms while only consuming a tenth of their time.

LQ optimal robust multivariable PID control for dynamic positioning of ships with norm-bounded parametric uncertainties

The design of a suitable controller for dynamic positioning (DP) of ships is a challenging task keeping the hydrodynamic uncertainties in mind. The magnitude and rate constraints on the actuators make the problem more difficult. The present work employs a two-degree-of-freedom (2-DOF), multivariable, proportional–integral–derivative (PID) control for DP of a ship. To design the controller, first, the ship dynamics with hydrodynamic parameter uncertainties is represented in an uncertain state-space descriptor form. Then, the design of feedback gains of 2-DOF PID controller for the uncertain descriptor plant is converted into a state feedback design for an augmented uncertain descriptor plant. Next, a linear matrix inequality-based condition is obtained to solve this state feedback problem in order to ensure a desired linear quadratic (LQ) performance, even in presence of uncertainties. Finally, to further improve the tracking performance, the feedforward gain of the 2-DOF PID controller is designed using H∞ approach. The DP performance of the proposed controller is tested for the considered ship model along with uncertainties, actuator constraints, and environmental disturbances. A comparison with the existing PID controllers is also carried out to show superiority of the proposed controller.

H ∞-PID distributed control for output leader-tracking and containment of heterogeneous MASs with external disturbances

This paper addresses both the output leader-tracking and output containment control problems for heterogeneous linear high-order Multi-Agent Systems (MASs) sharing information over a directed communication topology and subject to external unknown disturbances. To solve these control problems, we propose a robust fully distributed Proportional-Integral-Derivative (PID) control strategy, equipped with a filter on the derivative action that allows obtaining both a simpler closed-loop formulation, without the need of the descriptor transformation, and good tracking performances in the case of fast reference signal behaviours. The stability analysis is analytically proven via the Lyapunov theory and the approach. The derived robust stability conditions are expressed as a set Linear Matrix Inequalities (LMIs) whose solution provides the proper tuning of the robust PID control gains. Numerical simulations confirm the effectiveness and robustness of the proposed approach in solving both the output leader-tracking and output containment control problems.

On-Line Auto-Tuning Robust Nonlinear Neural PID Controller Design for Double-Pipe Heat Exchanger Based on Particle Swarm Optimization

The heat exchanger systems have highly nonlinear features between its variables and time delay. In addition, the parameters of them change constantly during operation. Therefore, the heat exchangers need nonlinear robust controller for obtaining required outlet temperature under all operating conditions.In this paper, a robust nonlinear neural PID (NLNPID) controller is proposed for temperature control of double-pipe heat exchanger. Particle swarm optimization (PSO) technique is used to determine on line auto tuning the optimal parameters of proportional-integral-derivative (PID) controller. Mathematical model of heat exchanger takes into account the changes in the density, thermal conductivity, viscosity, specific heat and convection heat transfer coefficient of the fluid which are resulting from changing in temperatures and the velocities with classical PID control. Simulations results show that the performance of the NLNPID controller produces good responses features and give desired outlet temperature under all operating conditions without over shoot and fluctuating.

Robust finite-time PID control for discrete-time large-scale interconnected uncertain system with discrete-delay

In this paper, we investigate the decentralized proportional–integral–derivative (PID) output-feedback control problem for a class of discrete-time uncertain large-scale systems with delayed interconnections. The robust finite time stabilization of the addressed uncertain large-scale system is attained via the feedback control with the prescribed extended passive performance index. Specifically, by using an appropriate Lyapunov–Krasovskii functional, a new set of sufficient conditions is derived on the basis of linear matrix inequality which ensures the finite-time boundedness of the augmented closed-loop uncertain large-scale systems. Furthermore, cone complementarity linearization (CCL) algorithm is developed to obtain the control gain parameters. Finally, a numerical example is provided to illustrate the effectiveness of proposed control design.

A new time-domain robust anti-windup PID control scheme for vibration suppression of building structure

Robustness is one of the main issues in the design of closed-loop control systems, and to provide it, considering the differences between the actual and mathematical models, effects of external disturbances and measurement noise is necessary. In this study, a new time-domain robust anti-windup proportional integral derivative (PID) control scheme for vibration suppression of building structure is introduced. Because of computational complexity in determining and tuning gains of the controller for each structural vibration modes in the conventional PID control methods. Such as modal analysis; the dynamic vibration equations of the structure are implemented in the state space form to consider the effects of all vibration modes of the structural system in the controlling gains simultaneously. Uncertainties in the structural stiffness parameter, sensing noise, input time-delay, and saturation windup of the actuator are considered in a new formulation. To ensure robust stability and performance of the proposed controller, the closed-loop transfer function from exogenous inputs to the controlled outputs is defined as in the standard form of the H∞ mixed sensitivity minimization criterion. The PID control gains are obtained by minimizing the infinity norm of the closed-loop transfer function of the control system in standard form. Based on numerical study of an 11-story benchmark structure equipped with an active tuned mass damper, the robustness and stability of the proposed controller are presented. This controller can reduce further the maximum displacement, velocity, and acceleration of the structure subjected to far field earthquakes with considering uncertainties by 16, 10, and 16%, respectively, compared to the conventional PID controller.

Robust optimal proportional–integral–derivative controller design for a class of uncertain second-order delay-free and time-delay systems

The second-order systems can capture the dynamics of a vast majority of industrial processes. However, the existence of uncertainty in second-order approximation of such processes is inevitable because the approximation may not be accurate or the operating condition changes, resulting in performance degradation or even instability. This article aims at designing a novel robust proportional–integral–derivative controller for the uncertain second-order delay-free and time-delay systems in an optimal manner. The method is simple, effective, and can efficiently improve the performance of the uncertain systems. The approach is based on the linear quadratic theory, in which by adding a new matrix in the quadratic cost function regarding the uncertainties, the stability of the perturbed system is guaranteed and proven for both time-delay and delay-free second-order cases. The comparison with the recent works in the literature supports the effectiveness of the proposed methodology.

A robust PID controller based on linear quadratic gaussian approach for improving frequency stability of power systems considering renewables

This paper proposes a novel robust controller for frequency stabilization of electrical systems taken into consideration a high renewable energy sources (RESs) penetration. The suggested controller, robust PID (RPID) controller, is combination of a proportional–integral–derivative (PID) controller and a linear quadratic gaussian (LQG) controller. Furthermore, the Improved lightning attachment procedure optimization (ILAPO) technique is applied for determining the optimal setting of the parameters of the introduced RPID controller. A studied power system with RESs is used as a test system to justify the performance of the RPID controller. The superiority of the introduced robust controller is verified through a comparison of its performance under system uncertainties with those of other robust controllers presented in the literature. The results elucidate that the RPID controller can effectually enhance frequency stability and guarantee reliable performance for power grids supplied with high share of renewable energy for all studied scenarios. Consequently, the proposed RPID controller purveys creditable for modern power systems considering RESs.

A novel tuning method of differential forward robust PID controller for integrating systems plus time delay based on direct synthesis method

ABSTRACT In this paper, a novel tuning rule of differential forward proportional–integral–derivative (PID) controller for integrating systems with time delay is proposed by using the direct synthesis method. This method is based on comparing the characteristic equation of the system which consists of the differential forward PID controller adding a lead-lag filter and the integrating time-delay process with the desired characteristic equation. The desired characteristic equation is obtained by placing a plurality of poles at the same desired position. Parameters of the differential forward PID are tuned to achieve the desired robustness. Tuning rules are given in terms of plant parameters for various kinds of integrating systems. Compared with the conventional PID controller, the proposed method is more effective for frequently changed set-points. Simulations for various integrating processes as well as the non-linear jacketed CSTR system illustrate the applicability and effectiveness of the proposed method.

Robust two-degree of freedom H∞ control design for a twin rotor multi-input multi-output system with external disturbances and model uncertainties

The control of twin rotor multi-input multi-output system is difficult as it is subjected to model uncertainties and external disturbances. Furthermore, there also exists a coupling between pitch and yaw positions, which makes the system more difficult to control them separately. Considering the above difficulties in twin rotor multi-input multi-output system control, [Formula: see text] robust controller is designed to handle the model uncertainties and external disturbances with two-degree-of-freedom. A mixed sensitivity approach is employed to represent the uncertainties and external disturbances arise in the twin rotor multi-input multi-output system. For performance analysis, first the proposed two-degree-of-freedom [Formula: see text] controller is compared with the ‘Linear Quadratic Regulator-Linear Matrix Inequality’–based robust proportional–integral–derivative controller and Internal Model Control–based proportional–integral–derivative controller in MATLAB/Simulation and then in experimentation. From the obtained results, it is confirmed that the proposed controller exhibits enhanced robustness, faster tracking performance and accurate disturbance attenuation, when compared with Linear Quadratic Regulator-Linear Matrix Inequality–based robust proportional–integral–derivative controller and Internal Model Control–based proportional–integral–derivative controller in face of external disturbances and uncertainties.

An Improved Analytical Tuning Rule of a Robust PID Controller for Integrating Systems with Time Delay Based on the Multiple Dominant Pole-Placement Method

An improved analytical tuning rule of a Proportional-Integral-Derivative (PID) controller for integrating systems with time delay is proposed using the direct synthesis method and multiple dominant pole-placement approach. Different from the traditional multiple dominant pole-placement method, the desired characteristic equation is obtained by placing the third-order dominant poles at −1/λ and placing the second-order non-dominant poles at −5/λ (λ is the tuning parameter). According to root locus theory, the third-order dominant poles and the second-order non-dominant poles are nearly symmetrically located at the two sides of the fifth-order dominant poles. This makes the third-order dominant poles closer to the imaginary axis than the fifth-order dominant poles, which means that, possibly, better performances can be achieved. Analytical formulas of a PID controller with a lead-lag filter are derived. Simple tuning rules are also given to achieve the desired robustness, which is measured by the maximum sensitivity (Ms) value. The proposed method can achieve better performances and maintain better performances when there exist parameters’ perturbation compared with other methods. Simulations for various integrating processes as well as the nonlinear continuous stirred tank reactor (CSTR) model illustrate the applicability and effectiveness of the proposed method.

RETRACTED: Artificial ecosystem-based optimization for optimal tuning of robust PID controllers in AVR systems with limited value of excitation voltage

This paper presents an application of a novel optimization method called Artificial Ecosystem-Based Optimization (AEO) to determine the optimal design parameters of the proportional-integral-derivative (PID) controller for an automatic voltage regulator (AVR) system. Unlike the previous studies presented in the literature, the proposed method takes into account the excitation voltage limit and therefore formulates a new objective function for optimal PID parameters design. The practical aspect of the proposed constraint is significant since the generator field winding can be seriously damaged in case of the large excitation voltage. The performance of the proposed controller using the solution methodology proposed in this study and its contribution to the robustness of the control system are investigated. Further, the obtained PID parameters are used to simulate the AVR dynamics for a large step change in the generator’s voltage set-point. Besides, the obtained step responses have been compared with the corresponding responses of the AVR system whose PID parameters are determined by using well-known methods presented in the literature. Also, the proposed AEO-PID controller shows superior performances in the case of uncertainties in the AVR system’s parameters, as well as in the presence of the different disturbances in the system. The results obtained show that the obtained parameters provided a more secure and stable machine operation even with changes of the reference, generator, or excitation voltage signals compared with the performance of the controllers obtained by the previous works presented in the literature. Furthermore, AEO has proven its ability to get optimal solutions in a fast and efficient manner in terms of accuracy and time spent per iteration.

Inverse Tangent Functional Nonlinear Feedback Control and Its Application to Water Tank Level Control

This paper explores the significance and feasibility of addressing a notion that the system error of a nonlinear feedback control can be decorated by an inverse tangent function in order to attain a sound energy-efficient performance. The related mathematical model and relevant evaluation of this concept are further illustrated by demonstrating a case study about the control performance of water tank level. The rationale of robust control and theoretical algorithm of Lyapunov stability theorem are outlined to evaluate the effectiveness of nonlinear feedback with inverse tangent function in terms of improving robustness of PID (Proportional–Integral–Derivative) controller and energy-saving capability. By demonstrating five simulations of different scenarios, it ultimately proves that the modified robust PID controller by inverse tangent function meets the requirement of energy-saving capacity. Comparing with the routine PID control, the mean control input of controlling water tank level can be reduced up to 39.2% by using modified nonlinear feedback controller. This nonlinear feedback PID controller is energy efficient and concise for its convenient use, which is feasible to expand its utility to other applications.