A Robust Auto-Tuning PID Controller Design based on S-Shaped Time Domain Response

The paper brings a comparison of two novel combinations of a relay identification test with a Time Delay System (TDS) model. Namely, the on/off plus saturation relay test and relay transient experiment are introduced. In the former case with more then two unknown model parameters, it is necessary to apply some technique for frequency response multiple-points identification; here, the Autotune Variation Plus (ATV+) procedure introducing an artificial delay is utilized. Limit cycles data are evaluated via an unordinary time-domain approach and using the Discrete Time Fourier Transform (DTFT), respectively. Both methodologies are verified and compared on a model of the circuit heating laboratory system with internal delays. INTRODUCTION By autotuning (i.e. automatic tuning), a set of methods which enable the controller to be tuned automatically on demand from an operator or an external signal is meant (Astrom and Hagglund 1984; Hang et al. 2002). Industrial experience has clearly indicated that this is highly desirable and useful feature. The whole procedure usually consists of two basic steps: Process model parameters identification followed by controller tuning; however, some approaches do not require explicit model identification. The relay feedback autotuning (identification) test performing limit cycle oscillations was successfully applied to the autotuning of PID controllers in (Astrom and Hagglund 1984) and it is widely used and in practice as a well applicable technique. It is robust, easy to implement, timesaving, easy to use and close-loop control which keeps the process close to the setpoint. The classical relay-feedback loop scheme with a symmetrical relay is depicted in Figure 1. If the process is stabilizable and has a phase lag of at least π radians, the process input ( ) t u and output ( ) t y are logged until the system reaches stationary oscillations, the amplitude A of error ( ) t e equals the amplitude of ( ) t y and the phase shift between ( ) t e and ( ) t y is –π. Figure 1: Relay-feedback Test Scheme However, the original relay feedback test sometimes called ATV (Autotune Variation) – see (Luyben 1987; Yu 2006) – has two basic drawbacks. First, due to an approximation, the estimation of the critical point is not accurate enough for some processes, such as those with large time delays (Wang et al. 1999). For example, there is an error of 23% for a first order unstable system with input-output delay. Second, the basic test enables to estimate only single point of the frequency characteristics. Hence, there have been investigated and developed many advanced techniques, which should eliminate the two mentioned deficiencies. Much research has been undertaken in identifying multiple points on the process frequency response, for instance, inserting of an integral or a delay element into the open loop (Scali et al. 1999; Tan et al. 1996). In this paper, the basic on/off relay test is improved by the use of a saturation relay (Shen et al. 1996; Yu 2006) for more accurate model parameters estimation followed by the above mentioned artificial-delay test. This approach is confronted with a technique utilizing the relay transient with DTFT (Hang et al 1995; Wang et al. 1997). The advantage of the latter method is that arbitrarily many points on the Nyquist curve can be estimated by a single test. The novel merit of this contribution is that the two methods are verified and benchmarked on a simulated identification relay experiment for a TDS model of a circuit heating laboratory plant (Dostalek et al. 2008; Pekař et al. 2009). Proceedings 27th European Conference on Modelling and Simulation ©ECMS Webjorn Rekdalsbakken, Robin T. Bye, Houxiang Zhang (Editors) ISBN: 978-0-9564944-6-7 / ISBN: 978-0-9564944-7-4 (CD) RELAY-FEEDBACK TEST Let us concisely describe model parameters estimation from the relay feedback test. Consider a simple feedback with and on/off relay as in Figure 1. If the process is stabilizable and has a phase lag of at least π radians, the process input ( ) t u and output ( ) t y are logged until the system reaches stationary oscillations, the amplitude A of error ( ) t e equals the amplitude of ( ) t y and the phase shift between ( ) t e and ( ) t y is –π. Hence, the ultimate period u T is obtained from oscillations, which gives the information about the critical point, together with the ultimate gain which is approximately given by

In this paper, a new method for process frequency response identification is proposed, which can identify multiple points on a process frequency response from a single relay feedback test. The process output and input responses to the test are decomposed into the transient parts and the stationary cycle parts. These parts are then transformed to their frequency responses using the DFT and digital integration, respectively, to obtain the process frequency response. Real-time tests of the method on various processes give quite accurate process frequency responses, especially in the important frequency range from zero to the critical frequency. The testing conditions for our identification are the same as those for the step testing. The multiple points on process frequency response identified are very useful for process modeling and controller tuning. A real-time PID controller auto-tuning experiment which employs the proposed identification method is presented to demonstrate the applicability of our method.

A robust adaptive PID controller design motivated from the sliding mode control is proposed for a class of uncertain chaotic systems in this paper. Three PID control gains, \(K_p, K_i\), and \(K_d\), are adjustable parameters and will be updated online with an adequate adaptation mechanism to minimize a previously designed sliding condition. By introducing a supervisory controller, the stability of the closed-loop PID control system under with the plant uncertainty and external disturbance can be guaranteed. Finally, a well-known Vanderpol oscillator is used as an illustrative to show the efectiveness of the proposed robust a PID controller.

A novel methodology is proposed for robust gain-scheduled PID controller design for uncertain LPV systems. The proposed design procedure is based on the parameter-dependent quadratic stability approach. A new uncertain LPV system model has been introduced in this paper. To access the performance quality the notion of a parameter varying guaranteed cost is used. Numerical examples show the benefit of the proposed method.

This work proposes the coefficient diagram method (CDM)-based two degree of freedom proportional integral (CDM-PI) controller tuning rules for stable and unstable first order plus time delay (FOPTD) processes and pure integrating processes with time delay (PIPTD). To derive the tuning rules, a general first order plus time delay (FOPTD) model, the first order Taylor denominator (TD) approximation technique and the pole allocation strategy named CDM is used. The tuning rules derived here are novel and they relate the controller parameters to the process model parameters directly. The performance of the CDM-PI controller utilising the proposed tuning rules is tested with numerical examples of stable, unstable and pure integrating processes with time delay models. The test results indicate that the proposed tuning rules yield promising results over the other PI controllers. Performance measures confirm the effectiveness of the proposed tuning method.

In this paper design and real-time implementation of robust PID controller is performed for a Twin Rotor MIMO System (TRMS) which is a laboratory scale set-up of a helicopter. The robust PID controller is designed employing Kharitonov stability theorem. The controller design is based on the linearised model of highly nonlinear TRMS. The performance of the compensated system is investigated through experimental results with the laboratory prototype model. The time domain responses for the main and tail rotors are found to be satisfactory. The controller provides comparable performances as presented in the literature for similar system.

Relay-based estimation of multiple points on process frequency response

Performance-guaranteed Robust PID Controller Design for Systems with Unstable Zero

There are more trends to use electromechanical actuator (EMA) in aerospace and robotics applications which require easy control, high efficiency, and high dynamics. This paper deals with experimental identification, uncertainty modeling, and robust PID controller design for position control of a real EMA. It is identified and modeled as linear system with parametric uncertainty and time delay by using its experimental input-output data. The captured data, related to several experiments at different conditions, were used for model estimation and validation purposes. Robust PID controller is designed by graphical findings of the regions of stability with pre-specified margins and bandwidth requirements and by applying the complex Kharitonov's theorem. This novel method enables designers to make the convenient trade-off between stability and performance by choosing the proper margins and bandwidth specifications. The simulation and test results prove the superiority of the performance of this new EMA over the original EMA control system; this preference is pertaining to its robustness to parametric uncertainties.

This paper presents the methodology of design of a nonlinear robust controller for attitude regulation and its implementation in an experimental platform of an unmanned aerial vehicle (UAV) quadrotor. Details on the kinematic and dynamic modeling based on the Euler-Lagrange formalism are provided, as well as the particulars of the design of a nonlinear robust H-infinity PID controller to regulate the rotational moments. The performance and effectiveness of the proposed controller are tested in a simulation and an experimental platform. The performance of the proposed controller is compared with a conventional PID controller by using the integral square error (ISE) as performance parameter. Experimental results help to demonstrate the correct operation of the system for real-time applications in the presence of unmodeled dynamics and the uncertainties of the parameters.

The Dual Active Bridge converter (DABC), known for its bidirectional power transfer capability and high efficiency, plays a crucial role in various applications, particularly in electric vehicles (EVs), where it facilitates energy storage, battery charging, and grid integration. The Dual Active Bridge Converter (DABC), when paired with a high-performance CLLC filter, is well-regarded for its ability to transfer power bidirectionally with high efficiency, making it valuable across a range of energy applications. While these features make the DABC highly efficient, they also complicate controller design due to nonlinear behavior, fast switching, and sensitivity to component variations. We have used a Fractional-order PID (FOPID) controller to benefit from the simple structure of classical PID controllers with lower complexity and improved flexibility because of additional filtering gains adopted in this method. However, for a FOPID controller to operate effectively under real-time conditions, its parameters must adapt continuously to changes in the system. To achieve this adaptability, a Multi-Agent Reinforcement Learning (MARL) approach is adopted, where each gain of the controller is tuned individually using the Deep Deterministic Policy Gradient (DDPG) algorithm. This structure enhances the controller’s ability to respond to external disturbances with greater robustness and adaptability. Meanwhile, finding the best initial gains in the RL structure can decrease the overall efficiency and tracking performance of the controller. To overcome this issue, Grey Wolf Optimization (GWO) algorithm is proposed to identify the most suitable initial gains for each agent, providing faster adaptation and consistent performance during the training process. The complete approach is tested using a Hardware-in-the-Loop (HIL) platform, where results confirm accurate voltage control and resilient dynamic behavior under practical conditions. In addition, the controller’s performance was validated under a battery management scenario where the DAB converter interacts with a nonlinear lithium-ion battery. The controller successfully regulated the State of Charge (SOC) through automated charging and discharging transitions, demonstrating its real-time adaptability for BMS-integrated EV systems. Consequently, the proposed MARL-FOPID controller reported better disturbance-rejection performance in different working cases compared to other conventional methods.

Liquid level control in multiple connected tanks is a nonlinear control problem and is centre to various diverse industrial establishments, especially chemical process industries. The quality of products and process safety greatly relies upon the quality of control. In this paper, robust liquid level control for coupled tank systems has been proposed using optimal fractional order PID controllers. The controller tuning has been posed as multi-objective mixed sensitivity minimisation problem and is solved using multi-objective variant of particle swarm optimisation (MOPSO). At the end of the optimisation process, Pareto optimal set (POS) of solutions has been obtained. For choosing an optimal solution from the POS, there are several graphical methods like Pareto front, scatter matrices, parallel coordinate representations, etc. But these methods offer a very compact representation of the data and the problem arises for m-dimensional problems. So to ease the decision-making process, the paper also explores the use of level diagrams for aiding the decision maker in choosing the ideal solution from the Pareto optimal set. The designed fractional order PID controllers present robust behaviour to external disturbances and efficient tracking of the reference signal and the results obtained has been compared with different controller tuning algorithms.

Delay time modelling (DTM) has been developed and applied to industrial asset maintenance over the last 20 years. One of the key elements in DTM is the concept of the delay time which recognises an interval between the earliest point of defect identification and failure. Obviously the delay time is a random variable which varies depending upon the nature of defects and nature of inspections. If such a delay time can be captured by an appropriate distribution, various modelling options can be explored. This paper introduces the concept of delay time and its modelling techniques in the context of risk analysis of maintenance problems. It highlights the modelling for minimising the likelihood and consequence of economic or human risks associated with failures and maintenance activities under various maintenance strategies. Examples are presented, and the paper concludes with a summary of the key points discussed.

A new tuning method for proportional-integral-derivative (PID) controller design is proposed for a class of unknown, stable, and minimum phase plants. We are able to design a PID controller to ensure that the phase Bode plot is flat, i.e., the phase derivative w.r.t. the frequency is zero, at a given frequency called the "tangent frequency" so that the closed-loop system is robust to gain variations and the step responses exhibit an iso-damping property. At the "tangent frequency," the Nyquist curve tangentially touches the sensitivity circle. Several relay feedback tests are used to identify the plant gain and phase at the tangent frequency in an iterative way. The identified plant gain and phase at the desired tangent frequency are used to estimate the derivatives of amplitude and phase of the plant with respect to frequency at the same frequency point by Bode's integral relationship. Then, these derivatives are used to design a PID controller for slope adjustment of the Nyquist plot to achieve the robustness of the system to gain variations. No plant model is assumed during the PID controller design. Only several relay tests are needed. Simulation examples illustrate the effectiveness and the simplicity of the proposed method for robust PID controller design with an iso-damping property.

Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way.