System Dead Time Research Articles

Stability-preserving automatic tuning of PID control with reinforcement learning

Proportional-Integral-Derivative (PID) control has been the dominant control strategy in the process industry due to its simplicity in design and effectiveness in controlling a wide range of processes. However, most traditional PID tuning methods rely on trial and error for complex processes where insights about the system are limited and may not yield the optimal PID parameters. To address the issue, this work proposes an automatic PID tuning framework based on reinforcement learning (RL), particularly the deterministic policy gradient (DPG) method. Different from existing studies on using RL for PID tuning, in this work, we explicitly consider the closed-loop stability throughout the RL-based tuning process. In particular, we propose a novel episodic tuning framework that allows for an episodic closed-loop operation under selected PID parameters where the actor and critic networks are updated once at the end of each episode. To ensure the closed-loop stability during the tuning, we initialize the training with a conservative but stable baseline PID controller and the resultant reward is used as a benchmark score. A supervisor mechanism is used to monitor the running reward (e.g., tracking error) at each step in the episode. As soon as the running reward exceeds the benchmark score, the underlying controller is replaced by the baseline controller as an early correction to prevent instability. Moreover, we use layer normalization to standardize the input to each layer in actor and critic networks to overcome the issue of policy saturation at action bounds, to ensure the convergence to the optimum. The developed methods are validated through setpoint tracking experiments on a second-order plus dead-time system. Simulation results show that with our scheme, the closed-loop stability can be maintained throughout RL explorations and the explored PID parameters by the RL agent converge quickly to the optimum. Moreover, through simulation verification, the developed RL-based PID tuning method can adapt the PID parameters to changes in the process model automatically without requiring any knowledge about the underlying operating condition, in contrast to other adaptive methods such as the gain scheduling control.

Decrease the order of nonlinear predictors based on generalized-Lipschitz condition

In this paper, a low order asymptotic predictor is introduced for nonlinear systems based on measurable outputs. Predictors play basic role in the control of dead-time systems having no delay-free input. Generalized Lipschitz Condition is employed here to develop Sequential Sub-Predictors (SSP) for complex nonlinear systems. Compared with existing synthesis methods which involve Lipschitz Condition, Generalized Lipschitz condition relaxes the conservative of the stability results using some modified matrix inequalities. The proposed idea can be more attractive for unstable systems due to reduce significantly the order of the predictors. Also using the proposed method, the effect of the external disturbance can be minimized based on H∞ index. Afterward, a SSP-based controller is presented to illustrate the effectiveness of proposed method to stabilize nonlinear systems with input-delay. Finally, the predictor capability is investigated by simulation examples.

Robust fractional-order [proportional integral derivative] controller design with specification constraints: more flat phase idea

This paper presents a new design methodology for robust fractional-order controllers with more than three parameters for first-order plus dead time systems using the synthesis scheme of the ‘more flat phase’ idea for a fractional-order controller with [proportional integral derivative] (FO[PID]) structure. The stability region of the FO[PID] controller and the synthesis scheme with the ‘more flat phase’ idea are illustrated through a simulation example. The corresponding pseudo-codes of the synthesis scheme are also summarised. The implementation and approximation error of the FO[PID] controller are also discussed. Likewise, the superiority of the FO[PID] controller designed with the ‘more flat phase’ idea is verified by additional simulations and experimental results where the closed-loop system with FO[PID] controller is not sensitive to the variations of the loop gain, ensuring satisfactory control performance. Obtained results show high potential in practical industrial applications.

An explicit robust stability condition for uncertain time-varying first-order plus dead-time systems

First-order plus dead-time (FOPDT) models are broadly used in process control to represent damped dynamic processes with time delays. An explicit condition for parameter- and delay-dependent robust stability of FOPDT systems with varying uncertain parameters and delay is derived in this paper. An internal model control (IMC) approach is proposed to parameterize stabilizing controllers that satisfy the output tracking objective in time-varying FOPDT systems represented by an uncertain first-order dynamic model with a time-varying delay in the control input. The small-gain theorem is used to derive an explicit necessary and sufficient parameter-dependent robust stability condition as a function of the nominal system gain, nominal varying delay, nominal time constant, and the bounds of the parameter uncertainties. An equivalent proportional-integral-derivative (PID) controller is then extracted to facilitate the implementation of the proposed IMC-based robust control. The application of the proposed explicit robust stability condition is studied in the context of air-fuel ratio (AFR) control in lean-burn spark ignition (SI) engines with a large time-varying transport delay in the control loop due to the placement of the universal exhaust gas oxygen (UEGO) sensor downstream the catalytic converter.

A predictor for dead-time systems based on the Kalman Filter for improved disturbance rejection and robustness

Dead-time processes are common in industry and represent a challenge for feedback control. The use of predictor structures with the controllers can attenuate this. A predictor is proposed here based on the Kalman filter, the Kalman Predictor (KP), that has only one tuning parameter for SISO systems and can be used with non-minimum phase, open-loop unstable and integrative systems of any order, and is also capable of estimating many types of disturbances. It is shown that the KP implicitly defines a Filtered Smith Predictor (FSP), hence, all tools used to analyze the closed-loop properties of the FSP can be used, including robustness and disturbance rejection. Tuning guidelines are provided and simulation examples with comparisons with other predictor structures are used to illustrate the advantages of the proposed KF.

Real-time implementation of QFT, GA, and BFTPSO controller for pH neutralization system

The control of a pH process is complex because of severe nonlinearities in its behavior. A continuous pH neutralization process is usually represented as a first-order plus dead time system, but its gain varies for different operating points. Therefore, a conventional linear controller cannot be used, and the pH system was thus represented as a linear state-space model around an equilibrium point. This linear model was then used to compute the PID controller gains using robust and optimization techniques like quantitative feedback theory, bacterial foraging technique-based particle swarm optimization algorithm, and genetic algorithm. The corresponding controller gains resulting from the three algorithms were used to control the pH using a reconfigurable I/O device, NI myRIO-1900. Finally, the output time domain specifications and the servo and regulatory responses, resulting from the three algorithms, were compared in simulation and in real-time to deduce the appropriate tuning algorithm for this system.

Tuning of linear active disturbance rejection controllers for second-order underdamped systems with time delay

Second-order plus dead-time systems are difficult to tune for a PID controller due to its special structure, especially the underdamped systems that have oscillatory responses. As a control technique with strong disturbance rejection ability, linear active disturbance rejection control is widely used as a substitute for PID controllers. This paper proposes a tuning formula for second-order linear active disturbance rejection control for underdamped second-order plus dead-time systems via internal model control. The formula is derived by minimizing the integral of time squared error index under certain robustness measure condition. Simulation results show that the linear active disturbance rejection controller tuned from the proposed formula can achieve satisfactory control performance for oscillatory systems.

Observation of photon antibunching with only one standard single-photon detector.

The second-order photon correlation function g2(τ) is of great importance in quantum optics. g2(τ) is typically measured with the Hanbury Brown and Twiss (HBT) interferometer, which employs a pair of single-photon detectors and a dual-channel time acquisition module. Here, we demonstrate a new method to measure and extract g2(τ) with a standard single-photon avalanche photodiode (dead-time = 22ns) and a single-channel time acquisition module. This is realized by shifting the coincidence counts of interest to a time window not affected by the dead-time and after-pulse of the detection system using a fiber-based delay line. The new scheme is verified by measuring g2(τ) from a single colloidal nanocrystal. Photon antibunching is unambiguously observed and agrees well with the result measured using the standard HBT setup. Our scheme simplifies the higher-order correlation technique and might be favored in cost-sensitive circumstances.

Control of non-minimum phase systems with dead time: a fractional system viewpoint

In this paper, a unified theory is proposed for the fractional control of the non-minimum phase plus dead time systems via the introduction of the fractional filter-PID control and stability inequality. More precisely, three Theorems from a fractional system-theoretic viewpoint are developed. The first Theorem deals with the controller design through the notion of generalised internal model control. That embeds the fractional filter-PID controller coupled with the unified form of the Smith predictor and the inverse compensator. The second Theorem designs a bound on the fractional filter parameter that leads to a stable feedback system. The third Theorem deals with the stability of a fractional quasi-characteristic polynomial of the commensurate order arising from the control of non-minimum phase systems with dead time. Two illustrative second-order plus dead time non-minimum phase systems are presented to test the efficacy of the resulting fractional filter-PID controller that obeys the stability bound. Controller and sensitivity performance indices for the nominal as well as mismatch cases reveal the effectiveness and robustness of the proposed method for enhanced closed-loop performance. Thus, the generalised inverse compensator, the fractional filter-PID controller and the stability inequality in corroboration with numerical simulations demonstrate the better control of non-minimum phase systems with dead time.

Virtual shaft: Robust coupling by bidirectional and distributed prediction of coupling values

Shifting automotive powertrain development tasks to earlier phases (frontloading) increases efficiency by utilizingtest-benches as opposed to prototype vehicles (road-to-rig approach). The coupling of distributed test-benches by a virtualized shaft connection is required to reproduce interactions of automotive powertrain components. A coupling algorithm simulates a rigid connection by synchronizing the torque and speed of two distributed test-bench’s electric motors. System dead-times lead to limited stability and reduced bandwidth of the coupling algorithm. In this study, a method for a stable bidirectional coupling of speed and torque of both subsystems is described analytically and verified by simulation. All component models are calibrated based on measurements using state-of-the-art test-bench equipment. A distributed prediction algorithm is proposed for the dead-time compensation. Four Kalman predictors estimate the coupling values of both subsystems at wall-clock-time without measurement and communication latencies. A detailed drive cycle analysis is performed through simulation. This enables a Virtual Shaft Algorithm to achieve a higher bandwidth and an improved coupling robustness.

Design of Optimal PID Control with a Sensitivity Function for Resonance Phenomenon-involved Second-order Plus Dead-time System

The present study discusses an optimal design method of a proportional integral derivative (PID) control system for a continuous-time second-order plus dead-time system (SOPDT) that includes an under-damping system. The proposed PID control system is designed to minimize the performance index defined as the tracking performance for the set-point or the regulation performance for the disturbance, where the assigned stability margin is also achieved in order to attain robust stability for the modeling error. Because of the relationship between tracking performance and robust stability, the PID parameters are decided such that the tracking performance is optimized subject to the user-specified stability margin. In the proposed design method, in order to obtain the most general optimal PID possible parameters for controlled plants, we discuss a design method based on a normalized plant model. As a result, the PID parameters are seamlessly optimized between over-damping, critical-damping, and under-damping systems. The effectiveness of the proposed method is demonstrated through numerical examples.

Effect of the Nature of the Carrier Gas on the Retention Characteristics of Sorbates

The possibility is verified of estimating the phase ratio of a chromatographic column chromatographically from the semi-empirical dependence of the retention time of homologs on the number of carbon atoms in their structure. The effect the nature and pressure of the carrier gas has on the phase ratio is considered. It is shown that the average value is 1190 ± 100 for the polar phase and 380 ± 50 for the non-polar phase, regardless of the nature of the carrier gas. An exception is СО2, where phase ratio β drops strongly after adjusting the initial data. At the same time, it is clear that the values of the phase ratio determined chromatographically differ strongly from those claimed by the manufacturers of columns, while system dead time tМ and energy $$\Delta {{G}_{{{\text{C}}{{{\text{H}}}_{{\text{2}}}}}}}$$ of the methylene unit sorption of a homological series can be reliably determined via extrapolation.

Design of fractional order controllers using constrained optimization and reference tracking method

In recent times, fractional order controllers are gaining more interest. There are several fractional order controllers are available in literature. Still, tuning of these controllers is one of the main issues which the control community is facing. In this paper, online tuning of five dierent fractional order controllers is discussed viz. tilted proportional-integral-derivative (T-PID) controller, fractional order proportional-integral (FO-PI) controller, fractional order proportional-derivative (FO-PD) controller, fractional order proportional-integral-derivative (FO-PID) controller. A reference tracking method is proposed for tuning of fractional order controllers. First order with dead time (FOWDT) system is used to check feasibility of the control strategy.

Analytical Tuning Method of MPC Controllers for MIMO First-Order Plus Fractional Dead Time Systems

An analytical model predictive control (MPC) tuning method for multivariable first-order plus fractional dead time systems is presented in this paper. First, the decoupling condition of the closed-loop system is derived, based on which the considered multivariable MPC tuning problem is simplified to a pole placement problem. Given such a simplification, an analytical tuning method guaranteeing the closed-loop stability as well as pre-specified time-domain performance is developed. Finally, simulation examples are provided to show the effectiveness of the proposed method.

An improved relay feedback identification technique for Hammerstein model

This research work is motivated by the success of relay feedback in linear system domain thereby an attempt has been made at extending it to nonlinear Hammerstein models. The identification is carried out using a relay with hysteresis in feedback, the parameter estimation is done solely by solving state-space equations at various points on one cycle of the input and output signals. The work considers not all but wider probable cases of the linear subsystem-real roots, complex conjugate roots, first order plus dead time systems, integrating second order plus dead time and repeated roots. The proposed method is simple compared to other works in literature as well as applicable to unstable systems. Simulations of various examples are carried out to demonstrate the efficacy of the proposed identification scheme.

Higher Order PD and iPD controller tuning

This paper deals with tuning of proportional-derivative (PD) and “intelligent” proportional-derivative (iPD) controller for the double integrator plus dead time (DIPDT) system. iPD corresponds to a PD controller augmented with a disturbance-observer based integral (I) action based on finite impulse response (FIR) filter. Since noise reduction requires working with as short sampling period as possible, the design of controllers is appropriate to implement in the continuous-time domain with a suitable discretization. In this way, when the noise attenuation filters are approximated by an equivalent dead time added to the plant dead time, stabilizing controllers with higher order output derivatives may be introduced simply and without excessive noise-induced control effort. Together with filtration of the reconstructed disturbance it significantly improves the overall loop performance.

A Novel Deadbeat Predictive Current Control Scheme for OEW-PMSM Drives

In an open-end winding permanent magnet synchronous machines drive with single dc voltage source, zero-sequence current (ZSC) can lead to a high current stress of power modules and loss of motor. Therefore, some modulation strategies have been employed to alleviate ZSC by suppressing zero-sequence voltage (ZSV). However, ZSC still exists in the system because ZSV can also be generated by other nonlinear factors of inverter such as the dead time of system. In addition, zero-sequence back electromotive force in the zero-sequence path can also enlarge torque ripple and ZSC. In order to deal with above problems, first, this paper proposes a full-order adaptive zero-sequence observer to estimate future ZSC and ZSV, which are able to compensate one-step control delay. Second, to achieve the maximum voltage dc bus utilization, this paper proposes a novel deadbeat predictive current control (DPCC) scheme with alternate sub-hexagonal center pulsewidth modulation strategy to suppress ZSC and torque ripple simultaneously. Finally, this paper presents a comparative study of two types of methods, namely traditional DPCC scheme and the proposed DPCC scheme. Simulation and experimental results are demonstrated to verify the effectiveness of the proposed DPCC scheme.

A New PID Controller Design with Constraints on Relative Delay Margin for First-Order Plus Dead-Time Systems

The maximum sensitivity function as the conventional robustness index is often used to test the robustness and cannot be used to tune the controller parameters directly. To reduce analytical difficulties in dealing with the maximum sensitivity function and improve the control performance of the proportional-integral-derivative controller, the relative delay margin as a good alternative is proposed to offer a simple robust analysis for the proportional-integral-derivative controller and the first-order plus dead-time systems. The relationship between the parameters of the proportional-integral-derivative controller and the new pair, e.g., the phase margin and the corresponding gain crossover frequency, is derived. Based on this work, the stability regions of the proportional-integral-derivative controller parameters, the proportional gain and the integral gain with a given derivative gain, are obtained in a simple way. The tuning of the proportional-integral-derivative controller with constraints on the relative delay margin is simplified into an optimal disturbance rejection problem and the tuning procedure is summarized. For convenience, the recommended parameters are also offered. Simulation results demonstrate that the proposed methodology has better tracking and disturbance rejection performance than other comparative design methodologies of the proportional-integral/proportional-integral-derivative controller. For example, the integrated absolute errors of the proposed proportional-integral-derivative controller for the tracking performance and disturbance rejection performance are less than 91.3% and 91.7% of the integrated absolute errors of other comparative controllers in Example 3, respectively. The proposed methodology shows great potential in industrial applications. Besides, the proposed method can be applied to the design of the proportional-integral-derivative controller with filtered derivative which is recommended for practical applications to weaken the adverse influence of the high-frequency measurement noise.

Controlling industrial dead-time systems: When to use a PID or an advanced controller

This work presents a comparative analysis of PID, DTC and MPC strategies when used to control SISO processes with dead time considering characteristics commonly found in industry, such as noisy measurements in the process output and modeling error. For unconstrained processes, it is shown that the performance improvement obtained by using a more advanced control strategy instead of a PID is small or nonexistent for cases which require high robustness. However, for cases with well-known process models it is shown that the improvement obtained by using a more complex control structure is justified even for small delays. For constrained processes it was demonstrated that a PID with anti-windup is able to provide similar or even better results than MPC when robust solutions are considered.

Discrete-Time First-Order Plus Dead-Time Model-Reference Trade-off PID Control Design

The present study proposes a novel proportional-integral-derivative (PID) control design method in discrete time. In the proposed method, a PID controller is designed for first-order plus dead-time (FOPDT) systems so that the prescribed robust stability is accomplished. Furthermore, based on the control performance, the relationship between the servo performance and the regulator performance is a trade-off relationship, and hence, these items are not simultaneously optimized. Therefore, the proposed method provides an optimal design method of the PID parameters for optimizing the reference tracking and disturbance rejection performances, respectively. Even though such a trade-off design method is being actively researched for continuous time, few studies have examined such a method for discrete time. In conventional discrete time methods, the robust stability is not directly prescribed or available systems are restricted to systems for which the dead-time in the continuous time model is an integer multiple of the sampling interval. On the other hand, in the proposed method, even when a discrete time zero is included in the controlled plant, the optimal PID parameters are obtained. In the present study, as well as the other plant parameters, a zero in the FOPDT system is newly normalized, and then, a universal design method is obtained for the FOPDT system with the zero. Finally, the effectiveness of the proposed method is demonstrated through numerical examples.