Non-minimum Phase Systems Research Articles

Tuning equations for sliding mode controllers: An optimal multi-objective approach for non-minimum phase systems

This article proposes an approach to create tuning equations based on optimal multi-objective design specifically for Sliding Mode controllers applied to non-minimum phase processes that can be approximated to a First-Order-Plus-Deadtime class of systems. This is achieved by taking into account various performance criteria, including tracking, controller action, and transient response. The proposed tuning equations aim to achieve a well-balanced trade-off among competing objectives. The paper presents a systematic framework for formulating the multi-objective optimization problem and derives analytical expressions for the tuning parameters based on the desired performance specifications. The effectiveness of the proposed method is demonstrated through simulations in three dynamic processes: a high-order linear system with dominant delay, a high-order plus inverse response system, and a nonlinear process with variable parameters and dominant deadtime. A comparative analysis with an existing SMC tuning technique shows superior performance results. This paper provides valuable information on the optimization procedure for obtaining analytical tuning equations for controllers, offering a promising avenue for improving the performance of control systems in practical applications.

Tracking control of non-minimum phase systems: a kernel-based approach

Feedforward control with model inversion is a widely-used solution for high-precision output tracking. However, because inverting a non-minimum phase model generates unbounded control input, model-inversion only applies to limited types of systems. This paper presents a new non-parametric pseudo-inversion approach to design bounded optimal control input with desirable properties for arbitrary types of systems. Closed-form equations are presented for the batch (full preview) and recursive (limited preview) implementations of this approach, and its performance is compared against existing pseudo-inversion methods in benchmark numerical examples. Furthermore, the practical implementation of the proposed method is demonstrated by designing a feedforward controller for a commercial 3-Dimensional (3D) printer. The results show that the proposed approach effectively compensates for the structural vibrations of the printer, preventing layer-shifting errors that usually happen during high-speed printing.

Optimal Controller Identification for multivariable non-minimum phase systems

This work deals with data-driven control for non-minimum phase (NMP) systems, where the goal is to design a controller for a plant whose model is unknown by using a batch of input–output data collected from it. The approach is based on the Model Reference paradigm, where a transfer function matrix – the reference model – is used to specify the desired closed-loop performance. The NMP systems issue in Model Reference approaches is a well-known problem in control literature and it is no different in data-driven methods. This work explains how to adapt the formulation of the Optimal Controller Identification (OCI) method to cope with this class of systems. Considering a convenient parametrization of the reference model and a flexible performance criterion, it is possible to identify the NMP transmission zeros of the plant along with the optimal controller parameters, as it will be shown. Both diagonal and block-triangular reference model structures are treated in detail. Simulation examples show the effectiveness of the proposed approach.

An Adjoint Based Data-Driven Inverse-Free Iterative Feedforward Control Method

Abstract Feedforward control is widely used in control systems since it can achieve high tracking performance by effectively compensating for known disturbances before they affect the system. For traditional feedforward control methods, the performance improvement highly depends on the model quality of the system model and the accuracy of the model-inversion. However, on one hand, in practice, the modeling error is inevitable, especially for precision motion systems with complex dynamics, on the other hand, the non-minimum phase (NPM) systems often lead to a problem that plant inversion is unstable and non- causal. Therefore, this paper proposes an approach that combines the bene?ts of data-driven feedforward control and the gradient descent method. This integrated approach aims to address challenges related to laborious model identi?cation and unstable plant inversion simultaneously. The main idea is to replace the model with dedicated experiments on the system and to avoid calculating plant inversion by applying the gradient descent method to the learning process. The simulation and experimental results show that the algorithm can achieve the optimal point-to-point tracking performance without relying on model-inversion.

Retrospective‐cost‐based model reference adaptive control of nonminimum‐phase systems

AbstractThis paper presents a novel approach to model reference adaptive control inspired by the adaptive pole‐placement controller (APPC) of Elliot and based on retrospective cost optimization. Retrospective cost model reference adaptive control (RC‐MRAC) is applicable to nonminimum‐phase (NMP) systems assuming that the NMP zeros are known. Under this assumption, the advantage of RC‐MRAC is a reduced need for persistency. The present paper compares APPC and RC‐MRAC under various levels of persistency in the command for minimum‐phase and NMP systems. It is shown numerically that the model‐following performance of RC‐MRAC is less sensitive to the persistency of the command compared to APPC at the cost of knowledge of the NMP zeros. RC‐MRAC is also shown to be applicable for disturbance rejection under unknown harmonic disturbances.

Optimizing the series cascade control structure for nonminimum phase system regulation

AbstractThis work elucidates the control of integrating a nonminimum phase system via a series cascade scheme with fractional‐order P.I. (Proportional–Integral) plus D (Derivative) controller. The traditional Internal Model Control (IMC) is adopted for inner loop controller design. The feedback D controller is synthesized with the outer loop process model, showing the proposed work's universality. The outer loop controller is suggested in the IMC framework after the accountability of fractional‐filter and inverse response compensator. This combination is revealed to enhance performance without compromising robustness. The Riemann sheet principle is explored to compute the stability of the suggested controller. The sensitivity analysis has asserted the robustness. More importantly, the optimal value of controller settings is achieved via the Teaching Learning Based Optimization (TLBO) algorithm. This TLBO algorithm uses an objective function that minimizes Integral Square Error. Two illustrative problems are utilized to examine the recommended control structure's virtue.

What are the key stability challenges in high-bandwidth, non-minimum phase systems with time-varying, and non-smooth delays?

The analysis and control of stability in high-bandwidth systems characterized by non-minimum phase delays represent a formidable challenge within the realm of control theory and engineering. This research aims to address the pivotal question of whether it is feasible to enhance the stability of such intricate systems. These systems inherently possess uncertain and swiftly changing delay characteristics, rendering them exceptionally demanding to control effectively. In the course of this investigation, we embark on a comprehensive exploration of the theoretical underpinnings of the stability of high-bandwidth, non-minimum phase delay systems. This encompassing inquiry encompasses a meticulous consideration of both derivative-delay and piecewise continuous delay components. To underpin our analysis, we judiciously incorporate feedback mechanisms, drawing upon mathematical tools such as the Jensen inequality and Lyapunov-based methodologies to rigorously establish stability conditions. Furthermore, our exploration extends to encompass the concept of input-output stability and complements it with the notion of asymptotic stability, thereby ensuring that the systems in question exhibit uniform stability across diverse temporal domains. The outcomes of our investigation furnish compelling evidence that by harnessing the power of discrete-time Lyapunov-Krasovskii functionals, it becomes conceivable to circumscribe the maximum delay within predefined thresholds. This achievement holds the promise of enhancing stability in non-minimum phase delay systems characterized by high bandwidth. These findings have far-reaching implications, profoundly influencing the design and control paradigms across a spectrum of engineering applications. Notably, this impact extends to areas such as communication networks, real-time control systems, and robotics, where the mitigation of instability due to non-minimum phase delays has been an enduring challenge.

Optimal fractional-order series cascade controller design: A refined Bode’s ideal transfer function perspective

In process industries, cascade control is comprehensively used for disturbance attenuation. This manuscript presents the control of the non-minimum phase (NMP) system with dead time via a series cascade scheme. An improved fractional-filter-based control scheme encompassing inverse and dead time compensators with analytical tuning is proposed. The design hinges on an enhanced Bode’s ideal transfer function, which incorporates delay and NMP zero to deal with the system’s NMP and dead time. Particle swarm optimization (PSO) is utilized to obtain the optimal assessment of fractional filter-based controller settings by minimizing an objective function. Two benchmark problems are presented to test the efficacy of the recommended controller. The Riemann surface and sensitivity examination are used to realize the stability and robustness of the feedback design. Numerical simulations exhibit the superiority of the suggested controller for closed-loop results.

Vibration control of an eleven-story structure with MR and TMD dampers using MAC predictive control, considering nonlinear behavior and time delay in the control system

In the last two decades, model predictive control has made significant progress in research and industrial process control. This achievement is through the high ability of model predictive control to solve industrial process control problems in the time domain. It is applicable in many control systems, such as delayed systems, univariate and multivariable systems, non-minimum phase systems, and unstable systems. This strategy is essentially for controlling delayed systems. It is possible to create a time delay in transmitting structure movement data in the systems that use sensors and receivers of sensor-recorded data, which is a factor to reduce the efficiency of the control system and even the instability of the structure. In this study, to investigate and compensate the effect of time delay in the hybrid vibration control system, an 11-story structure with a tuned mass damper and an MR damper was used by model algorithmic control (MAC) and nonlinear behavior of the structure in the control process. The linear and nonlinear structural models are created in OpenSEES, the control system in MATLAB; and the TCP/IP connection method was used to connect two programs. The fuzzy decision system is based on accelerating and decelerating movement. The structure has been subjected to seven earthquakes with maximum accelerations of 0.1 g to 1.0 g with an incremental step of 0.1 g. According to the results of incremental dynamic analysis, the average percentage of the fuzzy control system performance reduction, considering the time delay, is 15.28% and 38.57% for the maximum displacement and base shear in the linear structure, and 8.04% and 5.97% for the nonlinear structure, respectively. The assumed time delays have been effectively overcome using the model algorithmic control system (MAC), determining the prediction horizon of the structural behavior, and optimizing the control force in the control horizon. The performance of the MAC control system has been better than the fuzzy control system and the passive control with TMD. The average improvement percentage of the results of the maximum linear and nonlinear structural displacement response is respectively equal to 10.2 and 11.8 with the MAC control system, compared to the fuzzy control system without time delay. For the maximum base shear, it is equal to 8.32 and 2.49, respectively.

Online Adaptive Dynamic Programming for Optimal Self-Learning Control of VTOL Aircraft Systems With Disturbances

In this paper, a novel online adaptive dynamic programming control algorithm is developed to solve optimal control problems for Vertical Take Off and Landing (VTOL) aircraft systems. Considering the input disturbances of the VTOL systems and adding perturbation interference term to the value function, a robust policy iteration-based ADP method will be introduced to obtain the optimal controller with an approximate optimal control approach. The convergence property is developed to ensure the value function converges to a finite neighborhood of the optimal one. The uniform ultimate boundedness (UUB) of the iterative errors will be strictly proved by Lyapunov stability theory. Finally, we will give mathematical simulation results with the developed method. Compared with sliding mode control (SMC) and linear quadratic regulator (LQR) methods, the simulation results illustrate better overall performance of the novel method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —As an effective tool to solve the optimal control problem of complex nonlinear systems, adaptive dynamic programming (ADP) can effectively overcome the computational problems caused by “curse of dimensionality”. VTOL aircraft system is a typical strongly coupled, underactuate, non-minimum phase system, and the sudden random disturbances will make the control task more challenging. Aiming at the particularity of VTOL system, a new online policy iteration-based ADP algorithm is developed to obtain optimal control with an approximate optimal control strategy in this paper. Decoupling and disturbances can be effectively avoided. convergence will be analyzed to make sure the value function converge to a finite neighborhood of the optimal value function. The UUB property of the system will be proved by the Lyapunov approach.

On a Benchmark in Output Regulation of Non-Minimum Phase Systems

Output regulation for non-minimum phase nonlinear systems is still a largely open problem, with only a few results in the literature covering a few specific cases. This paper explores the design of a state-feedback regulator for a basic non-minimum phase benchmark system that, nevertheless, already embodies many of the challenges and difficulties linked to the lack of minimum phase. Although the developed solution only provides local regulation guarantees for this specific benchmark, it nonetheless suggests a path for the successive development of a systematic regulation theory for non-minimum phase systems in normal form. In turn, this paper is a first step in such a direction.

A Robust Hybrid MRAPID Controller Design for an Underactuated Nonlinear System With Parametric Uncertainty

This paper presents an optimal design and simulation of a novel enhanced model reference adaptive proportional–integral–derivative controller (MRAPIDC) for underactuated nonlinear systems with parametric uncertainty. The controller is applied to a cart‐inverted pendulum mechanical system, a fourth‐order, nonminimum phase system with pronounced nonlinearity and unstable dynamics. This system includes a single input and multiple outputs, specifically the position and velocity of the cart, as well as the angle and angular velocity of the inverted pendulum arm. The main objective of the proposed controller is to maintain the pendulum in an upright position against gravity while moving the cart to a desired displacement regardless of parametric uncertainty, input variation, external disturbances, and random noise. The hybrid controller design is developed by combining the MRAPIDC design with controller gains applied to each output and incorporating the system adaptation error in a specific configuration. Optimal performance is achieved through parameter tuning via the social spider optimization (SSO) algorithm, and MIT (Massachusetts Institute of Technology) rule criteria are employed to ensure system stability. Performance improvements are demonstrated through several error indices, showing enhanced transient and steady‐state responses compared to both the reference model adaptive controller and the conventional proportional–integral–derivative (PID) controller. The proposed controller can effectively achieve system stability with a short settling time of 2.55 s and establishes a steady swinging balance within 2.9 s.

Stabilization of second-order non-minimum phase system with delay via PI controllers. Spectral abscissa optimization

Stabilization of second-order non-minimum phase system with delay via PI controllers. Spectral abscissa optimization

A novel tuning approach with SSR algorithm for non-minimum phase system

A novel tuning approach with SSR algorithm for non-minimum phase system

Stabilizing ℓ-semioptimal fractional controller for discrete non-minimum phase system under unknown-but boundeddisturbance

We consider the problem of optimal stabilizing controller synthesis for a discrete non-minimum phase dynamic plant described by a linear difference equation with an additive unknown-but-bounded disturbance. When considering the ’worst’ case of disturbance, solving this optimization problem has combinatorial complexity. However, by choosing an appropriate sufficiently high sampling rate, it becomes possible to achieve an arbitrarily small level of suboptimality using a noncombinatorial algorithm. In this article, we propose using fractional delays to achieve a small level of suboptimality without significantly increasing the sampling rate. We approximate fractional delays by minimizing the ℓ1-norm of the objective function. The proposed approximation of the fractional delay allows obtaining zero additional error for many non-integer solutions. Furthermore, it is shown that with a non-zero approximation error, the resulting controller may have a smaller additional error than the controller obtained using integer optimization. The theoretical results are illustrated by simulation examples with non-minimum-phase plants of the second and third orders.

An Objective Holographic Feedback Linearization Based on a Sliding Mode Control for a Buck Converter with a Constant Power Load

As a typical load, the constant power load (CPL) has negative impedance characteristics. The stability of the buck converter system with a mixed load of CPL and resistive load is affected by the size of the CPL. When the resistive load is larger than the CPL, the buck converter with the output voltage as an output function is a non-minimum phase nonlinear system, because its linear approximation has a right-half-plane pole. The non-minimum phase characteristic limits the application of many control techniques, but the objective holographic feedback linearization control (OHFLC) method is a good control strategy that can bypass the non-minimum phase system and make the system stable. However, the traditional OHFLC method, in designing the controller, generally uses a linear optimal quadratic design method to obtain a linear feedback control law. It requires a state quantity component with a one-order relative degree to the system. But it is not easy to find such a suitable state quantity with a one-order relative degree to the system. In this paper, an improved OHFLC method is proposed for Buck converters with a mixed loads of CPL and resistive loads, using the sliding mode control (SMC) theory to design the controller, so that the output state quantity components with different relative degrees to the system can be used in the holographic feedback linearization method. Finally, the simulation and experimental results also demonstrate that this method has the same, or even better, dynamic response performance and robustness than the traditional OHFLC method.

Asymptotic output tracking in a class of non‐minimum phase nonlinear systems via learning‐based inversion

AbstractAsymptotic output tracking of non‐minimum phase (NMP) nonlinear systems has been a popular topic in control theory and applications. Many approaches have focused on finding solutions under minimal assumptions either in the target system or desired trajectories, as there is no general solution available. In this article, we propose a practical and simple solution for cases where the reference trajectory is periodic in time. Our approach employs a learning‐based scheme to iteratively determine the desired feedforward input. Unlike previous learning‐based frameworks, our method only requires the output tracking error to update the feedforward input iteratively and can be applicable to NMP systems. Our method retains the key advantages of the learning‐based framework, including robustness to parameter uncertainties and periodic disturbances. We evaluate the effectiveness of our algorithm using simulation results with an inverted pendulum on a cart, a typical NMP nonlinear system.

Sliding mode control design using generalized relative degree approach: Aerospace application

Relative degree (RD) approach is a powerful tool for obtaining system's input-output dynamics used for output tracking controller designs of minimum phase systems. Designs using the RD alone can fail due both to insufficient control authority in minimum phase systems, and instability of internal/zero dynamics attributed to nonminimum phase systems. A novel definition and a concept of Practical Generalized RD (PGRD) are proposed in this paper and are used in concert with Sliding Mode Control (SMC) to compensate for system perturbations in minimum phase systems. The use of known Generalized Relative Degree (GRD) in nonminimum phase systems allows for the elimination of internal dynamics. However, instability that emerges in the corresponding control dynamic extension is defeating any output tracking controller design. A novel methodology of using GRD for designing continuous SMC in nonminimum phase systems is presented. An algorithm for generating a bounded solution of the unstable dynamic extension is proposed and used in concert with SMC, allowing robust control design for nonminimum phase systems. The efficacy of the proposed GRD-based approaches is demonstrated on a minimum and nonminimum phase rocket attitude control problem both analytically and via simulation.

Tuning of controller parameters using Pythagorean fuzzy similarity measure for stable and time delayed unstable plants.

This paper proposes a tuning method based on the Pythagorean fuzzy similarity measure and multi-criteria decision-making to determine the most suitable controller parameters for Fractional-order Proportional Integral Derivative (FOPID) and Integer-order Proportional Integral-Proportional Derivative (PI-PD) controllers. Due to the power of the Pythagorean fuzzy approach to evaluate a phenomenon with two memberships known as membership and non-membership, a multi-objective cost function based on the Pythagorean similarity measure is defined. The transient and steady-state properties of the system output were used for the multi-objective cost function. Thus, the determination of the controller parameters was considered a multi-criteria decision-making problem. Ant colony optimization for continuous domains (ACOR) and artificial bee colony (ABC) optimization are utilized to minimize multi-objective cost functions. The proposed method in the study was applied to three different systems: a second-order non-minimum phase stable system, a first-order unstable system with time delay, and a fractional-order unstable system with time delay, to validate its effectiveness. The cost function utilized in the proposed method is compared with the performance measures widely used in the literature based on the integral of the error, such as IAE (Integral Absolute Error), ITAE (Integral Time Absolute Error), ISE (Integral Square Error), and ITSE (Integral Time Square Error). The proposed method provides a more effective control performance by improving the system response characteristics compared to other cost functions. With the proposed method, the undershoot rate could be significantly reduced in the non-minimum phase system. In the other two systems, significant improvements were achieved compared to other methods by reducing the overshoot rate and oscillation. The proposed method does not require knowing the mathematical model of the system and offers a solution that does not require complex calculations. The proposed method can be used alone. Or it can be used as a second and fine-tuning method after a tuning process.

Periodic-Disturbance Observer Using Spectrum-Selection Filtering Scheme for Cross-Coupling Suppression in Atomic Force Microscopy

Repetitive disturbances exist widely within the automation systems, which is one of the major issues that hinder the achievement of precision operations. Dedicated to mitigating these disturbances, a generalized periodic-disturbance observer (PDOB) using spectrum-selection filtering scheme is proposed in this paper with an application to a non-minimum phase system. The design process of the spectrum-selection filter that derives from the comb-like notch filter for the proposed PDOB is presented in detail. The variants of the proposed PDOB are also presented for the disturbances distributed only in odd-and even-harmonics. To achieve tracking of the desired trajectories, the proposed PDOB is combined in parallel with a baseline Proportional-Integral (PI) controller. The stability condition of the closed-loop system is derived to provide criteria for parameters selection. The experimental validation of the generalized PI+PDOB is conducted via real-time cross-coupling suppression in raster scanning of atomic force microscope (AFM), where the coupling induced root-mean-square tracking errors are reduced from 141.8 <inline-formula> <tex-math notation="LaTeX">$nm$</tex-math> </inline-formula> to 2.4 <inline-formula> <tex-math notation="LaTeX">$nm$</tex-math> </inline-formula> by employing the proposed PDOB for the scanning rate of 100 Hz. The results of convergence testing, raster scanning and AFM imaging are compared to illustrate the significant improvements achieved with the proposed PDOB. In addition, experimental results show that the tracking performance using PI+PDOB in the case of existing periodic disturbances resembles with that using PI control without periodic disturbances, which implies that the employment of the proposed PDOB does not interfere with the tracking-based control schemes for staircase trajectories, showing the advantages of the proposed PDOB. <i>Note to Practitioners</i>&#x2014;Repetitive motions facilitate diverse advanced functions within the automation systems, including the scanning electron microscope, atomic force microscope, manipulators, and robotic systems. These operations would unavoidably introduce repetitive disturbances, hindering its achievable precision. One particular issue is the widely existent cross-coupling effect that results in these repetitive disturbances. In this paper, a generalized periodic-disturbance observer (PDOB) based on the spectrum-selection filtering scheme is developed to address the repetitive disturbances for real-time coupling suppression in raster scanning of Atomic Force Microscopy. Consequently, the offline learning procedures required for other learning based schemes can be avoided. Without loss of generalization, the design of PDOB is demonstrated with an application to non-minimum phase systems. A scheme of the varying parameter is also developed to achieve both fast convergence and better performance at rejecting the periodic disturbances. The detailed variations to handle the disturbances distributed only in odd-and even-harmonics are also demonstrated. Different from other disturbance observers, the inclusion of the spectrum-selection filtering scheme provides more flexibility in rejecting specific repetitive disturbances. In terms of tracking accuracy and the simple structure, this development can also be easily implemented to other systems that suffer from repetitive disturbances.