Non-minimum Phase Zeros Research Articles

Improving the Step Response of Flexible Systems in the Presence of Real Nonminimum Phase Zeros

Abstract It is well known that real nonminimum phase (RNMP) zeros impose a tradeoff between the settling time and undershoot in the step response of flexible systems. Existing methods to alleviate this tradeoff predominantly rely on various advanced control strategies without delving into a broader mechatronic approach that combines physical system and control system design. To address this gap, this article proposes a proportional viscous damping-based physical system design in combination with feedback control and prefilter design. First, the effect of proportional viscous damping on RNMP zeros of flexible systems is established to propose a damping strategy that pushes all the RNMP zeros further away from the imaginary axis. Then, a step-by-step mechatronic system design process is presented to apply this damping strategy along with a full-state feedback control strategy and prefilter to a multi-degree-of-freedom (DoF) flexible system. The application of this design process yields simultaneous improvement in the settling time and undershoot in the step response of this flexible system.

Stabilizing regions of dominant pole placement for second order lead processes with time delay using filtered PID controllers.

In order to handle second order lead processes with time delay, this paper provides a unique dominant pole placement based filtered PID controller design approach. This method does not require any finite term approximation like Pade to obtain the quasi-polynomial characteristic polynomial, arising due to the presence of the time delay term. The continuous time second order plus time delay systems with zero (SOPTDZ) are discretized using a pole-zero matching method with specified sampling time, where the transcendental exponential delay terms are converted into a finite number of poles. The pole-zero matching discretization approach with a predetermined sampling period is also used to discretize the continuous time filtered PID controller. As a result, it is not necessary to use any approximate discretization technique, such as Euler or Tustin, to derive the corresponding discrete time PID controller from its continuous time counterpart. The analytical expressions for discrete time dominant pole placement based filtered PID controllers are obtained using the coefficient matching approach, while two distinct kinds of non-dominant poles, namely all real and all complex conjugate, have been taken into consideration. The stabilizable region in the controller and design parameter space for the chosen class of linear second order time delay systems with lead is numerically approximated using the particle swarm optimization (PSO) based random search technique. The efficacy of the proposed method has been validated on a class of SOPTDZ systems including stable, integrating, unstable processes with minimum as well as non-minimum phase zeros.

Effects of Two-Channel Noise and Packet Loss on Performance of Information Time Delay Systems.

The performance limitations of multiple-input multiple-output (MIMO) information time delay system (ITDS) with packet loss, codec and white Gaussian noise (WGN) are investigated in this article. By using the spectrum decomposition technique, inner-outer factorization, and partial factorization techniques, the expression of performance limitations is obtained under the two-degree-of-freedom (2DOF) compensator. The theoretical analysis results demonstrate that the system performance is related to the time delay, non-minimum phase (NMP) zeros, unstable zeros and their directions in a given device. In addition, WGN, packet loss and codec also impact the performance. Finally, the theoretical results are verified by simulation examples. Simulation results show that packet loss rate and encoding and decoding have a greater impact on system performance.

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.

Grey-Box Modeling and Decoupling Control of a Lab Setup of the Quadruple-Tank System

The quadruple-tank system (QTS) is a popular educational resource in universities for studying multivariable control systems. It enables the analysis of the interaction between variables and the limitations imposed by multivariable non-minimum phase zeros, as well as the evaluation of new multivariable control methodologies. The works utilizing this system present a theoretical model that may be too idealistic and based on erroneous assumptions in real-world implementations, such as the linear behavior of the actuators. In other cases, an identified linear model is directly provided. This study outlines the practical grey-box modeling procedure conducted for the QTS at the University of Cordoba and provides guidance for its implementation. A configurable nonlinear model was developed and controlled in a closed loop using different controllers. Specifically, decentralized control, static decoupling control, and simplified decoupling control were compared. The simulation designs were experimentally validated with high accuracy, demonstrating that the conclusions reached with the developed model can be extrapolated to the real system. The comparison of these three control designs illustrates the advantages and disadvantages of decoupling in certain situations, especially in the presence of non-minimum phase zeros.

Tracking Performance of Feedback Systems Over a Fading Channel With Limited Bandwidth.

In this study, we investigated the optimal tracking performance (OTP) of feedback control systems with limited bandwidth and colored noise in a fading channel. For the steady state of the feedback control systems, an equivalent average channel (EAC) model was developed by retaining the effects of the first and second moments of the multiplicative channel output, and on the basis of the coprime decomposition, all-pass factorization, and Youla parameterization of controllers, exact expressions for the OTP were derived by designing two compensators. The expressions quantitatively show the relationship between the OTP and inherent features of the plant. Specifically, the directions and locations of unstable poles (UPs) and nonminimum phase (NMP) zeros adversely affect the tracking performance. Furthermore, the bandwidth limitation and the presence of colored noise also degrade the tracking performance. Finally, our conclusions were verified by considering a numerical arithmetic example.

Optimal tracking performance of networked control systems under communication constraints

This paper investigates the optimal tracking performance (OTP) of multiple-input multiple-output discrete-time communication-constrained systems by thinking about Denial of Service (DoS) attacks, codecs and additive Gaussian white noise under energy constraints. The non-cooperative relationship between DoS attacks and intrusion detection systems (IDS) is analyzed using repeated game theory. A penalty mechanism is constructed to force the attackers to adopt a cooperative strategy, thus improving the system performance. Partial factorization and spectrum decomposition are used to provide the OTP for systems. The results demonstrate that the systems’ OTP are linked to intrinsic characteristics like non-minimum phase zeros and unstable poles. Finally, concrete examples are shown that the results are accurate.

Dynamic Virtual Power Plant Design for Fast Frequency Reserves: Coordinating Hydro and Wind

To ensure frequency stability in future low-inertia power grids, fast ancillary services such as fast frequency reserves (FFR) have been proposed. In this work, the coordination of conventional (slow) frequency containment reserves (FCR) with FFR is treated as a decentralized model matching problem. The design results in a dynamic virtual power plant (DVPP) whose aggregated output fulfills the system operator’s (SO’s) requirements in all time scales, while accounting for the capacity and bandwidth limitation of participating devices. This is illustrated in a 5-machine representation of the Nordic synchronous grid. In the Nordic grid, stability issues and bandwidth limitations associated with non-minimum phase zeros of hydropower is a well-known problem. By simulating the disconnection of a 1400 MW importing dc link, it is shown that the proposed DVPP design allows for coordinating fast FFR from wind, with slow FCR from hydro, while respecting dynamic limitations of all participating devices. The SO’s requirements are fulfilled in a realistic low-inertia scenario without the need to install battery storage or to waste wind energy by curtailing the wind turbines.

Damping analysis of floating offshore wind turbines (FOWTs): a new control strategy reducing the platform vibrations

Abstract. In this paper, the coupled dynamics of the floating platform and the wind turbine rotor are analyzed. In particular, the damping is explicitly derived from the coupled equations of the rotor and floating platform. The analysis of the damping leads to the study of instability phenomena, thus obtaining the explicit conditions that lead to the non-minimum phase zero (NMPZ). Two NMPZs are analyzed, one related to the rotor dynamics and the other one to the platform pitch dynamics. The latter introduces a novelty, and an explicit condition is provided in this work for its verification. In the second part of the paper, from the analysis of the damping of the floating platform, a new strategy for the control of floating offshore wind turbines (FOWTs) is proposed. This strategy allows one to impose on the controller an explicit level of damping in the platform pitch motion that adapts with wind speed and operating conditions without changing the period of platform pitching. Finally the new strategy is compared to one without compensation and one with a non-adapting compensation by performing aero-hydro-servo-elastic numerical simulations of a reference FOWT. Generated power, motions, blade pitch and tower base fatigue are compared, showing that the new control strategy can reduce fatigue in the structure without affecting the power production.

On the zeros of three-DoF damped flexible systems

This paper presents an investigation of the non-minimum phase (NMP) zeros in the single input single output (SISO) transfer function of three-DoF (degrees of freedom) damped flexible linear time-invariant (LTI) systems under the assumption of classical damping. It is well-known that when all the modal residue signs of any multi-DoF damped flexible LTI system are the same, NMP zeros never occur in the system dynamics for any value of system parameters including modal residue, modal frequency and modal damping ratio. However, when all the modal residue signs are not the same, then additional conditions in terms of the system parameters are required to guarantee the elimination of NMP zeros. In this paper, the zero loci of a three-DoF damped flexible LTI system are developed to derive the sufficient and necessary conditions for the elimination of all NMP zeros. These conditions can be employed in the robust physical design of flexible systems, i.e., as long as these conditions are satisfied, the elimination of NMP zeros is guaranteed even when the system parameters undergo variations.

Performance Analysis of MIMO Information Time-Delay System Under Bandwidth, Cyber-Attack, and Gaussian White Noise

The performance limitations of multi-input–multi-output (MIMO) information time-delay system (ITDS) due to bandwidth, additive white Gaussian noise (AWGN), and cyber-attacks are investigated in this work. The game theory methods are used to analyze the impact of intrusion detection systems on systems performance in order to deal with network attacks under incomplete information. The expression of performance limitations is obtained by using the spectrum decomposition technique, inner–outer factorization, and partial factorization techniques under the two-degree-of-freedom (2DOF) compensator. The results demonstrate that the performance limitations of the system are not only influenced by the inherent properties, including time delay, nonminimum phase (NMP) zeros, and unstable zeros and their directions of the given object but are also influenced by the communication parameters, such as AWGN and bandwidth. In addition, the results show that under the cyber-attacks with incomplete information, the performance is only influenced by the payoff matrix of the defense, and not by the payoff matrix of the offense. Finally, the theoretical results are verified by performing simulations.

PI Control of Loudspeakers Based on Linear Fractional Order Model

This paper aims at the proportional-integral (PI) control of the cone vibration of the electrodynamic loudspeakers system recently described using a linear fractional order model. After introducing the fractional order model of the circuit of these loudspeakers, firstly, a new method is developed to design a fractional order PI controller to place the poles of the system in a desired area of the complex plane which is called D-stabilizing. The design parameters of the method depend directly on the speed of the system output response, the cone vibration. Moreover, the offered fractional order controller avoids any non-minimum phase zero, which causes undesired undershoots in the output, for the closed-loop control system. Secondly, considering uncertainties in the coefficients of the model, a methodology is presented to determine up to how much the uncertainties can increase such that the controller is still able to maintain both D-stability and the absence of non-minimum phase zeros for the control system. Finally, the merit of the presented results and the superiority of the designed fractional order controller over its conventional integer order counterpart are illustrated through numerical simulations.

Tracking and Regulation Performance Limitations of Networked Control Systems Over Erasure Channel With Input Quantization

Limitations in communication networks, such as delays, packet dropouts, and insufficient bandwidth, create great challenges in the output tracking control and regulation performance of networked control systems. In this article, we propose some novel performance indexes for achieving the optimal tracking and regulation performance. By considering the influences of noise, packet dropouts, and logarithmic quantization on the channel input, the performance limitations of the networked systems are investigated by applying a two-degree-of-freedom (TDOF) control technique. The results obtained prove that the tracking and regulation performance is dependent not only on the position of the plant’s nonminimum phase zeros and unstable poles, but also on the directions. Furthermore, it is demonstrated that noise and quantization error have a negative influence on the control performance, and the design of the TDOF controller can eliminate the effect of packet dropouts on control performance. To validate the innovation results, several simulations are conducted.

DBFact: A better approach to calculate the minimum variance control law for nonminimum phase MIMO systems

AbstractThis paper presents the diagonal with Blaschke products factorization (DBFact) approach to factor out multivariable time delay and nonminimum phase zeros from multi‐input multi‐output (MIMO) systems. Based on that, a new output‐order independent minimum variance (MV) control law for MIMO systems is proposed. The DBFact method is a two‐step factorization procedure, relying on the diagonal and Blaschke factorization methods. This method has the advantage of being a direct and non‐iterative procedure. This new factorization approach allows the calculation of an MV control law considering the multivariable time delay as a limiting‐performance factor and the nonminimum phase zeros and their corresponding directions. Based on the proposed MV control law, a performance benchmark is introduced, which can be calculated by the DBFact filters and routine operating data. The DBFact methodology was applied to two control structures of the linear plant model of Linde's heat integrated air separation, in which the MV control law output‐order dependency property and the suitability of the performance benchmark were evaluated. Some results were compared with those obtained by admitting the generalized interactor matrix instead of the DBFact filters. The results show the capability of the DBFact methodology to factor the nonminimum phase terms to provide a reliable MIMO controller performance benchmark and illustrate the importance of considering the nonminimum phase zeros and their actual directionality in the MV control law.

LQ‐optimal 2‐rate control of a class of single‐input single‐output discrete‐time plants

AbstractUse of 2‐rate controllers for robust control via zero‐placement is a topic of interest in current literature. This paper proposes a 2‐degree‐of‐freedom, 2‐rate, dynamic, output feedback controller to compensate single‐input single‐output (SISO), unstable, non‐minimum phase (NMP), discrete, linear time‐invariant (LTI) plants of relative degree unity; and achieves exactly the same loop and input‐output properties as can be achieved by a linear quadratic regulator (LQR) for the equivalent lifted LTI plant using state feedback. Such LQR‐like loop and input‐output properties, it may be noted, cannot be attained by LTI or even the existing 2‐rate output feedback controllers for plants with NMP zeros. Examples presented illustrate the efficacy of the proposed controller with respect to the existing ones.

Optimal Tracking Control of Networked Systems Subject to Model Uncertainty and Additive Colored Gaussian Noise

For control systems with communication links, the various communication parameters can bring negative effects on system’s performance. Taking tracking problem for example, it is extremely difficult to achieve accurate tracking, that is, the tracking error can not be zero because of these communication constraints, such as model uncertainties originate from plant and channel, and additive colored Gaussian noise (ACGN). In this brief, by employing unit feedback controller (UFC) and two-parameters controller (TPC), the output tracking control (OTC) performance of networked systems subject to model uncertainties and ACGN is investigated. Consequently, two exact expressions of OTC performance limitations are derived with UFC and TPC. These results clearly show that OTC performance tightly depends on non-minimum phase zeros (NMPZs) and unstable poles (UPs) of the given plant. Moreover, the existence of uncertainties and ACGN can result in the poor OTC performance.

Optimal Tracking Performance Analysis of MIMO Control Systems Under Multiple Constraints

This article is concerned with the optimal tracking performance on linear time-invariant (LTI) discrete-time multi-input–multi-output (MIMO) systems based on quantization, channel noise, and encoding–decoding, as well as bandwidth constraints. Meanwhile, power constraint, feedback and feedforward loops constraints of the systems are also taken into account. The tracking performance expression is described by inner–outer factorization, which is obtained by all stabilizing two-degree-of-freedom compensators. The results will illustrate that tracking performance limitation is decided by nonminimum phase zeros and unstable poles of the given plant. Moreover, time delay, encoding–decoding, channel noise, quantization, and bandwidth constraints also impact the performance. The accuracy of the method is verified by simulation examples.

Geometrical Characterization of Sensor Placement for Cone-Invariant and Multi-Agent Systems against Undetectable Zero-Dynamics Attacks

Undetectable attacks are an important class of malicious attacks threatening the security of cyber-physical systems, which can modify a system's state but leave the system output measurements unaffected and hence cannot be detected from the output. This paper studies undetectable attacks on cone-invariant systems and multi-agent systems. We first provide a general characterization of zero-dynamics attacks, which characterizes fully undetectable attacks targeting the nonminimum phase zeros of a system. This geometrical characterization makes it possible to develop a defense strategy seeking to place a minimal number of sensors to detect and counter the zero-dynamics attacks on the system's actuators. The detect and defense scheme amounts to computing a set containing potentially vulnerable actuator locations and nodes and a defense union for feasible placement of sensors based on the geometrical properties of the cones under consideration.

Data-Driven Decomposition Control to Output Tracking With Nonperiodic Tracking–Transition Switching Under Input Constraint

AbstractThis paper is concerned with solving, from the learning-based decomposition control viewpoint, the problem of output tracking with nonperiodic tracking–transition switching. Such a nontraditional tracking problem occurs in applications where sessions for tracking a given desired trajectory are alternated with those for transiting the output with given boundary conditions. It is challenging to achieve precision tracking while maintaining smooth tracking–transition switching, as postswitching oscillations can be induced due to the mismatch of the boundary states at the switching instants, and the tracking performance can be limited by the nonminimum-phase (NMP) zeros of the system and effected by factors such as input constraints and external disturbances. Although recently an approach by combining the system-inversion with optimization techniques has been proposed to tackle these challenges, modeling of the system dynamics and complicated online computation are needed, and the controller obtained can be sensitive to model uncertainties. In this work, a learning-based decomposition control technique is developed to overcome these limitations. A dictionary of input–output bases is constructed offline a priori via data-driven iterative learning first. The input–output bases are used online to decompose the desired output in the tracking sessions and design an optimal desired transition trajectory with minimal transition time under input-amplitude constraint. Finally, the control input is synthesized based on the superpositioning principle and further optimized online to account for system variations and external disturbance. The proposed approach is illustrated through a nanopositioning control experiment on a piezoelectric actuator.

Variable-Speed Wind Turbine Control Designed for Coordinated Fast Frequency Reserves

Modern power systems present low levels of inertia due to the growing shares of converter-interfaced generation. Consequently, renewable energy sources are increasingly requested to provide frequency support. In addition, due to the inertia loss, the requirements regarding frequency containment reserves (FCR) are becoming tough to meet with traditional units such as hydro, whose non-minimum phase (NMP) characteristic reduces the closed-loop stability margins. The shortcomings of traditional synchronous generation motivates new protocols for fast frequency reserves (FFR). In this work, we design a wind turbine (WT) model useful for FFR. It is shown that the dynamical shortcomings of the WT, in providing steady-power or slow FCR support, are suitably described by a first-order transfer function with a slow NMP zero. The WT model is tested in a 5-machine representation of the Nordic synchronous grid. It is shown that the NMP model is useful for designing a controller that coordinates FFR from wind with slow FCR from hydro turbines. By simulating the disconnection of a 1400 MW importing dc link in a detailed nonlinear model, it is shown that the wind--hydro combination not only satisfies the latest regulations, but also presents a smooth response avoiding overshoot and secondary frequency dips during frequency recovery.