Initial Undershoot Research Articles

Demystifying Enigmatic Undershoot in Setpoint Command Following [Focus on Education

It often occurs in practice that the response to a step input or setpoint command moves initially in a direction that is opposite to the direction of the asymptotic response. In many real-world applications, this phenomenon—called initial undershoot—presents a fundamental limitation on control system performance. Although the basic mechanism responsible for initial undershoot, namely, an odd number of real, positive zeros, is well understood, it is surprising that, as the setpoint changes, initial undershoot may appear or disappear for the same plant dynamics. The goal of this tutorial note is to investigate the causes of this puzzling phenomenon. In particular, for setpoint command following with a changing setpoint, this article shows (spoiler alert) that the internal state when the setpoint changes determines the presence or absence of initial undershoot. Complete proofs for both initial and delayed undershoot in both continuous time and discrete-time systems are given to make the article self-contained and useful for students and instructors of systems and control theory

Achievable Thrust Expansion Control at Current Saturation of Variable-Pitch Propeller for Drones

On drones, which have become popular in recent years, the installation of variable pitch propellers is being considered and researched for industrial applications such as large drones and flying mobility. A variable-pitch propeller not only provides higher responsiveness than control by rotational speed alone, but also expands the thrust that can be achieved by optimizing the combination of rotational speed and pitch angle. In this paper, it is experimentally verified that, in the current saturation region of a variable-pitch propeller motor, the control method using simple frequency-separated command values for rotational speed and pitch angle causes an inverse response in transient conditions, and we attempt to explain this response by a linearized model including unstable zero. Basing on the model, we proposed a control method in which the pitch angle is controlled by a first-order filter while the maximum current is applied. The proposed method enables control design with a trade-off between initial undershoot and settling time with one parameter. The effectiveness of the proposed method is verified by simulation and experiment.

Vowel harmony and positional variation in Kyrgyz

While it is well known that the phonetic realization of a segment may differ by position, it is unclear how positional variation interacts with vowel harmony, the imperative that vowels be identical along some phonological dimension. Pearce (2008, 2012) contends that phonological harmony blocks phonetic reduction, suggesting that phonology dictates phonetic realization for this class of assimilatory patterns. This paper investigates harmony and vowel reduction in Kyrgyz, finding that non-initial vowels are more centralized than their initial-syllable counterparts. The potential sources for this reduction, including initial strengthening, supralaryngeal declination, predictability, and undershoot are discussed. The proposed predictability-based analysis provides an analysis of reduction based on phonological knowledge and representations.

Preactuated Multirate Feedforward Control for Independent Stable Inversion of Unstable Intrinsic and Discretization Zeros

A plant with unstable zeros is known to be difficult to be controlled, because of the initial undershoot of its step response and the unstable poles in its inversion system. There are two reasons why a plant has unstable zeros in the discrete-time domain: 1) noncollocation of actuators and sensors; and 2) discretization by the zero-order hold. Addressing problems 1) and 2) simultaneously, approximate and discrete-time stable inversion methods have been proposed as model-based feedforward control for unstable zeros. However, the approximated inversion methods compromise tracking performance, and the discrete-time stable inversion methods obtain oscillatory control inputs owing to the direct cancellation of discretization zeros. Therefore, this paper proposes a preactuated multirate feedforward control, which is an independent stable inversion of the two types of zeros: unstable intrinsic zeros from noncollocation and unstable discretization zeros from the zero-order hold. The proposed method combines both a state trajectory generation based on a time- and imaginary-axis reversal to address problem 1) and a multirate feedforward implementation to address problem 2), independently. Moreover, this study verifies the tracking performance improvement by performing simulations and experiments in comparison with well-known model-based feedforward control methods using a high-precision motion stage. The role and importance of multirate feedforward control are also demonstrated by comparing it to a continuous-time domain approach with preactuation.

Analysis of Effects due to Right Half Plane Zeros in PI Controller based Hydro Turbine

In this paper, some detrimental time-domain characteristics (zero-crossings, overshoot(due to zeros) and initial undershoot) are analyzed using the system transfer function model, for identifying the presence of these effects in step response of non-minimum phase linear systems. Moreover, regarding initial undershoot, a theorem for detection using the state-space model, without the need to obtain transfer function, is proposed. In addition to detection, a theorem for estimating the percentage of initial undershoots is also proposed. Applications of these theorems are carried out on load frequency control of hydro-electric power plant which employs hydro-turbine, a non-minimum phase system. Moreover, the effect of a PI controller on initial undershoot is discussed for a general non-minimum phase linear system and illustrated for load frequency control of hydro-electric power plant.

Preactuated multirate feedforward for a high-precision stage with continuous time unstable zeros

A plant with unstable zeros is known to be difficult to control because of initial undershoot of step response and unstable poles in its inversion system. There are two reasons why a plant has unstable zeros in the discrete time domain: 1) non-collocation of actuators and sensors, 2) discretization by zero-order-hold. Problem 2) has been solved by the perfect tracking control (PTC) method based on multirate feedforward control proposed by our research group. However, the conventional PTC method cannot achieve the perfect tracking for a plant with continuous time unstable zeros generated by 1) because of the divergence of the desired state trajectories. Preactuation Perfect Tracking Control (PPTC) method proposed by our group solves problem 1) by the state trajectory generation based on a time axis reversal. This paper verifies the effectiveness of PPTC by simulations and experiments in comparison with several single rate feedforward control methods.

Zero Crossings, Overshoot and Initial Undershoot in the Step and Impulse Responses of Linear Systems

We consider functions in the time and the frequency domain, and show how the occurrence of real zeros in the frequency domain is related to the number of zero crossings, overshoot and initial undershoot in the time domain. This leads to simple proofs of known facts for linear time-invariant systems and their generalizations in various directions. In particular, we consider time-delay systems and differential systems of fractional order.

Longitudinal Aircraft Dynamics and the Instantaneous Acceleration Center of Rotation

Nonminimum-phase zeros, that is, closed-right-half-plane (CRHP) zeros, affect both the open-and closed-loop behavior of continuous-time linear systems in undesirable ways [1]. For example, an asymptotically stable linear system with an odd number of positive zeros experiences initial undershoot to a step input (see “Initial Undershoot”). Moreover, under the rules of root locus, zeros in the open-right-half plane (ORHP) attract closed-loop poles, which limits the controller gain and thus the performance of the closed-loop system.

Improved Tank in Series Model for the Planar Solid Oxide Fuel Cell

Models of different complexity are required in the iterative process of designing a solid oxide fuel cell (SOFC). Models having less complexity and computational dexterity are the ideal ones at the early stages. This work presents the development of an improved tank in series reactor model of the SOFC operating in cocurrent, countercurrent, and cross-current flow directions. The model, which accounts for the charge balances in the electrodes and electrolyte in addition to the component balances and the energy balances, is used for simulating the potentiostatic operation of the cell. The simulation results from the TSR model indicate the influence of flow direction on the steady state and dynamic performances of the cell. Among different flow directions, the coflow case is the most favorable for the planar SOFC, with improved performance. In response to a voltage step increase, the coflow case provides the most uniform transient behavior at different points of the cell. Despite the coflow direction, in which temperature dominates the slow dynamics of the local current density, in the low temperature regions of the counterflow and cross-flow cases, the slow dynamics of the current density tends to be characterized by the initial undershoot followed by a slower transient response that is due to the combined effects of the diffusion resistance within the porous electrode, hydrogen accumulation toward the fuel outlets, and the influence of the PEN temperature.

Nonminimum-phase zeros - much to do about nothing - classical control - revisited part II

The purpose of this article is to illuminate the critical role of system zeros in control-system performance for the benefit of a wide audience both inside and outside the control systems community. Zeros are a fundamental aspect of systems and control theory; however, the causes and effects of zeros are more subtle than those of poles. In particular, positive zeros can cause initial undershoot (initial error growth), zero crossings, and overshoot in the step response of a system, whereas nonminimum-phase zeros limit bandwidth. Both of these aspects have real-world implications in many applications. Nonminimum-phase zeros exacerbate the tradeoff between the robustness and achievable performance of a feedback control system. From a control-theoretic point of view, a nonminimum-phase zero in the loop transfer function L is arguably the worst feature a system can possess. Every feedback synthesis methodology must accept limitations due to the presence of open-right-half-plane zeros, and the mark of a good analysis tool is the ability to capture the performance limitations arising from nonminimum-phase zeros.

On the Zeros, Initial Undershoot, and Relative Degree of Collinear Lumped-Parameter Structures

This paper considers collinear lumped-parameter structures where each mass in the structure has a single degree of freedom. Specifically, we analyze the zeros and relative degree of the single-input, single-output (SISO) transfer function from the force applied to an arbitrary mass to the position, velocity, or acceleration of another mass. In particular, we show that every SISO force-to-motion transfer function of a collinear lumped-parameter structure has no positive (real open-right-half-plane) zeros. In addition, every SISO force-to-position transfer function of a spring-connected collinear lumped-parameter structure has no non-negative (real closed-right-half-plane) zeros. As a consequence, the step response does not exhibit initial undershoot. In addition, we derive an expression for the relative degree of SISO force-to-position transfer functions. The formula depends on the placement of springs and dashpots, but is independent of the values of the spring constants and damping coefficients. Next, we consider the special case of serially connected collinear lumped-parameter structures. In this case, we show that every SISO force-to-position transfer function of a serially connected collinear lumped-parameter structure is minimum phase, that is, has no closed-right-half-plane zeros. The proofs of these results rely heavily on graph-theoretic techniques.

Tokamak Position Control Using Internal and External Magnetic Sensor Coils

A linear, second-order transformer model using magnetic sensor coils has been developed to describe the position control of a tokamak. This model is used to analyze the behavior of a proportional-derivative controller, which has been implemented on TEXT-Upgrade (TEXT-U). The magnetic sensor coils may be placed internal or external to the conducting vacuum vessel. If placed externally, however, eddy currents induced in the vessel wall introduce an error in the position measurement. It is found that this error signal introduces a positive zero in the system transfer function. The transfer function becomes a non-minimum-phase function, which restricts the response speed, stable area, and utilization of the power supply capability. Although the position control system is stabilized by use of a proportional-derivative controller, the controller cannot affect the positive zero. This analysis has been experimentally verified on TEXT-U. With external sensors, the stable operating area is small, and the sensors exhibit an initial undershoot to a step position change, as expected. The observed stable area is predicted by the model, although the model overestimates the size of the actual stable area.

Cerebral haemoglobin oxygenation during sustained visual stimulation--a near-infrared spectroscopy study.

Using near-infrared spectroscopy, we investigated the time-course of the concentrations of oxygenated haemoglobin, [oxy-Hb], and deoxygenated haemoglobin [deoxy-Hb], in the occipital cortex of healthy human adults during standard sustained visual stimulation. Within a few seconds after stimulation (by coloured dodecahedron), we observed a decrease in [deoxy-Hb], peaking after 13 s ('initial undershoot'). In the subsequent 1-2 min, in seven out of ten subjects, [deoxy-Hb] gradually returned to a plateau closer to the baseline level. After cessation of stimulation, there was a 'post-stimulus overshoot' in [deoxy-Hb]. There was a statistically significant correlation between the size of the 'initial undershoot' and the post-stimulus overshoot'. The concentration of oxyhaemoglobin increased upon functional activation. However, in the mean across all subjects there was no 'initial overshoot'. After approximately 19 s it reached a plateau and remained constantly elevated throughout the activation period. After cessation of activation there was a 'post-stimulus undershoot' of oxyhaemoglobin. It is important to consider the time-course of haemoglobin oxygenation when interpreting functional activation data, especially those data obtained with oxygenation-sensitive methods, such as BOLD-contrast fMRI.

Further studies on oscillating tubulo-glomerular feedback responses in the rat kidney.

Free flow microperfusion (closed feedback loop) experiments in halothane-nitrous oxide-anaesthetized rats demonstrate two distinct oscillating TGF pressure responses to pertubations in early distal tubular flow. Flow rate changes of 2-8 nl min-1 induced a 0.02-0.03 Hz rhythm, while changes above 25 nl min-1 often induced a 0.17-0.18 Hz rhythm in proximal tubular pressures. Following interruption of microperfusion, the intratubular pressure showed an initial undershoot and a gradual oscillating return to the control level. Each nephron responded individually to systemic changes with oscillations differing in frequency, phase and amplitude from that of its neighbour. Postglomerular vascular pressure (PSV) oscillated synchronous with the proximal luminal pressure (PProx) of the same nephron; the amplitude of the PSV might be larger than that of the corresponding PProx. The PSV decreased by 2-3 mmHg as Henle loop flow was increased by 7 nl min-1. Furosemide (0.1-0.2 mM) in the microperfusate abolished the oscillations, and caused PSV and PProx to increase by 3-4 mmHg. This effect was rapidly reversible.

Effect of repetitive activation on the afterhyperpolarization in dorsal spinocerebellar tract neurones.

1. The changes in the afterhyperpolarization (a.h.p.) with repetitive activation have been studied in dorsal spinocerebellar tract cells of the cat using intracellular recording techniques. 2. The a.h.p. following a single spike was conditioned at different interspike intervals by a single preceding spike. In the majority of neurones the a.h.p. following a spike added approximately linearly with that generated by a preceding spike. 3. In other cells the a.h.p. following a spike was instead depressed by a preceding spike. THis depression was approximately constant at interspike intervals less than the a.h.p. duration (50-100 msec). Thereafter the a.h.p. slowly recovered during the next 100-300 msec. There was no associated decrease in the initial brief hyperpolarizing undershoot. 4. With shortlasting repetitive activation at high frequency (greater than 100 impulses/sec) the a.h.p, peak amplitude increased progressively with successive spikes (5-15 spikes). No change in the time constant of decay was observed. A good correspondence was found between the observed increase in peak amplitude of the a.h.p.s and that given by a theoretical linear superposition of the successive a.h.p.s. 5. Changes in the brief hyperpolarizing undershoot with repetitive activation is also described.

Osmoregulation of the grass shrimp Palaemonetes pugio exposed to polychlorinated biphenyls (PCBs). I. Effecton chloride and osmotic concentrations and chloride-and water-exchange kinetics

Grass shrimp, Palaemonetes pugio, were capable of hypo- and hyper-osmotic regulation of body fluids. Hemolymph chloride and osmotic concentrations were maintained at relatively stable levels over a wide salinity range. Following an abrupt transfer from intermediate (14 and 17‰) to high (31 and 35‰) or low (1 and 2‰) salinities, hemolymph chloride levels exhibited initial overshoot and undershoot, respectively, of new steady-state levels. Osmotic concentrations exhibited an initial undershoot at low, but not overshoot at high salinity. Chloride space in salinity-acclimated shrimp was relatively stable at salinities from 1 to 35‰. Changes in chloride space following salinity transfer paralleled those of hemolymph chloride levels, and are discussed in the light of alterations in intracellular sodium concentrations reported earlier. Rate constants for chloride turnover indicated independent exchanges of sodium and chloride ions. Water-turnover measurements showed that permeability of P. pugio was greatest at the isosmotic salinity (17‰) and reduced at salinities which were associated with active osmoregulation. Exposure to sublethal and 96-h LC50 levels of Aroclor® 1254 did not seriously alter hemolymph chloride and osmotic concentrations, chloride space or chloride-exchange kinetics in adult shrimp. Disruption of hemolymph chloride regulation in juvenile shrimp was associated with large mortalities not observed in adults. Shrimp exposed to Aroclor 1254 at 17‰ S exhibited reduced water permeability similar to levels previously observed in controls at high and low salinities in response to osmotic or ionic gradients. Exposure to PCBs did not result in further reduction in permeability at the latter salinities.