Jaromír Fišer

Dominant three pole placement in PID control loop with delay

In applying the pole assignment to tuning the PID controllers the usual presence of delay in the control loop brings about an infinite order character of the system dynamics, i.e. an infinite spectrum of poles. Therefore any pole placement can result in the desired tuning of the control loop only if the prescribed and placed poles really become the dominant poles of the control loop dynamics. With respect to three parameters of PID controller just three poles can be placed by the assignment and a dominance guarantee of their prescription is crucial in this way of tuning. A novel method of selecting a trio of numbers with an equal real part to make them the dominant poles of the control loop is dealt with in the paper with an additional minimizing the absolute error integral. An original assessment is introduced to check the dominance of the pole placement and an optimum of relative damping of the response is assessed to minimize the control error integral. The quality of the disturbance rejection response is taken as the decisive criterion in the presented design of the time delay plant control.

IAE optimization of delayed PID control loops using dimensional analysis approach

A specific issue of the PID control loop with time delay is the contradiction of its infinite-order dynamics with the only three controller parameters used to adjusting its behaviour. For selecting the optimum PID parameters the IAE criterion has been used as performance measure of the disturbance rejection in the investigated control loop. In order to obtain the results in a generic form the dimensionless description of the control loop, originally introduced in [14], was applied. The plant model is based on the dimensional analysis, reducing the relevant parameters of the control loop to a pair of similarity numbers, namely the so-called laggardness (ϑ) and swingability (λ) numbers. The IAE optimum search is performed by means of the gradient-based method over a representative set of options of λ, ϑ, and the optimum PID controller parameters are assessed for the whole considered area of λ, ϑ. To each of the optimum rejection responses a characteristic quasi-polynomial corresponds and then the rightmost part of its spectrum can be evaluated. The large scale spectral analysis has shown that for all of the investigated options a double pair group of poles results as dominant in the control loop dynamics. The final result of the paper consists in summarizing the IAE optimum settings of PID and comparing the obtained natural frequency angles of the control responses and their damping.

Dominant Trio of Poles Assignment in Delayed PID Control Loop

Besides its original use in the state space based design the pole assignment is also applied to tuning the PID controllers. However, infinite spectrum of poles caused by the usual assumption of a time delay in the plant model highlights the requirement that the prescribed pole positions have to result in the dominant pole assignment to be effective in tuning the control loop with time delay. With respect to three parameters of PID controller just three poles can be placed by the assignment and a dominance guarantee of their prescription is crucial in this tuning method. A novel method of selecting a trio of numbers p 1,2,3 to make them the dominant poles of the control loop is dealt with in the chapter with an additional minimizing the absolute error integral. The dominance of each of the placement trials is checked by an argument increment criterion and an optimum of relative damping of the response is assessed to minimize the control error integral. The quality of the disturbance rejection response is taken as the decisive criterion in the presented design of the time delay plant control.

Anisochronic state feedback design compensating for system delays

Abstract An original extension of Ackermann formula has been developed to design the state feedback control in systems with delays described by convolution integrals. The used modelling approach called anisochronic promotes a useful potential to select the state variables primarily as available plant outputs, avoiding the usual need for state observers. In accordance with the functional nature of the plant model the state feedback is designed on the basis of convolution integrals too. Its delay distributions are designed to compensate for the plant model delays with the aim to endow the control system with a finite spectrum of eigenvalues in spite of the infinite original spectrum of the plant. Implementation problems of the feedback feasibility have been overcome by means of multi variable extension of Smith predictor scheme. The presented method is demonstrated on an application to real plant - heat transfer system where the plant delays result both from the transport delays and distributed parameters of heat transfer phenomena.

Oscillators for Modelling Circadian Rhythms in Cyanobacteria Growth

There is strong evidence that the behaviour of living systems is subject to biological clocks which can be considered as mutually coupled oscillators. These applications of oscillators were studied since very early (Minorsky, 1962; Pavlidis, 1973). In case of biological systems like algae populations or micro-organism cultures their varying growth rate and other living system activities are liable, first of all, to the diurnal cycles of light irradiation as the decisive model input. The living systems adopt these cyclic conditions as their inner circadian rhythms and exert a specific tendency to maintain their rhythm even if the cyclic external influences change their period or shape. In this way the model of system entrainment to circadian rhythms is based on the idea of nonlinear resonance phenomenon. The circadian rhythms, also referred to as internal biological rhythms, play a role as temporal regulatory pacemakers practically in any activity of living species, but their mechanism remains still largely unknown (Ditty et al., 2009). Experimental studies and mathematical modelling have demonstrated that circadian pacemakers working on periods close to 24 hours can be modelled as limit cycle oscillators (Pavlidis, 1973; Winfree, 1970; Wever, 1970). Typically a pacemaker model implementation involves a Van der Pol oscillator as a limit cycle generator influencing the model of population growth (Fiser et al., 2008). Then this circadian pacemaker structure can be identified with the experimentally obtained data. A part of the recent research in cyanobacteria growth modelling has been already described in the previous paper (Fiser et al., 2006), where an algae population growth is investigated.

Sliding Mode Control of Time Delay Systems

Abstract An original scheme based on sliding mode control principle has been developed for controlled processes with distributed parameters and after-effects. Application of functional process model called anisochronic allows to build up the model on extremely low number of state variables selected as identical with available outputs as a rule. Specific feature of the method is a cancellation of process model delays by the state feedback based on original functional extension of Ackermann formula. Integrating controller action is added to avoid steady state offset and the augmented state feedback is implemented on a framework of Smith predictor scheme. Real application to a heat transfer plant demonstrates practical merits of the presented method.

An approach to time delay system design based on unit-circle complex plane mapping

Abstract Scaling problems and missing ability to display improper points as well-known drawbacks of standard conformai mapping of characteristic function are overcome by an original non-conformal mapping onto unit circle region. The presented method proves to be particularly useful for analysis and design of time delay systems where rather involved forms of contours are encountered and extremely wide range of investigated frequencies is needed. The generally proposed method is demonstrated on a control design application.

Dominant root locus in state estimator design for material flow processes: A case study of hot strip rolling

The purpose of the paper is to achieve a constrained estimation of process state variables using the anisochronic state observer tuned by the dominant root locus technique. The anisochronic state observer is based on the state-space time delay model of the process. Moreover the process model is identified not only as delayed but also as non-linear. This model is developed to describe a material flow process. The root locus technique combined with the magnitude optimum method is utilized to investigate the estimation process. Resulting dominant roots location serves as a measure of estimation process performance. The higher the dominant (natural) frequency in the leftmost position of the complex plane the more enhanced performance with good robustness is achieved. Also the model based observer control methodology for material flow processes is provided by means of the separation principle. For demonstration purposes, the computer-based anisochronic state observer is applied to the strip temperatures estimation in the hot strip finishing mill composed of seven stands. This application was the original motivation to the presented research.

Dynamic similarity approach to control system design: delayed PID control loop

ABSTRACT The paper deals with a novel approach to dimensional analysis-based design of control systems. The case of delayed PID control loop is used to show the benefits of the dynamic similarity concept in providing the control with response resulting from a constrained Integral Absolute Error (IAE) optimisation. The particular aim is to achieve this response in disturbance rejection ability of the control loop. To reach a generalised description in evaluating the control loop dynamics, a dimensionless model is applied resulting from the dimensional analysis of the control loop parameters. By means of originally introduced similarity numbers of laggardness and swingability, explicit relationships between the dominant pole coordinates and the PID controller gains are obtained. The proposed optimum setting of control loop constrained in damping is referred to the ultimate oscillations of the plant.

Relay Feedback Oscillator Design for Modeling Circadian Rhythms in Cyanobacteria

The paper introduces a relay feedback oscillator for modeling circadian rhythms in cyanobacteria. The relay feedback oscillator is equipped with low pass filter F(jω), hysteresis-type relay and negative feedback. This negative feedback represents an autoregulatory mechanism of the circadian clock and the notion of this autoregulatory mechanism is based on the well-known Goodwin biochemical oscillator [1]. The relay is responsible for the mediation of both the activation and degradation of oscillator state variables (protein concentrations) and in this way the pacemaker is constituted. Later on, low pass filter poles are identified for the purpose of modeling auto-oscillations with the free running period of 24h and the method of the pole identification consists in an ultimate frequency test providing stability margin of a single-loop composed of the filter and the relay in the feedback. Next, a relay output / input ratio of amplitudes and hysteresis are found out by the graphical test of the single-loop on the stability margin which is carried out in Bode graph. Finally, the output correspondence of relay feedback oscillator model with Miyoshi oscillator [2] is provided because the Miyoshi oscillator is well recognized among biochemical oscillators for species of cyanobacteria.© 2011 ASME