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Proton Exchange Membrane (PEM) fuel cell is one of the possible green solutions for energy generation. The response of the fuel cell system depends on the air and hydrogen feed, flow and heat/ water management. The main control task is to maintain the supply manifold pressure at the cathode side of PEMFC. In this article, an approximation of First Order plus Time Delay (FOPTD) model is obtained from the PEMFC dynamic model by process reaction curve method. The analysis of uncertainty for the FOPTD model was carried out. From the FOPTD model, a simple PID controller was designed and compared with other techniques like: ZN-PI, Skogestad – Internal Model Control (SIMC)-PI, Improved SIMC-PID and Smith Predictor. Also, MPC and decoupler design were implemented. The simulation results are obtained using MATLAB® which shows that the response of supply manifold pressure is well controlled. The gain and phase margins were found out from the Bode plots of open loop transfer function and closed loop transfer...

This paper analyze the performance of the first-order zero crossing digital phase locked loops (FR-ZCDPLL) when fractional loop delay is added to loop. The nonlinear dynamics of the loop is presented, analyzed, and examined through bifurcation behaviour. Numerical simulation of the loop is conducted to proof the mathematical analysis of the loop operation. The results of the loop simulation show that the proposed FR-ZCDPLL has enhanced the performance compared to the conventional Zero Crossing DPLL (C-ZCDPLL) in terms of wider lock range, captured range and stable operation region. In addition, extensive experimental simulation was conducted to find the optimum loop parameters for different loop environmental conditions. The addition of the fractional loop delay network in the conventional loop also reduces the phase jitter and its variance especially when the Signal to Noise Ratio (SNR) is low.

Closed loop control systems are designed for Linear Time-Invariant (LTI) controllable and observable systems modelled by bond graph. Cascade and feedback interconnections of bond graph models are considered and are realized through active (signal) bonds with no loading effect. The use of active bonds may lead to non-conservation of energy and the overall system is modelled by proposed pseudo-junction structures. These structures are build by adding parasitic elements to the bond graph models which assure that each storage element is connected to a dissipative element and the overall system may become singularly perturbed. The structures for these interconnections can be seen as consisting of inner structures that satisfy energy conservation properties and outer structures including multiport-coupled dissipative fields. These structures are called pseudo due to the structural properties of power conservation not being satisfied in the outer structures. The multiport-coupled dissipative fields highlight energy properties like passivity. These properties are useful for control design. In both interconnections, junction structures and multiport-coupled dissipative fields for the controllers are proposed and passivity is guaranteed for the closed loop systems assuring robust stability. The pseudo-junction structure for the cascade interconnection is applied to the structural representation of the closed loop transfer functions, in a one-degree of freedom feedback configuration, when a controller from the parameterisation of all stabilizing controllers is applied to a given nominal plant. Applications are given when the plant and the controller are described by state-space realizations, in this case parasitic elements are not added. Moreover, the feedback interconnection is used and the controller is tuned getting necessary and sufficient stability conditions based on the characteristic polynomial of the closed loop transfer function, solving a pole-placement problem and achieving zero-stationary state error.

In this article, an existing exact algorithm for identifying a continuous transfer function from frequency data, giving major importance to phase data, is further developed. The new version allows identifying a digital transfer function, or a continuous fractional order transfer function with a fixed order of commensurability (i.e. a ratio of two polynomials in s α, α ∉ ℤ). This is straightforward if the desired transfer function is to have only zeros or only poles; if a transfer function with both zeros and poles is sought, however, a numerical method must be resorted to.

SUMMARY Measurement of the current waveform of time domain AEM systems which possess coincident-loop receivers can be problematic due to the high moment of the transmitter loop during the on-time. We present a simple method of measuring the transmitter current waveform by measuring the current induced in a closed, accurately laid out and surveyed multi-turn, insulated ground loop of known inductance L and resistance R. The current induced in the ground loop, measured by a 24-bit A/D converter capable of sampling at 96 kHz, is the convolution of the time derivative of the transmitter current waveform with an exponential decay of time constant equal to the L/R ratio of the wire loop. With an understanding of the A/D converter measuring the ground loop response, the transmitter waveform can be deconvolved from the ground loop decay.

The output of a singe-input-single-output linear feedback system with more than one pole in excess over the zeros in the loop transmission cannot track arbitrarily fast its input (by the root locus). In this work we extend the linear feedback so that some of the open loop poles may depend on the open loop gain; we call this new class quasi-linear feedback systems. We then derive time domain, pole-zero, and frequency domain conditions which ensure arbitrarily fast and robust tracking by quasi-linear feedback, for an arbitrary number of poles in excess over the zeros. We prove that in a particular case these conditions are equivalent, and that the boundedness in frequency of the closed loop transfer function is no longer necessary for achieving arbitrarily fast tracking. The robustness is to external disturbances and initial conditions, and the open loop has to be minimum phase. Some examples are presented which illustrate these results. They also show that this good performance can be obtained with a reduc...

In this paper we consider a mathematical model for open-channels that expresses the dynamic relationships, in terms of transcendental functions, between the gate opening sections and the corresponding stored water volume variations in the different canal reaches with respect to an initial reference configuration of uniform flow. Series expansion around s = 0 gives a state variable linear and time invariant model as well as the corresponding rational transfer matrix. Both a proportional and a proportional integral decentralized constant-volume control law are designed by solving a linear least squares problem. Such a procedure enables us to impose the desired structure on the feedback gain matrix by means of the optimization of the controller parameters. It makes the closed loop transfer function approach a target function as closely as possible over a specified frequency range. In this paper, the target closed loop functions are designed by means of an LQR technique applied to the linear time-invariant mo...

This paper describes a method of modelling the dynamic behaviour of thyristor controlled induction motors. Real time operational verification of the model is achieved by interfacing the drive system to a computer. The steady state torque produced by the motor is described as a function of firing angle, extinction angle and speed. The model concentrates on the open loop transfer functions of induction machines for small perturbations in motor speed and firing angles. The non-linear characteristics of the open loop transfer function has been discussed. The solutions presented here can lead to a comprehensive closed loop analysis for the dynamic behaviour of induction motors in industrial applications.

Variations of the standard H∞ and H 2 control problems are presented. State-space solutions are given for the problems of weighting the individual elements of a closed loop transfer function while (i) satisfying an H∞-Frobenius norm bound or (ii) minimizing the H 2 norm of the closed loop. The optimal H∞-Frobenius case is analysed and it is shown that the optimal controller is often unique, in contrast to the standard H∞ control problem. The solutions of an algebraic Riccati equation are examined in the H 2 case to reveal the structure of the H 2 optimal controller

The loop transfer property of a full state feedback control system can be recovered by an observer based controller, provided the system is minimum phase and the observer gain is chosen appropriately. It is possible that the closed loop transfer function from the disturbance to the controlled output cannot be recovered by the same observer as above, even if the loop transfer function is recovered. The disturbance attenuation property is shown to be recovered if the Riccati equation that is to be solved in designing the observer is modified appropriately. Although the observer succeeds in recovering the disturbance attenuation property, it may fail to recover the loop transfer property. Simultaneous recovery of both properties is, therefore, desirable in practical control system design. It is shown that simultaneous recovery is possible if the number of independent measurement outputs is greater than or equal to the number of independent control and disturbance inputs. The dual result is also given; both p...

A QFT type methodology is developed for an SISO system for maximizing the size of a persistent bounded disturbance in the presence of hard time domain constraints on system states, control input, output and the bandwidth. It is assumed that the system dynamics can be represented by a combination of structured uncertainty in the low frequencies and unstructured uncertainty in the high frequencies. The design procedure is based on mapping the time domain constraints into an equivalent set of frequency domain constraints which are then used to determine an allowed design region in the amplitude-phase plane, for the nominal loop transfer function. Once such a region is found, classical loop shaping determines a suitable loop transfer function. The pole-zero structure of the compensator is a natural consequence of loop shaping and is not preconceived. A procedure is also presented for determining the maximum size of the nominal persistent bounded disturbance that can be tolerated by the resulting closed loop s...

An adaptive second-order phase-locked loop is proposed which keeps the loop within the pseudo-linear region for a wide range of input frequency steps. FETs are used to vary the filter transfer function as well as the overall loop gain, thus yielding almost linear behaviour. Theoretical analysis of the proposed system has been carried out, and the system has been constructed to show that it is possible to increase this region for the loop from a frequency step of 353.5 rad/s to 3535 rad/s without affecting its noise-rejection property.

We study the multivariable loop-shaping problem when the plant is ill-conditioned and feedback properties at both plant input and output are of interest. Directionality and coupling are described, using the singular value decomposition and principal angles between high- and low-gain subspaces. Our first results are new first-order approximations to feedback properties at one loop-breaking point in terms of the open loop transfer function at that point. Secondly, we study the relation between feedback properties at different loop-breaking points in the event that the plant is ill-conditioned. The nominal relation is studied first. Then robustness is studied, using a first-order approximation to feedback properties at one point with respect to uncertainty at the other. This analysis yields conditions that must be satisfied by feedback properties at one point to prevent poor feedback properties at the other. Finally, these two sets of results are combined to show how feedback properties at both points may be...

A simple control algorithm is developed for application to digital control of industrial processes. The desired closed-loop transfer function is specified in terms of the transient response specifications, such as percent overshoot, peak time, settling time, etc. The process with zero-order hold and tho closed-loop transfer function are modelled in the z-domain to obtain the controller transfer function. The structure the controller is much simpler than the ones obtained by using either Dahlin, Kalman or the dead-beat algorithm.

Perturbation analysis has been used to obtain transfer functions for a.c. magnetic suspension systems. A simple transfer function relating r.m.s. values, and a more complex transfer function based upon suppressed carrier modulation theory are developed. Results obtained for these models are compared with those from a non-linear digital simulation.

For various loop configurations the long-term rms current required to produce a specified magnetic field is given by a simple formula, containing the average loop width as the only loop geometry parameter. A simplified and easy to use formula for loop inductance facilitates the determination of the impedance presented to the loop amplifier. The amplifier requirements are heavily dependent upon field strength specification and impedance of the loop. Taking these factors into account, an economical and correct dimensioning of the amplifier and loop system may be obtained.

The generation of wide-band FM using direct frequency modulation generally requires a subsidiary loop to stabilize the carrier frequency. The steady-state carrier drift is shown to depend on the loop gain as well as the reference frequency of the frequency demodulator used in the feedback loop. The transient behaviour of the loop may be controlled by the loop filter which is used to separate modulating signal from drift. An appropriate low-Q second-order filter is preferable to one with sharp roll-off rate. The consequent overlapping of drift and signal spectra is resolved by the use of a suitable pre-emphasis filter between the modulating signal and the FM loop. The demodulator non-linearity has significant effect on the transient response when the drift size is large. Two types of demodulator characteristics are used in evaluating the step response of the loop. It is found that the pulse-counting type provides a faster loop response at large drift steps than the ‘S’ type.

One method of model-based compensator design for linear systems consists of two stages: state feedback design and observer design. A key issue in recent work in multivariable synthesis involves selecting the observer (state feedback) gain so that the final loop transfer function is the same as the state feedback (observer) loop transfer function. This is called loop transfer recovery (LTR)(Athans and Stein 1987, Kazerooni and Houpt 1986, Kazerooni et al. 1985, Doyle and Stein 1981). This paper shows how identification of the internal mechanism of the LTR provides simple design rules with little algebra for single-input single-output (SISO) systems. In the SISO case, the LQG/LTR reduces to computation of a compensator that shapes the loop transfer function by (i) cancelling the zeros of the plant with the compensator poles, and (ii) locating a new set of zeros for the compensator to shape the loop transfer function.

A multivariable autoregressive (MAR) method is applied to the core stability estimation of a boiling water reactor-5 operation. Noise data measured during steady-state operations at startup tests are used. In this method, the closed loop transfer function from reactor pressure to reactor power is identified from reactor noise data and transformed into an impulse response function. The decay ratio representing stability characteristics is evaluated from this function. The variation range of decay ratio estimates obtained by this method is sufficiently small, if the analyzing conditions are appropriately selected. The value of the decay ratio is 0.23 during natural circulation and decreases with core flow, reaching close to zero at the rated power. A similar power dependence for the decay ratio is seen in results from a core stability analysis code. The MAR method is a useful tool for stability estimation, even if no external disturbance tests are conducted.

The problem of factorizing the transfer function of single-input single-output 2-D systems to a product of two 1-D transfer functions using state feedback is considered. It is shown that if the numerator of the open-loop transfer function can, by itself, be written as a product of two 1-D polynomials, then the closed-loop transfer function can, under certain conditions, be factorized as a product of two 1-D transfer functions. In cases where the numerator is not by itself factorizable, then the present results restrict themselves to factoring the denominator of the closed-loop system to a product of two 1-D polynomials.

This paper presents a frequency simplification technique for simplifying a transfer function of higher dimension to a transfer function of lower dimension. It tries to match the frequency response of the two transfer functions in the best possible manner in the frequency range of interest. It compares this method with the existing methods by means of a numerical example. The advantages of this method is that it aims at minimizing the error in the phase of two transfer functions together with magnitude and simplifies the complexity of work in solving for the unknown coefficient of the assumed simplified transfer function.

In a wide variety of physical problems a vector of inputs {x (t), x2(t) ... xk(t) } is fed into some system producing a vector of outputs {y (t), .. y,(t)}, where t represents time. When the differential equations relating inputs and outputs may be written down and solved it is possible to express the outputs explicitly as some function of all the past inputs. When the inputs are kept constant, this relationship does not depend on time and is usually called the steady state response surface by statisticians. For one input and output the time dependent relationship may be expressed as y(t) = ?(x(t)) + n(t) (1.1)

Exact and approximate methods of determination of limit cycles in a relay control system including an actuator with PI transfer function and limited output are given. No restrictions are imposed on the order of the plant transfer function. The stability of the limit cycles may be checked using one of the well-known methods, using the describing function concept. With the help of an example it is indicated that the methods suggested give reliable results in eases where the use of the conventional describing function fails.

Final value control of linear plants is discussed. An expression for the closed loop fundamental matrix in terms of the open loop transition matrix is presented. It is shown that for scalar minimum energy final value control the singularities in the feedback loops are inversely proportional to the order of the derivative being fed back. That is, the position loop has an ηth order singularity, the velocity loop an η—lth order singularity, etc. A partially closed loop control is developed that allows the highest-order feedback singularity to be set between η and zero. Those results are illustrated by examples.

The power-coolant temperature transfer functions obtained for cases of both distributed and lumped parameters are used to compare three models of the internal reactor loop. The power-coolant transfer functions are considered for two values of the coolant velocity, and the results as attenuation and phase characteristics are presented. After adding the attenuation and phase characteristics of the power-coolant temperature and zero power kinetics transfer functions, the open-loop frequency responses of the examples considered are obtained. Two examples of the closed loop transfer functions are also considered. It is concluded that the distributed model of the thermal processes in the reactor core allows the open loop gain coefficient to he greater in comparison with the lumped model at the same conditions of damping of the closed loop. (C.E.S.)

The second method of Liapunov is used to determine asymptotic stability of non-linear feedback control systems. The paper deals with second and third order control systems with hardspring characteristic as non-linear element and determines stability by simplified stability criteria from the development of Liapunov functions. The above systems have been simulated in an analogue computer and establish the results from stability criteria.The above Liapunov method fails in certain practical cases where the open loop transfer function of a control system has a pole at the origin. A pole shifting technique has been used in this paper to extend the applicability of the method.

For the majority of all dynamic systems open loop optimal control generally results from optimization theory whereas a closed loop control is normally desired. By accepting a degradation in performance a closed loop control termed ‘ specific optimal control ’ can be obtained. A series of experiments were performed investigating this type of sub-optimal control. It was found that although the proper choice of the control parameter leads to easily implemented closed loop controls, there are major drawbacks to its use. The coefficients in the control law ore highly dependent upon the final time T, the system parameters, and the initial state of the system. Furthermore, the performance of the system was found to be a function of the algebraic form chosen for the control law. To determine the most optimum control law entails performing a series of computer experiments. This last difficulty may be circumvented in part by embedding the specific optimal control problem in an identification problem. Resul...