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Calculation of controllability and observability Gramians, and the determination of balanced realizations are considered in a symbolic framework. Two different approaches to balancing transformations are explained step by step, and more efficient, simple and numerically stable algorithms based on these two approaches have been implemented. An example taken from the literature is presented in order to illustrate the superiority of the symbolic implementations.

Discusses the issues of robust machining process controller synthesis. In practical cases, operating conditions have strong effects on the process dynamics, and it is very difficult to use a fixed controller to achieve a satisfactory performance. Here, a fuzzy logic inference mechanism is introduced to adjust the controller according to the online measurements of operating variables which define the operating conditions or operating points of the processes, and a linear matrix inequality (LMI) based hybrid approach is developed to achieve robust properties against modelling errors, uncertainties, disturbances, and inaccuracy of online measurements. The simulation results of a design example (an end milling process control) shows that robust stability and satisfactory performance have been achieved.

Modern symbolic computational systems which perform automated manipulation of mathematical variables offer insights during modelling and problem solving which remain otherwise partially or wholly obscured to the analyst. The classic inverted pendulum model is re-visited, and previous work concerning the systems controllability is investigated. In particular, the ability of the software to factorise complicated multivariable polynomials is exploited to identify, in fully general form, the anticipated pole-zero term cancelling throughout the transfer functions of the system when it is in a state of un-controllability. All three balancing problems associated with the two link pendulum are treated, and the phenomenon of non-controllability is examined in this way along the entire `curve of non-controllability' which, within the approximation of linearity, theoretically exists for each when damping is present.

The problem of quality control of radiological systems is discussed in relation to the management and quantitative tools normally applied to it. An alternative approach is to consider the problem in terms of a control loop in which the sets of quality control measurements are compared with their previous values in the context of a numerical model of the system to produce quantitative corrections. A preliminary study of the method applied to mammographic quality control is discussed. By modelling the system with an appropriate set of simultaneous linear equations relating changes in the controlled parameters to the quality control measurements obtained, it is possible to identify quantitative changes in the controlled parameters from measured quality control data using the inverse of the model in matrix form. This method essentially encapsulates knowledge about the system in the form of the linear model. This is distinct from current work attempting to apply artificial intelligence knowledge-based techniques to the problem. (4 pages)

Sparse-matrix solution is a dominant part of execution time in simulating VLSI circuits by a detailed simulation program such as SPICE. The paper develops a parallel-block partitionable sparse-matrix-solution algorithm which exploits sparsity at the matrix block level as well as within a nonzero block. An efficient mapping scheme to assign different matrix blocks to processors is developed which maximises concurrency and minimises communication between processors. Associated reordering and efficient sparse storage schemes are also developed. Implementation of this parallel algorithm is carried out on a transputer processor array which plugs into a PC bus. The sparse matrix solver is tested on matrices generated from a transistor-level expansion of ISCAS-85 benchmark logic circuits. Good acceleration is obtained for all benchmark matrices up to the number of transputers available.

The capabilities of computer algebra extend well beyond the bounds of mathematics. The author describes some of the less obvious ways in which these powerful software packages can come to the help of the busy engineer in many fields, including text processing, graphics etc.

A method of controlling certain types of nonlinear dynamical systems whose dynamics can be modelled by a multilayer neural network is proposed. The control algorithm assumes that the plant equations are not known but the dimension of the system is known. The control input is derived by inversion of a forward neural network via the Newton Raphson method. During inversion of the multilayer neural network some optimal control senses are resolved. To suppress the control error due to the modelling error of the forward neural network, the inversion controller with a conventional feedback controller is proposed, which provides a better performance than a pure inversion controller. The proposed algorithm shows various advantages, and computer experiments on a bioreactor prove the effectiveness of this algorithm.

A new design method of discrete-time model reference adaptive control for nonminimum phase systems is presented, with disturbances using approximate inverse systems obtained from the long division method. It is assumed that the disturbances are described by a polynomial function of time with known degree and unknown coefficients. The proposed scheme uses only input and output data, and the existence of bounds for all signals is proved. This scheme also assures that plant output converges to the reference model output. The results of computer simulation are also presented to illustrate the effectiveness of the proposed method.

The A-function method is extended to analyse a relay feedback control system with additional nonlinearities in the feedback loops. The situation considered is where the output of the plant and/or its successive derivatives are fed back to the input through nonlinearities that are assumed to be of polynomial type. An example is also given to show that, in some cases, the method can be used in systems with saturation or a linear segmented nonlinearity in the feedback path.

This paper outlines the method of obtaining the frequency response functions of a rotor system with strong gyroscopic effect and simulating it using MATLAB. The rotating effect causes unsymmetry in the system matrices, resulting in complexity in decoupling the mathematical models of the system for the purpose of modal analysis. A different method is therefore required, which can handle general system matrices rather than symmetrical matrices, which is normal for passive structures. In this paper the mathematical model of an overhung rotor system with two degrees of freedom is expressed. This model is then used to extract the right and left eigenvalues/vectors and subsequently the frequency response functions are extracted and simulated. MATLAB is used to carry out such simulation, since it has good capability for eigen analysis and also good graphical facility. The reflection of splitting of critical speeds of a rotating rotor on the frequency response functions are shown.

This paper presents a new parallel-in-space algorithm for power system transient stability simulations. The nonlinear differential equations are discretized by applying the trapezoidal rule and solved together with the nonlinear algebraic equations for each time step. A network partitioning scheme, which is based on the subdivision of the factorization path tree of the network matrix, is proposed to exploit the parallelism-in-space of the transient stability problem. The parallel version of the very dishonest Newton (VDHN) method, in which the parallel algorithm for solving large sparse network matrix equations is incorporated, is developed and tested on a distributed memory message passing multicomputer. Test results on a sample power system are presented to show the performance of the proposed algorithm.

Local model networks represent a nonlinear dynamical system by a set of locally valid submodels across the operating range. Training such feedforward structures involves the combined estimation of the submodel parameters and those of the interpolation functions. The paper describes a new hybrid learning approach for local model networks that uses a combination of singular value decomposition and second order gradient optimization. A new nonlinear internal model control scheme is proposed which has the important property that the controller can be derived analytically. Simulation studies of a pH neutralization process confirm the excellent modelling and control performance using the local model approach. (3 pages)

The backstepping control design algorithms described by Krstic et al. (1995) provide a systematic framework for the design of regulating strategies suitable for large classes of nonlinear uncertain systems. However, the equations arising at the successive steps are usually too complicated to be computed by hand. We consider here a symbolic toolbox which implements a general algorithm for the design of dynamic adaptive controllers following the basic ideas of backstepping with tuning functions without transformation into canonical forms. This algorithm is applicable to observable minimum phase systems not necessarily in triangular form and also to uncertain nonlinear systems in triangular forms. Additionally the control can be generated by a sliding mode approach. (3 pages)

Geometrical symmetry often occurs in computational electromagnetics. However, it is generally not taken into account when the excitations do not share this symmetry. A rationale is required, and group theory gives the only valuable tool for this purpose. It provides a symmetric decomposition of any problem that allows its study on a reduced part of the initial geometry. This way of treating field problems generally leads to substantial computational savings. The symmetry concept is applied in the paper to 3-D linear eddy-current analysis, where the numerical scheme used is a mixed FEM–BEM method. Detailed examples are presented to demonstrate the efficiency of the use of symmetry.

This paper presents developments made to the TLM method in the time to frequency-domain transformation of the impulse response solution. An improvement to the technique by which the three-dimensional TLM time-domain method is post-processed is presented. It is shown that the selective choice of a particular data windowing profile plays a significant role in the accuracy of the results, clarity of output response and the extraction of the S-parameters.

This paper presents the method used to derive the oscillation condition by using symbolic calculus. The program is based on the full nonlinear Barkhausen criterion method. The behaviour of an oscillator is described by a complex polynomial called the characteristic polynomial. This polynomial enables us to calculate the steady state features of the oscillation as well as the differential equation for transient analysis in the time domain. The literal determination of this characteristic polynomial involves lengthy algebraic calculations and cannot be done by hand as the electronic oscillator circuit involves too many components. We recently developed a formal calculus program allowing to automatically obtain all necessary equations for oscillation analysis. We propose new methods to calculate them in an optimal form.

A conventional diagonal controller can be designed from very limited process information. In the SVD control structure, a diagonal controller is applied to orthogonal sums and differences of the inputs and outputs. In this work, the process knowledge needed for the design of an SVD controller is analyzed. In contrast to conventional decentralized controllers, the SVD controller can compensate for the process directionality of ill-conditioned processes. The dual composition control problem of distillation columns is of particular interest, since reliable models for these columns are quite hard to obtain.

The problem of stabilising a MIMO plant via output feedback controllers of a given degree was recently tackled via linearisation around some special degenerate compensators. This can be numerically implementated as an ɛ-perturbation method. The solution is in the form of a perturbation series which can be constructed by repetitively solving a set of linear equations, coming from the expansions (in ɛ and s) of the original pole placement equations. This expansion can be done almost trivially in any symbolic language using standard symbolic commands. The code is only a few lines long and can be done by the nonexpert. There is no need to understand the algebra of the problem, which involves tensor, polynomial algebra and some combinatorics since the load of the expansion is taken solely by the symbolic package. (4 pages)

This paper has dealt with the prototype symbolic manipulation CAD system-Symbolic Control Toolbox-running on MATLAB. The functions are implemented in the conventional numerical CAD system. Compared with numerical CAD our system has the following advantages: there exists no roundoff or truncated error; the adopted algorithms are so systematic and well-defined that functions are implemented easily over various rings corresponding to control theories by using the symbolic computation; and the results of the system can be extended easily to n-dimensional system theories.

A novel algorithm for solving nonlinear discrete time optimal control problems with model-reality differences is presented. The technique uses dynamic integrated system optimisation and parameter estimation (DISOPE) which achieves the correct optimal solution in spite of deficiencies in the mathematical model employed in the optimisation procedure. A new method for approximating some Jacobian trajectories required by the algorithm is introduced. It is shown that the iterative procedure associated with the algorithm naturally suits applications to batch chemical processes.

A robust fault isolation method is proposed in this paper, which uses state-space parity equations to isolate structured faults. Numerically robust calculations are used to find certain matrices. A workable sufficient condition is given for high threshold isolability. The method is applied to a lathe-spindle system which demonstrates the effectiveness of the method.

The authors propose a method of simulation of static power converters which uses the diagonalization technique of the dynamical matrix to decrease simulation time. This technique leads to a simple analytical solution, allowing a reduction in calculation time. This method allows new independent state variables, and a reduction in the system order. The authors verify that for the structures analyzed, they have observed, on average, a gain in calculation time of the matrix exponential of about 97% using the matrix diagonalization method. The developed algorithm is shown to be a powerful, efficient and, in certain cases, an essential tool for the study of applications in power electronics.

The control structure selected has a strong effect on the performance of a closed loop system with respect to its disturbance rejection capability. But the methods used for its selection usually result in a combinatorial problem. To overcome this difficulty, an input pre-screening criterion, the input-disturbance alignment (IDA) measure, is proposed. Some important features of the new indicator and its relationship with other well-known controllability measures are presented. The case study included shows that the IDA can be used for efficient selection of manipulated variables from a large number of candidate inputs. These then form the best control structure for disturbance rejection.

This paper reports the use of a general technique to combine several different methods to solve complex systems of algebraic equations in the context of load flow calculations of electrical power networks. Such a combinations of methods, referred to as 'team algorithms', seem specially well suited to be used with distributed memory computer systems, in an asynchronous environment. Experimental results solving example problems in a commercially available parallel computer system show that a 'synergetic effect' with considerable speedup can be obtained using these 'team algorithms'.

A solution to the standard H_{∞} optimal control problem is presented which is particularly useful in machine control applications where the output to be controlled is different from the signal for feedback. The solution is obtained in polynomial form and the plant structure which is assumed represents a range of applications in the manufacturing and process industries. The results are applied to the design of a thickness-control system for cold-rolling mills subject to a range of disturbance noise inputs.

The paper deals with the problem of converting a continuous-time uncertain linear system to an equivalent discrete-time uncertain model and its digital simulation. The system matrices characterising the state-space representation of the original uncertain system are assumed to be interval matrices. The geometric series method together with interval arithmetic is employed to obtain the approximate discrete-time interval models. A new technique is developed to estimate the modelling errors. These modelling errors are used to modify the approximate interval models obtained via the interval geometric-series method. The resulting interval models (the enclosing interval models) are able to tightly enclose the exact uncertain model. Also their approximate discrete-time interval solutions are able to tightly enclose the exact interval solution of the continuous-time uncertain state-space equation. The proposed digital uncertain models can be used for digital simulation and digital design of continuous-time uncertain systems.

The problem of crosstalk evaluation among PCB traces in a multilayer structure is studied by means of an S-matrix technique. Two bundles of parallel lines, perpendicularly crossing on different layers, are analysed, and this complex structure is decomposed into subsystems, each of them characterised by its S matrix. The elementary subsystems consist of bundles of parallel lines and crosses of two lines. A particular algorithm, developed to number the ports of each subsystem, allows one to easily recognise the interconnected ports and the external ports (I/O ports), to achieve the S matrix of the overall structure in terms of the I/O ports. The application of this technique to a 3*3 and a 7*7 configuration shows a good agreement between theoretical and experimental results. Finally, the analysis in the frequency domain is extended to the time domain by a standard FFT algorithm, and, as an example, an ESD current flowing along a line is examined, as well as its coupling toward an adjacent and crossing line.

Detailed accurate models for brushless alternator units are important for practice and design engineers. The paper presents a state space mathematical model for a standalone unit based on a circuit model in the direct-phase reference frame. Also, a state space mathematical model for an automatic voltage regulator is integrated with the machine model. Due to the existence of the rotating bridge rectifier of the exciter, the topology of the circuit is variable. Therefore, Kron's tensor technique is applied using a variable connection matrix to develop the mathematical model. The model is complete in the sense that it covers all modes of operation of the bridge rectifier. Consequently, it is capable of simulating any loading conditions. To enhance the model accuracy, electromagnetic nonlinearity effects are introduced through developed saturation factors which consider the mutual saturation effects between the main and leakage fluxes. Also, a new saturation concept using dynamic and static inductances is applied. Various transient responses are computed and compared with test results to verify the validity of the presented model.

This paper describes a method for predicting the operating performance of an array (input match, power split between the stacks and the voltages on components) based on a 4-port S-parameter measurement and on a model of the array's bay feeder network. The measured S-parameters are converted to an admittance matrix (Y-parameters) and used to terminate an equivalent circuit of the array's bay feeder. The resulting network is analysed using a method based on chain matrices. The correlation between measurement and prediction of the arrays performance in the straight ahead condition, and at small slew angles is good. However, this deteriorates as the angle of the slew is increased. The model is based on a balanced network and it is thought the errors between measurement and prediction, at large slew angles, can be attributed to unbalanced currents in the array. (7 pages)

The paper is devoted to the development of a neural network architecture which implements the Newton-Raphson algorithm for solving the set of nonlinear equations of power-system load-flow analysis. The principal context is that of online network analysis in energy management systems with particular reference to the optimal power-flow function. The author shows that the complete Newton-Raphson load-flow formulation maps into an array of two-layer neural networks. The development starts from a formulation for solving as a minimisation problem the linearised equation system to which the Newton-Raphson sequence leads at each iteration. For that purpose, an objective function in quadratic form is derived. A neural network structure is given which implements the steepest descent method for minimising this objective function. It is shown that the weighting coefficients of neural networks are formed from element values in the Jacobian matrix of Newton-Raphson load-flow analysis. When the Jacobian matrix is nonsingular, the quadratic objective function derived has a unique and global minimum. A principal feature of the extensive parallel processing capability of the architecture developed is that the computing time of load-flow analysis is independent of the number of nodes in the power network for which analysis is carried out. For a sample section of a power network, and by software simulation, the architecture which the paper seeks to report gives solutions which are identical with those from a standard sequential processor load-flow program.

The authors present the application of the technique of least absolute value (LAV) dynamic filtering to optimal tracking of power system harmonics. The proposed technique uses digitised samples of the voltage and current waveform at a power system bus, where a harmonics standard is contemplated. The proposed technique can easily handle time-varying harmonic parameters. Two models are developed and tested. In the first, the measurements matrix varies with time, with an identity transition matrix. In the second model, the state transition matrix is a function of the sampling rate and the number of harmonics chosen. The algorithm is tested using a simulated and actual recorded data set. A sample set of results is reported.

Voltage collapse is associated with a stress condition of power systems. Control actions must provide the desired results, otherwise a system may operate in an unknown condition. It has been shown that this unknown condition is associated with two regions of operation and the boundary between them. The boundary between the two regions is related to a singular load-flow Jacobian. To identify the critical bus, a reduction of the load-flow Jacobian in relation to each load bus is developed. To reduce the computational burden associated with large power networks, a network partitioning is proposed which is based on voltage variation at each load bus in relation to load variation at the other load busses. Comparison between the proposed method and network partitioning using Sanchis' method is presented. The weak area of a power system is identified in the network partitioning; a weak area is defined as the area that contains the critical bus of the power network. To calculate the margins for each load bus of the weak area, the relation between load variation at each load bus and voltage magnitude and phase angle variations at the critical bus is normalised, one by one. The busses strongly connected to the critical bus have smaller load variation in relation to the busses weakly connected to the critical bus. The proposed method has been tested using the IEEE 24-bus and 300-bus systems.

In this paper, multivariable quantitative feedback theory (QFT) techniques are used to synthesize a robust feedback system for an extractive distillation column with a vaporous sidestream. The simplified state space model of Gilles and Retzbach (1983) for an extractive distillation column exhibiting sharp temperature profiles is adopted here. Large uncertainty of +20% is assumed in three key matrix elements of this model. The feedback design problem is to find the controller and prefilter matrices such that the stated tracking specifications are met, over the entire range of model parametric uncertainty. The fourth MIMO QFT technique of Yaniv and Horowitz (1986) is used to solve the design problem. The design is verified through extensive simulations in both frequency and time domains, for several plants picked from the plant uncertainty set. In all cases, the results obtained are quite satisfactory.

This paper is concerned with the application of an algebraic language (known as Mathematica) to control engineering algorithmic problems. Several problems of significant interest to control engineers are considered. Two methods of computing the Smith-McMillan form are considered, and one approach to the creation of minimal state-space realizations is considered. Balanced minimal realizations, model-order reduction and minimal order matrix-fraction models are also examined. The algorithms implemented are all illustrated by examples.

This paper develops an economic dispatch algorithm for the determination of the global or near global optimum dispatch solution. The algorithm is based on the simulated annealing technique. In the algorithm, the load balance constraint and the operating limit constraints of the generators are fully accounted for. In the development of the algorithm, transmission losses are first discounted and they are subsequently incorporated in the algorithm through the use of the B-matrix loss formula. The algorithm is demonstrated by its application to a test system. The results determined by the new algorithm are compared to those found by dynamic programming with a zoom feature.

This paper reports the development and testing of a method for modelling distribution load impedances as unbalanced three phase impedance matrices for detailed harmonic studies. The experimental testing of this method was performed using data obtained from capacitor switching and transformer tap changing disturbances. This method is applicable even in situations where harmonic sources within the load exist provided that the output of these sources is effectively uncorrelated with the applied disturbance. Computer simulation results gave a measure of the self impedances, accurate to within 5% at the resonant peaks, but gave a considerably less accurate measure of the mutuals. The results obtained from measurements taken at the supply end of a distribution feeder show that the disturbance resulting from a transformer tap change event can give results consistent with those obtained from the much larger capacitor switching disturbance.

A flux linkage model for a three-phase five-legged transformer is derived for the first time. The effects of mutual and leakage inductances, saturation, eddy current loss, hysteresis and residual fluxes of the nonlinear iron core are included. A straightforward and novel method for determining both the apparent and incremental inductance matrices for a loaded two-winding five-legged transformer is derived. Simulations presented in this study are based on apparent inductance (λ/*i*) rather than incremental inductance (*d*λ/*di*) because this approach was found to be preferable for predicting the behaviour of the laboratory transformer used. Eddy current and hysteresis losses are combined into one core loss term which is accounted for by a voltage-dependent resistance load on the secondary. The flux in each segment of the iron core and in the air gaps and tank are computed at each step. The model thus yields a completely detailed description of the transformer's behaviour. A predictor-corrector algorithm written in FORTRAN is used to solve the set of nonlinear differential equations that describe the transformer. A 386-based computer has been used to generate voltage and current waveforms faithful to laboratory data. Transformer inrush current and ferroresonance are selected as test cases because of the extreme computational demands that these transients place on the computer model. The model can be implemented using only nameplate and conventional test data along with the *B*-*H* curve of the core material. The method is thus well suited to the practical needs of engineers in industry.

An adaptive inferential algorithm is developed for estimation and control of multirate systems. The output *y* is measured *J* times slower than the secondary process output *v* and the input *u*, but an output estimate *Y _{e}* is produced at each sampling interval of

*v*and

*u*. Compared with previous work on multirate inferential systems, the proposed algorithm has a more formal theoretical basis. For example, the output

*y*is related to the secondary output

*v*not only through external stochastic disturbances but also through the internal system structure. Convergence properties are formally proven for the case of zero external stochastic disturbances, and a simplified algorithm is proposed for practical applications. Simulated results illustrate the convergence properties of the algorithm and the improvement obtained in simple feedback control systems.

The dynamical performance of robot manipulators is greatly affected by the different payloads handled by the end-effector (hand). Hence, it is very important, especially for industrial applications, to study the different interconnected relationships between the manipulator's joints, speeds, loads and actuation forces. In the paper, a simplified semicustomised symbolic formulation of robot dynamics, based on the Lagrangian, is presented, with emphasis on the Coriolis and centripetal effects. The accuracy and computational efficiency of this new formulation is demonstrated by simulation of the Stanford and PUMA 560 robot manipulators. Useful quantitative measurements and error analysis are also included on the significance of Coriolis and centripetal terms, under different load and speed conditions.

An explicit formulation is presented for computing first- and second-order S-matrix sensitivities of microwave circuits. It can be very easily programmed in conjunction with a subnetwork growth approach to reduce CPU time and memory space requirements.

The state-space model of a synchronous phase convertor is formulated by using the nonlinear dynamic equations of both the pilot (synchronous) and the load (induction) motors. A design procedure is presented for computing the parameters of a linear controller for the field exciter of the synchronous motor. The controller parameters are updated according to the selected steady-state operating conditions. The proposed controller is suitable for microcomputer implementation. The validity of the controller design method is further verified by using computer simulation. Numerical experiments indicate that the synchronous phase convertor system can be operated stably under different load and fault conditions.

The state vector of a power system varies with time owing to the dynamic nature of system loads. Therefore, it is necessary to establish a dynamic model for the time evolution of the state vector. The dynamic state estimation approach consists of predicting the state vector based on past estimations, followed by a filtering process performed when a new set of measurements is available. This paper presents a new algorithm for forecasting and filtering the state vector, using exponential smoothing and least-squares estimation techniques. The proposed algorithm is compared with another one based on standard Kalman filtering theory. Numerical results showing the performance for both dynamic estimators under different operational conditions are presented and discussed. Detection and identification of multiple bad data are also included. The new dynamic estimator exploiting state forecasting is extremely useful to real-time monitoring of power systems.

A digital computer method is presented for the accurated evaluation of temporary overvoltage transients on a series-compensated transmission line. The synchronous generator is represented by a five-coil dynamic model, and the effect of the excitation control system associated with the generator is included in the analysis. The transmission network is represented by differential equations involving the network state variables. This kind of study is essential in choosing economical voltage ratings for surge arresters protecting high-voltage station equipment, in view of the modern practice of allowing surge diverters to reseal after temporary overvoltages exceeding their voltage ratings for a limited duration of time. These studies, when conducted for the case of series-compensated transmission lines, can reveal the limit of compensation level beyond which the danger of self-excited oscillation can occur in a particular system.

A mathematical model for the type of transformer/rectifier unit used in aircraft power-supply systems is described. The model is suitable for investigating both the effects of variations in the transformer parameters on the unit performance and the interaction of the unit with other items of plant in the supply system. The experimental unit on which the model was validated exhibits a marked imbalance between the two diode bridges fed from the transformer secondary windings, and this is satisfactorily predicted by the model. Introducing even a moderate harmonic content into the supply voltage to the transfer increases considerably the imbalance, which under full-load conditions could result in overloading and possible failure of the diodes of one of the bridges.

Livsic's theory on the decomposition of the state space into invariant subspaces is combined with the *QR* algorithm for finding the eigenvalues of the state-variable matrix to yield an efficient method for computer-aided synthesis of passive networks. The synthesis is presented for scattering and transfer-scattering matrices which are widely used in practical problems.

The method of the busbar impedance matrix is applied to the solution of asymmetrical faults through the use of sequence impedances and Kron's constraint matrix. An algorithm is proposed to facilitate the routine solution of complex configurations including simultaneous and cross-country faults.

A systematic Laplace method is presented for the determination of transient-current propagation in a single transmission line. Earth distortion and skin effects are considered in the analysis. The method permits a quick and accurate evaluation of these currents without using numerical-integration techniques. From the theoretical study, an efficient method of digital simulation is also derived for a simpler current evaluation in computers. Both analytical and simulation expressions are in terms of line and earth parameters, which enables an easy computation of transient current propagation in different transmission lines to be carried out. Results are in good agreement with those obtained by other investigations. Part 1 of the paper deals with a basic single line which would represent either an a.c. or a d.c. scheme. Part 2 of the paper deals with polyphase systems with lossy earth returns (see p. 137).

In this study, an operational Laplace method is presented for analysing current propagation in 3-phase lines. Basic results obtained in Part 1(see p. 133)are systematically extended to polyphase-line investigations. Modified operational Laplace impedances are used and results can be carried out by a direct use of Laplace transforms and matrix theory. An efficient simulation method is also presented, which permits an easier computation of current surges in computers. Earth distortion and skin effects of conductors are included in the study. Analytical and simulation expressions are in terms of line and earth parameters. Therefore they make easy numerical computations of currents of different lines. Other advantages mentioned in Part 1 of the paper are also found in Part 2 of the paper, e.g. no numerical integration is needed.

Models in many disciplines are specified by algebraic equations. In filter engineering, they are always specified by differential equations (DEs), and in this chapter we develop the necessary background to enable us to use DEs as models. For each DE we will see that there is a unique transition matrix, and it is through the transition matrix that the DE is actually implemented. Our discussion is thus about DEs and their transition matrices, and how such matrices are derived.