Some applications of matrices to location of zeros of polynomials†

In previous papers classical theorems on location of zeros of a polynomial with respect to the left half plane Γ₁ or the unit circle Γ₂ have been reformulated more simply in terms of appropriate companion matrices. It is shown how this work can be extended to the problem of zero location with respect to more general regions Γ of the complex plane. The first approach is to apply the bilinear transformation to the given polynomial, so that for example Γ₁ can be mapped into Γ₂ and a matrix representation of this is derived. An alternative method is discussed which relies on transformation of Γ into Γ₂. Some examples illustrate how any theorem involving Hurwitz-typc minors can be expressed in companion matrix terms, with a consequent halving of the orders of the determinants involved.     

Some properties of cyclic codes†

This paper will investigate the properties of cyclic codes under the transformation,$ It is discovered that for all the non-shortened cyclic codes (binary or (/-nary), if (g(x), (x?1)) =1 then the set of all the codewords is invariant under the transformation f_b. If (g(x), Lx?1)) = (x; ? 1), then the set of all the code words is generated by g(x)/(x ? 1). Next this paper shows a systematic method for the construction of orthogonal cyclic codes in which all the code vectors are orthogonal to each other even to itself     

Additional properties of an extended maximum principle†

This paper establishes conditions of optimality for linear systems whoso control functions are quantized either in magnitude or in switching instants, or both. Optimization of such systems hitherto has been limited primarily to dynamic programming techniques and direct enumeration. The results are derived from the total variations of the cost functional. The main theorem provides sufficient conditions of optimality for linear systems with quadratic and positive semi-definite performance index. The results are significant for digital control systems, pulse-code-modulated control systems and some types of pulse-frequency-modulated systems which are subject to quantized variations only. For the special case where the control functions are not quantized, the optimality conditions provided by the extended maximum principle are shown to be necessary and sufficient.     

A chain of factored matrices for Routh array inversion and continued fraction inversion†

A chain of factored matrices is derived for formulating the Routh array if the first column of the array is known. The factored matrices may be used to perform the conversion of a continued fraction into a rational function. A set of tables based on the second Cauer form of continued fraction expansion is also included. These tables are the approximated rational functions for the commonly used irrational function √ and the transcendental function e ^s .     

Enhancement of robustness in observer-based fault detection†

A prerequisite for the feasibility of the technique of observer-based fault detection and isolation (FDI) in dynamic systems is a satisfactory robustness with respect to modelling uncertainties. This paper surveys the most relevant methods to increase the robustness in both the stage of residual generation and residual evaluation. Among these methods are the generalized observer scheme, the robust parity space check, the unknown input and H _∞ observer scheme, the decorrelation filter, and the concept of adaptive threshold selection. It is pointed out that the unknown input observer concept, which provides perfect decoupling from the modelling errors or its optimal approximation with the aid of H _∞ techniques, constitutes a general framework of robust residual generation that generalizes and unifies numerous other approaches, among them the parity space and detection filter approach. It is further shown that this FDI method can even be applied to a certain class of nonlinear systems.     

Comparison of numerical methods in stability analysis†

The results of a numerical investigation of a number of techniques for stability analysis of non-linear systems are presented. The methods compared are the Ingwerson and Zubov procedures for construction of Liapounov functions, a number of methods for obtaining optimum Liapounov functions, tracking functions, the alternating extreme radius path, and the method of Luus and Lapidus.     

Stability properties of discrete-continuous feedback control systems with signal dependent sampling III. Stability properties†

The nature and existence of recurrent trajectories have been considered in this final paper on the stability properties of discrete-continuous systems with signal dependent sampling. Simulation results illustrate the complexity which may arise in recurrent trajectories and furthermore demonstrate the effect of ‘stability control’ in a signal dependent sampling system. These results lead to the concepts of ‘controllable stability’ and ‘controllable quality’ of which the former has been considered in some detail.     

Minimization of transmission cost in decentralized control systems†

This paper considers the problem of stabilizing a linear time-invariant multivariable system by using local feedback controllers and some limited information exchange among local stations. The problem of achieving a given degree of stability with minimum transmission cost is solved.     

Preservation of controllability in linear time-invariant perturbed systems†

The controllability of systems with weak connections is studied. A necessary and sufficient condition for a singularly perturbed system to be strongly controllable is obtained. The controllability invariance of the slow subsystem of a singularly perturbed system due to a class of fast feedback controls is shown.     

Parameter Estimation in the Presence of Measurement Noise†

The parameter estimation problem is formulated for the situation in which the measurement of every variable is subject to noise. The method of maximum likelihood is shown to produce estimators for which the amount of computation is prohibitive but these estimators fire used as a standard against which the accuracy of more easily computed methods may be judged, A number of alternative methods are introduced, a form of weighted least squares being shown to require the least computation and to approximate the accuracy of the maximum likelihood estimators. An example of the appli-cation of this method is given.     

Partial avoidance of jump resonance in feedback systems†

For a specified maximum value of input amplitude, partial avoidance of jump resonance in the frequency response of certain non-linear feedback systems can be achieved by shaping the open-loop frequency response of the linear part of the system. The procedure is illustrated with an example of feedback amplifier circuit.     

Solution of non-linear optimization problems in power systems†

The application of differential dynamic programming or hybrid quasilinearization technique to the solution of non-linear optimization problems in power systems has encountered the problem of computational instability, particularly in higher order systems. This paper describes the application of a continuation procedure to alleviate this difficulty. Sixth order non-linear systems have been optimized with and without constraints on control variables. Both open-loop and, for the first time, closed-loop systems including both exciter and governor dynamics, are analysed. The studies presented show that this technique is quite effective in obtaining accurate solutions for non-linear boundary-value-problems in power systems.     

Determination of backscatter sources in surface type targets†

A high-resolution X-band FM-CW radar system has been used to investigate the sources of backscatter in various types of crops, trees and surfaces. This system provided a range resolution in air of 6·67 cm and gave a 16 cm diameter illumination area at the target. The surface targets included concrete walkways, grass, soil and asphalt pavements. Although the surfaces of these targets were the primary sources of backscatter, some subsurface boundaries were located and a volume scatter contribution was evident for some targets.     

Sensitivity in the decoupling of non-linear control systems†

The problem of decoupling the inputs and outputs of non-linear control systems is considered for the situation where the system parameters are subject to variations or disturbances. The exact decoupling is obtained by an adaptive state feedback controller which requires full knowledge of the instantaneous values of the system parameters. By utilizing the parameter sensitivity concept an approximate adaptive decoupling controller is synthesized in the present paper which is shown to decouple the system to first (or higher) order in the variations of the system parameters from their known nominal values. The sensitivity equations which generate the state sensitivity function required for the synthesis are derived. The results of the paper actually provide the tools for reducing the system sensitivity to parameter variations when the final goal is the improvement of decoupling.     

Multistage estimation of bias states in linear systems†

This paper provides an alternative, constructive derivation of Friedland's (1966) method for recursive bias filtering ; and, extends his method to the case where we may wish to increase (or decrease) the number of biases. We show that it is possible to add (or delete) bias states in such a manner that previously computed quantities can be used to obtain new estimates of the dynamical state vector and the now larger bias vector. Adding (or deleting) bias states is important when, for example, the bias states are used to model constant but unknown instrumentation error sources, of which there can be a large number.     

Functional relationships between the conventional steady-state error characteristics and the weighting matrices in the quadratic performance index†

The requirements for achieving a desired value of steady-state error for an optimal control system utilizing the quadratic performance index with a linear plant arc derived for step, ramp and parabolic inputs. These requirements result in a number of the elements of the Riccati matrix being specified in terms of known parameters. It is shown that for an all-pole plant the control system cannot follow a parabolic input. For the stop input the q ₁₁ element of the Q matrix is specified as one for an all-pole plant or in terms of the fixed plant parameters for a pole-zero plant. Depending on the number of requirements and the order of the system it may be necessary to assume values for the remaining elements of the Q matrix in order to solve the reduced Riccati equation.     

Application of mathematical programming in the design of optimal control systems†

The problem of applying the various computational methods of mathematical programming in the design of an optimal control system is discussed. A general case of non-linear, non-autonomous, state equations, subject to inequality constraints on both state and control variables, is considered. Both continuous and discrete time systems are investigated. In case of discrete time systems, the sampling intervals are assumed generally unequal and aperiodic, with inequality constraints imposed upon them.

Systems like these impose considerable computational difficulties when treated by the maximum principle or dynamie programming. Using mathematical programming, one may simplify a wide class of those computational problems.

Several examples of applying mathematical programming to particular control problems are presented.     

H∞ interpolation of rational matrices

The problem of minimizing the L^∞-norm of some stable rational matrix E(s) subject to two basic types of matrix interpolation constraints is considered: given anti-stable rational matrices N¯(s) and M¯(s), the interpolating matrix £(s) is required to satisfy either the condition N¯(s)E(s) = M¯(s) or the condition that N¯(s)E(s) has an unstable projection equal to M¯(s). These two kinds of constraints are directly related to model-matching problems arising from H^∞ optimal control. Specific conditions on N¯(s) and M¯(s) for the interpolation problems to be well-posed are discussed and closed-form characterizations for both optimal and suboptimal solutions in E(s) are provided. The analysis is based on a state-space setting and the results are suitable for computational purposes.     

Expanded subalphabets in the theories of languages and semigroups†

A new type of code, called an expanded subalphabet, is introduced. It is shown that the following four conditions on a subset L of the free monoid A ^? over a finite or infinite alphabet A are equivalent: (1) L is the submonoid generated by an expanded subalphabet; (2) L is a retract; (3) L is the fixed language of an endomorphism; (4) L is the stationary language of an endomorphism. Expanded subalphabets are used as a tool for the investigation of fixed languages (=retracts). Special results for the case of a finite alphabet are given and the relationship with the theory of L-systems is indicated.