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UBC Theses and Dissertations

Application of eigenvalue techniques in the analysis : compensation and synthesis of linear multivariable feedback control systems Kasvand, Tonis


Systematic numerical and graphical techniques have been developed for analysing,and to some extent synthesizing,linear or linearized multiple input-output (multivariable) feedback control systems. A class of multivariable systems is studied which can be considered to consist of two parts. The first part of the system consists of the system configuration and certain elements characteristic to the system, i.e. the interaction or the system dynamics, which are considered to be unalterable. The other part of the system consists of elements which the designer may modify to ensure a satisfactory system performance. These have been called the variable elements, consisting usually of power amplifying devices and compensating networks. The variable elements are represented by one common variable element. This allows the unalterable part of the system dynamics to be represented independently of the variable elements. This method is suitable for automatic high speed computing and plotting facilities, or it may serve as a guide while simulating multivariable systems on analog computers. Sets of curves can be computed which represent the critical behaviour of the unalterable part of the system, for example its instability and null response, with respect to the elements whose influence on the system is being studied (the variable elements). Thus, when the stability of the system is being studied, the corresponding critical curves behave as the -1 point in the study of single-variable control systems. Hence the concepts developed for single-variable systems, such as the Nyquist and Nichols plots and root-locus methods, may be extended to the study of multivariable systems. The curves representing system instability are the eigenvalues of the loop-matrix in the system, and those representing null response have been called "eigenzeros". Any transfer function in the system may then be represented graphically in terms of the loci of the eigenvalues and the eigenzeros (which represent the unalterable elements), and the locus representing the elements whose influence on the system is being determined (the variable elements). Analogously to single-variable systems, compensation for improved stability and frequency response consists of modifying the locus of the variable elements with respect to the loci representing the fixed part of the system. Furthermore, additional compensation techniques exist which transform the entire eigenvalue spectrum, but leave the locus of the variable elements unchanged. In the synthesis of linear multivariable feedback control systems, it is often difficult to obtain stable elements, i.e. elements with only left-half s-plane poles, when the overall system response is given. For certain classes of multivariable systems,eigenvalue techniques may be used to predict the stability of the elements in certain configurations before the values for the elements have been calculated. As a special case, the stability of two-variable systems (systems with two inputs and outputs) have been formulated so that the -1 point of single-variable systems transforms to one critical locus. Numerous examples are used to illustrate the theory and its applications.

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