Electrical and Electronic Engineering - Theses
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ItemDecentralised stabilisability of LTI control systems( 2014)Decentralised control systems comprise some local control channels; each one uses its own local control inputs/ outputs and no information exchange exists among them. Many control systems in our daily life are working based on decentralised control concepts. However, this kind of system is vulnerable to having fixed modes, which may cause significant impact on the system stability. In this thesis, we will present a comprehensive study of classifications, characterisations and eliminations of these undesirable modes including decenetralised fixed modes (DFMs) and quotient fixed modes (QFMs). It should be noted that DFMs are those modes of a LTI system, which cannot be displaced by any decentralised static or dynamic output feedback controller. However, QFMs are those modes of the system, which are not movable by any decentralised control scheme, be nonlinear or time varying controllers. In terms of classification, we derive a sufficient condition for which unstructured decentralised fixed modes (UDFMs) of a LTI system cannot be stabilised by any form of decentralised output feedback controllers. The sufficient condition establishes the link between special UDFMs (SUDFMs) and QFMs. The link between UDFMs and QFMs provides an explicit explanation as to why some UDFMs cannot be eliminated by sampled-data controllers. In addition, we use this link to highlight the role system parameter uncertainties can play in shifting stabilisable DFMs to UDMFs or QFMs and vice versa. On the other hand, to complete the existing algebraic characterisations we derive sufficient conditions and provide constructive proofs for DFMs to be/ not to be QFMs of LTI systems, and thus are not/ are stabilisable. The conditions are stated in terms of algebraic properties of the system transfer function matrix (TFM) and whether the DFMs are transmission zeros (TZs) of the quotient subsystems or not. The characterisation of DFMs proposed in this thesis offers some advantages over existing characterisations. Furthermore, the existing graph theoretic characterisation of fixed modes becomes more complete as we derive sufficient conditions for such systems with DFMs to be stabilisable regardless of whether the DFMs are distinct or repeated eigenvalues of the system. The sufficient conditions relate to whether the DFMs are QFMs of the system and thus can or cannot be shifted by any type of controller. It is shown that a DFM is a QFM when some graphical properties, expressed in terms of newly introduced concepts of imaginary inputs (outputs), auxiliary graphs, complementary cut sets, and system boxes exist. The sufficient conditions are stated in terms of some new theorems with constructive proofs. In addition, to clarify the existing uncertainty in DFMs elimination by sample data control we illuminate the reason behind an ambiguity relating to the existence of continuous LTI systems with repeated non-cyclic eigenvalues with a DFM, which may or may not be eliminated by discrete-time controllers. We identify the reason why sampling is or is not able to eliminate DFMs and provide precise conditions for elimination of DFMs by sample data control. The above findings are then used to introduce an algorithm for full decentralised decomposition of LTI control systems. The algorithm identifies a decentralised control structure that preserves the individual controllability and observability as well as strong connectedness among control channels. Such decomposition avoids introducing DFMs and QFMs and hence ensures existence of stabilisable decentralised output feedback controllers.