Equivalence transformations in linear systems theory

There is growing interest in infinite frequency structure of linear systems, and transformations preserving this type of structure. Most work has been centred around Generalised State Space (GSS) systems. Two constant equivalence transformations for such systems are Rosenbrock's Restricted System Equivalence (RSE) and Verghese's Strong Equivalence (str.eq.). Both preserve finite and infinite frequency system structure. RSE is over restrictive in that it is constrained to act between systems of the same dimension. While overcoming this basic difficulty str.eq. on the other hand has no closed form description. In this work all these difficulties have been overcome. A constant pencil transformation termed Complete Equivalence (CE) is proposed, this preserves finite elementary divisors and non-unity infinite elementary divisors. Applied to GSS systems CE yields Complete System Equivalence (CSE) which is shown to be a closed form description of str.eq. and is more general than RSE as it relates systems of different dimensions. Equivalence can be described in terms of mappings of the solution sets of the describing differential equations together with mappings of the constrained initial conditions. This provides a conceptually pleasing definition of equivalence. The new equivalence is termed Fundamental Equivalence (FE) and CSE is shown to be a matrix characterisation of it. A polynomial system matrix transformation termed Full Equivalence (fll.e.) is proposed. This relates general matrix polynomials of different dimensions while preserving finite and infinite frequency structure. A definition of infinite zeros is also proposed along with a generalisation of the concept of infinite elementary divisors (IED) from matrix pencils to general polynomial matrices. The IED provide an additional method of dealing with infinite zeros

Pole and zero placement in multivariable control systems

A method is proposed for designing multivariable systems based on an alternate derivation of Davison's theorem on pole placement and the solution of the nonlinear equations for the feedback gains by the least square error method. Output feedback is used to control a complex dynamical system. The freedom in design, after allocating poles, is used to place zeros and/or satisfy other design objectives. This method results in algorithms which are computationally attractive. However, this is done at a considerable sacrifice in terms of the design freedom available. For a system with m inputs and p outputs only m + p variables are available instead of mp variables

Research on optimal control, stabilization and computational algorithms for aerospace applications

The research carried out in the areas of optimal control and estimation theory and its applications under this grant is reviewed. A listing of the 257 publications that document the research results is presented

Finite settling time stabilization for linear multivariable time-invariant discrete-time systems: An algebraic approach

The problem of Total Finite Settling Time Stabilization of linear time-invariant discrete-time systems is investigated in this thesis. This problem falls within the same area of the well-known deadbeat (time-optimal) control and in particular, constitutes a generalization of this problem. That is, instead of seeking time-optimum performance, it is required that all internal and external variables (signals) of the closed-loop system settle to a new steady state after a finite time from the application of a step change to any of its inputs and for every initial condition. The state/output deadbeat control is a special case of the Total FSTS problem. Using a mathematical and system theory framework based on sequences and the polynomial equation (algebraic) approach, we are able to tackle the FSTS problem in a unifying manner. The one-parameter (unity) feedback configuration is mainly used for the solution of the FSTS problem and FSTS related control strategies. The whole problem is reduced to the solution of a polynomial matrix Diophantine equation which guarantees not only internal stability but also internal FSTS and is further reduced to the solution of a linear algebra problem over R. This approach enables the parametrizat ion of the family of all FSTS controllers, as well as those which are causal, in a Youla-Bongiorno-Kucera type parametrization. The minimal McMillan degree FSTS problem is completely solved for vector plants and a parametrization of the FSTS controllers according to their McMillan degree is obtained. In the MIMO case bounds of the minimum McMillan degree controllers are derived and families of FSTS controllers with given lower/upper McMillan degree bounds are provided in parametric form. Having parametrized the family of all FSTS controllers, the state deadbeat regulation is treated as a special case of FSTS and complete parametrization of all the deadbeat regulators is presented. In addition, further performance criteria, or design constraints are imposed such as, FSTS tracking and/or disturbance rejection, partial assignment of controller dynamics, l1-, l∞-optimization and robustness to plant parameter variations. Finally, the Simultaneous-FSTS problem is formulated, and necessary as well as sufficient conditions for its solution are derived. Also, a two-parameter control scheme is introduced to alleviate some of the drawbacks of the one-parameter control. A parametrization of the family of FSTS controllers as well as the FSTS controllers for tracking and/or disturbance rejection is given as an illustration of the particular advantages of the two-parameter FSTS controllers

Covariance and Gramian matrices in control and systems theory.

Covariance and Gramian matrices in control and systems theory and pattern recognition are studied in the context of reduction of dimensionality and hence complexity of large-scale systems. This is achieved by the removal of redundant or 'almost' redundant information contained in the covariance and Grarrdan matrices. The Karhunen-Loeve expansion (principal component analysis) and its extensions and the singular value decomposition of matrices provide the framework for the work presented in the thesis. The results given for linear dynamical systems are based on controllability and observability Gramians and some new developments in singular perturbational analysis are also presented

Safety system design optimisation

This thesis investigates the efficiency of a design optimisation scheme that is appropriate for systems which require a high likelihood of functioning on demand. Traditional approaches to the design of safety critical systems follow the preliminary design, analysis, appraisal and redesign stages until what is regarded as an acceptable design is achieved. For safety systems whose failure could result in loss of life it is imperative that the best use of the available resources is made and a system which is optimal, not just adequate, is produced. The object of the design optimisation problem is to minimise system unavailability through manipulation of the design variables, such that limitations placed on them by constraints are not violated. Commonly, with mathematical optimisation problem; there will be an explicit objective function which defines how the characteristic to be minimised is related to the variables. As regards the safety system problem, an explicit objective function cannot be formulated, and as such, system performance is assessed using the fault tree method. By the use of house events a single fault tree is constructed to represent the failure causes of each potential design to overcome the time consuming task of constructing a fault tree for each design investigated during the optimisation procedure. Once the fault tree has been constructed for the design in question it is converted to a BDD for analysis. A genetic algorithm is first employed to perform the system optimisation, where the practicality of this approach is demonstrated initially through application to a High-Integrity Protection System (HIPS) and subsequently a more complex Firewater Deluge System (FDS). An alternative optimisation scheme achieves the final design specification by solving a sequence of optimisation problems. Each of these problems are defined by assuming some form of the objective function and specifying a sub-region of the design space over which this function will be representative of the system unavailability. The thesis concludes with attention to various optimisation techniques, which possess features able to address difficulties in the optimisation of safety critical systems. Specifically, consideration is given to the use of a statistically designed experiment and a logical search approach