The implementation issues of digital controllers with finite word length (FWL) considerations are addressed. Both the shift and delta operator parameterisations of a general controller structure are considered. A unified formulation is adopted to derive a computationally tractable stability related measure that describes FWL closed-loop stability characteristics of different controller realisations. Within a given operator parameterisation, the optimal FWL controller realisation, which maximises the proposed stability related measure, is the solution of a nonlinear optimisation problem. The relationship between the z-operator and d-operator controller parameterisations is analysed, and it is shown that the d parameterisation has a better FWL closedloop stability margin than the z-domain approach under a mild condition. A design example is included to verify the theoretical analysis and to illustrate the proposed optimisation procedure.

The scalar mixed H2=l1 problem for discrete-time systems is considered. The continuity property of the optimal value with respect to changes in the l1 constraint is studied. An upper approximation method and a lower approximation method of the optimal value are given. Suboptimal values and superoptimal values of the problem can be obtained by solving a sequence of finite-dimensional quadratic programming problems.

The closed-loop stability issue of finite word length (FWL) realizations is investigated for digital controllers implemented in floating-point arithmetic. Unlike the existing methods which only address the effect of the mantissa bits in floatingpoint implementation to the sensitivity of closed-loop stability, the sensitivity of closed-loop stability is analysed with respect to both the mantissa and exponent bits of floating-point implementation. A computationally tractable FWL closed-loop stability measure is then defined, and the method of computing the value of this measure is given. The optimal controller realization problem is posed as searching for a floating-point realization that maximizes the proposed FWL closed-loop stability measure, and a numerical optimization technique is adopted to solve for the resulting optimization problem. Simulation results show that the proposed design procedure yields computationally efficient controller realizations with enhanced FWL closed-loop stability performance.

This note presents a method for the reduction of controller fragility. The method is based on the sensitivity of closed-loop poles to perturbations in the controller parameters. By means of a state space parameterization of the controller, the closed-loop pole sensitivity can be reduced. A controller fragility measure based on the closed-loop pole sensitivity is proposed. Conditions for the optimal state-space realization of the controller are presented, along with a numerical method for obtaining the solution.

The effect ofroundoffnoise in a digital controller is analysed for a sampled-data system in which the digital controller is implemented in a state-space realisation. A new measure, called averaged roundoffnoise gain, is derived. Unlike the traditionally used measure, where the analysis is performed based on an equivalent digital control system, this newly defined averaged roundoff noise gain allows one to take consideration of the inter-sample behaviour. It is shown that this measure is a function of the state-space realisation. Noting the fact that the state-space realisationsof a digital controller are not unique, the problem of optimum controller structure is to identifythose realisations that minimise the averaged roundoff noise gain subject to the 2-scaling onstraint which is for preventing the signals in the controller from overflow. An analytical solution to the problem is presented and a design example is given. Both theoretical analysis and simulation results show that the optimum controller realisations obtained with the proposed approach are superior to those obtained with the traditional analysis based on a digital control system.