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.

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.

The effect of roundoff noise in a digital controller is analysed for a digital feedback control system. An analytical expression for the roundoff noise gain, defined as the ratio between the variances of the output error and the rounding error, is obtained. The problem of identifying the minimum roundoff noise realisations can be solved using an existing procedure. Noting that the optimal realisations are fully parametrised, based on a polynomial operator approach a new sparse controller realisation is derived. This realisation is a generalisation of the direct forms in the classical shift operator and the prevailing delta-operator. It provides more degrees of freedom with which to reduce the roundoff noise. The problem of finding optimal polynomial operators can be solved by an exhaustive search, and a design example is given. It is shown that with the proposed sparse realisation the optimal polynomial operators can outperform the shift- and delta-operators.

The pole-sensitivity approach is employed to investigate the stability issue of the discrete-time control system, where a digital controller, implemented with finiteword length (FWL), is used. A new stability related measure is derived, which is more accurate in estimating the closed-loop stability robustness of an FWL implemented controller than some existing measures for the pole-sensitivity analysis. This improved stability measure thus provides a better criterion to find the optimal realizations for a generic controller structure that includes output-feedback and observer-based controllers. A numerical example is used to verify the theoretical analysis and to illustrate the design procedure.

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.