Multirate Design Research Articles

Rational sampling rate conversion structures with minimum delay requirements

Recent developments on multirate system design show that a rational (L/M) sampling rate conversion system can be reconfigured into a three-dimensional polyphase structure. However, the delay requirements associated with each subfilter path in such a structure have not sufficiently been considered. The paper presents a detailed analysis of the delay requirements, and shows that a reduction by a factor of L on the number of data stores can be achieved. A design approach is introduced for efficient structures based on the minimisation of the total number of delay units. Two transformation rules are derived which directly lead to a homogeneous system configuration. This approach is general and simple so that any rational sampling rate conversion system can be specified by a set of parameters calculated by a small computer program. A hardware implementation is also presented, which demonstrates the usefulness of the present approach. >

Multirate control: A new approach

A new multirate digital control of linear periodic and time-invariant systems is proposed. An explicit multirate law is given which transforms the overall control design into that of a relatively small-dimensional discrete time-invariant system for which standard design techniques can be used. It is shown that the proposed multirate scheme is, in general, more effective as a design tool than current multirate designs. The controllability properties of the multirate system are studied. Consequently, necessary and sufficient conditions for pole placement and stabilizability of the multirate system are given in terms of the continuous plant matrices. Moreover, the multirate LQR problem is solved. In this case, the solution is obtained by solving an algebraic discrete Riccati equation. Here, necessary and sufficient conditions for the existence of a unique nonnegative-definite solution of the Riccati equation and the asymptotic stability of the corresponding closed-loop multirate system are obtained. These conditions are given in terms of the continuous-time parameters.

A time invariant approach to multirate optimal regulator design

The scope of this paper is to develop a methodology for a time invariant, single rate, approach to optimal multirate regulator design. The proposed method is developed employing a time invariant expanded discrete time model for the multirate system capable of providing all the information between sampling instants needed to compute the corresponding control weighting matrices for each base rate interval and then to accumulate all the control effort at the last base rate sampling interval for which all sampling switches are synchronized. In this way, the design of an optimal multirate regulator reduces to an equivalent single rate design of the same dimensionality as the original sampled system

A new optimal multirate control of linear periodic and time-invariant systems

The optimal multirate design of linear, continuous-time, periodic and time-invariant systems is considered. It is based on solving the continuous linear quadratic regulation (LQR) problem with the control being constrained to a certain piecewise constant feedback. Necessary and sufficient conditions for the asymptotic stability of the resulting closed-loop system are given. An explicit multirate feedback law that requires the solution of an algebraic discrete Riccati equation is presented. Such control is simple and can be easily implemented by digital computers. When applied to linear time-invariant systems, multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly. Compared to the classical single-rate sampled-data feedback in which the state is always sampled at the same rate, the multirate system can provide a better response with a considerable reduction in the optimal cost. In general, the multirate scheme offers more flexibility in choosing the sampling rates.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multirate observer design via singular perturbation theory

Abstract In this paper multirate observer design is considered via singular perturbation theory. Following a discussion of the design of single-rate observers, i.e. fast- and slow-sampling observers, multirate observer design is developed within the framework of a decomposition-coordination principle. Problems of asymptotic stability and the responses of an error system are investigated, and the relation between single-rate and multirate observers is clarified.

A New Technique for Multirate Digital Control Design and Sample Rate Selection

A new approach for designing multirate digital flight control systems based on optimal control theory is described. The error rejection properties of control systems designed by the, new technique are investigated through a case study. The example considered is the F-14 aircraft controlled by a multirate proportional-plusintegral controller. The aircraft/controller system is flown through a turbulent atmosphere; a covariance analysis of state variable errors determines the ability of the controller to reject turbulence-induced errors. The controller sample rate and control sequence are varied to determine their influence on the disturbance rejection properties of the controller. Finally, a sample rate optimization scheme based on performance/c omputation tradeoff is presented. HE practical need for multiple sample rate flight control systems is a consequence of the finite computing capabilities of onboard digital computers and the multiple information rates of onboard instruments. Although digital computer technology continues to advance significantly, new and expanded software requirements for such functions as navigation, display, and control manage to keep pace with improvements in computational capability. Accordingly, the flight control system designer is always allocated a fixed (and usually limiting) amount of computational capability to implement a control design. In modern flight control system applications, the problem of implementing a desired algorithm within a limited computational capability is compounded by a need to accommodate high-frequency effects such as vehicle flexibility. Functions associated with bending effects (i.e., instrument output filtering or active structural control) may demand sample rates an order of magnitude higher than is necessary for suitable control of rigid body modes of vehicle motion. Faced with widely varying sample rate requirements among the dynamic modes of the vehicle; a multirate control structure is the solution to computational limitations and multiple information-rates. Synthesizing a multirate control system to meet desired specifications has been a difficult task. Ad hoc approaches have typically been used in which a suitable analog design is converted to a digital design via Tustin transform techniques. In this case multirate designs are generated by a trial-anderror process of running the low bandwidth compensating elements at low sample rates and evaluating the resulting system performance via simulation. Classically based techniques for multirate system analysis are available l~3; however, these techniques are limited in application due to the significant growth of dimensionalit y that they entail. Such techniques are also analysis, as opposed to synthesis, approaches; hence creating a control design is still an iterative procedure. In the present paper, a new multirate design technique4'5 based on optimal control is described which obviates dimensionality problems characteristic of classical techniques. In addition, this technique offers a systematic method for converting a desired analog control design to an equivalent multirate digital design without approximation.

Simplification of the characteristic equation of multirate sampled data control systems

A method of simplifying the characteristic equation of two classes of linear time invariant multirate sampled data systems to a form similar to that of the analogous single-rate system is developed. The resulting equations are useful for multirate system design.