Algorithmic Techniques in Estimation and Control for Multirate Sampled Digital Control Systems

Ioannis S. Apostolakis    Department of Electrical & Systems Engineering, University of Connecticut, U-157 Storrs, CT 06269*
* The author is currently with ICONICS, Inc., 100 Foxborough Blvd, Foxborough, MA 02035

I INTRODUCTION

Multirate sampled data systems were first introduced in the 1950’s by the pioneering works of Kranc [1], Jury [2]-[3], Kalman et al. [4], Sklansky et al. [5]. Since then, they did not receive much attention until the last 15 years when the number of researchers in the field of multirate systems increased significantly. One of the reasons for the increase in interest is that multirate sampled data systems operate with reduced sampling information which, in turn, reduces the implementation cost and frees valuable processor time from A/D and D/A conversions without significant loss of performance. On the other hand, multirate controllers are a special class of time varying controllers, which have been shown in the past by several authors to improve performance and robustness properties [6]. Today, multirate controllers are employed in a variety of applications, with flight control systems taking the lead [7],[8]. Multirate sampled data systems may be categorized based on their sampling structure into two basic categories :

• Synchronous multirate sampled data systems

• Nonsynchronous multirate sampled data systems

In a synchronous multirate system, all sampling intervals are considered to be integer multiples of a base rate sampling interval, which in turn is the least common divisor of all. As such, all sampling devices are synchronized at a specific sampling interval called the model rate sampling interval. Consequently, the sampling structure is periodic with the period equal to the model rate sampling interval. In a nonsynchronous multirate sampled data system, all sampling devices can operate asynchronously. In this presentation, we will be considering synchronous multirate sampled data systems mainly because of the advantages their implementation and performance evaluation posseses.

The problem of designing a state estimator and control law for multirate systems for the general stochastic case is closely related to the modeling approach employed in multirate systems. In general, we can distinguish between time varying and time invariant modeling approaches for multirate systems. In the past Amit et al. [5], Glasson [7],[8], and Berg et al. [10], developed LQG designs for multirate systems utilizing a time varying modeling approach. Recently, Al-Rahmani et al. [11] presented a time invariant formulation of the multirate optimal control problem. Apostolakis et al. [12],[13] proposed a time invariant approach to optimal control design of multirate systems which utilizes periodically time varying control gains and requires knowledge of the state of the system only when all sampling devices within the system are synchronized. Another problem is that even in the case that the continuous time process and measurement noises are uncorrelated, the resulting multirate discrete expanded process and measurement noises are correlated. The last issue is the causality of the multirate estimator and conroller implementation. A feasible multirate estimator design should satisfy the causality condition.

To overcome the above difficulties we will develop a model rate estimation and control law which will estimate the state of the multirate system when all sampling devices within the system are synchronized while ensuring convergence between synchronization instants. In addition, the state dimension of the above controller will be equal to the state dimension of the original sampled system and the proposed structure will also ensure the causality of our design.

II MULTIRATE SYSTEM MODELING

The multirate system model we will develop is related to the model presented by Araki [14] and then expanded by Godbout et al. [15] to include a more general sampling structure. We first consider the continuous plant which may include any filter or sensor dynamics appended to it. We assume that the plant control inputs are preceded by zero order hold (ZOH) devices. The multirate system will be modeled as a single rate, time invariant system capable of providing all information between sampling instants. Let us consider the continuous time system depicted in Figure 1. We assume that the plant has (p) outputs and (l) control inputs. Let (n) be the order of the plant. We define T to be the base rate sampling interval of our system. The base rate sampling interval T can be thought of as being the sampling interval of a single rate system discretized at T. The sampling intervals at the system’s outputs ¯t and control inputs ¯t are assumed to be integer multiples of the base rate sampling interval (T) with integer multiplicities qyj, quj. Let Tyj, Tuj represent the sampling intervals at each system output yj and control input uj respectively and let (q) denote the least common multiple of all the qyj, quj defined previously.

Let us define as To the model rate sampling interval for which all the sampling devices are synchronized. That is :

o=qT

with

yj=qyjTyj,Tuj=qujTuj

The multirate system will be modeled as a single rate, shift invariant system at the model rate sampling interval To. Let the continuous time plant have the state-space representation :

¯t=Apx¯t+Bpu¯t+Hpξ¯t

¯t=Cx¯t+Du¯t+ω¯t

where

¯t~N0¯Ξp,Ξp≥0

¯t~N0¯Ωp,Ωp>0

We assume that both process noise ¯t and measurement noise ¯t are zero mean white random processes and uncorrelated, that is ξ¯ω¯T=0. In the above equations, Ξp, Ωp are the process noise and measurement noise covariance matrices respectively and they are assumed to be positive (semi) definite and positive definite. We also assume that the noise correlation times are shorter than the base rate sampling interval (T). The discretized plant equations at the base rate sampling interval T are given by :

¯k+1T=Ax¯kT+Bu¯kT+Hξ¯kT

¯kT=Cx¯kT+Du¯kT+ωkT

where, the discretized system matrices are given by:

=expApT,B=∫0TexpAptBpdτ,H=∫0TexpAptHpdτ

In addition to the above we have that the base rate discretized noise characteristics of the plant are given by

¯kT~N0¯ΞB,ΞB≥0

¯kT~N0¯ΩB,ΩB>0

and, as before, Eξ¯ω¯T=0. The noise covariance matrices discretized at the base rate (T) are given by the equations [16]

B≅ΞPT,ΩB≅ΩpT

Note that for the previous equations to hold, we have assumed that the base rate sampling interval (T) is small compared to the system’s time constants. The above base rate model of the plant will be the building block in mapping a single rate system to a multirate one. The multirate system model will be developed at the model rate interval (To) and will include all state, input and output information for each base rate interval (T) within the model rate interval (To). Note that the expanded multirate model in general is a ficticious model which will be utilized in designing the multirate estimator. The expanded discrete time representation of the plant is given by :

¯Ek+1To]=AEx¯EkTo+BEu¯EkTo+HEξ¯EkTo

¯EkTo=CE1x¯EkTo+CE2x¯Ek+1To+DEu¯EkTo+ω¯EkTo

The matrices AE, BE, CE1, CE2, DE, HE are built from the base rate state-space realization of the plant equations. The effective order of the expanded discrete time model for the plant is (n * q) with state vector ¯EkTo while the control input vector...