Dynamics and Control 2, 113-129 (1992) © 1992 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Vector-Channel Lattice Filters for the Delta Operator: Complete Derivation FARYAR JABBARI Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92717 Received May 22, 1990; revised November 12, 1990, August 29, 1991 Editor: H. Stalford Abstract. The numerical difficulties associated with high sampling rates have been the main motivationfor the recent interest in application of delta-operator formulation in adaptive identification and control techniques, as an alternativeto the traditional shift operator. In this article, the derivationof the vector-channellattice algorithm for least-squares parameter estimation of an input/output model based on the delta operator is presented. Since the vector-channelformulationis a generalizationof the standard mtdtichannelforms, results presented here apply to the multichannel lattices as a special case. In addition to the basic derivation, algorithms for residual errors and autoregressive coefficients, a simple example, and issues regarding implementation are discussed. 1. Introduction In this article, the complete derivation of the vector-channel lattice filters for the delta- operator-based least-squares estimation is presented. The main motivation is the combina- tion of superior numerical characteristics of lattices with advantages of the delta-operator approach. Furthermore, lattices allow order-recursive implementation, which provides the opportunity for on-line order identification and/or change of order. The derivation here treats the vector-channel case in which there could bep measurement channels, each with m independent measurements. The vector-channel structure forces the sensors (or measurements) in each channel to "see" the same dynamics (see [31 and [81 for details). The vector-channel structure is a generalization of the standard multichannel forms. In- deed, by setting m = 1 in the derivation, the standard least-squares lattice is obtained. In [4], the derivation for a single scalar-channel delta lattice is presented. Though the derivation in that reference captures the essence of the delta lattice, it is not suited for con- trol and identification applications, since even a SISO system requires a two-channel struc- ture, one for the input and one for the output. This article presents the complete derivation for the vector-channel lattice filter, which reduces to the standard multichannel form as a special case. Algorithms for the residual filters and the AR coefficients are included also. The resulting algorithms can be used for MIMO system as a special case (i.e., m = 1). A simple example and some details regarding the embedding technique and other implemen- tation issues are provided in Section 2.4. Since the main contribution of this article is the derivation itself, much of the discussion on advantages of the lattices and delta-operator approach are left to references such as [3] and [6]. In the following, however, the motivation behind this work is briefly outlined.