Yet Another Discrete-time H ∞ Fixed-lag Smoothing Solution

A solution of the discrete H ∞ smoothing problem is presented that lends itself well for generalization to sampled-data problems. The solution is complete and requires two sign-definite spectral factoriza-tions and one Nehari extension problem. A state-space equivalent is derived that is believed to be minimal. w e y v u H G v G y S-(a) w e y v u K G v G y-(b) FIG. 1.1. (a) Hold design. (b) Discrete filter design 1. Introduction. The H ∞ problem hardly needs an introduction, and one might argue that it definitely needs no other solution. However, for certain sampled data problems , matters are still not settled. One case (and the motivation for this work) is shown in Fig. 1.1(a). Here the game is to design a stable hold H that renders the L ∞-norm of the error system G v − H SG y less than some given bound. For causal holds this problem is elegantly solved in [10] using the machinery of systems with jumps but if the hold H is allowed a given amount of preview then [8] is the first solution. The sampled-data solution put forward in [8] is, largely, a careful translation to sampled-data systems of a pure discrete H ∞ fixed-lag smoothing solution. It is this discrete solution that we present in this note. The discrete fixed-lag smoothing problem that we consider is depicted in Fig. 1.1(b). Here G v and G y are given discrete and causal LTI systems and the problem is to find a filter K that is stable and causal up to some given degree of preview and that renders the L ∞-norm of the error system G e := G v − K G y smaller than a given bound γ. A preview of means that the impulse response k[n] of K is zero for discrete time less than −, say