Fuzzy control was originally designed to control systems whose state s can be described by several parameters a, ..., b. Often, for such a system, there exist reasonable control strategies u(a,b), ..., v(a,b), each of which work for some states. Experts can express when each strategy works: “if a is small, ..., and b is small, then u(a,b)”; “if a is large, ..., and b is medium, then v(a,b)”; and so on.

Fuzzy control formalizes the meaning of “small,” “medium,” and so on, and explains how to properly interpolate the strategies u(a,b), ..., v(a,b) into a single control strategy that is applicable for all possible combinations of a, ..., b. Similar rules describe how, under different controls, the state s(t) of the system changes with time t.

Many practical systems are spatially distributed. This paper considers the simplest case of 1D spatial systems (such as a multi-layer heater) in which the state s depends on the spatial coordinate x. For such a system, the state s(x,t+1), at the next moment of time t+1, depends not only on previous state s(x,t), control u(x,t), and external effects w(x,t), but also on the nearby state s(x-1,t) and s(x-1,t+1) (for example, when a hot surface at x-1 heats the surface at x). In principle, we can list the rules corresponding to all possible combinations of values (small, medium, and so on) of s(x,t), u(x,t), w(x,t), s(x-1,t), and s(x-1,t+1). However, there are so many such combinations that this approach is not very feasible. The authors show that for many practical systems, both to describe the system dynamics and to describe an appropriate control, it is sufficient to enumerate conditions based only on s(x,t), s(x-1,t), s(x-1,t+1). This fact drastically decreases the number of rules while retaining the ability to have a stable controller.