Controller Synthesis subject to Logical and Structural Constraints: A Satisfiability Modulo Theories (SMT) Approach

We report on a simple approach to use satisfiability modulo theories (SMT) solvers to synthesize stabilizing controllers subject to logical and structural constraints. Examples of logical/structural specifications allowed by our methodology include the transitive property of the connectivity of a networked system, and the mutually exclusive use of inputs or sensors, to name a few. The aforementioned structural constraints can also impose the sparsity pattern and linear dependency restrictions prevailing in the decentralized control literature. The main goal of this article is to discuss preliminary results and examples in which both the plant and the controller are linear time-invariant (LTI). Our approach consists of encoding the stability conditions as well as the logical and structural constraints as an SMT instance. We illustrate our methodology on two classes of problems: (i) full state feedback design for positive systems, with applications to combination drug therapy and transportation network design, and (ii) static output feedback (SOF) design. The article includes numerical examples for each of these applications computed using a freely available SMT solver. It is noteworthy that the examples of positive systems mentioned above, in particular, can be solved in less than four minutes even when the dimension of the state is one thousand.