This paper proposes a framework for automatic formal controller synthesis for general hybrid systems with a subset of safety and reachability specifications. The framework uses genetic programming to automatically co-synthesize controllers and candidate Lyapunov-like functions. These candidate Lyapunov-like functions are used to formally verify the control specification, and their correctness is proven using a Satisfiability Modulo Theories solver. The advantages of this approach are: no restriction is made to polynomial systems, the synthesized controllers are expressed as compact expressions, and no explicit solution structure has to be specified beforehand. We demonstrate the effectiveness of the proposed framework in several case studies, including nonpolynomial systems, sampled-data systems, systems with bounded uncertainties, switched systems, and systems with jumps.
This paper presents an automatic formal controller synthesis method for nonlinear sampled-data systems with safety and reachability specifications. Fundamentally, the presented method is not restricted to polynomial systems and controllers. We consider periodically switched controllers based on a Control Lyapunov Barrier-like functions. The proposed method utilizes genetic programming to synthesize these functions as well as the controller modes. Correctness of the controller are subsequently verified by means of a Satisfiability Modulo Theories solver. Effectiveness of the proposed methodology is demonstrated on multiple systems.
Controller synthesis techniques based on symbolic abstractions appeal by producing correct-by-design controllers, under intricate behavioural constraints. Yet, being relations between abstract states and inputs, such controllers are immense in size, which makes them futile for embedded platforms. Control-synthesis tools such as PESSOA, SCOTS, and CoSyMA tackle the problem by storing controllers as binary decision diagrams (BDDs). However, due to redundantly keeping multiple inputs per-state, the resulting controllers are still too large. In this work, we first show that choosing an optimal controller determinization is an NPcomplete problem. Further, we consider the previously known controller determinization technique and discuss its weaknesses. We suggest several new approaches to the problem, based on greedy algorithms, symbolic regression, and (muli-terminal) BDDs. Finally, we empirically compare the techniques and show that some of the new algorithms can produce up to ≈ 85% smaller controllers than those obtained with the previous technique.
Preliminaries
Minimum set coverThe minimum set cover problem (MSC) is formulated as:Problem 2.1 (MSC). Given a set X and a cover {S j } j∈I , i.e. X ⊆ j∈I S j , where |X|, |I| < ∞, find the smallest subcover I * ⊆ I : X ⊆ j∈I * S j .Both, the decision and selection versions of (MSC, are known to be NPcomplete. The first approximate poly-nomial-time solution for MSC was given by [12]. Later, [5] suggested an approximate poly-nomial-time solution for the generalized "minimum set weight cover problem" (MWSC); which extends MSC by that each set S k is assigned a weight s k ≥ 0 and the question is to find the smallest sub-cover with the minimum total weight. According to [6], the Chvátal's algorithm time complexity is: O (|I| · |X| · min (|I|, |X|)).1 Instead of storing the control law as an explicit map, we search for a symbolic function that for a given state computes the input value.2 Up to the found optimal BDD variable reordering.
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