Abstract
KeYmaeraX is a Hoare-style theorem prover for hybrid systems. A hybrid system can be seen as an aggregation of both discrete and continuous variables, whose values can change abruptly or continuously, respectively. KeYmaeraX supports only variables having the primitive type bool or real. Due to the mixture of discrete and continuous system elements, one promising application area for KeYmaeraX are closed-loop control systems. A closed-loop control system consists of a plant and a controller. While the plant is basically an aggregation of continuous variables whose values change over time accordingly to physical laws, the controller can be seen as an algorithm formulated in a classical programming language. In this paper, we review some recent extensions of the proof calculus applied by KeYmaeraX that make formal proofs on the stability of dynamic systems more feasible. Based on an example, we first introduce to the topic and prove asymptotic stability of a given system in a hand-written mathematical style. This approach is then compared with a formal encoding of the problem and a formal proof established in KeYmaeraX. We also discuss open problems such as the formalization of asymptotic stability.
Highlights
To formulate asymptotic stability, one would need to encode a situation that is far in the future ( →∞) as asymptotic stability means that there will be a point in time, a er which the system will always remain within an region
A stability proof for switched system o en requires to nd more sophisticated Lyapunov functions taken all di erent system modes into account
We report on experiences we gained when merging veri cation techniques from two engineering disciplines: control theory and so ware engineering
Summary
Given is a periodic linear time invariant (LTI) dynamic system in state space form [10]. One can clearly see that this system is asymptotic stable for the given initial condition, i.e. lim. . By quadrant switching, two periodic systems are joined to form an asymptotically stable system. 1 is enforced in the right-upper and le -lower quadrants (note that h( ) ≥ 0 holds) to have a greater value change for 2 than for 1. Whenever the trajectory of 1 crosses the diagram axes ( 1 = 0 or 2 = 0), the non-zero value of the coordinate (alternating 1, 2) form a monotonically decreasing series. To sum up, combining two periodic systems 1, 2 to a switched system can result both in an asymptotic stable system. 2. e di erence in the de nition of 2 is rather marginal. erefore, it would be very helpful to have a veri cation tool able to check formally, whether the resulting system is stable or not
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