Hybridization for stability verification of nonlinear switched systems

We propose a novel hybridization method for stability analysis that over-approximates nonlinear dynamical systems by switched systems with linear inclusion dynamics. We observe that existing hybridization techniques for safety analysis that over-approximate nonlinear dynamical systems by switched affine inclusion dynamics and provide fixed approximation error, do not suffice for stability analysis. Hence, we propose a hybridization method that provides a state-dependent error which converges to zero as the state tends to the equilibrium point. The crux of our hybridization computation is an elegant recursive algorithm that uses partial derivatives of a given function to obtain upper and lower bound matrices for the over-approximating linear inclusion. We illustrate our method on some examples to demonstrate the application of the theory for stability analysis. In particular, our method is able to establish stability of a nonlinear system which does not admit a polynomial Lyapunov function

Robust Stability Analysis of Nonlinear Hybrid Systems

We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems

On the Trade-off Between Efficiency and Precision of Neural Abstraction

Neural abstractions have been recently introduced as formal approximations of complex, nonlinear dynamical models. They comprise a neural ODE and a certified upper bound on the error between the abstract neural network and the concrete dynamical model. So far neural abstractions have exclusively been obtained as neural networks consisting entirely of $ReLU$ activation functions, resulting in neural ODE models that have piecewise affine dynamics, and which can be equivalently interpreted as linear hybrid automata. In this work, we observe that the utility of an abstraction depends on its use: some scenarios might require coarse abstractions that are easier to analyse, whereas others might require more complex, refined abstractions. We therefore consider neural abstractions of alternative shapes, namely either piecewise constant or nonlinear non-polynomial (specifically, obtained via sigmoidal activations). We employ formal inductive synthesis procedures to generate neural abstractions that result in dynamical models with these semantics. Empirically, we demonstrate the trade-off that these different neural abstraction templates have vis-a-vis their precision and synthesis time, as well as the time required for their safety verification (done via reachability computation). We improve existing synthesis techniques to enable abstraction of higher-dimensional models, and additionally discuss the abstraction of complex neural ODEs to improve the efficiency of reachability analysis for these models.Comment: To appear at QEST 202

Formal synthesis of stabilizing controllers for periodically controlled linear switched systems

In this paper, we address the problem of synthesizing periodic switching controllers for stabilizing a family of linear systems. Our broad approach consists of constructing a finite game graph based on the family of linear systems such that every winning strategy on the game graph corresponds to a stabilizing switching controller for the family of linear systems. The construction of a (finite) game graph, the synthesis of a winning strategy and the extraction of a stabilizing controller are all computationally feasible. We illustrate our method on an example

Characterization of well-posedness of piecewise linear systems

One of the basic issues in the study of hybrid systems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. The paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Caratheodory. The concepts of jump solutions or of sliding modes are not considered here. In this sense, the problem to be discussed is one of the most basic problems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and sufficient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multimodal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterion depending on the state, we give a characterization of all admissible state feedback gains for which the closed loop system remains well-pose

Model-based compositional verification approaches and tools development for cyber-physical systems

The model-based design for embedded real-time systems utilizes the veriable reusable components and proper architectures, to deal with the verification scalability problem caused by state-explosion. In this thesis, we address verification approaches for both low-level individual component correctness and high-level system correctness, which are equally important under this scheme. Three prototype tools are developed, implementing our approaches and algorithms accordingly. For the component-level design-time verification, we developed a symbolic verifier, LhaVrf, for the reachability verification of concurrent linear hybrid systems (LHA). It is unique in translating a hybrid automaton into a transition system that preserves the discrete transition structure, possesses no continuous dynamics, and preserves reachability of discrete states. Afterward, model-checking is interleaved in the counterexample fragment based specification relaxation framework. We next present a simulation-based bounded-horizon reachability analysis approach for the reachability verification of systems modeled by hybrid automata (HA) on a run-time basis. This framework applies a dynamic, on-the-fly, repartition-based error propagation control method with the mild requirement of Lipschitz continuity on the continuous dynamics. The novel features allow state-triggered discrete jumps and provide eventually constant over-approximation error bound for incremental stable dynamics. The above approaches are implemented in our prototype verifier called HS3V. Once the component properties are established, the next thing is to establish the system-level properties through compositional verication. We present our work on the role and integration of quantier elimination (QE) for property composition and verication. In our approach, we derive in a single step, the strongest system property from the given component properties for both time-independent and time-dependent scenarios. The system initial condition can also be composed, which, alongside the strongest system property, are used to verify a postulated system property through induction. The above approaches are implemented in our prototype tool called ReLIC