Simulation-based reachability analysis for nonlinear systems using componentwise contraction properties

A shortcoming of existing reachability approaches for nonlinear systems is the poor scalability with the number of continuous state variables. To mitigate this problem we present a simulation-based approach where we first sample a number of trajectories of the system and next establish bounds on the convergence or divergence between the samples and neighboring trajectories. We compute these bounds using contraction theory and reduce the conservatism by partitioning the state vector into several components and analyzing contraction properties separately in each direction. Among other benefits this allows us to analyze the effect of constant but uncertain parameters by treating them as state variables and partitioning them into a separate direction. We next present a numerical procedure to search for weighted norms that yield a prescribed contraction rate, which can be incorporated in the reachability algorithm to adjust the weights to minimize the growth of the reachable set

Approximate reduction of heterogenous nonlinear models with differential hulls

We present a model reduction technique for a class of nonlinear ordinary differential equation (ODE) models of heterogeneous systems, where heterogeneity is expressed in terms of classes of state variables having the same dynamics structurally, but which are characterized by distinct parameters. To this end, we first build a system of differential inequalities that provides lower and upper bounds for each original state variable, but such that it is homogeneous in its parameters. Then, we use two methods for exact aggregation of ODEs to exploit this homogeneity, yielding a smaller model of size independent of the number of heterogeneous classes. We apply this technique to two case studies: a multiclass queuing network and a model of epidemics spread

A Robust Scenario MPC Approach for Uncertain Multi-modal Obstacles

Motion planning and control algorithms for autonomous vehicles need to be safe, and consider future movements of other road users to ensure collision-free trajectories. In this letter, we present a control scheme based on Model Predictive Control (MPC) with robust constraint satisfaction where the constraint uncertainty, stemming from the road users\u27 behavior, is multimodal. The method combines ideas from tube-based and scenario-based MPC strategies in order to approximate the expected cost and to guarantee robust state and input constraint satisfaction. In particular, we design a feedback policy that is a function of the disturbance mode and allows the controller to take less conservative actions. The effectiveness of the proposed approach is illustrated through two numerical simulations, where we compare it against a standard robust MPC formulation

Interval Reachability Analysis using Second-Order Sensitivity

We propose a new approach to compute an interval over-approximation of the finite time reachable set for a large class of nonlinear systems. This approach relies on the notions of sensitivity matrices, which are the partial derivatives representing the variations of the system trajectories in response to variations of the initial states. Using interval arithmetics, we first over-approximate the possible values of the second-order sensitivity at the final time of the reachability problem. Then we exploit these bounds and the evaluation of the first-order sensitivity matrices at a few sampled initial states to obtain an over-approximation of the first-order sensitivity, which is in turn used to over-approximate the reachable set of the initial system. Unlike existing methods relying only on the first-order sensitivity matrix, this new approach provides guaranteed over-approximations of the first-order sensitivity and can also provide such over-approximations with an arbitrary precision by increasing the number of samples

Sensitivity analysis of uncertain dynamic systems using set-valued integration

We present an extension of set-valued integration to enable efficient sensitivity analysis of parameter-dependent ordinary differential equation (ODE) systems, using both the forward and adjoint methods. The focus is on continuous-time set-valued integration, whereby auxiliary ODE systems are derived whose solutions describe high-order inclusions of the parametric trajectories in the form of polynomial models. The forward and adjoint auxiliary ODE systems treat the parameterization error of the original differential variables as a time-varying uncertainty, and propagate the sensitivity bounds forward and backward in time, respectively. This construction enables building on the sensitivity analysis capabilities of state-of-the-art solvers, such as CVODES in the SUNDIALS suite. Several numerical case studies are presented to assess the performance and accuracy of these set-valued sensitivity integrators

Functional sets with typed symbols: Framework and mixed Polynotopes for hybrid nonlinear reachability and filtering

Verification and synthesis of Cyber-Physical Systems (CPS) are challenging and still raise numerous issues so far. In this paper, an original framework with mixed sets defined as function images of symbol type domains is first proposed. Syntax and semantics are explicitly distinguished. Then, both continuous (interval) and discrete (signed, boolean) symbol types are used to model dependencies through linear and polynomial functions, so leading to mixed zonotopic and polynotopic sets. Polynotopes extend sparse polynomial zonotopes with typed symbols. Polynotopes can both propagate a mixed encoding of intervals and describe the behavior of logic gates. A functional completeness result is given, as well as an inclusion method for elementary nonlinear and switching functions. A Polynotopic Kalman Filter (PKF) is then proposed as a hybrid nonlinear extension of Zonotopic Kalman Filters (ZKF). Bridges with a stochastic uncertainty paradigm are outlined. Finally, several discrete, continuous and hybrid numerical examples including comparisons illustrate the effectiveness of the theoretical results.Comment: 21 pages, 8 figure