This paper presents TIRA, a Matlab library gathering several methods for the computation of interval over-approximations of the reachable sets for both continuous-and discrete-time nonlinear systems. Unlike other existing tools, the main strength of intervalbased reachability analysis is its simplicity and scalability, rather than the accuracy of the over-approximations. The current implementation of TIRA contains four reachability methods covering wide classes of nonlinear systems, handled with recent results relying on contraction/growth bounds and monotonicity concepts. TIRA's architecture features a central function working as a hub between the user-defined reachability problem and the library of available reachability methods. This design choice offers increased extensibility of the library, where users can define their own method in a separate function and add the function call in the hub function.

Complex-valued Gaussian processes are commonly used in Bayesian frequencydomain system identification as prior models for regression. If each realization of such a process were an H∞ function with probability one, then the same model could be used for probabilistic robust control, allowing for robustly safe learning. We investigate sufficient conditions for a general complex-domain Gaussian process to have this property. For the special case of processes whose Hermitian covariance is stationary, we provide an explicit parameterization of the covariance structure in terms of a summable sequence of nonnegative numbers. We then establish how an H∞ Gaussian process can serve as a prior for Bayesian system identification and as a probabilistic uncertainty model for probabilistic robust control. In particular, we compute formulas for refining the uncertainty model by conditioning on frequency-domain data and for upper-bounding the probability that the realizations of the process satisfy a given integral quadratic constraint.

We propose an iterative method to safely learn the unmodeled dynamics of a nonlinear system using Bayesian Gaussian process (GP) models with polynomial kernel functions. The method maintains safety by ensuring that the system state stays within the region of attraction (ROA) of a stabilizing control policy while collecting data. A quadratic programming based exploration control policy is computed to keep the exploration trajectory inside an inner-approximation of the ROA and to maximize the information gained from the trajectory. A prior GP model, which incorporates prior information about the unknown dynamics, is used to construct an initial stabilizing policy. As the GP model is updated with data, it is used to synthesize a new policy and a larger ROA, which increases the range of safe exploration. The use of polynomial kernels allows us to compute ROA inner-approximations and stabilizing control laws for the model using sum-of-squares programming. We also provide a probabilistic guarantee of safety which ensures that the policy computed using the learned model stabilizes the true dynamics with high confidence.

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