  Among the major challenges in neural control system technology is the
validation and certification of the safety and robustness of neural network
(NN) controllers against various uncertainties including unmodelled dynamics,
nonlinearities, and time delays. One way in providing such validation
guarantees is to maintain the closed-loop system output with a NN controller
when its input changes within a bounded set, close to the output of a robustly
performing closed-loop reference model. This paper presents a novel approach to
analysing the performance and robustness of uncertain feedback systems with NN
controllers. Due to the complexity of analysing such systems, the problem is
reformulated as the problem of dynamical tracking errors between the
closed-loop system with a neural controller and an ideal closed-loop reference
model. Then, the approximation of the controller error is characterised by
adopting the differential mean value theorem (DMV) and the Integral Quadratic
Constraints (IQCs) technique. Moreover, the Relative Integral Square Error
(RISE) and the Supreme Square Error (SSE) bounded set are derived for the
output of the error dynamical system. The analysis is then performed by
integrating Lyapunov theory with the IQCs-based technique. The resulting
worst-case analysis provides the user a prior knowledge about the worst case of
RISE and SSE between the reference closed-loop model and the uncertain system
controlled by the neural controller.
