Metric Temporal Logic (MTL) specifications can capture complex state and timing requirements. Given a nonlinear dynamical system and an MTL specification for that system, our goal is to find a trajectory that violates or satisfies the specification. This trajectory can be used as a concrete feedback to the system designer in the case of violation or as a trajectory to be tracked in the case of satisfaction. The search for such a trajectory is conducted over the space of initial conditions, system parameters and input signals. We convert the trajectory search problem into an optimization problem through MTL robust semantics. Robustness quantifies how close the trajectory is to violating or satisfying a specification. Starting from some arbitrary initial condition and parameter and given an input signal, we compute a descent direction in the search space, which leads to a trajectory that optimizes the MTL robustness. This process can be iterated to reach local optima (min or max). We demonstrate the method on examples from the literature.

Abstract-The aim of the safety controller synthesis problem is to synthesize a feedback controller that results in closedloop trajectories that meet certain criteria, namely, the state or output trajectories terminate in a Goal set without entering an Unsafe set. We propose a formal method for synthesizing such a controller using finitely many human generated trajectories. The main theoretical idea behind our results is the concept of trajectory robustness, which is established using the theory of approximate bisimulation. Approximate bisimulation has been used to establish robustness (in the L∞ sense) of execution trajectories of dynamical systems and hybrid systems, resulting in trajectory-based safety verification procedures.The work reported in this paper builds on our earlier work where the dynamics of the system is assumed to be affine linear. We extend the existing results to special classes of nonlinear dynamical systems, feedback linearizable and differentially flat systems. For both cases, we present some examples where it is possible to synthesize the controller using human generated trajectories, which are obtained through interactive computer programs with graphical interface (computer games).

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This paper focuses on the task of safety controller synthesis, that is, designing a controller that will take a system from any point within a compact set of initial states to a point inside a set of acceptable goal states, while never entering any state that is deemed unsafe. To do this we use a human generated trajectory-based approach. We introduce the control autobisimulation function, which is the analog of the control Lyapunov function for approximate bisimulation. We consider a class of hybrid systems and use this function to determine a set of admissible feedback control laws that guarantee trajectory robustness for underlying dynamics that are linear affine, feedback linearizable, and differentially flat. This property ensures that any trajectory of the closed-loop system that is initialized within some neighborhood of a nominal trajectory will stay within some tube of the nominal trajectory when given the same input. This feedback control and input can be used as the controller for some subset of the initial states. We demonstrate how to combine multiple trajectories into a synthesized controller that satisfies the safety problem.

Although the entropy of a given signal-type waveform is technically zero, it is nonetheless desirable to use entropic measures to quantify the associated information. Several such prescriptions have been advanced in the literature but none are generally successful. Here, we report that the Fourier-conjugated 'total entropy' associated with quantum-mechanical probabilistic amplitude functions (PAFs) is a meaningful measure of information in non-probabilistic real waveforms, with either the waveform itself or its (normalized) analytic representation acting in the role of the PAF. Detailed numerical calculations are presented for both adaptations, showing the expected informatic behaviours in a variety of rudimentary scenarios. Particularly noteworthy are the sensitivity to the degree of randomness in a sequence of pulses and potential for detection of weak signals.