We propose a novel strategy to construct optimal controllers for continuous-time nonlinear systems by means of linear-like techniques, provided that the optimal value function is differentiable and quadratic-like. This assumption covers a wide range of cases and holds locally around an equilibrium under mild assumptions. The proposed strategy does not require solving the Hamilton–Jacobi–Bellman equation, i.e., a nonlinear partial differential equation, which is known to be hard or impossible to solve. Instead, the Hamilton–Jacobi–Bellman equation is replaced with an easy-solvable state-dependent Lyapunov matrix equation. We exploit a linear-like factorization of the underlying nonlinear system and a policy-iteration algorithm to yield a linear-like policy-iteration for nonlinear systems. The proposed control strategy solves optimal nonlinear control problems in an asymptotically exact, yet still linear-like manner. We prove optimality of the resulting solution and illustrate the results via four examples.

ABSTRACT Multirotor Aerial Vehicles may be fault-tolerant by design when rotor-failure is possible to measure or identify, especially when a large number of rotors are used. For instance, an octocopter can be capable to complete some missions even when a double-rotor fault occurs during the execution. In this paper, we study how a rotor-failure reduces the vehicle control admissible set and its importance with respect to the selected mission, i.e. we perform mission-related fault-tolerant analysis. Furthermore, we propose a risk-sensitive motion-planning algorithm capable to take into account the risks during the planning stage by means of mission-related fault-tolerant analysis. We show that the proposed approach is much less conservative in terms of selected performance measures than a conservative risk planner that assumes that the considered fault will certainly occur during the mission execution. As expected, the proposed risk-sensitive motion planner is also readier for accepting failures during the mission execution than the risk-insensitive approach that assumes no failure will occur.

We propose a novel strategy to construct optimal controllers for continuous-time nonlinear systems by means of linear-like techniques, provided that the optimal value function is differentiable and quadratic-like. This assumption covers a wide range of cases and holds locally in general. The proposed strategy avoids solving the Hamilton-Jacobi-Bellman (HJB) equation, that is a nonlinear partial differential equation, which is known to be hard or impossible to solve. Instead, the HJB equation is replaced with an easy-solvable state- dependent Lyapunov matrix equation without introducing any approximation. We achieve this exploiting a linear-factorization of the underlying nonlinear system and a policy-iteration algorithm (PI) to yield a linear-like PI for nonlinear systems. The proposed control strategy solves optimal nonlinear control problems in an exact, yet still linear-like manner. We prove optimality of the resulting solution and illustrate the results via two examples.