This note is concerned with the problem of stabilizing a class of linear continuous-time systems with completely unknown system dynamics subject to input energy constraint. To deal with this problem, a model-based low gain feedback law is designed firstly by establishing a special algebraic Riccati equation. Such a low gain feedback law can semiglobally stabilize the linear systems subject to input energy constraint with the exact system model. In order to relax the assumption that the system model is exactly known, an off-policy reinforcement learning approach is designed to solve the same problem without requiring the completely knowledge of the system dynamics. Finally, in order to verify the effectiveness of the proposed model-free approach, simulation result on the spacecraft rendezvous problem is introduced. K E Y W O R D Sinput energy constraint, low gain feedback, reinforcement learning, spacecraft rendezvous 1 948

With inherent ‘S’ characteristics and the one-tunnel-with-two-units arrangement of the pump-turbine, hydraulic transient changes in the successive start-up process are complex, and the optimal control is difficult. This paper aims to study the dynamic characteristics and successive start-up control strategy optimization of two hydraulic couplings pumped storage units (PSUs) under low-head extreme conditions. Firstly, an accurate model of two hydraulic coupling PSUs’ successive start-up is established. Based on this model, the influence of the interval time of successive start-up on the dynamic characteristics of PSUs is carried out. It is shown that the change of the interval time of the successive start-up (ΔT) of the two PSUs has a significant impact on the dynamic response stability of the low-head start-up. If ΔT is more than 40 s, the hydraulic oscillation and speed fluctuation of the PSUs deteriorate. Secondly, with the different controller parameters for the two PSUs, a novel multi-objective optimization scheme with fractional order PID controller (FOPID) is proposed to figure out the best control scheme for the successive start-up. Furthermore, selecting the sum of the rise time (Tr) of the rotating speed of two PSUs and the sum of the integral time absolute error (ITAE) of two PSUs is the objective. Meanwhile, the optimization scheme of PID with different parameters (PIDDP) is used to compare and verify the optimization method proposed in this paper. The results for this extreme condition indicate that FOPID has more significant advantages in optimizing the instability of the successive start-up process, with the better Pareto front, and the optimized scheme has a more stable dynamic transition process of flow, water hammer pressure, and rotational speed.