The increasing penetration of inverter-based resources (IBRs) calls for an advanced active and reactive power (PQ) control strategy in microgrids. To enhance the controllability and flexibility of the IBRs, this paper proposed an adaptive PQ control method with trajectory tracking capability, combining model-based analysis, physics-informed reinforcement learning (RL), and power hardware-in-the-loop (HIL) experiments. First, model-based analysis proves that there exists an adaptive proportional-integral controller with time-varying gains that can ensure any exponential PQ output trajectory of IBRs. These gains consist of a constant factor and an exponentially decaying factor, which are then obtained using a model-free deep reinforcement learning approach known as the twin delayed deeper deterministic policy gradient. With the model-based derivation, the learning space of the RL agent is narrowed down from a function space to a real space, which reduces the training complexity significantly. Finally, the proposed method is verified through numerical simulation in MATLAB-Simulink and power HIL experiments in the CURENT center.With the physics-informed learning method, exponential response time constants can be freely assigned to IBRs, and they can follow any predefined trajectory without complicated gain tuning.