Globally Stable Discrete Time PID Passivity-based Control of Power Converters: Simulation and Experimental Results

The key idea behind PID Passivity-based Control (PID-PBC) is to leverage the passivity property of PIDs (for all positive gains) and wrap the PID controller around a passive output to ensure global stability in closed-loop. However, the practical applicability of PID-PBC is stymied by two key facts: (i) the vast majority of practical implementations of PIDs is carried-out in discrete time -- discretizing the continuous time dynamical system of the PID; (ii) the well-known problem that passivity is not preserved upon discretization, even with small sampling times. Therefore, two aspects of the PID-PBC must be revisited for its safe practical application. First, we propose a discretization of the PID that ensures its passivity. Second, since the output that is identified as passive for the continuous time system is not necessarily passive for its discrete time version, we construct a new output that ensures the passivity property for the discretization of the system. In this paper, we provide a constructive answer to both issues for the case of power converter models. Instrumental to achieve this objective is the use of the implicit midpoint discretization method -- which is a symplectic integration technique that preserves system invariants. Since the reference value for the output to be regulated in power converters is non-zero, we are henceforth interested in the property of passivity of the incremental model -- currently known as shifted passivity. Therefore, we demonstrate that the resulting discrete-time PID-PBC defines a passive map for the incremental model and establish shifted passivity for the discretized power converter model. Combining these properties, we prove global stability for the feedback interconnection of the power converter with the discretized PID-PBC. The paper also presents simulations and experiments that demonstrate the performance of the proposed discretization.