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

Alessio Moreschini, Wei He, Romeo Ortega, Yiheng Lu, Tao Li

TL;DR

The paper addresses enabling globally stable discrete-time PID-PBC for power converters by designing a passivity-preserving discretization. It employs the implicit midpoint method to maintain the Hamiltonian structure and introduces a shifted-passive output for the discrete model, ensuring that the incremental DT-PID controller remains passive. The main result proves global stability of the DT interconnection with the power converter and shows convergence to the desired equilibrium under a damping-injection condition. Validation through Buck-Boost simulations and experiments demonstrates improved stability and tracking performance over Euler discretization, highlighting practical impact for digital control of nonlinear power electronics.

Abstract

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.

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

TL;DR

The paper addresses enabling globally stable discrete-time PID-PBC for power converters by designing a passivity-preserving discretization. It employs the implicit midpoint method to maintain the Hamiltonian structure and introduces a shifted-passive output for the discrete model, ensuring that the incremental DT-PID controller remains passive. The main result proves global stability of the DT interconnection with the power converter and shows convergence to the desired equilibrium under a damping-injection condition. Validation through Buck-Boost simulations and experiments demonstrates improved stability and tracking performance over Euler discretization, highlighting practical impact for digital control of nonlinear power electronics.

Abstract

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.

Paper Structure

This paper contains 14 sections, 4 theorems, 50 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CT power converter models
  4. Definition of an output $y$ such that $\tilde{u} \mapsto \tilde{y}$ is passive
  5. Global stabilization of the PID-PBC
  6. Midpoint discretization
  7. Shifted Passivity of the Discretized Power Converter Model
  8. Shifted Passivity of the DT PID
  9. Closed-Loop Stability
  10. Simulation and Experimental Results
  11. DT model of Buck-Boost converter and problem formulation
  12. Simulation results
  13. Experimental results
  14. Concluding Remarks and Future Research

Key Result

Proposition 2.1

Consider the system eqn:power_converter_compact, with the desired state equilibrium $x^\star$, in closed-loop with the CT PID-PBC eqn:PID, where $\tilde{y}$ is given by ynew. (i) The equilibrium $(x^\star,\xi^\star )$ of the closed-loop system is globally stable for any $K_P>0$, $K_I>0$ and $K_D \g we have that concon holds for all initial conditions.

Figures (7)

  • Figure 1: The circuit topology of buck-boost converter.
  • Figure 2: The response curves of the system with a step change in reference and different gains.
  • Figure 3: The response curves of the system with different sampling time $\delta$.
  • Figure 4: The response curves of the system under Euler discretization and the proposed midpoint method.
  • Figure 5: The experimental response curves of the system with step changes in reference from $15$ V to $22$ V.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • proof