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Model-Free Design and Analysis of 2DOF PI Controller for Noisy LTI Systems

Taiga Kiyota, Kazuhiro Sato

TL;DR

This work develops a model-free design for a 2DOF PI controller for MIMO LTI systems under Gaussian disturbances, combining a data-driven feedforward design with PI gain tuning. It provides nonasymptotic bounds on the required data horizon for feedforward estimation and analyzes how feedforward errors propagate to the PI stage, including a zero-order gradient approach for PI tuning. Compared with model-based methods, the proposed approach offers favorable data-efficiency for moderate feedforward accuracy and substantially lower per-iteration computation, albeit with larger data needs for high-accuracy feedforward. Numerical experiments demonstrate improved control performance over model-based design across multiple systems, validating the theoretical claims and highlighting practicality for data-driven control of uncertain systems.

Abstract

Set-point tracking for systems with unknown model parameters is a fundamental problem in control, and two-degree-of-freedom (2DOF) Proportional-Integral (PI) controllers -- consisting of a feedforward controller and PI controller -- are widely employed for this task. In this paper, we propose a model-free design of 2DOF PI controllers, establish its theoretical properties, and compare them with a model-based method from both theoretical and numerical perspectives. For the feedforward design, we extend an existing model-free algorithm to systems subject to Gaussian process and measurement noises. We derive a nonasymptotic lower bound on the required control input/output data length and characterize the resulting estimation error. For PI gain tuning, we formulate a constrained optimization problem and establish sample complexity of a zeroth-order optimization method. Moreover, we quantify how inaccuracies in the feedforward design propagate to the performance of the PI controller, highlighting an interaction that has not been examined in prior work. We further provide a theoretical comparison between the proposed method and the model-based method. In particular, for PI gain tuning, the proposed method is computationally more efficient by avoiding explicit gradient computations. Numerical experiments demonstrate that the 2DOF PI controller designed by the proposed method exhibits better control performance than the model-based method.

Model-Free Design and Analysis of 2DOF PI Controller for Noisy LTI Systems

TL;DR

This work develops a model-free design for a 2DOF PI controller for MIMO LTI systems under Gaussian disturbances, combining a data-driven feedforward design with PI gain tuning. It provides nonasymptotic bounds on the required data horizon for feedforward estimation and analyzes how feedforward errors propagate to the PI stage, including a zero-order gradient approach for PI tuning. Compared with model-based methods, the proposed approach offers favorable data-efficiency for moderate feedforward accuracy and substantially lower per-iteration computation, albeit with larger data needs for high-accuracy feedforward. Numerical experiments demonstrate improved control performance over model-based design across multiple systems, validating the theoretical claims and highlighting practicality for data-driven control of uncertain systems.

Abstract

Set-point tracking for systems with unknown model parameters is a fundamental problem in control, and two-degree-of-freedom (2DOF) Proportional-Integral (PI) controllers -- consisting of a feedforward controller and PI controller -- are widely employed for this task. In this paper, we propose a model-free design of 2DOF PI controllers, establish its theoretical properties, and compare them with a model-based method from both theoretical and numerical perspectives. For the feedforward design, we extend an existing model-free algorithm to systems subject to Gaussian process and measurement noises. We derive a nonasymptotic lower bound on the required control input/output data length and characterize the resulting estimation error. For PI gain tuning, we formulate a constrained optimization problem and establish sample complexity of a zeroth-order optimization method. Moreover, we quantify how inaccuracies in the feedforward design propagate to the performance of the PI controller, highlighting an interaction that has not been examined in prior work. We further provide a theoretical comparison between the proposed method and the model-based method. In particular, for PI gain tuning, the proposed method is computationally more efficient by avoiding explicit gradient computations. Numerical experiments demonstrate that the 2DOF PI controller designed by the proposed method exhibits better control performance than the model-based method.
Paper Structure (21 sections, 9 theorems, 83 equations, 5 figures, 1 table, 3 algorithms)

This paper contains 21 sections, 9 theorems, 83 equations, 5 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem formulation
  4. Feedforward Design
  5. PI Gain Tuning
  6. proposed method
  7. Feedforward Design
  8. PI Gain Tuning
  9. Comparison with model-based method
  10. Feedforward Design
  11. PI gain tuning
  12. numerical experiments
  13. Feedforward Design
  14. PI Gain Tuning
  15. Results across multiple systems
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose that $\sigma_p(C A_K^{-1} B)\geq 4\sqrt{2m\mathrm{tr}(C\Sigma C^\top) }$ holds, where $\Sigma$ is the solution to (eq:Pcont_Lyapunov). For any $\epsilon_u \geq 0$, set where $M_1$, $M_2$, and $M_3$ are defined in (eq:def_M1), (eq:def_M2), and (eq:def_M3), respectively, and $Z$ is the unique solution to the Lyapunov equation $A_K^\top Z +Z A_K+I=O$. Then, for any $\delta_u>0$, we have wit

Figures (5)

  • Figure 1: Boxplots of the steady-state output relative errors $\frac{\|y^\star -\mathbb{E}[y_{u_0}(\infty)]\|}{\|y^\star \|}$ over 10 trials.
  • Figure 2: Boxplots of $\bar{f}(K)$ over 10 trials with $N_\mathrm{eval}=200,\,\tau_\mathrm{eval}=300$.
  • Figure 3: Trajectories of the output $y(t)$. The red line represents the trajectory under the controller designed by the proposed method. The blue line shows the trajectory under the controller designed by the model-based method. The black line corresponds to the desired output setpoint $y^\star$. (a) The first component of $y(t)$. (b) The second component of $y(t)$.
  • Figure 4: Boxplots of the average steady-state output relative errors across 10 systems.
  • Figure 5: Boxplot of the ratio of the PI gain performance metric $\frac{\bar{f}_\mathrm{MF}(K) }{ \bar{f}_\mathrm{MB}(K)}$ across 10 systems.

Theorems & Definitions (30)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 2
  • ...and 20 more