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Model Error Compensator vs Disturbance Observer: A Structural Comparison

Eng: Control Engineering Eng: Control Theory

This article provides a detailed comparison between the Model Error Compensator (MEC) and the Disturbance Observer (DOB), two approaches for enhancing the robustness of control systems against model uncertainty and external disturbances. While both methods share the goal of making the closed-loop system behave like a nominal model, their structures, assumptions, and applicability differ in important ways. Related articles, research papers, and MATLAB links are placed at the bottom.

Author: Hiroshi Okajima, Associate Professor, Kumamoto University, Japan — 20 years of control engineering research

For the comprehensive guide on MEC, see: Model Error Compensator (MEC): Enhance the Robustness of Existing Control Systems with Simple Compensation

Why Compare MEC and Disturbance Observer?

In practical control systems, the plant model used for controller design is never perfectly accurate. Model uncertainty, unmodeled dynamics, parameter variations, and external disturbances can all degrade control performance. Two well-known approaches to address this problem are:

  • The Model Error Compensator (MEC), which feeds back the difference between the actual plant output and the nominal model output to make the effective plant dynamics match the nominal model.
  • The Disturbance Observer (DOB), which estimates an equivalent disturbance signal that accounts for the gap between the nominal model and the actual plant, and then cancels it through feedforward compensation.

Both methods aim to make the controlled system behave as if the plant were the nominal model, and in certain configurations they can achieve equivalent disturbance rejection performance. However, their internal structures, design procedures, and practical applicability differ, particularly for complex plant classes such as nonlinear systems, non-minimum phase systems, and multi-rate systems. This article clarifies these differences to help practitioners choose the most appropriate method for their application.

Disturbance Observer: Basic Structure

The disturbance observer was originally proposed by Ohnishi (1983) for motion control of servo systems and has since been widely applied in robotics, hard disk drives, and mechatronic systems.

Concept

The DOB treats all discrepancies between the nominal model P_{n} and the actual plant P as an equivalent input disturbance d_{eq}. If this equivalent disturbance can be estimated, it can be subtracted from the control input to cancel the model error.

The equivalent disturbance is defined as:

\displaystyle d_{eq}(s) = \left(\frac{1}{P_n(s)} - \frac{1}{P(s)}\right) u(s) + \frac{1}{P(s)} d(s)

where d(s) is the actual external disturbance. The DOB estimates d_{eq} using the measured output y and the nominal model inverse P_{n}^{-1}:

\displaystyle \hat{d}_{eq}(s) = Q(s)\left(\frac{y(s)}{P_n(s)} - u(s)\right)

where Q(s) is a low-pass filter (called the Q-filter) that makes the estimation proper and realizable. The estimated disturbance is subtracted from the control input.

Key Requirement: Inverse Model

The DOB inherently requires the inverse of the nominal model P_{n}^{-1}(s). This poses challenges when:

  • The nominal model has unstable zeros (non-minimum phase system), making the inverse unstable
  • The system has a relative degree greater than one, requiring the Q-filter to provide sufficient roll-off
  • The system is nonlinear, where a simple transfer function inverse does not exist

Model Error Compensator: Basic Structure

The MEC was proposed by Okajima et al. (2013) and developed into a comprehensive framework for robustifying existing control systems.

Concept

The MEC places the nominal model P_{n} in parallel with the actual plant P and feeds back the output error between them through an error compensator D:

\displaystyle u_c(s) = u(s) + D(s)\bigl(y(s) - P_n(s) u_c(s)\bigr)

where u_{c} is the compensated control input applied to the actual plant and u is the original control input from the controller. The error compensator D is designed as a high-gain feedback element that drives the output of the actual plant toward the output of the nominal model.

Key Feature: No Inverse Model Required

Unlike the DOB, the MEC does not require the inverse of the nominal model. The compensator D operates on the output error signal directly, using high-gain feedback to suppress the model error. This makes the MEC naturally applicable to:

  • Non-minimum phase systems (unstable zeros do not cause problems)
  • Nonlinear systems (the MEC structure works without transfer function inversion)
  • Unstable systems (the high-gain feedback can stabilize the error dynamics)

For the simplest case of a SISO minimum-phase system with relative degree 1, the error compensator D can be a simple high-gain proportional or PI controller. For more complex system classes, the design methodology has been extended using parallel feedforward filters, LMI-based optimization, and other techniques.

Structural Comparison

Block Diagram Perspective

The fundamental difference is in how the model error is handled:

  • DOB: Estimates the equivalent disturbance at the input side using the inverse model, then cancels it via feedforward.
  • MEC: Corrects the output discrepancy between the plant and the model via feedback, without estimating the disturbance explicitly.

Transfer Function Equivalence

For the nominal case (minimum phase, relative degree 1, SISO), when the Q-filter bandwidth in the DOB and the error compensator gain in the MEC are tuned to achieve the same disturbance rejection bandwidth, both methods produce equivalent closed-loop transfer functions. This equivalence has been analyzed in the MEC literature.

However, the structural difference becomes significant in the following situations:

  • Design procedure: In MEC, the error compensator D is designed for a simple feedback loop (plant + error compensator), where standard robust control tools (e.g., LMI, loop shaping) can be directly applied. In DOB, the Q-filter design must account for the interaction with the existing controller.
  • Noise sensitivity: Both methods amplify sensor noise at high frequencies. In MEC, the noise-performance trade-off is controlled by the gain of D. In DOB, it is controlled by the Q-filter bandwidth. The trade-off structure is similar but the design perspectives differ.

Detailed Comparison Table

Aspect Model Error Compensator (MEC) Disturbance Observer (DOB)
Core mechanism Output error feedback Input disturbance estimation and cancellation
Requires inverse model No Yes ( P_{n}^{-1})
Minimum phase systems Applicable (high-gain feedback) Applicable (standard Q-filter design)
Non-minimum phase systems Applicable (with parallel feedforward filter) Requires special treatment (approximate inverse or modified structure)
Nonlinear systems Applicable (no inversion needed) Challenging (no simple inverse for nonlinear models)
Unstable systems Applicable Applicable (with stabilizing Q-filter)
MIMO systems Applicable Applicable (but inverse becomes matrix inversion)
Design tool High-gain feedback design; compatible with LMI, loop shaping, robust control Q-filter design; frequency-domain shaping
Combination with existing controllers Add-on compensator; existing controller remains unchanged Add-on compensator; existing controller remains unchanged
Disturbance rejection performance Equivalent (for same bandwidth) Equivalent (for same bandwidth)
Noise-performance trade-off Controlled by error compensator gain Controlled by Q-filter bandwidth
Historical origin Okajima et al., SICE JCMSI (2013) Ohnishi, IO Division of JIEE (1983)

When to Use Which?

Use MEC when:

  • The control object is non-minimum phase or has unstable zeros, where the model inverse is problematic
  • The system is nonlinear, and a transfer function inverse cannot be easily computed
  • You want to apply standard robust control design tools (LMI, loop shaping) to the error compensator design
  • You want a unified framework that applies to a wide range of system classes without changing the compensator structure

Use DOB when:

  • The system is minimum phase with low relative degree, where the inverse model is straightforward
  • The application is a servo/mechatronic system where the DOB framework is well established and extensively validated
  • Frequency-domain design of the Q-filter is preferred, leveraging the rich existing literature on DOB tuning
  • The team has existing expertise in DOB-based control design

Practical recommendation:

In many cases, it is rational to implement both methods and compare their performance for the specific application. Since both methods can achieve equivalent disturbance rejection for minimum-phase systems, the choice often comes down to the specific system class, the designer's expertise, and the available design tools.

Application to Non-Minimum Phase Systems

One of the most significant practical differences between MEC and DOB appears in non-minimum phase systems (systems with unstable zeros).

For a plant with an unstable zero at s = z_{0} (where z_{0} > 0 in continuous time), the inverse model P_{n}^{-1}(s) contains an unstable pole, making direct DOB implementation impossible. Several workarounds exist for DOB (e.g., approximate inversion, modified Q-filter design), but they introduce additional complexity and conservatism.

In contrast, the MEC approach handles non-minimum phase systems through a parallel feedforward filter that avoids the need for model inversion. The parallel feedforward compensator shapes the effective system zeros so that the error compensator can operate effectively without requiring high-gain feedback through the non-minimum phase dynamics. This approach has been validated for both continuous-time and discrete-time non-minimum phase systems with polytopic-type uncertainties.

For details, see the paper: H. Okajima and N. Matsunaga, Model Error Compensator Design for Continuous- and Discrete-Time Non-minimum Phase Systems with Polytopic-Type Uncertainties, SICE JCMSI (2022)

MEC Origin Paper — H. Okajima, H. Umei, N. Matsunaga and T. Asai, A Design Method of Compensator to Minimize Model Error, SICE JCMSI, Vol. 6, No. 4, pp. 267–275 (2013). The foundational paper establishing the MEC framework.

MEC for Polytopic Uncertainty — H. Okajima and N. Matsunaga, A design method of model error compensator for systems with polytopic-type uncertainty and disturbances, SICE JCMSI (2021). Extends MEC design to systems with structured parameter uncertainty using LMI optimization.

MEC with IFAC World Congress — H. Okajima, Model Error Compensator for adding Robustness toward Existing Control Systems, IFAC PaperOnLine, Vol. 56, Issue 2, pp. 3998–4005 (2023). Comprehensive overview presented at IFAC World Congress 2023.

DOB Survey — E. Sariyildiz, R. Oboe and K. Ohnishi, Disturbance Observer-based Robust Control and Its Applications: 35th Anniversary Overview, IEEE Trans. Ind. Electron., Vol. 67, No. 3, pp. 2042–2053 (2020). A comprehensive survey of DOB theory and applications.

MATLAB Code

Blog Articles (blog.control-theory.com)

Research Web Pages (www.control-theory.com)

Video

Self-Introduction

Hiroshi Okajima — Associate Professor, Graduate School of Science and Technology, Kumamoto University. Member of SICE, ISCIE, and IEEE.

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