This article explains how to apply the Model Error Compensator (MEC) to non-minimum phase systems using a parallel feedforward compensator approach. Non-minimum phase systems have unstable zeros, which make high-gain feedback compensation difficult. The parallel feedforward compensator (PFC) converts the effective system to a minimum phase system, enabling MEC to achieve robust performance even for this challenging class of plants. Related articles, related papers, and MATLAB links are placed at the bottom.

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

This article is based on the following papers.

Yoshida, H. Okajima and T. Sato, Model error compensator design for continuous- and discrete-time non-minimum phase systems with polytopic-type uncertainties, SICE Journal of Control, Measurement, and System Integration, Vol. 15, Issue 2, pp. 141–153 (2022) (Open Access)

Ichimasa, H. Okajima, K. Okumura and N. Matsunaga, Model Error Compensator with Parallel Feed-Forward Filter, SICE Journal of Control, Measurement, and System Integration, Vol. 10, No. 5, pp. 468–475 (2017) (Open Access)

Co-authors: Ryuichiro Yoshida and Takumi Sato (Kumamoto University) for the JCMSI 2022 paper; Gou Ichimasa, Kousuke Okumura, and Nobutomo Matsunaga (Kumamoto University) for the JCMSI 2017 paper.

Why Non-Minimum Phase Systems Are Challenging for MEC

What Are Non-Minimum Phase Systems?

A non-minimum phase system is a system whose transfer function has zeros in the right-half of the complex plane (unstable zeros). For example, the transfer function:

$\displaystyle P(s) = \frac{s - z_0}{(s + p_1)(s + p_2)}$

with $z_{0} > 0$ is non-minimum phase because it has a zero at $s = z_{0}$ in the right-half plane. Non-minimum phase systems arise naturally in many practical applications, such as flexible structures, some aircraft dynamics, and systems with time delay (when approximated by Padé approximation).

The High-Gain Limitation

The basic MEC relies on high-gain feedback through the error compensator $D(s)$ to drive the plant output toward the model output. For minimum phase systems, increasing the gain of $D(s)$ monotonically improves the model error suppression.

However, for non-minimum phase systems, the feedback loop between the plant and the error compensator (the "inner loop" of MEC) has an open-loop transfer function $P(s) D(s)$ that inherits the unstable zeros of $P(s)$ . Due to these unstable zeros, arbitrarily increasing the gain of $D(s)$ can destabilize the inner loop. This is a fundamental limitation imposed by the non-minimum phase structure, not a limitation of the MEC framework itself.

The Solution: Parallel Feedforward Compensator

To overcome this limitation, a parallel feedforward compensator (PFC) $F(s)$ is added in parallel with the actual plant. The PFC is designed so that the combined system $P(s) + F(s)$ becomes effectively minimum phase, enabling high-gain error compensation.

MEC with Parallel Feedforward Compensator

Structure

The MEC with PFC has the following structure:

Actual plant $P(s)$ — the non-minimum phase plant to be controlled
Parallel feedforward compensator $F(s)$ — added in parallel to cancel or move the unstable zeros
Modified nominal model $P_{n}(s) + F(s)$ — the model is also augmented with the same PFC
Error compensator $D(s)$ — the high-gain feedback compensator

The compensated control input becomes:

$\displaystyle u_c(s) = u(s) + D(s)\left\{(y(s) + F(s) u_c(s)) - (P_n(s) + F(s)) u_c(s)\right\}$

The error signal is now $(y(s) + F(s) u_{c}(s)) - (P_{n}(s) + F(s)) u_{c}(s) = y(s) - P_{n}(s) u_{c}(s)$ , which is the same model error signal as the standard MEC. The crucial difference is that the inner loop now involves $(P(s) + F(s)) D(s)$ instead of $P(s) D(s)$ , and $P(s) + F(s)$ can be designed to be minimum phase.

Model Error Compensator with Parallel Feedforward Filter

Design of the Parallel Feedforward Compensator

The PFC $F(s)$ must be designed so that:

The combined transfer function $P(s) + F(s)$ is minimum phase (no unstable zeros)
$F(s)$ is stable and proper
The effect of $F(s)$ on the overall control performance is acceptable

The zero placement of $P(s) + F(s)$ depends on the choice of $F(s)$ . The JCMSI 2017 paper by Ichimasa et al. provides systematic design guidelines for choosing $F(s)$ to achieve minimum phase characteristics.

Design Methodology for Non-Minimum Phase MEC

Polytopic Uncertainty Framework

The JCMSI 2022 paper by Yoshida et al. extends the non-minimum phase MEC design to systems with polytopic-type uncertainties. The actual plant is assumed to belong to a polytopic set:

$\displaystyle P(s) \in \mathrm{Co}\{P_1(s), P_2(s), \ldots, P_N(s)\}$

where $\mathrm{Co}$ denotes the convex hull and $P_{1}(s), \ldots, P_{N}(s)$ are the vertex plants. The design goal is to find the error compensator $D(s)$ and the PFC $F(s)$ that guarantee robust performance for all plants in the polytopic set.

LMI-Based Design

The error compensator design is formulated as an LMI (Linear Matrix Inequality) optimization problem based on a common Lyapunov function. The design procedure consists of:

Choose the PFC $F(s)$ to make $P_{i}(s) + F(s)$ minimum phase for all vertex plants $i = 1, \ldots, N$
Formulate the LMI for the error compensator $D(s)$ that minimizes the model error under the polytopic uncertainty
Solve using iterative design or PSO — since the LMI is bilinear in the Lyapunov matrix and the compensator parameters, an iterative approach alternating between the two is used

Particle Swarm Optimization (PSO)

For the initial design or when the iterative LMI approach does not converge well, particle swarm optimization can be used to search for good error compensator parameters. The PSO result can then be refined using the iterative LMI design.

The MATLAB code implementing both the PSO and iterative design is available on GitHub (see MATLAB Code section below).

Comparison: MEC vs Disturbance Observer for Non-Minimum Phase Systems

A key advantage of MEC over the Disturbance Observer (DOB) for non-minimum phase systems is:

DOB requires the inverse model $P_{n}^{-1}(s)$ , which is unstable for non-minimum phase systems (the inverse has poles at the locations of the original unstable zeros). This makes the standard DOB structure difficult to apply.
MEC does not require any inverse model. The PFC approach resolves the non-minimum phase issue within the MEC framework without introducing unstable elements.

For a detailed structural comparison between MEC and DOB, see Model Error Compensator vs Disturbance Observer.

Applications

Continuous-Time and Discrete-Time Systems

The JCMSI 2022 paper presents the MEC + PFC design for both continuous-time and discrete-time non-minimum phase MIMO systems. The discrete-time extension is important for digital implementation of the compensator.

Time-Delay Systems

Systems with time delay, when approximated by Padé approximation, become non-minimum phase. The PFC approach can be applied to these systems as well. The JCMSI 2017 paper by Ichimasa et al. specifically addresses MEC with PFC for time-delay and non-minimum phase plants.

Integration with Various Controllers

Just as with the standard MEC, the non-minimum phase MEC can be combined with any controller: PID, MPC, state feedback, feedback linearization, etc. The MEC + PFC is placed between the controller output and the plant input, independent of the controller design.

Model Error Compensator: Comprehensive Guide — For the full overview of MEC including basic structure and design philosophy, see the MEC hub article.

MEC + PID Control — For the simplest integration of MEC with PID control (minimum phase case), see MEC + PID Control: Adding Robustness to the Most Widely Used Controller.

MEC vs Disturbance Observer — For a structural comparison with the Disturbance Observer, see MEC vs DOB: A Structural Comparison. The DOB's need for an inverse model makes it particularly difficult for non-minimum phase systems.

MEC for Nonlinear Systems — Nonlinear systems present another class of challenging plants for MEC. See MEC for Nonlinear Systems: Robust Feedback Linearization.

MEC Design with LMI — The polytopic uncertainty framework and LMI-based design used for non-minimum phase MEC is also applicable to minimum phase systems. The JCMSI 2021 paper provides the foundation: R. Yoshida, Y. Tanigawa, H. Okajima and N. Matsunaga, A design method of model error compensator for systems with polytopic-type uncertainty and disturbances, SICE JCMSI, Vol. 14, Issue 2 (2021) (Open Access).

LMIs and Controller Design — For background on LMI optimization in control engineering, see Linear Matrix Inequalities (LMIs) and Controller Design.

MATLAB Code

GitHub (Polytopic MEC + Non-Min Phase): Robust-control-MATLAB_MEC01 — Includes pso.m for PSO-based design and iterative_design.m for LMI-based iterative design. The stable_nmp.mat file contains the non-minimum phase plant used in the JCMSI 2022 numerical examples.
GitHub (PFC for Time-Delay and NMP Systems): MATLAB_MEC03_withPFC

Blog Articles (blog.control-theory.com)

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

Model Error Compensator / Publications / LMI

Video

Paper Information

Yoshida, H. Okajima and T. Sato, "Model error compensator design for continuous- and discrete-time non-minimum phase systems with polytopic-type uncertainties", SICE Journal of Control, Measurement, and System Integration, Vol. 15, Issue 2, pp. 141–153, 2022. DOI: 10.1080/18824889.2022.2052628 (Open Access)
Ichimasa, H. Okajima, K. Okumura and N. Matsunaga, "Model Error Compensator with Parallel Feed-Forward Filter", SICE Journal of Control, Measurement, and System Integration, Vol. 10, No. 5, pp. 468–475, 2017. DOI: 10.9746/jcmsi.10.468 (Open Access)

Related Japanese Paper: 岡島寛, 一政豪, 松永信智, 非最小位相系に対するモデル誤差抑制補償器の設計, 計測自動制御学会論文集, Vol. 51, No. 11, pp. 794–801, 2015.

Self-Introduction

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

www.control-theory.com / Blog / YouTube / GitHub

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