This article provides a detailed explanation of an extended Model Error Compensator (MEC) that uses a parallel feed-forward filter (PFF) to handle non-minimum-phase plants. The standard MEC achieves high model error suppression for minimum-phase plants but faces difficulty when the plant has unstable zeros or time delays. This paper proposes a new structure that overcomes these limitations. 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 paper.

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)

MATLAB Code (GitHub)

This paper is co-work with Gou Ichimasa (Master's student, Kumamoto University), Kosuke Okumura (Master's student, Kumamoto University), and Prof. Nobutomo Matsunaga (Professor, Kumamoto University).

Contents

Why Non-Minimum-Phase Systems Are Challenging for MEC

In model-based control, if the nominal model accurately represents the plant, the designed controller performs well. However, when a modeling error exists, the actual output can deviate significantly from the intended behavior. The Model Error Compensator (MEC) was proposed to suppress this modeling error by feeding back the difference between the plant output and the model output.

The standard MEC works well for many types of control systems, including unstable systems, nonlinear systems, and MIMO systems. However, non-minimum-phase plants — those with unstable zeros or time delays — pose a significant challenge. The standard MEC requires a high-gain compensator to achieve strong model error suppression, but designing a high-gain compensator for a non-minimum-phase system is inherently difficult because it can destabilize the closed-loop system.

Non-minimum-phase characteristics appear in many practical systems. For example, systems with right-half-plane zeros exhibit inverse response behavior, and systems with time delays introduce phase lag that limits achievable bandwidth. This paper proposes a new MEC structure with a parallel feed-forward filter (PFF) that overcomes these non-minimum-phase characteristics, enabling effective model error suppression even for these difficult plant classes.

Review of the Standard MEC

The standard MEC structure, proposed in the foundational paper (SICE JCMSI, 2013), is shown conceptually as follows. The plant $P(s)$ is augmented with a compensator that includes the nominal model $P _m(s)$ and a differential compensator $D(s)$ . The feedback signal is the difference between the plant output $y$ and the model output $y _m$ .

The transfer function of the compensated system $P _c(s)$ from the input $u _c$ to the output $y$ is:

$\displaystyle P_c(s) = P(s) \frac{1 + P_m(s) D(s)}{1 + P(s) D(s)}$

When $P(s) = P _m(s)$ , the compensated system reduces to $P _m(s)$ regardless of $D(s)$ . When a model error $\Delta P(s)$ exists, with $P(s) = P _m(s) + \Delta P(s)$ , the difference between $P _c(s)$ and $P _m(s)$ is:

$\displaystyle P_c(s) - P_m(s) = \frac{1}{1 + P(s) D(s)} \Delta P(s)$

This shows that a high-gain compensator $D(s)$ can suppress the model error. However, when $P(s)$ is a non-minimum-phase system, a high-gain $D(s)$ that stabilizes the loop $P(s)D(s)$ is difficult to design.

Non-Minimum-Phase Plants and Model

The paper considers plants with non-minimum-phase characteristics described by:

$\displaystyle P(s) = P_0(s) e^{-Ls} \prod_{t=1}^{N} \frac{s - z_i}{s + \bar{z}_i} + \Delta P(s)$

where $P _0(s)$ is a stable minimum-phase transfer function, $z _i$ represents unstable zeros, $\bar{z} _i$ is the complex conjugate of $z _i$ , $L$ is the time delay, and $\Delta P(s)$ is the additive model error. The nominal model is:

$\displaystyle P_m(s) = P_0(s) e^{-Ls} \prod_{t=1}^{N} \frac{s - z_i}{s + \bar{z}_i}$

This formulation covers three important cases: (i) plants with unstable zeros only ( $N \neq 0, L = 0$ ), (ii) plants with time delay only ( $N = 0, L \neq 0$ ), and (iii) plants with both unstable zeros and time delay ( $N \neq 0, L \neq 0$ ).

Proposed MEC with Parallel Feed-Forward Filter

The key idea of this paper is to introduce a parallel feed-forward filter (PFF) $F(s)$ into the MEC structure. Instead of using the difference between $y$ and $y _m$ as the feedback signal, the proposed structure uses the difference between $y _f = y + F(s) u$ and $y _{mf} = y _m + F(s) u _c$ as the feedback signal.

The transfer function of the compensated system becomes:

$\displaystyle P_c(s) = P(s) \frac{1 + (P_m(s) + F(s)) D(s)}{1 + (P(s) + F(s)) D(s)}$

When $P(s) = P _m(s)$ , this reduces to $P(s)$ for any $D(s)$ , preserving the fundamental property of the MEC. The critical advantage is that if $P _m(s) + F(s)$ becomes a stable minimum-phase system, the compensator $D(s)$ can be designed using the same framework as the standard MEC for minimum-phase plants.

When a model error $\Delta P(s)$ exists, the error is expressed through:

$\displaystyle P_{cf}(s) - P_m = \gamma(s) \Delta P(s)$

where $\gamma(s)$ is:

$\displaystyle \gamma(s) = \frac{1 + F(s) D(s)}{1 + (P(s) + F(s)) D(s)}$

The design conditions are: (C1) suppress the difference between $P _{cf}(s)$ and $P _m(s)$ , and (C2) ensure stability of $P _{cf}(s)$ for any given $P(s)$ .

Design of the PFF and Compensator

Design of the PFF

The PFF $F(s)$ is designed differently depending on the type of non-minimum-phase characteristics.

Case (i): Unstable zeros only ( $N \neq 0, L = 0$ ). The filter $F(s)$ is designed by minimizing the evaluation function:

$\displaystyle \mathcal{J} = \left\lVert W_e(s) \frac{F(s)}{P_m(s) + F(s)} \right\rVert_\infty$

under the condition that $P _m(s) + F(s)$ is a stable minimum-phase system. This optimization is performed using particle swarm optimization (PSO).

Case (ii): Time delay only ( $N = 0, L \neq 0$ ). The filter $F(s)$ takes the form of a Smith predictor:

$\displaystyle F(s) = P_0(s)(1 - e^{-Ls})$

Case (iii): Both unstable zeros and time delay ( $N \neq 0, L \neq 0$ ). The filter is constructed by combining the above two approaches. First, $F _1(s)$ handles the time delay component, and then $F _2(s)$ handles the remaining undershoot characteristics. The total PFF is $F(s) = F _1(s) + F _2(s)$ .

Design of the Compensator

Once $F(s)$ is designed so that $P _m(s) + F(s)$ is a stable minimum-phase system, the compensator $D(s)$ is designed by solving the following problem.

Problem 1: Find $D(s)$ that minimizes:

$\displaystyle \Gamma_1 = \sup_{\Delta P(s)} \left\lVert W_e(s) \frac{1}{1 + (P(s) + F(s)) D(s)} \right\rVert_\infty$

subject to $D(s)$ being a robustly stabilizing controller for the loop transfer function $(P(s) + F(s))D(s)$ . This is a standard $\mu$ synthesis problem and can be solved numerically with MATLAB.

Elimination of Steady-State Errors

An important property is established for the elimination of constant disturbances and steady-state errors. When $D(s)$ has an integrator and $F(s)$ has a zero at $s = 0$ , i.e.,

$\displaystyle D(s) = \frac{1}{s} D_0(s), \quad F(s) = s F_0(s)$

the error signal $\gamma(s)$ has a zero at $s = 0$ (as long as $P(s)$ does not have a zero at the origin). This ensures that the steady-state error vanishes for step inputs. Furthermore, the transfer function from a step disturbance at the plant input to the output also has a zero at $s = 0$ , confirming that step disturbances are rejected.

When $F(s)$ is designed based on the Smith predictor in the time-delay case, it automatically has a zero at $s = 0$ , so steady-state error elimination is achieved without additional design effort.

Numerical Examples

The paper demonstrates the method on a non-minimum-phase plant with both an unstable zero and a time delay:

$\displaystyle P(s) = \left( \frac{-(s-3)}{(s+1)(s+2)} + \frac{0.1}{(s+1)} \Delta(s) \right) e^{-0.5s}$

where the infinity norm of $\Delta(s)$ is less than 1. The nominal model is:

$\displaystyle P_m(s) = \frac{-(s-3)}{(s+1)(s+2)} e^{-0.5s}$

The PFF is constructed in two parts. The first part $F _1(s)$ handles the time delay component, and the second part $F _2(s)$ handles the undershoot, designed via PSO. The resulting $P _m(s) + F(s)$ behaves as a minimum-phase system, with the undershoot and time-delay effects compensated.

The compensator $D(s)$ is then designed by solving the $\mu$ synthesis problem. Simulation results confirm effectiveness for both step and sinusoidal inputs. The outputs of the compensated system $P _c(s)$ closely track those of the nominal model $P _m(s)$ , demonstrating that the model error is effectively suppressed despite the non-minimum-phase characteristics and the presence of plant uncertainty.

Extension to MIMO Systems

The paper also extends the proposed method to MIMO (multiple-input/multiple-output) systems. For an $n$ -input $n$ -output plant, the PFF $F(s)$ is designed so that $P _M(s) + F(s)$ has minimum-phase characteristics, and the compensator $D(s)$ is designed by solving the MIMO version of the optimization problem:

$\displaystyle \Gamma_2 = \sup_{\Delta P(s)} \left\lVert W_e(s) (I + (P(s) + F(s)) D(s))^{-1} \right\rVert_\infty$

where $I$ is the identity matrix. The MIMO extension builds on the existing MEC design for MIMO systems, making the proposed approach applicable to a wide class of practical multi-variable control problems.

This work is part of a broader research program on the Model Error Compensator:

Foundational MEC 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 original paper that proposed the MEC structure and its design method. The present paper extends this work to non-minimum-phase plants. See also the blog article.

MEC for Polytopic Uncertainty — 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 (2021). Extends the MEC to handle polytopic-type uncertainty using LMI-based design and common Lyapunov functions.

MEC for Non-Minimum Phase with Polytopic Uncertainty — R. Yoshida, H. Okajima and T. Sato, Model Error Compensator Design for Continuous- and Discrete-Time Non-minimum Phase Systems with Polytopic-Type Uncertainties, SICE JCMSI (2022). Further extends the PFF-based MEC approach to handle polytopic uncertainties for both continuous- and discrete-time non-minimum-phase MIMO systems.

MEC Overview (IFAC) — H. Okajima, Model Error Compensator for adding Robustness toward Existing Control Systems, IFAC PapersOnLine, Vol. 56, Issue 2, pp. 3998–4005 (2023). A comprehensive overview of the MEC framework presented at the IFAC World Congress.

MATLAB Code

GitHub: MATLAB_MEC03_withPFC — MATLAB code for the numerical examples in this paper

Blog Articles (blog.control-theory.com)

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

Model Error Compensator / Publications / LMI

Video

YouTube: Control Engineering Channel / Portal

Paper Information

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)

Co-authors: Gou Ichimasa (Master's student, Kumamoto University), Kosuke Okumura (Master's student, Kumamoto University), Nobutomo Matsunaga (Professor, Kumamoto University)

This work was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) 16K06419.

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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