This article provides a detailed explanation of a compensator design method that minimizes the modeling gap between a nominal model and the actual plant. This paper is the foundational work that later developed into a broader research program known as the Model Error Compensator (MEC). 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.

Okajima, H. Umei, N. Matsunaga and T. Asai, A Design Method of Compensator to Minimize Model Error, SICE Journal of Control, Measurement, and System Integration, Vol. 6, No. 4, pp. 267–275 (2013)

This paper is co-work with Hironori Umei (Master's student, Kumamoto University), Prof. Nobutomo Matsunaga (Professor, Kumamoto University), and Prof. Toru Asai (Osaka University at the time of publication; currently Professor, Chubu University).

Contents

Why Minimizing Model Error Matters

In model-based control system design, a controller is designed using a nominal model $P _{n}$ of the actual plant $P$ . If the model is accurate, the controller performs as expected. However, in practice, there is always a modeling gap between $P$ and $P _{n}$ due to parameter variations, unmodeled dynamics, and other uncertainties.

Traditional robust control methods address this problem by designing a controller that achieves desired performance for all plants in a given uncertainty set. While effective, this approach has an inherent limitation: because the controller must work for all possible plants, the resulting performance tends to be conservative (e.g., slower response). Moreover, important specifications such as rise time and overshoot cannot always be handled directly within the robust control framework.

This paper takes a fundamentally different approach. Instead of designing a robust controller, it proposes to attach a compensator to the plant itself so that the compensated plant behaves like the nominal model. If the compensated plant dynamics is close to $P _{n}$ , then any controller designed for the nominal model will work well, regardless of the actual plant uncertainty. This idea separates the robustness design from the control performance design, providing great flexibility in the overall control system design.

Basic Idea

Consider the standard model-based control design flow. A nominal model $P _{n}$ is obtained, a controller is designed using $P _{n}$ , and then the controller is applied to the actual plant $P$ . If the model error

$\displaystyle \Delta P = P - P_n$

is large, the control system may not achieve the expected performance.

The key idea is to introduce a compensator $H$ that wraps around the actual plant $P$ to produce a compensated system $P^{\prime}$ whose dynamics is close to $P _{n}$ . The compensated system $P^{\prime}$ is then used in the control system instead of the plant itself. The design specifications for $H$ are:

C1: The model error of the compensated system $\Delta P^{\prime} = P^{\prime} - P _{n}$ should be smaller than the original $\Delta P$ in terms of the input-output relation.

C2: When $P = P _{n}$ (no model error), the compensated system should satisfy $P^{\prime} = P _{n}$ exactly.

Proposed Compensator Structure

To satisfy the specification C2, the paper proposes a compensator structure that includes the nominal model $P _{n}$ internally. The compensator measures the response difference between the actual plant output $y$ and the nominal model output $y _{n}$ , and feeds this difference back through a differential compensator $D$ to correct the control input.

The structure works as follows: the same input $u$ is applied to both the actual plant $P$ and the nominal model $P _{n}$ . The output difference $y - y _{n}$ is processed by $D$ and subtracted from the input. This creates a feedback loop that drives the plant output toward the nominal model output.

Since the feedback signal $y - y _{n}$ is zero when $P = P _{n}$ , the specification C2 is automatically satisfied for any choice of $D$ .

Transfer Function of the Compensated System

For a SISO linear time-invariant plant, the transfer function of the compensated system is:

$\displaystyle P'(s) = \frac{1 + P_n(s)D(s)}{1 + (P_n(s) + \Delta P(s))D(s)} (P_n(s) + \Delta P(s))$

From this expression, we can verify that $P^{\prime}(s) = P _{n}(s)$ holds when $\Delta P(s) = 0$ for any $D(s)$ . Furthermore, if $D(s)$ provides high-gain feedback, the effect of the model error is suppressed, and the dynamics of $P^{\prime}(s)$ approaches that of $P _{n}(s)$ .

The model error of the compensated system is:

$\displaystyle \Delta P'(s) = \frac{1}{1 + (P_n(s) + \Delta P(s))D(s)} \Delta P(s)$

This has the form of the original model error $\Delta P(s)$ multiplied by the sensitivity function of a unity feedback system. By designing $D(s)$ appropriately, the compensated model error $\Delta P^{\prime}(s)$ can be made smaller than the original $\Delta P(s)$ .

Design of the Differential Compensator for Stable Plants

Uncertainty Model

The model error is assumed to have a multiplicative structure:

$\displaystyle \Delta P(s) = \Delta(s) W(s) P_n(s), \quad \Delta(s) \in S_\Delta$

where $W(s)$ is a weighting function that characterizes the frequency-dependent magnitude of the uncertainty, and $S _{\Delta}$ is the set of stable transfer functions with H-infinity norm bounded by 1.

Design Problem (Problem 1)

The design of $D(s)$ is formulated as the minimization of the worst-case weighted sensitivity:

$\displaystyle \Gamma = \min_{D(s)} \sup_{\Delta(s)} \lVert W_e(s) \gamma(s) \rVert_\infty$

where $W _{e}(s)$ is an evaluation weighting function (typically a low-pass filter), and

$\displaystyle \gamma(s) = \frac{1}{1 + P_n(s)D(s)(1 + W(s)\Delta(s))}$

subject to the robust stability condition:

$\displaystyle \left\lVert W(s) \frac{P_n(s)D(s)}{1 + P_n(s)D(s)} \right\rVert_\infty < 1$

This robust stability condition requires the complementary sensitivity function $T(s) = P _{n}(s) D(s) / (1 + P _{n}(s) D(s))$ to be sufficiently small relative to the uncertainty weight $W(s)$ .

The design problem is a robust performance problem and can be solved by standard mu synthesis using the main loop theorem. If a small value of $\Gamma$ is achieved, the compensated model error is small at low frequencies (where $W _{e}$ has large gain), meaning the compensated plant dynamics is close to the nominal model dynamics.

When $\Gamma$ is less than 1, the set of compensated plants is strictly smaller than the original set of plants in the sense of the H-infinity norm, confirming that the specification C1 is satisfied.

Design for Feedback Control Systems

Extension to General Plants Including Unstable Systems

The compensator can also be used in feedback control systems, allowing the method to handle unstable plants. Suppose a feedback controller $C _{FB}$ is already designed for the nominal plant $P _{n}$ to achieve good nominal performance. The proposed compensator is then added to the plant, and the difference between the nominal closed-loop output and the compensated closed-loop output is characterized by:

$\displaystyle \frac{\hat{y} - \hat{y}_n}{\hat{r}} = \gamma^*(s) \Delta P(s)$

where $\gamma^*(s)$ depends on $P _{n}(s)$ , $C _{FB}(s)$ , $D(s)$ , and $\Delta P(s)$ . For the explicit form, see Eq. (28) in the paper.

The internal stability of the feedback control system with the compensated plant is equivalent to the stability of a feedback system with loop transfer function $L(s) = P _{n}(s) D(s)$ . This result (proved in the paper's Appendix A) means that the compensator design reduces to stabilizing a loop around the nominal model rather than the uncertain plant.

The design problem (Problem 2) minimizes the evaluation function while ensuring both internal stability and robust stability of the closed-loop system. The differential compensator $D(s)$ provides an additional design degree of freedom for achieving robust stability while maintaining the same nominal performance.

Extension to Nonlinear Plants

An important advantage of the proposed method is its applicability to nonlinear plants. Two cases are discussed:

Input-Affine Nonlinear Systems

For a nonlinear plant of the form $\dot{x} = f(x) + g(x)u$ , $y = h(x)$ , the design proceeds by first applying input-output approximate linearization to obtain a linearized model, then designing $D$ using the linear design method, and finally applying the designed $D$ to the original nonlinear plant.

Hammerstein Systems

For Hammerstein models, which consist of a static nonlinear gain $I(u)$ followed by linear dynamics $T(s)$ , the compensator structure is applied after the nonlinear gain. Because the difference between the plant and the nominal model lies in the linear dynamic part, the right-hand side of the compensator structure can be treated as a linear time-invariant system, and the design method for linear plants applies directly.

Numerical Examples

Feedforward Control Example

The paper demonstrates the method with a first-order minimum-phase plant $P(s) = K _{1} / (T _{1} s + 1)$ with parameter variation $T _{1} \in \lbrack 0.8, 1.2 \rbrack$ and $K _{1} \in \lbrack 0.9, 1.1 \rbrack$ , and nominal model $P _{n}(s) = 1/(s+1)$ .

A feedforward controller $C(s) = (s+1)/(0.2s+1)$ is used. First, a simple PI-type differential compensator $D(s) = (10s+3)/s$ is applied. The simulation results show that the response variation with the compensator is smaller than without it, and all outputs are close to the nominal response.

Then, $D(s)$ is designed using Problem 1 with weighting functions $W(s) = (0.375s + 0.1)/(s+1)$ and $W _{e}(s) = 1/(5s+1)$ . The resulting performance index is:

$\displaystyle \Gamma < \mu = 0.001913$

This very small value confirms that the compensated plant dynamics is extremely close to the nominal model. The output responses with the designed $D(s)$ show nearly zero variation — the individual responses are indistinguishable from each other.

Discussion Points

The paper also discusses several practical considerations: the difficulty of designing $D(s)$ for non-minimum phase plants (where high-gain feedback is limited due to stability constraints), the issue of input signal constraints (the compensator modifies the input signal, so large model errors lead to large input variations), and comparisons with traditional robust control and disturbance observers.

A key advantage over disturbance observers is that the proposed structure does not require an inverse model of the plant, making it applicable to systems where computing the inverse is difficult (e.g., nonlinear systems).

Position as the Origin of the Model Error Compensator (MEC)

Although this 2013 paper does not use the term "Model Error Compensator" (MEC), it is the foundational paper that introduced the core concept and structure of the MEC. The idea of attaching an internal-model-type compensator to the plant to minimize the modeling gap, using the difference between the actual output and the nominal model output as a feedback signal, is precisely the MEC structure that was subsequently refined, extended, and applied to a wide range of control problems.

Later research built upon this work to develop the MEC framework for MIMO systems, polytopic-type uncertainties, non-minimum phase systems with parallel feedforward compensators, and various practical applications including vehicle control, wheelchair control, and integration with model predictive control. The name "Model Error Compensator" was adopted in subsequent publications to describe this family of methods, and this 2013 SICE JCMSI paper serves as the origin of the entire MEC research program.

Model Error Compensator (MEC) — Overview — The MEC research has grown into a comprehensive framework. For a detailed summary covering the basic structure, design methods, applications to nonlinear systems, integration with PID control and MPC, and comparisons with disturbance observers, see the MEC overview article and the IFAC paper: H. Okajima, Model Error Compensator for adding Robustness toward Existing Control Systems, IFAC PapersOnLine, Vol. 56, Issue 2, pp. 3998–4005 (2023).

MEC for Polytopic-Type 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 design to systems with polytopic uncertainty descriptions using LMI-based methods.

MEC for Non-Minimum Phase Systems — H. Okajima and T. Sato, Model Error Compensator Design for Continuous- and Discrete-Time Non-minimum Phase Systems with Polytopic-Type Uncertainties, SICE JCMSI, Vol. 15, No. 2, pp. 37–49 (2022). Addresses the non-minimum phase limitation discussed in this paper by introducing a parallel feedforward compensator.

MEC with Parallel Feed-Forward Filter — G. Ichimasa, H. Okajima, K. Okumura and N. Matsunaga, Model Error Compensator with Parallel Feed-Forward Filter, SICE JCMSI, Vol. 10, No. 5, pp. 468–475 (2017). Enhances the MEC structure with a parallel feedforward filter for improved performance.

MATLAB Code

GitHub (MEC design): Robust-control-MATLAB_MEC01 — MATLAB codes for MEC design with polytopic-type uncertainty (PSO-based and iterative design methods)
GitHub (MEC with sensor noise): MATLAB_MEC02_sensor_noise — MEC design using S(s) and T(s)
GitHub (MEC for nonlinear systems): non_linear_control_MATLAB_MEC04 — MEC applied to nonlinear systems

Note: These GitHub repositories correspond to later MEC papers that extended the method proposed in this foundational paper.

Blog Articles (blog.control-theory.com)

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

Model Error Compensator / Publications

Video

YouTube: Control Engineering Channel / Portal

Paper Information

Okajima, H. Umei, N. Matsunaga and T. Asai, "A Design Method of Compensator to Minimize Model Error", SICE Journal of Control, Measurement, and System Integration, Vol. 6, No. 4, pp. 267–275, 2013. DOI: 10.9746/jcmsi.6.267

Co-authors: Hironori Umei (Master's student, Kumamoto University), Nobutomo Matsunaga (Professor, Kumamoto University), Toru Asai (Graduate School of Engineering, Osaka University at the time of publication; currently Professor, Department of Mechanical Engineering, Chubu University)

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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