Internal Model Control and Model Error Compensator: From IMC to Add-On Robustness

This article provides a detailed comparison between Internal Model Control (IMC) and the Model Error Compensator (MEC), two control methods that share the same core principle — using the difference between a plant output and a model output as a feedback signal — but differ in their architecture and role within a control system. This article clarifies the structural relationship, the key similarities, and where the two methods diverge. The relationship to the Disturbance Observer (DOB) and GIMC is also discussed. 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 IMC and MEC?

Internal Model Control (IMC) is one of the most well-known model-based control strategies, originating from the work of Garcia and Morari (1982) and extensively developed in the textbook by Morari and Zafiriou (1989). The IMC framework provides a transparent design procedure that directly relates controller tuning to the process model, and its connection to PID tuning has made it widely adopted in the process industry.

The Model Error Compensator (MEC), developed by Okajima et al. (2013), addresses a similar problem — making the controlled system robust against model uncertainty — but takes a fundamentally different architectural approach. While IMC integrates the model into the controller structure, MEC adds a compensator to an existing control system to suppress the effect of model errors.

Despite this architectural difference, the two methods share a deep structural connection: both place the nominal model $P_{M}$ inside the feedback loop and use the difference between the actual plant output and the nominal model output as a key feedback signal. In fact, the foundational JCMSI 2013 paper on MEC explicitly describes the MEC structure as an "internal model type compensator structure." Understanding this connection — and where the two methods diverge — helps practitioners choose the right approach and appreciate how MEC extends the ideas behind IMC to a broader class of problems.

Internal Model Control: Basic Structure

The IMC Architecture

The IMC structure, proposed by Morari, consists of the following components:

The actual plant $P$
A nominal model $P_{M}$ of the plant (the "internal model")
An IMC controller $C$ for shaping the transient response

The IMC controller $C$ and the model $P_{M}$ are combined into a single controller unit (denoted $H$ in the original literature). The model $P_{M}$ is placed in parallel with the actual plant $P$ inside the feedback loop. The model error signal — the difference between the actual output $y$ and the model output $P_{M} u$ — is fed back to the controller.

Important: The IMC structure itself does not use the inverse model $P_{M}^{-1}$ . The model $P_{M}$ appears directly (not inverted) in the IMC architecture. This is a key distinction from the Disturbance Observer (DOB), which requires the inverse model in its compensator structure.

The closed-loop input-output relationship and disturbance response of IMC are given by:

$\displaystyle y = \frac{PC}{1 + (P - P_M)C}\,r + \frac{1 - P_M C}{1 + (P - P_M)C}\,Pd$

When the model is perfect ( $P = P_{M}$ ) and no disturbance exists ( $d = 0$ ):

$\displaystyle y = P_M C\,r$

By choosing $C = T / P_{M}$ , any desired input-output transfer function $T$ can be realized, just as in feedforward control. However, unlike feedforward control, IMC also feeds back the model error and disturbance, providing robustness against model uncertainty and disturbance rejection.

IMC Design Procedure

The standard IMC design chooses the controller $C$ to achieve a desired closed-loop transfer function $T$ . In the simplest case:

$\displaystyle C = \frac{T}{P_M}$

For this to be realizable, $T / P_{M}$ must be stable and proper. When $P_{M}$ has non-minimum-phase zeros, $T$ must also contain those zeros (so that the unstable poles in $P_{M}^{-1}$ are canceled). Additionally, a low-pass filter is typically introduced to ensure robustness against high-frequency model uncertainty.

Key Properties of IMC

Perfect model → ideal response: When $P = P_{M}$ and $d = 0$ , the output exactly equals $Tr$ .
Feedback of model error: Differences between $P$ and $P_{M}$ , and external disturbances, are automatically fed back for correction.
Equivalence with Smith Predictor: When $P$ includes time delays, IMC and the Smith Predictor are structurally equivalent.
Standard IMC assumes stable plants: The basic IMC requires $P$ and $P_{M}$ to be stable. For unstable plants, extensions such as 2-DOF structures are needed.

GIMC: Generalized Internal Model Control

IMC was extended to a 2-DOF structure known as GIMC (Generalized IMC) by Zhou and Ren (2001). In GIMC, the disturbance rejection performance can be tuned independently from the reference tracking performance, providing greater design flexibility than standard IMC. This 2-DOF extension addresses one of the key limitations of basic IMC, where performance and robustness are coupled through a single filter parameter. GIMC has been applied to magnetic levitation systems and fault-tolerant control.

It is also worth noting that IMC, MEC, and DOB can all be understood within the broader framework of 2-DOF control. The relationship between MEC and 2-DOF conditional feedback structures is discussed in a separate article (planned).

Model Error Compensator: Basic Structure

The MEC Architecture

The MEC has a fundamentally different role from IMC. Instead of being the controller, MEC is an add-on compensator that wraps around the existing plant to make the effective plant dynamics closer to the nominal model. The existing controller is left unchanged.

The MEC structure consists of:

The actual plant $P$
A nominal model $P_{M}$
An error compensator $D$
An existing controller (designed for $P_{M}$ )

The compensated control input $u_{c}$ is formed by adding the output of the error compensator to the original control input $u'$ :

$\displaystyle u_c = u' + D(y - y_M)$

where $y_{M}$ is the output of the nominal model $P_{M}$ driven by the same compensated input $u_{c}$ . The signal $y - y_{M}$ is the model error signal — the same type of signal used in IMC. The error compensator $D$ feeds this signal back with high gain to drive the plant output toward the model output.

Key feature: Like IMC, the MEC structure does not use the inverse model $P_{M}^{-1}$ . Both IMC and MEC place $P_{M}$ directly (not inverted) inside the feedback loop. This is in contrast to the DOB, which requires the inverse model.

Design of the Error Compensator

In the foundational JCMSI 2013 paper, the design of the error compensator $D$ is formulated as an $H\_{\infty}$ control problem. The evaluation function is:

$\displaystyle \Gamma = \min_D \sup_\Delta \left\| W_e \frac{1}{1 + P_M D(1 + W\Delta)}\,DQ \right\|_\infty$

where $W_{e}$ is a frequency weighting function and $\Delta$ represents the model uncertainty. For constant disturbance rejection and zero steady-state error, $D(j\omega) \to \infty$ as $\omega \to 0$ , which requires an integrator in $D$ .

For the simplest case (SISO, minimum phase, relative degree 1), the error compensator $D$ can be a high-gain PI controller applied to the model error signal. As the gain of $D$ increases, the effective plant dynamics converges to $P_{M}$ .

Key Properties of MEC

Add-on structure: The existing controller is not modified.
No inverse model required: The compensator $D$ operates on the output error directly.
High-gain feedback principle: The model error is suppressed by high-gain feedback.
Applicable to non-minimum phase and nonlinear systems: No model factorization or inversion is needed.

The Common Principle: Model Error Signal

IMC, MEC, and DOB all use the difference between the actual plant output and the nominal model output as a feedback signal. This model error signal captures the discrepancy due to model uncertainty, parameter variations, and external disturbances. The three methods differ in how they process this signal:

IMC: The model error signal modifies the reference input to the IMC controller $C$ . The controller and the model-based compensation are integrated into a single unit $H$ .
MEC: The model error signal is fed back through the error compensator $D$ to correct the control input. The existing controller is separate and unchanged.
DOB: The model error signal is processed through the inverse model $P_{M}^{-1}$ and a low-pass filter to estimate an equivalent disturbance, which is then subtracted from the input.

IMC and DOB: Similar Goal, Different Structure

IMC and DOB have similar feedback structures, but the DOB uses the inverse model $P_{M}^{-1}$ in its compensator, while IMC does not. Specifically:

In IMC, the feedback signal is $y - P_{M} u$ (model error), and the controller $C$ processes this signal directly.
In DOB, the feedback signal involves $P_{M}^{-1} y - u$ (inverse-model-based equivalent disturbance estimation), and a filter $D$ with $D(0) = 1$ is used.

When constructing the inverse model $P_{M}^{-1}$ is difficult (e.g., for non-minimum-phase systems), the DOB requires a filter designed to approximate the inverse at low frequencies. This is a structural limitation that neither IMC nor MEC shares.

MEC and IMC: Same Signal, Different Architecture

Both IMC and MEC use the model error signal $y - P_{M} u$ without requiring the inverse model. The foundational JCMSI 2013 paper explicitly calls the MEC structure an "internal model type compensator structure." However, IMC integrates the model-based compensation into the controller design, while MEC separates it as an independent add-on.

For a detailed comparison between MEC and DOB, see: Model Error Compensator vs Disturbance Observer: A Structural Comparison

Key Differences: Where IMC and MEC Diverge

Despite sharing the model error signal and both avoiding the inverse model, IMC and MEC differ in several fundamental ways.

1. Role in the Control System

IMC is the controller itself. The IMC controller $C$ (combined with the internal model $P_{M}$ ) determines the closed-loop transfer function. There is no separate "existing controller" — IMC replaces it.

MEC is an add-on compensator. It does not replace or modify the existing controller. The existing controller determines the control performance (tracking, regulation), while MEC handles the robustness. This separation — performance by the controller, robustness by MEC — is a central feature.

This distinction has a practical consequence: MEC can be attached to any existing control system (PID, state feedback, MPC, feedback linearization, etc.) without redesigning the controller. IMC requires designing the controller from scratch within the IMC framework.

2. Stability Requirements

Basic IMC requires both the plant $P$ and the model $P_{M}$ to be stable. For unstable plants, extensions such as GIMC or the 2-DOF conditional feedback structure are needed.

MEC can handle unstable plants directly, because the existing feedback controller stabilizes the plant, and MEC operates as an add-on within the stabilized loop.

3. Non-Minimum Phase Systems

When the plant has non-minimum-phase zeros, the IMC controller design requires $T$ to contain the same zeros as $P_{M}$ , which limits the achievable performance. The desired transfer function cannot be chosen freely.

MEC addresses non-minimum-phase systems through the parallel feedforward compensator (PFC) extension, without requiring model factorization or constraining the desired transfer function.

4. Nonlinear Systems

IMC is formulated within a transfer function framework and is difficult to extend directly to nonlinear systems. MEC, on the other hand, has been extended to nonlinear systems through robust feedback linearization. For details, see: MEC for Nonlinear Systems: Robust Feedback Linearization

5. Design Philosophy

IMC follows a unified design philosophy: the controller is designed together with the model-based compensation. The performance-robustness trade-off is managed by a single filter parameter.

MEC follows a separation design philosophy: performance is handled by the existing controller, and robustness is handled by the error compensator. This allows independent tuning — a high-performance nominal controller can be designed first, and then MEC can be added to ensure robustness without degrading the nominal performance.

Comparison Table

Aspect	IMC	MEC
Core signal	Model error $y - P_{M} u$	Model error $y - P_{M} u_{c}$
Role	Controller (replaces existing)	Add-on compensator (preserves existing)
Inverse model	Not required in structure	Not required
Inverse model in design	$C = T/P_{M}$ requires factorization	Not required
Stability requirement	$P$ and $P_{M}$ must be stable	Unstable plants handled via existing controller
Performance-robustness	Coupled (single parameter)	Separated (controller vs. compensator)
Non-minimum phase	$T$ must share zeros with $P_{M}$	PFC extension, no factorization
Nonlinear systems	Difficult	Applicable (details)
Equivalent to PID	Yes, for simple models	Can be combined with PID
Origin	Garcia and Morari (1982)	Okajima, Umei, Matsunaga, Asai (2013)

When to Use Which?

Use IMC when:

The system is a stable, linear process (typical in process control)
You are designing the controller from scratch (no pre-existing controller to preserve)
You want a transparent PID tuning rule derived from the process model
The plant includes time delays (IMC-Smith Predictor equivalence applies)

Use MEC when:

An existing controller is already in place and should not be modified
The system is non-minimum phase, nonlinear, or unstable
You want to separate performance design from robustness design
The system involves multiple control methods (PID, MPC, state feedback) that should each retain their original design
You want a general-purpose add-on that can be applied across different control systems

Practical recommendation:

For practitioners who have an existing control system that works well under nominal conditions but degrades under model uncertainty or parameter variations, MEC provides a direct solution: add MEC without touching the existing controller. For new control system designs where no controller exists yet, IMC provides an elegant starting point, particularly for SISO stable processes.

Historical Connection: From 2-DOF IMC to MEC

The connection between IMC and MEC is not only structural but also historical. The author's earlier work on 2-DOF Internal Model Control with dynamic quantizers (Okajima et al., JCMSI, 2011) used the IMC framework extended with two degrees of freedom. The development of MEC (JCMSI, 2013) can be viewed as an evolution from this 2-DOF IMC work, where the model-based compensation was separated from the controller design and reformulated as an independent add-on structure.

This evolution reflects a broader insight: while IMC couples the controller and the model-based compensation into a single entity, many practical situations benefit from decoupling them. MEC achieves this decoupling while retaining the core principle of using the model error signal for compensation.

Control Using the Signal Difference Between a Model and the Actual Plant — H. Okajima and N. Matsunaga, Systems, Control and Information (ISCIE), Vol. 60, No. 2, pp. 60–65 (2016). A Japanese-language overview that comprehensively discusses IMC, 2-DOF conditional feedback, DOB, MEC, and their structural relationships. This article is largely based on this overview paper.

MEC: The Foundational Paper — H. 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). The original paper proposing the MEC structure, describing it as an "internal model type compensator structure."

IFAC 2023 Overview — H. Okajima, Model Error Compensator for adding Robustness toward Existing Control Systems, IFAC PapersOnLine, Vol. 56, Issue 2, pp. 3998–4005 (2023). A comprehensive English-language overview of MEC, including comparisons with existing model-based compensation methods.

MEC vs Disturbance Observer — For a detailed comparison between MEC and DOB, see: Model Error Compensator vs Disturbance Observer: A Structural Comparison

MEC for Non-Minimum Phase Systems — MEC for Non-Minimum Phase Systems: PFC Approach. Extends MEC to systems where IMC's factorization-based controller design faces fundamental limitations.

MEC for Nonlinear Systems — MEC for Nonlinear Systems: Robust Feedback Linearization. Extends MEC to nonlinear systems via output-feedback linearization using the model error signal.

MEC + PID Control — MEC + PID Control: Adding Robustness to the Most Widely Used Controller. A practical demonstration of MEC's add-on philosophy: enhancing an existing PID controller without modifying it.

MATLAB Code

GitHub (MEC with LMI/PSO design): Robust-control-MATLAB_MEC01
GitHub (MEC with PFC): MATLAB_MEC03_withPFC
GitHub (Nonlinear control with MEC): non_linear_control_MATLAB_MEC04
GitHub (MEC + PID): MEC05-rengo2022

Blog Articles (blog.control-theory.com)

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

Model Error Compensator / Publications / LMI

Video

YouTube: Control Engineering Channel / Portal

Key References

C.E. Garcia and M. Morari, "Internal Model Control. 1. A Unifying Review and Some New Results," Ind. Eng. Chem. Process Des. Dev., Vol. 21, pp. 308–323, 1982.
1. Morari and E. Zafiriou, Robust Process Control, Prentice Hall, 1989.
D.E. Rivera, M. Morari, and S. Skogestad, "Internal Model Control. 4. PID Controller Design," Ind. Eng. Chem. Process Des. Dev., Vol. 25, pp. 252–265, 1986.
1. Zhou and Z. Ren, "A New Controller Architecture for High Performance, Robust, and Fault-Tolerant Control," IEEE Trans. Automatic Control, Vol. 46, No. 10, pp. 1613–1618, 2001.
1. 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.

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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