This article provides a comprehensive overview of model-based compensation methods in control engineering — control structures that place a plant model inside the feedback loop and use the difference between the model output and the actual plant output as a feedback signal. These methods share a common principle but differ in their architecture, assumptions, and applicability. This article explains each method, clarifies the structural relationships among them, and provides guidelines for choosing the right approach.

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

This article is based on the following survey paper by the author.

Okajima and N. Matsunaga, Control Using the Signal Difference Between the Model and the Actual Plant (モデルと実対象の信号差を利用した制御), Systems, Control and Information (システム／制御／情報), Vol. 60, No. 2, pp. 60–65, 2016.

Okajima, Model Error Compensator for adding Robustness toward Existing Control Systems, IFAC PapersOnLine, Vol. 56, Issue 2, pp. 3998–4005 (2023) (Free access, Elsevier)

Why Model-Based Compensation Matters
The Common Principle: Using the Model Error Signal
Internal Model Control (IMC)
Smith Predictor: Dead-Time Compensation via Internal Model
Two-Degree-of-Freedom Control and the Conditional Feedback Structure
Disturbance Observer (DOB)
Model Error Compensator (MEC)
Structural Comparison: How the Methods Relate
- Structural Relationships
- When to Use Which?
Connections to Related Research
Related Articles and Videos
Key References
ModelBasedCompensation #InternalModelControl #IMC #SmithPredictor #DisturbanceObserver #ModelErrorCompensator #MEC #2DOFControl #RobustControl #ControlEngineering #MATLAB

Why Model-Based Compensation Matters

In model-based control, a mathematical model of the plant is used to design the controller. When the model is accurate, the designed controller achieves the desired performance. However, in practice, model uncertainty, parameter variations, and external disturbances inevitably cause a gap between the model and the actual plant. This gap degrades control performance and can even destabilize the system.

A natural idea to address this problem is to place the model inside the feedback loop and use the signal difference between the model output and the actual plant output as a feedback signal. When the model matches the plant perfectly and no disturbance exists, this signal difference is zero and the compensation is inactive. When a gap exists, the compensation activates to suppress its effect. This conditional activation is the common principle shared by a family of model-based compensation methods.

Several well-known control methods employ this principle, including:

Internal Model Control (IMC) — a controller design framework that directly incorporates the plant model
Smith Predictor — a dead-time compensation strategy equivalent to IMC for time-delay systems
Two-Degree-of-Freedom (2-DOF) Conditional Feedback — a control architecture that independently adjusts tracking and robustness
Disturbance Observer (DOB) — an equivalent-disturbance estimation and cancellation approach
Model Error Compensator (MEC) — an add-on compensator that enhances robustness of existing control systems

Each method uses the model–plant signal difference in a structurally different way, leading to different design procedures, applicability constraints, and practical advantages. Understanding these relationships helps practitioners choose the most appropriate method for their application.

The Common Principle: Using the Model Error Signal

All model-based compensation methods considered in this article share the following structural feature: the nominal model $P_{M}$ is placed in parallel with the actual plant $P$ inside the feedback loop, and the model error signal

$\displaystyle e = y - y_M = y - P_M u$

is used as a key feedback signal. Here, $y$ is the actual plant output and $y_{M}$ is the model output. When the model is perfect ( $P = P_{M}$ ) and no disturbance exists, $e = 0$ and the compensation is inactive. When a gap exists, $e \neq 0$ and the compensation activates.

The methods differ in how this signal is processed:

IMC uses $e$ to modulate the controller output through a model-based controller $C$ .
DOB estimates an equivalent input disturbance from $e$ using the inverse model $P_{M}^{-1}$ and cancels it.
MEC applies high-gain feedback on $e$ through an error compensator $D$ , without using the inverse model.
2-DOF conditional feedback uses $e$ to activate feedback only when the model–plant gap or disturbance exists.

These processing differences lead to fundamentally different design philosophies, stability requirements, and applicable system classes.

Internal Model Control (IMC)

Internal Model Control (IMC) was proposed by Garcia and Morari (1982) and comprehensively developed in the textbook by Morari and Zafiriou (1989). IMC is one of the most widely used model-based control frameworks, particularly in the process industry.

Architecture

The IMC structure consists of the actual plant $P$ , a nominal model $P_{M}$ , and an IMC controller $C$ . The controller and model together form a single controller unit $H$ . The model $P_{M}$ is placed in parallel with the plant inside the feedback loop. The model error signal $e = y - P_{M} u$ is fed back to the controller $C$ .

When the model is perfect ( $P = P_{M}$ ) and no disturbance exists, the closed-loop response is:

$\displaystyle y = P_M C \, r$

By choosing $C = T / P_{M}$ , the desired transfer function $T$ can be achieved — the same as feedforward control. However, unlike feedforward control, IMC feeds back the model error and disturbance, providing robustness.

Key Properties

Transparent design: The IMC controller $C$ is directly related to the plant model $P_{M}$ and the desired transfer function $T$ , making the tuning transparent.
Connection to PID: The standard IMC design produces controllers equivalent to well-tuned PID controllers for many common plant models. This connection has made IMC-based PID tuning popular in the process industry.
Stability assumption: In its basic form, IMC requires the plant $P$ and the model $P_{M}$ to be stable. For unstable plants, the basic IMC structure cannot be directly applied.

GIMC: Generalized Internal Model Control

The Generalized IMC (GIMC) framework extends IMC to a 2-degree-of-freedom structure that can handle unstable plants and provides independent adjustment of disturbance rejection performance. GIMC separates the stabilizing controller from the performance-shaping controller, offering greater design flexibility while retaining the transparency of the IMC philosophy.

For a detailed structural comparison between IMC and MEC, including the GIMC interpretation, see: Internal Model Control and MEC: From IMC to Add-On Robustness

Smith Predictor: Dead-Time Compensation via Internal Model

The Smith Predictor is a control structure specifically designed for plants with time delay (dead time). It can be viewed as a special case of the internal model principle applied to time-delay systems.

Architecture

The Smith Predictor places the delay-free part of the model inside a feedback loop, so that the controller "sees" the delay-free plant response rather than the delayed response. The controller can then be designed as if the plant had no delay.

The Smith Predictor structure uses a signal:

$\displaystyle e_{\mathrm{Smith}} = y - P_M e^{-Ls} u + P_{M0} u$

where $P_{M} e^{-Ls}$ is the full model including delay and $P_{M0}$ is the delay-free part of the model. This effectively subtracts the predicted delayed output and adds the predicted delay-free output, allowing the controller to respond to model errors and disturbances without being limited by the dead time.

Connection to IMC

It is well known that the IMC structure and the Smith Predictor are equivalent for time-delay systems. When the plant contains dead time, the IMC framework naturally produces the Smith Predictor structure. This equivalence was established in the IMC literature and provides a unified understanding of both approaches.

Limitations

The Smith Predictor shares the limitations of basic IMC:

It requires the plant to be open-loop stable (for the basic form).
It requires an accurate estimate of the dead time $L$ . Mismatch in the dead-time estimate can significantly degrade performance.
It is difficult to extend to nonlinear or non-minimum phase systems.

Two-Degree-of-Freedom Control and the Conditional Feedback Structure

Two-degree-of-freedom (2-DOF) control resolves the fundamental conflict in single-degree-of-freedom control between reference tracking and disturbance rejection by providing two independent design parameters.

The Conditional Feedback Structure

The specific 2-DOF structure relevant to model-based compensation is the conditional feedback structure, analyzed by Sugie and Yoshikawa (1986). This structure consists of:

A feedforward path $T / P_{M}$ that shapes the reference response
A feedback controller $C_{1}$ that handles disturbance rejection and robustness
The desired transfer function $T$ placed in the feedback path

When the model matches the plant ( $P = P_{M}$ ), the closed-loop transfer function is:

$\displaystyle \frac{y}{r} = T$

regardless of the feedback controller $C_{1}$ . The feedback activates only when a model–plant gap or disturbance exists. This is why the structure is called "conditional feedback."

Key Properties

Clean separation: Reference tracking is determined by $T$ alone; disturbance rejection is determined by $C_{1}$ alone. The two objectives can be designed independently.
Unstable plants: Unlike basic IMC, this structure can handle unstable plants — the feedback controller $C_{1}$ provides the necessary stabilization.
Non-minimum phase constraint: When the plant is non-minimum phase, the desired transfer function $T$ must contain the same unstable zeros as the plant, since $T / P_{M}$ must be stable and proper.

Structural Origin of MEC

The conditional feedback structure is the direct structural origin of the Model Error Compensator. Setting the desired transfer function to $T = P_{M}$ in the 2-DOF structure produces the MEC architecture. For linear systems with a feedforward input, MEC and the 2-DOF conditional feedback structure become equivalent.

For the detailed derivation and extensions, see: Two-Degree-of-Freedom Control and MEC: Origin and Extensions

Disturbance Observer (DOB)

The Disturbance Observer (DOB) was proposed by Ohnishi (1983) for motion control of servo systems and has since been widely adopted in robotics, mechatronics, and hard disk drive control.

Architecture

The DOB treats all discrepancies between the nominal model $P_{M}$ and the actual plant $P$ as an equivalent input disturbance $d_{eq}$ . By estimating and canceling this equivalent disturbance, the plant appears to behave like the nominal model.

The estimated equivalent disturbance is:

$\displaystyle \hat{d}_{eq} = Q \left( P_M^{-1} y - u \right)$

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

Key Requirement: Inverse Model

The DOB structure inherently requires the inverse model $P_{M}^{-1}$ . This poses fundamental challenges for:

Non-minimum phase systems: The inverse model has unstable poles, making the DOB unstable.
Systems with high relative degree: The Q-filter must provide sufficient roll-off to make the inverse proper.
Nonlinear systems: A transfer function inverse does not exist in a straightforward sense.

Connection to Other Methods

The DOB has a structural similarity to IMC — both feed back the model error signal. However, the DOB processes this signal through the inverse model, while IMC uses a model-based controller. When the model is minimum phase and the Q-filter bandwidth and the IMC controller bandwidth are matched, the two methods can achieve equivalent disturbance rejection performance.

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

Model Error Compensator (MEC)

The Model Error Compensator (MEC) was proposed by Okajima et al. (2013) as a general-purpose method for enhancing the robustness of existing control systems. Unlike IMC (which redesigns the controller) or DOB (which requires the inverse model), MEC adds a simple compensator to an existing control system without modifying the original controller and without requiring an inverse model.

Architecture

The MEC structure consists of a nominal model $P_{M}$ and an error compensator $D$ . The model is placed in parallel with the actual plant, and the error compensator feeds back the model error signal $e = y - P_{M} u_{c}$ as high-gain feedback:

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

where $u_{c}$ is the compensated control input applied to the actual plant and $u$ is the original control input from the controller. By designing $D$ with sufficiently high gain, the model error is driven toward zero, and the effective plant dynamics approximate the nominal model.

Key Advantages

No inverse model required: MEC uses high-gain feedback on the output error, not the inverse model. This makes it naturally applicable to non-minimum phase systems, nonlinear systems, and unstable systems.
Add-on structure: MEC is added to the existing control system without modifying the original controller. This is valuable when the existing controller is already tuned, certified, or difficult to change.
Simple error compensator: For minimum-phase systems with relative degree 1, the error compensator $D$ can be as simple as a PI controller with high gain.
Broad applicability: MEC has been extended to MIMO systems, systems with polytopic uncertainty (via LMI design), non-minimum phase systems, and nonlinear systems. For details on each extension, see the MEC Hub article below.

Connection to 2-DOF Control

As described above, the MEC structure corresponds to setting $T = P_{M}$ in the 2-DOF conditional feedback structure. For linear systems with a feedforward input, the two are equivalent. However, MEC extends beyond this equivalence through freedom in signal selection, the PFC extension for non-minimum phase systems, and application to nonlinear systems.

For the comprehensive guide on MEC, including design methods, system class extensions, MATLAB code, and application examples, see: Model Error Compensator (MEC): Enhance the Robustness of Existing Control Systems

Structural Comparison: How the Methods Relate

The following table summarizes the key structural differences among the model-based compensation methods discussed in this article.

Feature	IMC	Smith Predictor	2-DOF Conditional FB	DOB	MEC
Inverse model required	No (in the structure itself)	No	No	Yes	No
Existing controller preserved	No (redesigns controller)	No	No	Yes (add-on)	Yes (add-on)
Unstable plants	Not in basic form	Not in basic form	Yes	Yes (with modifications)	Yes
Non-minimum phase	Limited (T must share zeros)	Limited	Limited (T must share zeros)	Difficult (inverse model unstable)	Yes (with PFC extension)
Nonlinear systems	Limited	No	Limited	Limited	Yes (see MEC Hub)
Dead-time systems	Yes (equivalent to Smith)	Specialized for this	Yes	Yes (with Q-filter design)	Yes (with PFC extension)
Design philosophy	Model-based controller design	Dead-time separation	Independent tracking/robustness	Equivalent disturbance cancellation	Model error suppression via high-gain FB

Structural Relationships

The methods are not independent — they are connected through a network of structural relationships:

IMC ↔ Smith Predictor: Equivalent for time-delay systems.
IMC → GIMC: GIMC extends IMC to a 2-DOF structure with independent disturbance rejection.
2-DOF conditional feedback → MEC: Setting $T = P_{M}$ in the conditional feedback structure yields MEC.
DOB ↔ MEC: For minimum-phase SISO systems, equivalent disturbance rejection can be achieved when the DOB Q-filter bandwidth and the MEC error compensator gain are matched.
IMC → MEC: MEC can be interpreted as an add-on robustness enhancement within the GIMC framework.

When to Use Which?

Use IMC when designing a controller from scratch for a stable plant where transparent tuning and PID equivalence are valued.
Use Smith Predictor when the dominant challenge is a known dead time in a stable plant.
Use 2-DOF conditional feedback when independent adjustment of tracking and robustness is the primary design goal.
Use DOB when the plant is minimum phase, the inverse model is readily available, and the existing servo control infrastructure supports it (common in mechatronics).
Use MEC when you want to add robustness to an existing control system without modifying the controller, or when the plant is non-minimum phase, nonlinear, or unstable — situations where DOB and basic IMC are difficult to apply.
Try both DOB and MEC when both are applicable — for minimum-phase linear systems, both can achieve equivalent performance, and the better choice may depend on the specific noise environment and implementation constraints.

Model-based compensation methods interact closely with other areas of control engineering:

System Identification — All methods require a nominal model $P_{M}$ . The quality of the model directly affects compensation performance. For a comprehensive guide to system identification methods including parametric, subspace, and multirate approaches, see System Identification: From Data to Dynamical Models.

State Observer and State Estimation — Observer-based control and model-based compensation are complementary: the observer provides state estimates from output measurements, while the compensator handles model uncertainty. For the state estimation guide, see State Observer and State Estimation.

State Feedback Control — State feedback design provides the nominal controller that can be combined with model-based compensation methods. For the comprehensive guide, see State Feedback Control and State-Space Design.

LMI-Based Design — LMI optimization provides systematic design tools for the error compensator in MEC, particularly for systems with polytopic-type uncertainty. See Linear Matrix Inequalities (LMIs) and Controller Design.

Blog Articles (blog.control-theory.com)

Key References

1. Morari and E. Zafiriou, Robust Process Control, Prentice-Hall, Englewood Cliffs (1989) — The foundational IMC textbook.
1. Sugie and T. Yoshikawa, General Solution of Robust Tracking Problem in Two-Degree-of-Freedom Control Systems, Trans. SICE, Vol. 22, No. 2, pp. 156–160 (1986) — The conditional feedback structure.
1. Ohnishi, A New Servo Method in Mechatronics, Trans. Japanese Society of Electrical Engineers, Vol. 107-D, pp. 83–86 (1987) — The disturbance observer.
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) (Open Access) — The foundational MEC paper.
1. Okajima and N. Matsunaga, Control Using the Signal Difference Between the Model and the Actual Plant, Systems, Control and Information, Vol. 60, No. 2, pp. 60–65 (2016) — The survey paper covering IMC, 2-DOF, DOB, and MEC.
1. Okajima, Model Error Compensator for adding Robustness toward Existing Control Systems, IFAC PapersOnLine, Vol. 56, Issue 2, pp. 3998–4005 (2023) (Free access) — Comprehensive MEC overview including PID integration.

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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ModelBasedCompensation #InternalModelControl #IMC #SmithPredictor #DisturbanceObserver #ModelErrorCompensator #MEC #2DOFControl #RobustControl #ControlEngineering #MATLAB

制御工学ブログ / Control Engineering Blog

Model-Based Compensation Methods in Control Engineering: IMC, DOB, 2-DOF, and MEC

Why Model-Based Compensation Matters

The Common Principle: Using the Model Error Signal

Internal Model Control (IMC)

Architecture

Key Properties

GIMC: Generalized Internal Model Control

Smith Predictor: Dead-Time Compensation via Internal Model

Architecture

Connection to IMC

Limitations

Two-Degree-of-Freedom Control and the Conditional Feedback Structure

The Conditional Feedback Structure

Key Properties

Structural Origin of MEC

Disturbance Observer (DOB)

Architecture

Key Requirement: Inverse Model

Connection to Other Methods

Model Error Compensator (MEC)

Architecture

Key Advantages

Connection to 2-DOF Control

Structural Comparison: How the Methods Relate

Structural Relationships

When to Use Which?

Blog Articles (blog.control-theory.com)

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

Video

Video Portal

Key References

ModelBasedCompensation #InternalModelControl #IMC #SmithPredictor #DisturbanceObserver #ModelErrorCompensator #MEC #2DOFControl #RobustControl #ControlEngineering #MATLAB

Why Model-Based Compensation Matters

The Common Principle: Using the Model Error Signal

Internal Model Control (IMC)

Architecture

Key Properties

GIMC: Generalized Internal Model Control

Smith Predictor: Dead-Time Compensation via Internal Model

Architecture

Connection to IMC

Limitations

Two-Degree-of-Freedom Control and the Conditional Feedback Structure

The Conditional Feedback Structure

Key Properties

Structural Origin of MEC

Disturbance Observer (DOB)

Architecture

Key Requirement: Inverse Model

Connection to Other Methods

Model Error Compensator (MEC)

Architecture

Key Advantages

Connection to 2-DOF Control

Structural Comparison: How the Methods Relate

Structural Relationships

When to Use Which?

Connections to Related Research

Related Articles and Videos

Blog Articles (blog.control-theory.com)

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

Video

Video Portal

Key References

ModelBasedCompensation #InternalModelControl #IMC #SmithPredictor #DisturbanceObserver #ModelErrorCompensator #MEC #2DOFControl #RobustControl #ControlEngineering #MATLAB