This article explains how the conditional feedback structure in two-degree-of-freedom (2-DOF) control is the structural origin of the Model Error Compensator (MEC). The 2-DOF conditional feedback structure, studied by Sugie and Yoshikawa (1986) among others, independently adjusts reference tracking and disturbance rejection by placing the plant model inside the feedback loop. MEC arises from this structure by setting the desired transfer function equal to the plant model: $T = P_{M}$ . For linear systems with a feedforward input, MEC and the 2-DOF conditional feedback structure become equivalent. However, MEC extends beyond this equivalence through the freedom in choosing which signals to feed back, the parallel feedforward compensator (PFC) for non-minimum-phase systems, and other design extensions. 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 2-DOF Control Matters
The Conditional Feedback Structure
From Conditional Feedback to MEC: Setting
Beyond Equivalence: MEC Extensions
Comparison Table
Practical Examples of MEC's 2-DOF Nature
Historical Context
Connections to Related Research
MATLAB Code
Related Articles and Videos
Key References
TwoDegreesOfFreedom #2DOFControl #ModelErrorCompensator #MEC #RobustControl #ControlEngineering #ConditionalFeedback #AddOnControl #MATLAB

Why 2-DOF Control Matters

In standard single-degree-of-freedom (1-DOF) feedback control, a single controller must simultaneously achieve two competing objectives: good reference tracking and good disturbance rejection / robustness. These objectives typically conflict — improving one degrades the other.

Two-degree-of-freedom (2-DOF) control resolves this conflict by providing two independent design parameters: one for shaping the reference response and one for shaping the feedback properties. This independence makes the design more transparent and flexible.

The concept of 2-DOF control has a long history in control engineering. The specific conditional feedback structure discussed in this article — where the model is placed inside the feedback loop and the feedback activates only when model error or disturbance exists — was analyzed in detail by Sugie and Yoshikawa (1986). This structure turns out to be the direct structural origin of the Model Error Compensator.

The Conditional Feedback Structure

Architecture

The 2-DOF conditional feedback structure consists of:

The actual plant $P$
A nominal model $P_{M}$
A desired input-output transfer function $T$
A feedforward path $T / P_{M}$
A feedback controller $C_{1}$ for disturbance rejection and robustness

The reference signal $r$ is processed through the feedforward path $T / P_{M}$ , and the feedback controller $C_{1}$ operates on the difference between the actual output $y$ and the output predicted by the desired transfer function $T$ .

Key Property

When the model matches the plant ( $P = P_{M}$ ):

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

The closed-loop transfer function equals $T$ regardless of the feedback controller $C_{1}$ . The feedback controller only activates when a discrepancy between the plant and the model exists, or when disturbances are present. This is why the structure is called conditional feedback — the feedback operates only when needed.

Consequence

This structure provides a clean separation:

Reference tracking: Determined by $T$ and the feedforward path. Independent of $C_{1}$ .
Disturbance rejection / robustness: Determined by $C_{1}$ . Independent of $T$ .

Unlike basic IMC, this structure can also handle unstable plants — the feedback controller $C_{1}$ can be designed to stabilize the closed loop, which is not possible in standard IMC where both $P$ and $P_{M}$ must be stable.

Constraint

For $T / P_{M}$ to be stable and proper, the desired transfer function $T$ must contain the same non-minimum-phase zeros as $P_{M}$ . The designer cannot freely choose $T$ when the plant has unstable zeros.

From Conditional Feedback to MEC: Setting $T = P_{M}$

The Origin of MEC

The Model Error Compensator arises from the conditional feedback structure by making a specific and natural choice for the desired transfer function:

$\displaystyle T = P_M$

This choice has immediate structural consequences:

The feedforward path becomes $T / P_{M} = P_{M} / P_{M} = 1$ — the identity. The original control input passes through without modification.
The feedback signal becomes the difference between the plant output $y$ and the model output $y_{M} = P_{M} u_{c}$ — the model error signal.
The feedback controller $C_{1}$ becomes the error compensator $D$ of MEC.

The objective is no longer to achieve a specific desired closed-loop response $T$ , but to make the compensated plant behave like the nominal model $P_{M}$ . Any controller subsequently designed for $P_{M}$ will then work as intended, because the effective plant dynamics is close to $P_{M}$ .

Equivalence for Linear Systems with Feedforward Input

For linear time-invariant systems where MEC uses a feedforward input structure, MEC and the conditional feedback structure with $T = P_{M}$ are structurally equivalent. This equivalence was noted in the overview paper by Okajima and Matsunaga (2016), which explicitly states that the MEC structure corresponds to the 2-DOF conditional feedback structure with $T = P_{M}$ .

The CCTA 2017 paper by Endo et al. also explicitly identifies this connection: "MEC is one of the 2-DOF control system."

Why This Matters

This 2-DOF interpretation explains the defining properties of MEC:

Controller independence: Because the feedforward path is the identity, the existing controller operates on the compensated plant just as it would on the model. The existing controller does not need to be modified or even known to MEC.
Separation of concerns: The existing controller handles performance (tracking, regulation), and the error compensator $D$ handles robustness. These are the two independent degrees of freedom.
Conditional feedback: When $P = P_{M}$ and no disturbance exists, the model error signal is zero and the compensator does nothing. MEC activates only when compensation is needed.

Beyond Equivalence: MEC Extensions

While the basic MEC for LTI systems with feedforward input is equivalent to the conditional feedback structure with $T = P_{M}$ , MEC extends beyond this equivalence in several important directions.

Freedom in Signal Selection

In the conditional feedback structure, the feedback signal is determined by the fixed architecture. In MEC, there is freedom in choosing which signals to feed back through the error compensator $D$ . For example, the compensated input $u_{c}$ (not the original input $u$ ) is used to drive the internal model, which creates a different feedback structure from the standard conditional feedback. This difference provides additional design flexibility.

Non-Minimum Phase Extension (PFC)

The conditional feedback structure requires $T$ to share the non-minimum-phase zeros of $P_{M}$ , and $T / P_{M}$ must be stable. When $T = P_{M}$ , the feedforward path is trivially stable, but the error compensator $D$ operates through high-gain feedback — and high-gain feedback is problematic for non-minimum-phase systems because it destabilizes the internal dynamics.

MEC addresses this through the parallel feedforward compensator (PFC), which modifies the effective system seen by the error compensator to have minimum-phase characteristics, enabling high-gain error compensation even for non-minimum-phase plants. This extension is not available in the standard conditional feedback framework. For details, see: MEC for Non-Minimum Phase Systems: PFC Approach

Nonlinear Systems

The conditional feedback structure is formulated for linear transfer function models. MEC has been extended to nonlinear systems through robust feedback linearization, where the nonlinear model is placed inside the compensator and the model error signal drives the compensation. For details, see: MEC for Nonlinear Systems: Robust Feedback Linearization

Error Compensator Design Methods

The design of the error compensator $D$ has been developed using several methodologies beyond what standard 2-DOF design offers:

$H_{\infty}$ optimization (JCMSI 2013)
LMI-based design with polytopic uncertainty (JCMSI 2021) → details
High-gain PI tuning for simple cases
Data-driven tuning via FRIT (Endo et al., CCTA 2017)

Comparison Table

Aspect	Conventional 2-DOF	MEC
Desired transfer function	Freely chosen $T$	$T = P_{M}$ (model matching)
Feedforward path	$T / P_{M}$ (explicit shaping)	Identity (no additional shaping)
Feedback controller	Designed within 2-DOF framework	Error compensator $D$ (add-on)
Existing controller	Replaced by 2-DOF design	Preserved unchanged
Non-minimum phase	$T$ must share zeros with $P_{M}$	PFC extension available
Nonlinear systems	Not directly applicable	Extended via robust feedback linearization
Equivalence	—	Equivalent to 2-DOF with $T = P_{M}$ for LTI + FF input

Practical Examples of MEC's 2-DOF Nature

The 2-DOF interpretation directly explains why MEC works as an add-on to various control methods:

MEC + PID: The PID controller handles reference tracking and basic regulation. MEC adds robustness without changing the PID gains. → MEC + PID Control
MEC + MPC: Model Predictive Control handles constraint satisfaction and optimization. MEC compensates for the model error that degrades MPC performance.
MEC + state feedback: State feedback handles pole placement. MEC ensures the effective plant matches the model used for pole placement.
MEC + feedback linearization: Feedback linearization handles nonlinearity. MEC compensates for the model error in the linearization. → MEC for Nonlinear Systems

In all cases, the existing controller handles performance (the first degree of freedom), and MEC handles robustness (the second degree of freedom).

Historical Context

The development of MEC has a direct connection to the 2-DOF conditional feedback structure:

Sugie and Yoshikawa (1986) established the conditional feedback structure for 2-DOF control, demonstrating the independence of reference tracking and feedback properties.
Okajima et al. (JCMSI, 2011) worked on 2-DOF Internal Model Control with dynamic quantizers, using the IMC framework extended with two degrees of freedom.
Okajima et al. (JCMSI, 2013) proposed MEC, which separates the robustness function from the controller design and implements it as an independent add-on compensator. The structural equivalence with the conditional feedback structure (for $T = P_{M}$ ) was recognized from the beginning.
Subsequent extensions (PFC for non-minimum phase, LMI design, nonlinear extensions, data-driven tuning) developed MEC beyond the scope of the original conditional feedback framework.

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 MEC, recognizing its 2-DOF structure.

MEC + FRIT for Quadcopter — H. Endo, R. Aramaki, K. Sekiguchi and K. Nonaka, Application of model error compensator based on FRIT to quadcopter, 2017 IEEE Conference on Control Technology and Applications (CCTA), pp. 1724–1729 (2017). Explicitly states "MEC is one of the 2-DOF control system."

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). Comprehensive overview including the relationship to 2-DOF control and other model-based compensation methods.

IMC and MEC — Internal Model Control and Model Error Compensator: From IMC to Add-On Robustness. Discusses IMC, GIMC (a 2-DOF extension of IMC), and MEC.

MEC + PID Control — MEC + PID Control: Adding Robustness to the Most Widely Used Controller. A practical example of MEC's 2-DOF nature.

MEC vs Disturbance Observer — Model Error Compensator vs Disturbance Observer: A Structural Comparison. DOB can also be viewed as an add-on compensator but requires the inverse model.

MATLAB Code

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

Key References

D.C. Youla, H.A. Jabr, and J.J. Bongiorno, "Modern Wiener-Hopf Design of Optimal Controllers—Part II: The Multivariable Case," IEEE Trans. Automatic Control, Vol. 21, No. 3, pp. 319–338, 1976.
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.
1. Endo, R. Aramaki, K. Sekiguchi, and K. Nonaka, "Application of model error compensator based on FRIT to quadcopter," 2017 IEEE CCTA, pp. 1724–1729, 2017.
1. Okajima, "Model Error Compensator for adding Robustness toward Existing Control Systems," IFAC PapersOnLine, Vol. 56, Issue 2, pp. 3998–4005, 2023.

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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TwoDegreesOfFreedom #2DOFControl #ModelErrorCompensator #MEC #RobustControl #ControlEngineering #ConditionalFeedback #AddOnControl #MATLAB

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Two-Degree-of-Freedom Control and the Model Error Compensator: Origin and Extensions

Why 2-DOF Control Matters