This article provides a detailed explanation of a design method for the Model Error Compensator (MEC) with a parallel feedforward compensator (PFC) for non-minimum phase MIMO systems with polytopic-type uncertainties. This work unifies two previous developments — the PFC-based approach for non-minimum-phase systems and the LMI/PSO-based design for polytopic uncertainties — into a single state-space framework applicable to both continuous-time and discrete-time systems. 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.

Yoshida, H. Okajima and T. Sato, Model error compensator design for continuous- and discrete-time non-minimum phase systems with polytopic-type uncertainties, SICE Journal of Control, Measurement, and System Integration, Vol. 15, No. 2, pp. 37–49 (2022) (Open Access)

MATLAB Code (GitHub)

This paper is co-work with Ryuichiro Yoshida (Master's student, Kumamoto University) and Takumi Sato (Undergraduate student, Kumamoto University).

Contents

Why This Unification Matters

The Model Error Compensator (MEC) has been developed through a series of papers, each addressing a specific challenge. The foundational paper (2013) introduced the MEC for minimum-phase SISO systems. The PFF paper (2017) extended MEC to non-minimum-phase systems using a parallel feed-forward filter designed in the frequency domain. The polytopic uncertainty paper (2021) introduced LMI-based analysis with PSO for systems with polytopic-type uncertainty, but was limited to minimum-phase plants.

When the previous polytopic design method is applied to non-minimum-phase MIMO systems, two problems arise. First, finding initial parameters for the differential compensator that stabilize the generalized plant becomes very difficult. Second, the resulting design performance tends to be poor. This is because high-gain feedback, which is the core mechanism of the MEC, is inherently challenging for non-minimum-phase systems.

This paper addresses both problems by introducing the parallel feedforward compensator (PFC) into the state-space polytopic framework. The PFC changes the apparent dynamics of the plant from the viewpoint of the MEC, making it appear as a minimum-phase system. The resulting framework handles MIMO non-minimum-phase plants with polytopic-type uncertainties in both continuous-time and discrete-time settings — a significant unification of previous results.

Plant and Model with Polytopic Uncertainty

The plant $P$ is described by:

$\displaystyle \delta x(t) = Ax(t) + B\tilde{u}(t) + B_w w_u(t)$

$\displaystyle y(t) = Cx(t) + D_w w_y(t)$

where $\delta$ denotes the derivative operator $\dot{x}(t)$ for continuous-time systems and the forward-shift operator $x(k+1)$ for discrete-time systems. The matrices $A$ , $B$ , $C$ have polytopic-type uncertainties:

$\displaystyle A = \sum_{i=1}^{N} \lambda_i A_i, \quad B = \sum_{i=1}^{N} \lambda_i B_i, \quad C = \sum_{i=1}^{N} \lambda_i C_i$

where $\lambda _i \geq 0$ and $\sum \lambda _i = 1$ . The nominal model $P _m$ is:

$\displaystyle \delta x_m(t) = A_m x_m(t) + B_m u_m(t), \quad y_m(t) = C_m x_m(t)$

One natural choice for the model parameters is the center of the polytope ( $\lambda _i = 1/N$ ).

Non-Minimum-Phase Characteristics

A non-minimum-phase system has unstable zeros — complex numbers $s$ satisfying $\mathrm{Re}(s) > 0$ (continuous-time) or $\lvert s \rvert > 1$ (discrete-time) — that reduce the rank of the Rosenbrock system matrix:

$\displaystyle \mathcal{P}(s) = \begin{pmatrix} sI - A & B \cr -C & 0 \end{pmatrix}$

Non-minimum-phase systems exhibit undershoots in their step responses and have fundamental control performance limitations. Designing a high-gain MEC for such systems is difficult because the feedback can destabilize the closed loop or produce poor transient behavior.

MEC with Parallel Feedforward Compensator

To overcome the non-minimum-phase difficulty, a PFC $F$ is added in parallel with the plant and model. The PFC changes the apparent plant dynamics as seen by the differential compensator $D$ . The key idea is that if the combined system $P + F$ behaves as a minimum-phase system, the MEC design proceeds in the same manner as for minimum-phase plants.

The PFC for the plant and model share the same state-space matrices $(A _f, B _f, C _f)$ :

$\displaystyle F: \quad \delta x_f(t) = A_f x_f(t) + B_f \tilde{u}, \quad y_f(t) = C_f x_f(t)$

$\displaystyle F_m: \quad \delta x_{fm}(t) = A_f x_{fm}(t) + B_f u_m(t), \quad y_{fm}(t) = C_f x_{fm}(t)$

The differential compensator $D$ now receives the combined output difference $y _F(t) - y _{Fm}(t)$ as its input, where $y _F = y + y _f$ and $y _{Fm} = y _m + y _{fm}$ .

An important advantage over previous approaches: the evaluation output $e _y(t) = Cx(t) - C _m x _m(t)$ directly measures the error between the plant output $y$ and the model output $y _m$ , not the augmented outputs.

Apparent Dynamics of the Plant with PFC

Defining the augmented states $x _F(t) = \lbrack x(t)^{T}, x _f(t)^{T} \rbrack^{T}$ and $x _{Fm}(t) = \lbrack x _m(t)^{T}, x _{fm}(t)^{T} \rbrack^{T}$ , the apparent plant $P _F$ and model $P _{Fm}$ from the viewpoint of $D$ are:

$\displaystyle A_F = \begin{pmatrix} A & 0 \cr 0 & A_f \end{pmatrix}, \quad B_F = \begin{pmatrix} B \cr B_f \end{pmatrix}, \quad C_F = (C, \; C_f)$

$\displaystyle A_{Fm} = \begin{pmatrix} A_m & 0 \cr 0 & A_f \end{pmatrix}, \quad B_{Fm} = \begin{pmatrix} B_m \cr B_f \end{pmatrix}, \quad C_{Fm} = (C_m, \; C_f)$

A critical observation: the augmented plant $P _F$ can be represented as a matrix polytope:

$\displaystyle A_F = \sum_{i=1}^{N} \lambda_i A_{Fi}, \quad B_F = \sum_{i=1}^{N} \lambda_i B_{Fi}, \quad C_F = \sum_{i=1}^{N} \lambda_i C_{Fi}$

This polytope structure is preserved because the PFC matrices $(A _f, B _f, C _f)$ are the same for all vertices.

Generalized Plant and Evaluation Output

By defining the error state $e _F(t) = x _F(t) - x _{Fm}(t)$ and the augmented state $\xi(t) = \lbrack e _F(t)^{T}, x _d(t)^{T}, x _{Fm}(t)^{T} \rbrack^{T}$ , the generalized plant dynamics is:

$\displaystyle \delta \xi(t) = \bar{A} \xi(t) + \bar{B} v(t)$

where $v(t) = \lbrack w _u(t)^{T}, w _y(t)^{T}, u(t)^{T} \rbrack^{T}$ . The matrices $\bar{A}$ and $\bar{B}$ are 3-by-3 block matrices involving the augmented plant, PFC, and differential compensator parameters. For the explicit form, see Eqs. (22)–(23) in the paper.

The evaluation output is:

$\displaystyle e_y(t) = Cx(t) - C_m x_m(t) = \bar{E} \xi(t)$

Under the assumption that $\Delta B _F = 0$ , $\Delta C _F = 0$ , or $D _d = 0$ , the generalized plant $\Phi$ can be represented as a matrix polytope:

$\displaystyle \bar{A} = \sum_{i=1}^{N} \lambda_i \bar{A}_i, \quad \bar{B} = \sum_{i=1}^{N} \lambda_i \bar{B}_i, \quad \bar{E} = \sum_{i=1}^{N} \lambda_i \bar{E}_i$

The system from $v(t)$ to $e _y(t)$ is denoted $\Phi$ .

H-infinity Performance Analysis (Continuous-Time)

For vertex matrices $(\bar{A} _i, \bar{B} _i, \bar{E} _i)$ and a given $\gamma _\infty > 0$ , find $X > 0$ such that:

$\displaystyle \begin{pmatrix} \bar{A}_i X + X \bar{A}_i^T & X \bar{E}_i^T & \bar{B}_i \cr \bar{E}_i X & -\gamma_\infty^2 I & 0 \cr \bar{B}_i^T & 0 & -I \end{pmatrix} < 0$

for all $i = 1, \ldots, N$ . If a common $X > 0$ satisfying all vertex LMIs is found, then the $H _\infty$ norm is bounded as $\lVert \Phi \rVert _\infty \leq \gamma _\infty$ .

H-infinity Performance Analysis (Discrete-Time)

For discrete-time systems, the LMI condition becomes:

$\displaystyle \begin{pmatrix} -X + \bar{A}_i X \bar{A}_i^T + \bar{B}_i \bar{B}_i^T & \bar{A}_i X \bar{E}_i^T \cr \bar{E}_i X \bar{A}_i^T & \bar{E}_i X \bar{E}_i^T - \gamma_\infty^2 I \end{pmatrix} < 0$

for all $i = 1, \ldots, N$ . Similarly, if $X > 0$ satisfying these LMIs is found, then $\lVert \Phi \rVert _\infty \leq \gamma _\infty$ .

The unified framework allows both continuous-time and discrete-time designs to be handled by simply switching the LMI conditions. The minimum $\gamma _\infty$ satisfying the LMIs is found using standard SDP solvers such as MATLAB.

Design of PFC and Differential Compensator

Design of the PFC

The PFC $F$ is designed so that the apparent dynamics $P _m + F$ becomes a minimum-phase system. One approach is the iteration-based method: first fix $D$ and optimize $F$ using PSO with the LMI analysis as evaluation, then fix $F$ and optimize $D$ , repeating until convergence. Alternatively, if the number of design variables is small, $F$ and $D$ can be designed simultaneously.

An important structural requirement: the PFC must have a zero at the origin (differentiator) and the differential compensator $D$ must have a pole at the origin (integrator) to ensure elimination of constant disturbances and steady-state errors. In controllable canonical form, these constraints are imposed by fixing specific matrix entries.

Design of the Differential Compensator

The differential compensator $D$ is designed using PSO combined with the iteration-based method: first, PSO finds initial parameters, then the iteration-based method refines them by alternating between optimizing the common Lyapunov matrix and the compensator parameters. This combined approach saves time in finding good initial values and typically produces better results.

The compensator $D$ is parameterized in controllable canonical form to minimize the number of free parameters. For a 4-state, 2-input, 2-output compensator, the integrator constraint is imposed through specific structure in $A _d$ and $B _d$ . For the explicit form, see Eq. (37) in the paper.

Numerical Examples

The paper demonstrates the method on a 2-input, 2-output continuous-time non-minimum-phase plant. The nominal model transfer function is:

$\displaystyle P_m(s) = \begin{pmatrix} \frac{-(s-2)}{s^2+2s+1} & \frac{-(2s^2-1)}{2s^2+3s+1} \cr \frac{-(2s-1)}{s^2+4s+3} & \frac{-(s-2)}{s^2+3s+2} \end{pmatrix}$

The plant has $N = 4$ vertex matrices, with the vertex matrices varying up to 16.5% from the nominal model.

The PFC is designed so that $P _m + F$ becomes minimum-phase. PSO settings: 50 particles, $k _{\max} = 100$ , $\rho = 1$ , $c _1 = c _2 = 0.8$ .

The designed MEC achieves $\gamma _\infty = 0.2404$ , meaning:

$\displaystyle \lVert \Phi \rVert_\infty \leq 0.2404$

Three approaches are compared via simulation with step inputs and step disturbances:

Method	Description	Performance
Without MEC	Plant with modeling errors, no compensation	Large deviations from ideal
MEC only (2021 method)	MEC without PFC for non-minimum-phase plant	Some improvement, but limited
Proposed (MEC with PFC)	MEC with PFC, polytopic LMI design	Superior suppression of errors and disturbances

The proposed method with PFC significantly outperforms the MEC-only approach for non-minimum-phase systems, confirming the effectiveness of combining the PFC with the polytopic uncertainty framework.

This work unifies two lines of MEC research:

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 MEC proposal. See also the blog article.

MEC with Parallel Feed-Forward Filter (SISO) — 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). First introduced the PFC approach for non-minimum-phase plants in the SISO transfer-function setting.

MEC for Polytopic Uncertainty (Minimum Phase) — 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, Vol. 14, No. 2, pp. 119–127 (2021). Introduced the PSO + LMI framework for polytopic uncertainty. The present paper extends this to non-minimum-phase 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.

Performance Limitation — H. Okajima and T. Asai, Performance Limitation of Tracking Control Problem for a Class of References, IEEE Trans. Automatic Control, Vol. 56, No. 11, pp. 2723–2727 (2011). Provides theoretical background on fundamental performance limitations of non-minimum-phase systems.

MATLAB Code

GitHub: Robust-control-MATLAB_MEC01 — MATLAB code for PSO-based and iteration-based design of MEC (covers both this paper and the 2021 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

Yoshida, H. Okajima and T. Sato, "Model error compensator design for continuous- and discrete-time non-minimum phase systems with polytopic-type uncertainties", SICE Journal of Control, Measurement, and System Integration, Vol. 15, No. 2, pp. 37–49, 2022. DOI: 10.1080/18824889.2022.2052628 (Open Access)

Co-authors: Ryuichiro Yoshida (Master's student, Kumamoto University), Takumi Sato (Undergraduate student, Kumamoto University)

This work was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) 21K04111.

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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