This article provides a detailed explanation of a design method for the Model Error Compensator (MEC) that handles polytopic-type uncertainty and disturbances. While the MEC was originally designed using frequency-domain methods (e.g., $\mu$ synthesis), this paper proposes a new state-space approach combining LMI-based performance analysis with particle swarm optimization (PSO), enabling flexible design for various evaluation criteria. 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, Y. Tanigawa, H. Okajima and N. Matsunaga, A design method of model error compensator for systems with polytopic-type uncertainty and disturbances, SICE Journal of Control, Measurement, and System Integration, Vol. 14, No. 2, pp. 119–127 (2021) (Open Access)

MATLAB Code (GitHub)

This paper is co-work with Ryuichiro Yoshida (Master's student, Kumamoto University), Yuki Tanigawa (Master's student, Kumamoto University), and Prof. Nobutomo Matsunaga (Professor, Kumamoto University).

Contents

Why a New Design Framework for MEC Is Needed

The Model Error Compensator (MEC) suppresses the modeling error between the actual plant and its nominal model by feeding back the output difference. Previous design methods for MEC were primarily based on frequency-domain uncertainty representations such as additive and multiplicative uncertainty, and the compensator was designed using $\mu$ synthesis.

However, different plants require different uncertainty representations. In particular, when plant dynamics are described in state-space form with polytopic-type uncertainty — a common representation in the robust control literature — a different design framework is needed. Polytopic uncertainty represents the actual plant as a convex combination of known endpoint systems, which naturally arises in many practical situations.

Furthermore, previous design methods were limited to $H _\infty$ performance in the frequency domain. In practice, one may want to evaluate performance using other criteria such as $H _2$ norm, peak value of impulse response, or pole placement. This paper addresses both challenges by proposing a state-space approach that combines LMI-based analysis with particle swarm optimization (PSO), providing the flexibility to use any LMI-representable evaluation criterion.

Review of the Model Error Compensator

The MEC, proposed in the foundational paper (SICE JCMSI, 2013), consists of a nominal model $P _m$ and a differential compensator $D$ . The MEC includes the nominal model internally and feeds back the output difference between the actual plant $P$ and the nominal model $P _m$ .

The compensated system dynamics from input $u _c$ to output $y$ is:

$\displaystyle P^{\ast}(s) = P(s) \frac{1 + P_m(s) D(s)}{1 + P(s) D(s)}$

When $P(s) = P _m(s)$ , this reduces to $P _m(s)$ regardless of $D(s)$ . The difference between $P^{\ast}(s)$ and $P _m(s)$ is:

$\displaystyle P^{\ast}(s) - P_m(s) = \frac{1}{1 + P(s) D(s)} (P(s) - P_m(s))$

A high-gain differential compensator $D(s)$ reduces this gap. The MEC is unique in that it can be used in conjunction with various existing control systems — feedback controllers, state feedback, model predictive control, and so on. The MEC manages robustness, while the controller can be designed independently without considering modeling errors.

State-Space Representation of MEC Components

Plant with Polytopic Uncertainty

The actual plant $P$ is described in state-space form as:

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

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

where $x(t)$ is the state, $\tilde{u}(t)$ is the control input, $w _u(t)$ is the disturbance input, $y(t)$ is the plant output, and $w _y(t)$ is the observation noise.

The matrices $A$ , $B$ , and $C$ 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 endpoint matrices $(A _i, B _i, C _i)$ define the vertices of the uncertainty polytope. The plant is assumed to be controllable and observable for all $\lambda$ in the polytope.

Nominal Model

The nominal model $P _m$ is:

$\displaystyle \dot{x}_m(t) = A_m x_m(t) + B_m u_m(t)$

$\displaystyle y_m(t) = C_m x_m(t)$

Differential Compensator

The differential compensator $D$ is given in state-space form as:

$\displaystyle \dot{x}_d(t) = A_d x_d(t) + B_d (y(t) - y_m(t))$

$\displaystyle y_d(t) = C_d x_d(t) + D_d (y(t) - y_m(t))$

The input to the nominal model is $u _m(t) = u(t)$ , and the input to the actual plant is $\tilde{u}(t) = u(t) - y _d(t)$ , where $u(t)$ is the controller output.

Generalized Plant and Evaluation Output

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

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

where $v(t) = \lbrack w _u(t)^{T},\; w _y(t)^{T},\; u(t)^{T} \rbrack^{T}$ is the combined input.

The matrices $\bar{A}$ and $\bar{B}$ are 3-by-3 block matrices involving $A$ , $B$ , $C$ , $A _m$ , $B _m$ , $A _d$ , $B _d$ , $C _d$ , $D _d$ , $B _w$ , and $D _w$ . For the explicit form, see Eqs. (14)–(15) in the paper.

The evaluation output is:

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

where $\bar{E} = \lbrack C,\; 0,\; \Delta C \rbrack$ with $\Delta C = C - C _m$ . Note that $e _y(t)$ is the output error excluding observation noise.

Under the assumption that $\Delta B = 0$ , $\Delta C = 0$ , or $D _d = 0$ , the generalized plant matrices can be represented as a matrix polytope, enabling LMI-based analysis. The condition $D _d = 0$ (no direct term in the differential compensator) is commonly used in practice, so this assumption is not restrictive.

Performance Analysis Using LMIs (H-infinity)

The system from $v(t)$ to $e _y(t)$ is denoted as $G _e$ . For given differential compensator parameters, the $H _\infty$ performance of $G _e$ can be analyzed using the following LMI conditions.

For all endpoint matrices $(\bar{A} _i, \bar{B} _i, \bar{E} _i)$ with $i = 1, \ldots, N$ , 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 $X > 0$ satisfying these LMIs exists, then the $H _\infty$ norm of the generalized plant is bounded:

$\displaystyle \lVert G_e \rVert_\infty \leq \gamma_\infty$

Since the $H _\infty$ norm equals the $L _2$ -induced norm, this provides a bound on the worst-case amplification from disturbances to the evaluation output:

$\displaystyle \lVert G_e \rVert_\infty = \sup_{v \in L_2,\; \lVert v \rVert_2 \neq 0} \frac{\lVert e_y \rVert_2}{\lVert v \rVert_2}$

The key advantage is that only vertex systems need to be checked, making the analysis computationally tractable even for complex polytopic uncertainty.

The paper notes that the same LMI framework can also accommodate $H _2$ performance, the peak value of impulse response, and pole placement constraints, enabling multi-objective design.

Design Method Using PSO

Since the LMI-based analysis requires the differential compensator parameters to be given, the paper uses particle swarm optimization (PSO) to search for the best parameters, with the LMI analysis providing the evaluation function.

The design procedure is:

Initialize particles with random positions (compensator parameters) and velocities. Solve the LMIs for each particle; if infeasible, re-initialize.
Evaluate each particle using the LMI analysis. If the LMIs are infeasible (indicating instability), assign a large penalty value.
Update particle positions and velocities using the standard PSO update rules with personal best $p _i$ and global best $g$ .
Repeat until the maximum iteration count is reached. Output the global best parameters.

The PSO update law is:

$\displaystyle z^i_x(k+1) = z^i_x(k) + z^i_v(k)$

$\displaystyle z^i_v(k+1) = \rho z^i_v(k) + r_{1,i} c_1 (p_i - z^i_x(k)) + r_{2,i} c_2 (g - z^i_x(k))$

where $\rho$ is the inertia weight, $c _1$ and $c _2$ are weighting factors, and $r _{1,i}$ and $r _{2,i}$ are random numbers in $\lbrack 0, 1 \rbrack$ .

To reduce the search dimensionality, the differential compensator $D$ is parameterized in controllable canonical form, which minimizes the number of free parameters. The evaluation function can be flexibly changed — not only $H _\infty$ norm but also $H _2$ norm, peak of impulse response, and pole placement, all of which have LMI representations.

Numerical Examples

The paper demonstrates the method on a MIMO system with 3 states, 2 inputs, and 2 outputs. The polytope has $N = 4$ vertices, representing uncertainty in the system matrices $A$ and $B$ (with $\Delta C = 0$ ). The nominal model corresponds to the center of the polytope ( $\lambda _i = 0.25$ for all $i$ ).

The differential compensator $D$ is designed in controllable canonical form with 17 parameters. PSO settings are: maximum update count $k _{\max} = 100$ , number of particles $m = 50$ , and weighting factors $\rho = 0.8$ , $c _1 = c _2 = 1$ .

The designed MEC achieves $\gamma _\infty = 0.0167$ , meaning:

$\displaystyle \lVert G_e \rVert_\infty \leq 0.0167$

This indicates very strong suppression of the model error effect.

Simulation results are shown for two types of disturbances:

Disturbance type	Actual ratio $\lVert e _y \rVert _2 / \lVert v \rVert _2$	Bound $\gamma _\infty$
Random noise	0.0070	0.0167
Step disturbance	0.0049	0.0167

In both cases, the actual performance is well within the guaranteed bound. The simulations confirm that the designed MEC effectively reduces the influence of disturbances and modeling errors for all plants within the polytope.

This work is part of a broader research program on the Model Error Compensator:

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 paper that proposed the MEC and its design based on $\mu$ synthesis. See also the blog article.

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). Extends MEC to non-minimum-phase plants using a parallel feed-forward filter.

MEC for Non-Minimum Phase with Polytopic Uncertainty — R. Yoshida, H. Okajima and T. Sato, Model Error Compensator Design for Continuous- and Discrete-Time Non-minimum Phase Systems with Polytopic-Type Uncertainties, SICE JCMSI (2022). Combines the PFF approach with the polytopic uncertainty framework from this paper.

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 presented at the IFAC World Congress.

Robust Performance Analysis (Japanese) — H. Okajima, Analysis of Robust Performance of Model Error Compensator for Polytopic-Type Uncertain Continuous-Time Linear Time Invariant Systems, Trans. SICE, Vol. 55, No. 12, pp. 800–807 (2019). The LMI-based analysis method used in this paper was originally developed in this Japanese paper.

MATLAB Code

GitHub: Robust-control-MATLAB_MEC01 — MATLAB code for PSO-based design and iterative design of MEC

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, Y. Tanigawa, H. Okajima and N. Matsunaga, "A design method of model error compensator for systems with polytopic-type uncertainty and disturbances", SICE Journal of Control, Measurement, and System Integration, Vol. 14, No. 2, pp. 119–127, 2021. DOI: 10.1080/18824889.2021.1918392 (Open Access)

Co-authors: Ryuichiro Yoshida (Master's student, Kumamoto University), Yuki Tanigawa (Master's student, Kumamoto University), Nobutomo Matsunaga (Professor, Kumamoto University)

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

Earlier Japanese paper: H. Okajima, "ポリトープ型不確かさを有する連続時間線形時不変システムに対するモデル誤差抑制補償器のロバスト性能解析," 計測自動制御学会論文集, Vol. 55, No. 12, pp. 800–807 (2019)

Self-Introduction

Hiroshi Okajima — Associate Professor, Graduate School of Science and Technology, Kumamoto University. Member of SICE, ISCIE, and IEEE.