<figure><img src="image_1.png" /></figure>

HAL
open science

# A Robust Discrete-Time Internal Model-based Controller for Continuous-Time LTI Systems with Uncertain Periodic Disturbances

Atabak Azimi, Stefan Koch, Denis Efimov, Markus Reichhartinger

► To cite this version:

Atabak Azimi, Stefan Koch, Denis Efimov, Markus Reichhartinger. A Robust Discrete-Time Internal Model-based Controller for Continuous-Time LTI Systems with Uncertain Periodic Disturbances. CDC 2025 - 64th IEEE Conference on Decision and Control, Dec 2025, Rio De Janeiro, Brazil. hal-05435635

HAL Id: hal-05435635

https://inria.hal.science/hal-05435635v1

Submitted on 30 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# A Robust Discrete-Time Internal Model-based Controller for Continuous-Time LTI Systems with Uncertain Periodic Disturbances

Atabak Azimi<fnref n="1" />, Stefan Koch<fnref n="1" />, Denis Efimov<fnref n="2" />, and Markus Reichhartinger<fnref n="1" />

**Abstract**— This paper introduces a robust discrete-time controller that combines internal model-based control with sliding mode techniques to reject periodic and matched disturbances in linear time-invariant systems. By explicitly accounting for discretization effects inherent in digital implementations, the controller bridges continuous-time dynamics and sampled-data operation. A systematic design approach is developed to handle uncertainties in the exosystem, ensuring bounded tracking error and improved disturbance rejection. Simulation studies on an active suspension system illustrate that the proposed method outperforms traditional internal model-based and sliding mode controllers, yielding reduced overshoot and tighter error bounds.

## I. INTRODUCTION

Periodic disturbances frequently arise in diverse industrial and control engineering settings, such as aircraft [1], vehicle suspension systems [2], and motion control systems [3]. One of the well-established approaches to reject such disturbances is the internal model-based control (IMC) [4], [5]. By generating a disturbance-related signal that replicates the periodic disturbances produced by an exogenous system (exosystem) and injecting it into the plant, IMC asymptotically cancels the undesired effect of disturbance.

However, this fails if the exosystem is uncertain. When the disturbance frequency is uncertain, IMC performance degrades, causing non-decaying outputs or persistent tracking error. To address this, adaptive IMC [6], [7], [8] was introduced. By tuning the parameters in the exosystem (i.e., adjusting the disturbance frequency), adaptive IMC improves overall controller performance. However, it inevitably introduces additional complexity and requires precise tuning, particularly for time-varying frequencies.

As an alternative, recent advances have combined IMC with sliding mode control (SMC) to handle both structured and unstructured disturbances more robustly. Integrating IMC and SMC has been shown to enhance servo motor control [9] and improve high-frequency disturbance rejection in electro-optical tracking systems [10]. In a complementary development, [11] proposes an internal-model-based SMC formulated in the continuous-time domain that remains robust to both matched and mismatched parametric uncertainties, thereby widening the practical scope of IMSMC applications. Another line of research in IMC and SMC [12]

introduces an additional term to IMC to compensate for non-decaying tracking error, preserving a simpler structure than adaptive IMC. As shown in [12], a comparison between the internal model-based sliding mode control (IMSMC) and adaptive IMC demonstrates the superior performance of IMSMC.

In this paper, the approach from [12], [13] is adopted to develop a robust discrete-time controller for a continuous-time plant that is discretized using sampling-and-hold elements, subject to continuous disturbances. In addition, this paper extends and refines the strategies in [12], [13] by focusing on the practical implementation aspects of sampled-data controllers in continuous-time systems. The key novelty lies in systematically handling the extra discretization terms that appear when transitioning to a real-time sampling framework, thereby bridging continuous and discrete settings without significantly increasing the controller's complexity. Unlike the conventional approach where either purely continuous-time or purely discrete-time controllers are investigated, this hybrid formulation is less commonly considered in the literature. This approach offers a more realistic model for industrial real-time scenarios and expands the applicability of the method in microprocessor-based (sampled) control systems.

The rest of this paper is organized as follows. Section II presents the problem formulation, outlining the assumptions and system setup, including details on the exosystem. Section III details the design of the robust discrete-time controller (IMSMC), emphasizing how the discretization terms are incorporated. Section IV provides a simulation study on an active suspension system to demonstrate the controller's effectiveness. Finally, Section V concludes the paper by summarizing the main findings.

## II. PROBLEM FORMULATION

Consider the linear continuous-time system governed by

$$ \dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{b}(u(t) + \delta(t)) + \mathbf{P}_{\eta}\eta(t), \quad (1a) $$

$$ y(t) = \mathbf{c}^{\mathrm{T}}\mathbf{x}(t), \quad \mathbf{x}(0) = \mathbf{x}_{0} \quad (1b) $$

where $\mathbf{x}(t) \in \mathbb{R}^n$ is the state vector, $u(t) \in \mathbb{R}$ is the control input, and $y(t) \in \mathbb{R}$ is the output of the system. In (1a), $\delta(t)$ represents an unstructured matched bounded disturbance for which the upper bound $\Delta_{\delta} > 0$ is known, i.e., $|\delta(t)| < \Delta_{\delta}, \forall t \ge 0$. The vectors $\mathbf{b}, \mathbf{c}$ and the matrices $\mathbf{A}, \mathbf{P}_{\eta}$ are known having the corresponding dimensions. Additionally, $\eta(t) \in \mathbb{R}^p$ in (1a) represents a bounded disturbance that is generated by the so-called disturbance exosystem

$$ \dot{\eta} = (\mathbf{S}_{\eta_n} + \mathbf{S}_{\eta_{\Delta}}(t))\eta = \mathbf{S}_{\eta}\eta, \quad \eta(0) = \eta_0. \quad (2) $$

<footnote n="1">¹A. Azimi, S. Koch, and M. Reichhartinger are with the Institute of Automation and Control at Graz University of Technology, Graz, Austria. {atabak.azimi, stefan.koch, markus.reichhartinger}@tugraz.at</footnote>

<footnote n="2">²D. Efimov is with Inria, Univ. Lille, CNRS UMR 9189 - CRISTAL, F-59000 Lille, France, Denis.Efimov@inria.fr.</footnote>
In (2), $S_{η_0} \in \mathbb{R}^{p \times p}$ is a known matrix capturing the nominal behavior which is characterized by $S_{η_Δ}(t) = \mathbf{0}_{p \times p}$. Hence, the uncertain part of the exosystem is introduced by $S_{η_Δ}(t) \in \mathbb{R}^{p \times p}$. In addition to $η(t)$, a separate exosystem is assumed to generate the reference $y_{ref}(t) \in \mathbb{R}$ for the output $y(t)$:

$$ \dot{\mathbf{z}}(t) = \mathbf{S}_r \mathbf{z}(t), \quad \mathbf{z}(0) = \mathbf{z}_0, \qquad (3a) $$

$$ y_{ref}(t) = -\mathbf{q}_r^T \mathbf{z}(t), \qquad (3b) $$

where $\mathbf{z}(t) \in \mathbb{R}^r$ is the state of the reference exosystem and $\mathbf{S}_r \in \mathbb{R}^{r \times r}$ is the dynamic matrix of the reference exosystem, which may also be unknown.

**Assumption 1:** All eigenvalues of the matrices $S_η, S_{η_0}$ and $S_r$ have zero real part for all time $t \ge 0$.

This ensures that both exosystems can generate periodic/constant signals. Note that although Assumption 1 allows the exosystems to generate ramp signals, in this paper the structured disturbance $η(t)$ and state of the reference exosystem $\mathbf{z}(t)$ are assumed to remain bounded.

The exosystems (2) and (3a) can be integrated in

$$ \begin{aligned} \dot{\mathbf{w}}(t) &= \begin{bmatrix} \mathbf{S}_{\eta} & \mathbf{0}_{p \times r} \\ \mathbf{0}_{r \times p} & \mathbf{S}_r \end{bmatrix} \mathbf{w}(t) = \mathbf{S} \mathbf{w}(t) = (\mathbf{S}_n + \mathbf{S}_{\Delta}(t)) \mathbf{w}(t), \\ \mathbf{w}(0) &= \mathbf{w}_0 = \begin{bmatrix} \eta_0 \\ \mathbf{z}_0 \end{bmatrix}, \end{aligned} \qquad (4) $$

where

$$ \mathbf{w}(t) = [\eta(t), \mathbf{z}(t)]^T, \quad \mathbf{S}_n = \begin{bmatrix} \mathbf{S}_{\eta_0} & \mathbf{0}_{p \times r} \\ \mathbf{0}_{r \times p} & \mathbf{S}_r \end{bmatrix}, \qquad (5) $$

$$ \mathbf{S}_{\Delta}(t) = \begin{bmatrix} \mathbf{S}_{\eta_{\Delta}}(t) & \mathbf{0}_{p \times r} \\ \mathbf{0}_{r \times p} & \mathbf{0}_{r \times r} \end{bmatrix}. $$

The exosystem state is defined as $\mathbf{w}(t)$ which combines the disturbance and reference information. In this manner, the plant (1a) can be rewritten as

$$ \dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{b}(u(t) + \delta(t)) + \mathbf{P}\mathbf{w}(t), \qquad (6a) $$

$$ y(t) = \mathbf{c}^{\mathrm{T}}\mathbf{x}(t), \quad \mathbf{x}(0) = \mathbf{x}_0, \qquad (6b) $$

where $\mathbf{P} = [\mathbf{P}_η, \mathbf{0}_{n \times r}]$. The tracking error is defined as

$$ \sigma(t) = \mathbf{c}^{\mathrm{T}}\mathbf{x}(t) + \mathbf{q}^{\mathrm{T}}\mathbf{w}(t), \qquad (7) $$

with $\mathbf{q}^T = [\mathbf{0}_{1 \times p}, \mathbf{q}_r^T]$.

**Remark 1:** The matrix $S_Δ(t)$ is time-varying, which can alter the exosystem's dynamics and is intended to produce periodic disturbances with time-varying frequency. Although $S_n$ alone can yield periodic signals, $S_Δ(t)$ further broadens the range of possible disturbance profiles.

**Remark 2:** In many cases, the matrix $S_r$ in (4) is known. However, it is also possible for the reference to be generated by the same exosystem as disturbance, i.e., $S_r = S_η$. In this scenario, the reference and disturbance share identical dynamics.

The primary objective is to design a robust controller that compensates for uncertainties in the exosystem (4) and $\delta(t)$, keeping the tracking error $\sigma(t)$ at zero. In [12], a continuous controller is introduced for this purpose. The present work

focuses on developing a discrete-time controller. Note that, in the presence of persistent disturbances, a discrete-time controller cannot fully compensate for inter-sampling effects and keep the tracking error identically at zero. Let $k = 0, 1, 2, ...$ denote the sampling index, set $(\cdot)_k = (\cdot)(kT)$ for sampling period $T > 0$ and define

$$ \sigma_k = \mathbf{c}^{\mathrm{T}}\mathbf{x}_k + \mathbf{q}^{\mathrm{T}}\mathbf{w}_k. \qquad (8) $$

The objective is to make $|\sigma_k|$ small as $k \to \infty$. It should be noted that increasing the sampling interval may result in significant deviations of $\sigma(t)$ to $\sigma_k$ in between samples, potentially exceeding predefined physical or practical thresholds for the error.

### III. CONTROLLER DESIGN

The controller is designed for the full information case where state $\mathbf{x}(t)$ and exosystem state $\mathbf{w}(t)$ are available in real time. The first step before designing the controller is to discretize $\mathbf{x}(t)$ and $\mathbf{w}(t)$. The solution of the plant (1a) and exosystem (4) are

$$ \begin{aligned} \mathbf{x}(t) &= e^{\mathbf{A}(t-t_0)}\mathbf{x}(t_0) + \int_{t_0}^{t} e^{\mathbf{A}(t-\tau)}\mathbf{b}u(\tau) \, d\tau \\ &\quad + \int_{t_0}^{t} e^{\mathbf{A}(t-\tau)}\mathbf{P}\mathbf{w}(\tau) \, d\tau + \int_{t_0}^{t} e^{\mathbf{A}(t-\tau)}\mathbf{b}\delta(\tau) \, d\tau, \end{aligned} \qquad (9) $$
$$ \mathbf{w}(t) = \Phi_{\mathbf{w}}(t, t_0)\mathbf{w}(t_0), $$

where $\Phi_{\mathbf{w}}(t, t_0)$ is the fundamental matrix solution [14] for (4), i.e.,

$$ \dot{\Phi}_{\mathbf{w}}(t, t_0) = \mathbf{S}(t)\Phi_{\mathbf{w}}(t, t_0) \quad \text{with} \quad \Phi_{\mathbf{w}}(t_0, t_0) = \mathbf{I}_{(p+r) \times (p+r)}. \qquad (10) $$

**Assumption 2:** The time-varying exosystem in (4) has a uniformly bounded fundamental matrix. In other words, there exists a constant $B > 0$ such that

$$ \|\Phi_{\mathbf{w}}(t, s)\| \le B, \quad \forall t \ge s \ge 0. \qquad (11) $$

Assumption 2 ensures boundedness of $\mathbf{w}(t)$. The dynamics of state $\mathbf{x}$ and exosystem state $\mathbf{w}$ in discrete-time domain can be calculated by replacing $t_0 = kT$ and $t = (k+1)T$ in (9), i.e.,

$$ \begin{aligned} \mathbf{x}_{k+1} &= e^{\mathbf{A}T} \mathbf{x}_k + \int_{kT}^{kT+T} e^{\mathbf{A}(kT+T-\tau)} \mathbf{b}u(\tau) \, d\tau \\ &\quad + \int_{kT}^{kT+T} e^{\mathbf{A}(kT+T-\tau)} \mathbf{P}\mathbf{w}(\tau) \, d\tau \\ &\quad + \int_{kT}^{kT+T} e^{\mathbf{A}(kT+T-\tau)} \mathbf{b}\delta(\tau) \, d\tau, \end{aligned} \qquad (12) $$
$$ \mathbf{w}_{k+1} = \Phi_{\mathbf{w}}(kT+T, T)\mathbf{w}_k = (e^{\mathbf{S}_n T} + \mathbf{D}_k)\mathbf{w}_k, $$

where $\mathbf{D}_k$ is the uncertainty matrix for the discrete-time exosystem in (12). The idea behind representing matrix $\mathbf{D}_k$ in (12) is the key step proposed in this paper. This formulation captures the exosystem's uncertainty as an additive component in the dynamic matrix. The matrix $\mathbf{D}_k$ can be calculated as

$$ \mathbf{D}_k = \Phi_{\mathbf{w}}(kT + T, T) - e^{\mathbf{S}_n T}, \qquad (13) $$
provided that $\mathbf{S}$ is known, though this is usually not the case in practice. Note that, in this paper, the actual value of the matrix $\mathbf{D}_k$ is not required, as we will explain in detail later how much information is needed for the controller design.

In (12), the controller $u(\tau)$ is a piecewise continuous signal and $u(\tau) = u_k$, $kT \le \tau < kT + T$. On the other hand, the exosystem state $\mathbf{w}(\tau)$ in general is a time-varying signal in between the samples. One can utilize the change of variable method as $\zeta = kT + T - \tau$ in (12) and simplify the terms including $u(\tau)$, i.e.,

$$
\begin{aligned}
\mathbf{x}_{k+1} &= \mathbf{A}_d \mathbf{x}_k + \mathbf{b}_d u_k \\
&\quad + \int_{kT}^{kT+T} e^{\mathbf{A}(kT+T-\tau)} \mathbf{P}(\mathbf{w}_k + \mathbf{w}_{\Delta}(\tau)) d\tau \\
&\quad + \int_{kT}^{kT+T} e^{\mathbf{A}(kT+T-\tau)} \mathbf{b}\delta(\tau) d\tau, \\
&= \mathbf{A}_d \mathbf{x}_k + \mathbf{b}_d u_k + \mathbf{P}_d \mathbf{w}_k \\
&\quad + \underbrace{\int_{kT}^{kT+T} e^{\mathbf{A}(kT+T-\tau)} \mathbf{P} \mathbf{w}_{\Delta}(\tau) d\tau}_{\mathbf{d}_{1k}} \quad (14) \\
&\quad + \underbrace{\int_{kT}^{kT+T} e^{\mathbf{A}(kT+T-\tau)} \mathbf{b}\delta(\tau) d\tau}_{\mathbf{d}_{2k}}, \\
&= \mathbf{A}_d \mathbf{x}_k + \mathbf{b}_d u_k + \mathbf{P}_d \mathbf{w}_k + \bar{\mathbf{d}}_{1k} + \bar{\mathbf{d}}_{2k}, \\
\mathbf{w}_{k+1} &= (e^{\mathbf{S}_n T} + \mathbf{D}_k) \mathbf{w}_k.
\end{aligned}
$$

In (14), $\mathbf{A}_d = e^{\mathbf{A}T}$, $\mathbf{b}_d = \int_0^T e^{\mathbf{A}\zeta} \mathbf{b} d\zeta$, $\mathbf{P}_d = \int_0^T e^{\mathbf{A}\zeta} \mathbf{P} d\zeta$, and $\mathbf{w}_{\Delta}(\tau) = \mathbf{w}(\tau) - \mathbf{w}_k$. The expressions for $\mathbf{x}_{k+1}$ and $\mathbf{w}_{k+1}$ are the exact discretization for the state $\mathbf{x}(t)$ and exosystem state $\mathbf{w}(t)$, hence in each sampling instance $kT$, the continuous state $\mathbf{x}(t)$ and the exosystem state $\mathbf{w}(t)$ are equal to $\mathbf{x}_k$ and $\mathbf{w}_k$, respectively.

The discrete-time plant with the exosystem can be summarized as

$$ \mathbf{x}_{k+1} = \mathbf{A}_d \mathbf{x}_k + \mathbf{b}_d u_k + \mathbf{P}_d \mathbf{w}_k + \bar{\mathbf{d}}_{1k} + \bar{\mathbf{d}}_{2k}, \quad (15a) $$

$$ \mathbf{w}_{k+1} = (e^{\mathbf{S}_n T} + \mathbf{D}_k) \mathbf{w}_k, \quad (15b) $$

$$ \sigma_k = \mathbf{c}^T \mathbf{x}_k + \mathbf{q}^T \mathbf{w}_k. \quad (15c) $$

**Assumption 3:** The pair $(\mathbf{A}_d, \mathbf{b}_d)$ is stabilizable and the pair $(\mathbf{A}_d, \mathbf{c})$ is detectable.

In the realm of disturbance rejection, internal model-based controller (IMC) is a powerful method [4]. The IMC controller is in the form

$$ u_k = -\mathbf{k}^T \mathbf{x}_k + \mathbf{l}^T \mathbf{w}_k, \quad (16) $$

where $\mathbf{k} \in \mathbb{R}^n$ is designed to place the eigenvalues of $\mathbf{A}_d - \mathbf{b}_d \mathbf{k}^T$ at desired places inside the unity circle in the complex plane. The second part, i.e., $\mathbf{l} \in \mathbb{R}^{r+p}$ is responsible to mitigate the effect of the disturbance on $\sigma_k$ in (15c). The nominal case is defined as $\mathbf{D}_k = \mathbf{0}_{(r+p) \times (r+p)}$ and $\bar{\mathbf{d}}_{1k} = \bar{\mathbf{d}}_{2k} = \mathbf{0}_{n \times 1}$, where the system is a discrete-time plant facing piecewise constant $\mathbf{w}_k$. In this scenario, if there

<figure><img src="image_3.png" /><figcaption>Fig. 1. Block diagram of the proposed discrete-time internal model control scheme. The exosystem block generates the continuous-time disturbance $\mathbf{w}(t)$ and $\mathbf{w}_k$. The discrete controller (22) processes $\mathbf{w}_k$, $y_k$ and $\mathbf{x}_k$ to compute the control input $u_k$. This input is then converted back to a continuous signal $u(t)$ via a zero-order hold (ZOH) element and applied with matched disturbance $\delta(t)$ to the continuous-time plant (6). A second ADC samples the plant output $y(t)$ and state $\mathbf{x}(t)$ to provide $y_k$ and $\mathbf{x}_k$ for feedback.</figcaption></figure>

exists $\mathbf{\Pi} \in \mathbb{R}^{n \times (p+r)}$ such that

$$ \begin{aligned} \mathbf{\Pi} e^{\mathbf{s}_n T} &= (\mathbf{A}_d - \mathbf{b}_d \mathbf{k}^T) \mathbf{\Pi} + \mathbf{b}_d \mathbf{l}^T + \mathbf{P}_d, \quad (17) \\ 0 &= \mathbf{c}^T \mathbf{\Pi} + \mathbf{q}^T, \end{aligned} $$

holds then

$$ \mathbf{x}_{k,ss} := \lim_{k \to \infty} \mathbf{x}_k = \mathbf{\Pi} \mathbf{w}_k, \quad (18) $$

and $\sigma_{k,ss} = \lim_{k \to \infty} \sigma_k = 0$. Note that by selecting the eigenvalues of $\mathbf{A}_d - \mathbf{b}_d \mathbf{k}^T$ different than $e^{\mathbf{s}_n T}$, the unique solution for $\mathbf{\Pi}$ and $\mathbf{l}$ is guaranteed [4] (Lemma 1.4.2).

To show that (18) holds, let us assume there is an error

$$ \tilde{\mathbf{x}}_k = \mathbf{x}_k - \mathbf{\Pi} \mathbf{w}_k. \quad (19) $$

The dynamic of this error can be calculated as

$$ \tilde{\mathbf{x}}_{k+1} = [(\mathbf{A}_d - \mathbf{b}_d \mathbf{k}^T) \mathbf{\Pi} + \mathbf{b}_d \mathbf{l}^T + \mathbf{P}_d - \mathbf{\Pi} e^{\mathbf{s}_n T}] \mathbf{w}_k + (\mathbf{A}_d - \mathbf{b}_d \mathbf{k}^T) \tilde{\mathbf{x}}_k. \quad (20) $$

The first part in (20) is zero by virtue of the regulator equation (17), and the second part contains the dynamic matrix of error $\tilde{\mathbf{x}}_k$ which is a Schur stable matrix by design of $\mathbf{k}$. Consequently, the error in (19) converges to zero asymptotically. Utilizing (18) for $\sigma_k$ in (15c), in conjunction with the last equation in (17), implies that $\sigma_{k,ss} = 0$.

In reality, the nominal case rarely occurs, i.e., $\mathbf{D}_k \neq \mathbf{0}_{p \times p}$, and the error from (19) does not vanish. In [13], a robust internal model-based sliding mode controller is proposed for uncertain exosystems. However, the plant in [13] is defined purely in discrete time, and no additional discretization terms $\bar{\mathbf{d}}_{1k}$ or $\bar{\mathbf{d}}_{2k}$ are introduced. In this study, these extra terms are accounted for, and the resulting controller ultimately acts on the continuous plant (6). The structure of the proposed controller is

$$ u_k = -\mathbf{k}^T \mathbf{x}_k + \mathbf{l}^T \mathbf{w}_k + v_k, \quad (21) $$

with the sliding mode part $v_k$. In order to visualize the interplay between the continuous-time system, the exosystem, and the discrete-time controller, a schematic diagram is provided in Fig. 1.

**Theorem 1:** Given the minimum phase system (15a), (15c) and a relative degree one from $\sigma_k$ to input $u_k$. Let
the exosystem state be generated by (15b). Then the control input $u_k$ defined as

$$ 
\begin{aligned}
u_k &= -\mathbf{k}^\mathrm{T}\mathbf{x}_k + \mathbf{l}^\mathrm{T}\mathbf{w}_k + v_k, \\
v_k &= \frac{1}{\mathbf{c}^\mathrm{T}\mathbf{b}_d}(-\mathbf{c}^\mathrm{T}(\mathbf{A}_d - \mathbf{b}_d\mathbf{k}^\mathrm{T})(\mathbf{x}_k - \mathbf{\Pi}\mathbf{w}_k) + v_k^* + \alpha\sigma_k), \\
v_k^* &= \begin{cases} -k_s & \text{if } \alpha\sigma_k \ge k_s \\ k_s & \text{if } \alpha\sigma_k \le -k_s \\ -\alpha\sigma_k & \text{if } \alpha\sigma_k \in [-k_s, k_s], \end{cases} \quad (22)
\end{aligned}
$$

ensures that there exists a $k^* > 0$ such that $|\sigma_k| \le M$, $\forall k \ge k^*$ with

$$ M = \sup_k |\mathbf{q}^\mathrm{T}\mathbf{D}_k\mathbf{w}_k + \mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{1_k} + \mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{2_k}|, \quad (23) $$

as long as the vector $\mathbf{k}$ and $\mathbf{l}$ are selected such that $\mathbf{A}_d - \mathbf{b}_d\mathbf{k}^\mathrm{T}$ is a Schur matrix, the regulator equation (17) is satisfied, the scalar $\alpha \in (0, 1)$, and positive constant $k_s$ is selected such that $k_s > M$. Besides, the elements of $\mathbf{x}_k$ are bounded.

*Proof:* Consider $\sigma_k$ as the sliding variable. From the discrete-time plant (15a), the tracking reference error (15c), the error defined in (19), and the proposed controller (22), the dynamic equation for $\sigma_k$ is

$$ 
\begin{aligned}
\sigma_{k+1} &= \mathbf{c}^\mathrm{T} [(\mathbf{A}_d - \mathbf{b}_d\mathbf{k}^\mathrm{T})\mathbf{x}_k + (\mathbf{b}_d\mathbf{l}^\mathrm{T} + \mathbf{P}_d)\mathbf{w}_k + \mathbf{b}_d v_k] \\
&\quad + \mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{1_k} + \mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{2_k} + \mathbf{q}^\mathrm{T} [(e^{S_n T} + \mathbf{D}_k) \mathbf{w}_k] \\
&= \alpha\sigma_k + \mathbf{q}^\mathrm{T}\mathbf{D}_k\mathbf{w}_k + \mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{1_k} + \mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{2_k} + v_k^*.
\end{aligned} \quad (24) $$

The unknown terms $(\mathbf{q}^\mathrm{T}\mathbf{D}_k\mathbf{w}_k + \mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{1_k} + \mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{2_k})$ in (24) include both constant weighting vectors $\mathbf{c}^\mathrm{T}$, $\mathbf{q}^\mathrm{T}$ and time-varying elements $\mathbf{D}_k\mathbf{w}_k$ with disturbances $\bar{\mathbf{d}}_{1_k}, \bar{\mathbf{d}}_{2_k}$. However, under Assumption 2, each of these quantities remain finite. Specifically, $\mathbf{D}_k$ in (13) results from subtracting two bounded matrices, and both $\mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{1_k}$ and $\mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{2_k}$ have bounded integrals with known upper and lower limits (see (14)). As a result, $\mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{1_k} + \mathbf{c}^\mathrm{T}\bar{\mathbf{d}}_{2_k}$ is also bounded. By selecting $k_s > M$ and substituting $v_k^*$ from (22) into (24), references [15], [16] imply that, under this control action, $\sigma_{k+1}$ converges to a neighborhood around zero whose width is bounded by $M$ in (23).

To investigate the boundedness of state $\mathbf{x}_k$, the closed-loop analysis in performed in this study. Applying the controller (22) on plant (15a) leads to

$$ 
\begin{aligned}
\mathbf{x}_{k+1} &= \left( \mathbf{A}_d - \mathbf{b}_d \mathbf{k}^\mathrm{T} + \frac{\mathbf{b}_d}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} \left[ -\mathbf{c}^\mathrm{T} (\mathbf{A}_d - \mathbf{b}_d \mathbf{k}^\mathrm{T}) + \alpha \mathbf{c}^\mathrm{T} \right] \right) \mathbf{x}_k \\
&\quad + \left( \mathbf{b}_d \mathbf{l}^\mathrm{T} + \mathbf{P}_d + \frac{\mathbf{b}_d}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} \left[ \mathbf{c}^\mathrm{T} (\mathbf{A}_d - \mathbf{b}_d \mathbf{k}^\mathrm{T}) \mathbf{\Pi} + \alpha \mathbf{q}^\mathrm{T} \right] \right) \mathbf{w}_k \\
&\quad + \frac{\mathbf{b}_d^\mathrm{T}}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} v_k^* + \bar{\mathbf{d}}_{1_k} + \bar{\mathbf{d}}_{2_k}.
\end{aligned} \quad (25) $$

The dynamic matrix of the closed-loop (25), i.e.,

$$ (\mathbf{A}_d - \mathbf{b}_d \mathbf{k}^\mathrm{T}) + \frac{\mathbf{b}_d}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} [-\mathbf{c}^\mathrm{T} (\mathbf{A}_d - \mathbf{b}_d \mathbf{k}^\mathrm{T}) + \alpha \mathbf{c}^\mathrm{T}], \quad (26) $$

can be rearranged and written as

$$ \left[ \mathbf{I}_n - \frac{\mathbf{b}_d \mathbf{c}^\mathrm{T}}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} \right] \mathbf{A}_d + \alpha \frac{\mathbf{b}_d \mathbf{c}^\mathrm{T}}{\mathbf{c}^\mathrm{T} \mathbf{b}_d}. \quad (27) $$

To evaluate the behavior of $\mathbf{x}_k$ in (25) we should find the eigenvalues of the close-loop system. Hence

$$ \det \left( z \mathbf{I}_n - \left( \left[ \mathbf{I}_n - \frac{\mathbf{b}_d \mathbf{c}^\mathrm{T}}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} \right] \mathbf{A}_d + \alpha \frac{\mathbf{b}_d \mathbf{c}^\mathrm{T}}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} \right) \right) = 0, \quad (28) $$

is needed to be solved to find the eigenvalues of the dynamic matrix (27). Utilizing the lemma

$$ \det(\mathbf{I}_n + \mathbf{u}\mathbf{v}^\mathrm{T}) = 1 + \mathbf{v}^\mathrm{T}\mathbf{u}, \quad (29) $$

(28) is simplified as

$$ 
\begin{aligned}
\det \left( z \mathbf{I}_n - \left( \left[ \mathbf{I}_n - \frac{\mathbf{b}_d \mathbf{c}^\mathrm{T}}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} \right] \mathbf{A}_d + \alpha \frac{\mathbf{b}_d \mathbf{c}^\mathrm{T}}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} \right) \right) &= \\
&= \left[ 1 + \frac{\mathbf{c}^\mathrm{T} \mathbf{A}_d (z \mathbf{I}_n - \mathbf{A}_d)^{-1} \mathbf{b}_d}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} - \alpha \frac{\mathbf{c}^\mathrm{T} (z \mathbf{I}_n - \mathbf{A}_d)^{-1} \mathbf{b}_d}{\mathbf{c}^\mathrm{T} \mathbf{b}_d} \right] \\
&\qquad \det(z \mathbf{I}_n - \mathbf{A}_d).
\end{aligned} \quad (30) $$

The transfer function for the system $(\mathbf{A}_d, \mathbf{b}_d, \mathbf{c}^\mathrm{T})$ is defined as

$$ 
\begin{aligned}
G(z) &= \frac{Y(z)}{U(z)} = \frac{N(z)}{\det(z\mathbf{I}_n - \mathbf{A}_d)} = \mathbf{c}^\mathrm{T}(z\mathbf{I}_n - \mathbf{A}_d)^{-1}\mathbf{b}_d, \\
zG(z) &= \frac{zY(z)}{U(z)} = \mathbf{c}^\mathrm{T}\mathbf{A}_d(z\mathbf{I}_n - \mathbf{A}_d)^{-1}\mathbf{b}_d + \mathbf{c}^\mathrm{T}\mathbf{b}_d, \quad (31)
\end{aligned} $$

where $Y(z)$ and $U(z)$ are the z-transforms of $y_k = \mathbf{c}^\mathrm{T}x_k$ and $u_k$, respectively, and $N(z)$ is the numerator polynomial of the transfer function $G(z)$. The determinant in (30) using the defined transfer function (31) is

$$ 
\begin{aligned}
& \left[ 1 + \frac{\mathbf{c}^\mathrm{T}\mathbf{A}_d(z\mathbf{I}_n - \mathbf{A}_d)^{-1}\mathbf{b}_d}{\mathbf{c}^\mathrm{T}\mathbf{b}_d} - \alpha \frac{\mathbf{c}^\mathrm{T}(z\mathbf{I}_n - \mathbf{A}_d)^{-1}\mathbf{b}_d}{\mathbf{c}^\mathrm{T}\mathbf{b}_d} \right] \\
& \det(z\mathbf{I}_n - \mathbf{A}_d) = \frac{(z - \alpha)N(z)}{\mathbf{c}^\mathrm{T}\mathbf{b}_d}.
\end{aligned} \quad (32) $$

The system $(\mathbf{A}_d, \mathbf{b}_d, \mathbf{c}^\mathrm{T})$ is minimum phase based on the assumption in Theorem 1, hence $N(z)$ is a Schur polynomial. The other eigenvalue of dymanic matrix (27) is $\alpha \in (0, 1)$. Hence, the elements of $\mathbf{x}_k$ are bounded. ■

*Remark 3:* The proposed controller makes the IMC robust against the uncertainty in the exosystem while only two extra parameters, $k_s$ and $\alpha$, are introduced. This demonstrates the simplicity of the controller compared to other methods such as adaptive IMC.

*Remark 4:* One could apply an explicitly discretized sliding mode controller instead the implicit one in (22), but it generally exhibits more pronounced chattering compared to the implicit discretized approach used here [17].

In the next section, the performance of the proposed controller is analyzed with an example from active suspension system.

## IV. SIMULATION RESULT

This section demonstrates the proposed method on an integrated active suspension system for a quarter-car model. To illustrate the approach, a similar simulation example from [12] is employed. Fig. 2 shows the schematic of the suspension arrangement, where the upper mass $M_s$, represents the car body (sprung mass) and the lower mass $M_{us}$
represents the tire (unsprung mass). Both masses are con-
nected to each other—and the tire is connected to the bumpy
ground—through spring–damper elements with respective
damping coefficients B_s and B_us, and spring constants K_s
and K_us.

A DC motor placed between M_s and M_us serves as the system's control input F_c. The primary goal is to adjust this motor's torque so as to diminish the disturbance from uneven road conditions on the car body, thereby minimizing the movement of the upper mass. To avoid redundancy, only the final plant equation in a suitable format as (6a) and (7) is presented here, while the detailed derivation can be found in [12]. The plant equation can be written as

$$
\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{b}\mathbf{u} + \mathbf{P}\mathbf{w}, \quad (33a)
$$

$$
\sigma = \mathbf{c}^{\mathrm{T}}\mathbf{x} + \mathbf{q}^{\mathrm{T}}\mathbf{w}, \quad (33\mathrm{b})
$$

where

$$
\mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & -1 \\ -520 & -1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 904.35 & 1.74 & -2000 & -2.61 \end{bmatrix}, \mathbf{b} = \begin{bmatrix} 0 \\ 0.5 \\ 0 \\ -0.87 \end{bmatrix},
$$

$$
\mathbf{P} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ -0.003 & 0 \\ 0.0026 & 0 \end{bmatrix}, \mathbf{c}^{\mathrm{T}} = \begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix}, \mathbf{q}^{\mathrm{T}} = \begin{bmatrix} 0.1 & 0.1 \end{bmatrix},
$$

$$
\mathbf{x} = [x_1, x_2, x_3, x_4]^T. \tag{34}
$$

Here, the state vector **x** is selected so that it captures the

<figure><img src="image_4.png" /><figcaption>Fig. 2. The active suspension system employed for the simulation studies is configured so that the upper mass represents the vehicle body, and the lower mass represents the tire. These two masses are connected by a simple spring–damper assembly, while the control input exerts an equal and opposite force on both masses. Additionally, the interaction between the tire and the bumpy road is also modeled via a spring–damper system.</figcaption></figure>

deviations of both car body and tire from their respective
equilibrium positions. Specifically, x₁ and x₂ represent the
displacement and velocity of the car body (sprung mass)
relative to an equilibrium point, while x₃ and x₄ reflect the
displacement and velocity of the tire (unsprung mass). These
offsets from equilibrium effectively remove the constant

gravitational terms from the equations of motion. Details on
the derivation of each xᵢ and the underlying physics can be
found in [12].

The sampling time is set to T = 5 milliseconds, which is suitable for most DC motor position and velocity control applications [18]. In generating the exosystem state inside the IMSMC controller, the dynamic matrix

$$
\mathbf{S}_n = \begin{bmatrix} 0 & 6\pi \\ -6\pi & 0 \end{bmatrix}, \quad e^{\mathbf{S}_n T} = \begin{bmatrix} 0.9956 & 0.0941 \\ -0.0941 & 0.9956 \end{bmatrix}, \quad (35)
$$

has been used. The actual exosystem state generated using
dynamic matrix

$$
\mathbf{S} = \begin{bmatrix} 0 & 14\pi \\ -14\pi & 0 \end{bmatrix}, \quad e^{\mathbf{S}T} = \begin{bmatrix} 0.9759 & 0.2181 \\ -0.2181 & 0.9759 \end{bmatrix}. \quad (36)
$$

Note that the reference signal is originated from the distur-
bance here and **S** = **S**<sub>η</sub>. The vector **k** in (22) is selected to
place the eigenvalues of the discrete plant at {-0.4, -0.3, 0.2,
0.5}. Solving the regulator equation leads to

$$
\mathbf{l} = [-9574, 5318]^\mathrm{T}, \quad \mathbf{\Pi} = \begin{bmatrix} -0.0033 & 0.0035 \\ -0.1 & -0.1 \\ -0.002 & 0.002 \\ -0.0343 & -0.0374 \end{bmatrix}. \tag{37}
$$

Under the IMSMC controller, the absolute value of unknown
elements in the dynamics of σ_k have an upper bound given
by

$$
|\mathbf{q}^{\mathrm{T}} \mathbf{D}_k \mathbf{w}_k + \mathbf{c}^{\mathrm{T}} \bar{\mathbf{d}}_{1_k}| \le 0.0178. \quad (38)
$$

The performance of the IMSMC controller is compared to IMC and sliding mode controller (SMC) methods. For the IMC controller, **k** and **l** are the same as IMSMC method. The controller structure for the SMC is

$$
\begin{align}
u_k &= -\mathbf{k}^\mathrm{T}\mathbf{x}_k + \frac{1}{\mathbf{c}^\mathrm{T}\mathbf{b}_d}(-\mathbf{c}^\mathrm{T}(\mathbf{A}_d - \mathbf{b}_d\mathbf{k}^\mathrm{T})\mathbf{x}_k + v_k^* + \alpha\sigma_k) \nonumber \\
v_k^* &= \begin{cases} -k_{\mathrm{SMC}} & \text{if } \alpha\sigma_k \ge k_{\mathrm{SMC}} \\ k_{\mathrm{SMC}} & \text{if } \alpha\sigma_k \le -k_{\mathrm{SMC}} \\ -\alpha\sigma_k & \text{if } \alpha\sigma_k \in [-k_{\mathrm{SMC}}, k_{\mathrm{SMC}}], \end{cases} \tag{39}
\end{align}
$$

where **k** is similar to the IMSMC method. Applying SMC
controller into the system, the unknown parts in the dynamic
of σ_k have different terms and the upper bound of the
absolute value of these unknown terms is

$$
| (\mathbf{q}^{\mathrm{T}}(e^{\mathbf{S}_n T} + \mathbf{D}_k) + \mathbf{c}^{\mathrm{T}}\mathbf{P}_d)\mathbf{w}_k + \mathbf{c}^{\mathrm{T}}\bar{\mathbf{d}}_{1_k}| \le 0.142. \quad (40)
$$

For the IMSMC, the gain k_s = 0.0179 and for the SMC,
k_SMC = 0.179 are selected. Both gains k_s and k_SMC must
be greater than the corresponding upper bounds given in
(38) and (40). The tuning scalar parameter α is selected as
0.8 in this simulation for both IMSMC and SMC methods.
Fig. 3 shows the reference tracking error σ_k evolution for
three different controller over the sampling index. At the
beginning, the IMC approach exhibits a significant overshoot
before stabilizing. This is primarily due to the sampling time,
and increasing the sampling time mitigates this overshoot.
Both the IMC and SMC methods have a similar steady-
state error bound of approximately 100 mm. Although this
<figure><img src="image_5.png" /><figcaption>Fig. 3. Comparison of IMC, SMC, and IMSMC methods in terms of reference tracking performance. The IMC approach exhibits a significant overshoot at the beginning before stabilizing, primarily due to the sampling time. Both the IMC and SMC methods have a similar steady-state error bound of approximately 100 mm. In contrast, the IMSMC starts without any overshoot, and over time, the reference tracking error oscillates around 20 mm, demonstrating its superior performance compared to the other two methods.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Control inputs of the IMC, SMC, and IMSMC controllers, corresponding to the tracking error in Fig. 3. The IMC control input exhibits large initial values due to the sampling time and overshoot in the tracking error. Over time, its amplitude decreases, which limits its ability to further reduce the tracking error in Fig. 3. The SMC control input closely resembles that of IMSMC but lacks sufficient amplitude to fully compensate for disturbance effects, whereas the IMSMC maintains better control performance.</figcaption></figure>

error bound is calculated for this specific simulation condition, it is noticeably large from a practical standpoint. In contrast, the IMSMC starts without any overshoot, and over time, the reference tracking error oscillates around 20 mm. This demonstrates the superior performance of the IMSMC compared to the other two methods.

Fig. 4 illustrates the control input for all three controllers, corresponding to the tracking error shown in Fig. 3. The large initial values of the IMC control input are due to the sampling time and the overshoot in the tracking error. Over time, the amplitude of the IMC control input becomes smaller than that of the IMSMC, indicating that it lacks sufficient actuation to effectively reduce the tracking error in Fig. 3. The SMC control input is similar to that of IMSMC but does not have a large enough amplitude to fully mitigate the disturbance effects on the tracking error.

## V. CONCLUSIONS

The developed robust discrete-time sliding mode controller effectively compensates for uncertainties and discretization effects in LTI systems subject to periodic disturbances.

Simulation results on an active suspension model confirm that the controller maintains bounded tracking error and significantly improves performance compared to conventional IMC and SMC methods. Overall, the approach successfully bridges continuous and digital control realms, addressing the currently existing gap of discrete-time robust internal-model based concepts for the operation of continuous-time systems, which is highly relevant for industrial applications.

## REFERENCES

[1] N. Dalwadi, D. Deb, M. Kothari, and S. Ozana, "Disturbance observer-based backstepping control of tail-sitter UAVs," in *Actuators*, vol. 10, p. 119, MDPI, 2021.

[2] T. A. Nguyen, "Active disturbance rejection control for an automotive suspension system based on parameter tuning using a fuzzy technique," *PloS one*, vol. 20, no. 1, p. e0313104, 2025.

[3] Y.-S. Lu and S.-H. Chang, "An infinite-order integral disturbance observer for improved estimation of periodic disturbances in motion control systems," *Journal of the Chinese Institute of Engineers*, vol. 46, no. 4, pp. 367-379, 2023.

[4] A. Isidori, L. Marconi, and A. Serrani, *Robust autonomous guidance: an internal model approach*. Springer Science & Business Media, 2003.

[5] B. A. Francis and W. M. Wonham, "The internal model principle of control theory," *Automatica*, vol. 12, no. 5, pp. 457-465, 1976.

[6] G. J. Silva and A. Datta, "Adaptive internal model control: the discrete-time case," *International Journal of Adaptive Control and Signal Processing*, vol. 15, no. 1, pp. 15-36, 2001.

[7] M. Bin, P. Bernard, and L. Marconi, "Approximate nonlinear regulation via identification-based adaptive internal models," *IEEE Transactions on Automatic Control*, vol. 66, no. 8, pp. 3534-3549, 2020.

[8] Z. Zhang and A. Serrani, "Adaptive robust output regulation of uncertain linear periodic systems," *IEEE Transactions on automatic control*, vol. 54, no. 2, pp. 266-278, 2009.

[9] P. Li and G. Zhu, "Robust internal model control of servo motor based on sliding mode control approach," *ISA transactions*, vol. 93, pp. 199-208, 2019.

[10] B. Zhang, K. Nie, X. Chen, and Y. Mao, "Development of sliding mode controller based on internal model controller for higher precision electro-optical tracking system," in *Actuators*, vol. 11, p. 16, MDPI, 2022.

[11] S. Koo and H. Shim, "Reference-tracking sliding-mode control robust to mismatched exogenous disturbances," in *2024 14th Asian Control Conference (ASCC)*, pp. 1-6, IEEE, 2024.

[12] A. Azimi, S. Koch, and M. Reichhartinger, "Robust internal model-based control for linear-time-invariant systems," *International Journal of Robust and Nonlinear Control*, vol. 34, no. 18, pp. 12476-12496, 2024.

[13] A. Azimi, S. Koch, and M. Reichhartinger, "Robust internal model-based control for linear-time-invariant systems in discrete-time domain," in *2024 IEEE 63rd Conference on Decision and Control*, CDC 2024, 2024.

[14] W. J. Rugh, *Linear system theory*. Prentice-Hall, Inc., 1996.

[15] B. Brogliato and A. Polyakov, "Digital implementation of sliding-mode control via the implicit method: A tutorial," *International Journal of Robust and Nonlinear Control*, vol. 31, no. 9, pp. 3528-3586, 2021.

[16] V. Acary, B. Brogliato, and Y. V. Orlov, "Chattering-free digital sliding-mode control with state observer and disturbance rejection," *IEEE Transactions on Automatic Control*, vol. 57, no. 5, pp. 1087-1101, 2011.

[17] B. Andritsch, V. Holzner, S. Koch, M. Reichhartinger, and M. Horn, "Experimental comparison of sliding mode control dedicated discretization approaches," in *2024 17th International Workshop on Variable Structure Systems (VSS)*, pp. 40-45, IEEE, 2024.

[18] M. Kuczmann, "Review of DC motor modeling and linear control: Theory with laboratory tests," *Electronics*, vol. 13, no. 11, p. 2225, 2024.