TELKOMNIKA Telecommunication Computing Electronics and Control

Vol. 21, No. 5, October 2023, pp. 1156~1162

ISSN: 1693-6930, DOI: 10.12928/TELKOMNIKA.v21i5.23157  1156

Brief note on match and miss-match uncertainties

Muhammad Nizam Kamarudin1, Sahazati Md Rozali2, Mohd H. Hairi1 Muhammad I. Zakaria3

1
Center for Robotics and Industrial Automation, Faculty of Electrical Engineering, Universiti Teknikal Malaysia Melaka (UTeM),
Melaka, Malaysia
2
Faculty of Electrical and Electronic Engineering Technology, Universiti Teknikal Malaysia Melaka (UTeM), Hang Tuah Jaya, 76100
Durian Tunggal, Melaka, Malaysia
3
School of Electrical Engineering, College of Engineering, Universiti Teknologi MARA, Shah Alam, Malaysia

Article Info ABSTRACT

Article history: The appearance of uncertainties and disturbances often effects the
characteristics of either linear or nonlinear systems. Plus, the stabilization
Received Jan 24, 2022 process may be deteriorated thus incurring a catastrophic effect to the system
Revised May 10, 2023 performance. As such, this manuscript addresses the concept of matching
Accepted Jun 05, 2023 condition for the systems that are suffering from miss-match uncertainties and
exogeneous disturbances. The perturbation towards the system at hand is
assumed to be known and unbounded. To reach this outcome, uncertainties
Keywords: and their classifications are reviewed thoroughly. The structural matching
condition is proposed and tabulated in the proposition 1. Two types of
Disturbance mathematical expressions are presented to distinguish the system with
Linear systems matched uncertainty and the system with miss-matched uncertainty. Lastly,
Nonlinear systems two-dimensional numerical expressions are provided to practice the proposed
Uncertainty proposition. The outcome shows that matching condition has the ability to
change the system to a design-friendly model for asymptotic stabilization.
This is an open access article under the CC BY-SA license.

Corresponding Author:
Muhammad Nizam Kamarudin
Center for Robotics and Industrial Automation, Faculty of Electrical Engineering
Universiti Teknikal Malaysia Melaka (UTeM), Hang Tuah Jaya, 76100 Durian Tunggal, Melaka, Malaysia
Email: nizamkamarudin@utem.edu.my

1. INTRODUCTION
Dealing with dynamical systems with uncertainties is trivial. Uncertainties may introduce nonlinarity
and need linearization to facilitate stabilization process [1]. Considering a mechanical rotational system in
practice, the uncertainties in the modeling may be contributed by the backlash in the gearing teeth [2]. Many
stabilization approaches available in the literature can be opted to perform tracking tasks as well as regulation
tasks for such systems. However, the presence of uncertainties and disturbances make the objectives a
challenging task. Among the approaches, a classical [3], [4], optimal, and robust approaches are common.
Among the robust stabilization approaches, backstepping [5], [6] and variable structure system
(VSS) [7]-[9] are known to be prevalent. Both techniques adopt the Lyapunov stability criteria [10]. VSS is a
discontinuous nonlinear system. As a time-invariant system is normally denoted as 𝑥̇ = 𝜑(𝑥) with a state vector
𝑥 ∈ ℛ 𝑛 , VSS on the other hand can be denoted by a piecewise function 𝑥̇ = 𝜑(𝑥, 𝑡) with 𝑡 ∈ ℛ. A concept
discussed in [8], [9] and the references therein state that a variable structure control has become the established
method to control or regulate VSS. A sliding mode approach has become the main subset to a variable structure
operation. Sliding mode is one of the established nonlinear management methods which is robust to parameter
uncertainties. As such, sliding mode concept is suitable to stabilize a system with uncertain parameters [11].
Nevertheless, few shortcomings of sliding mode that can be compromised by designers are the chattering
phenomenon, and it difficultness in handling miss-match uncertainties [12]. To overcome miss-match uncertainties
in the mathematical expression, one may augment back-stepping calculation to the sliding mode approach [13].

Journal homepage: http://telkomnika.uad.ac.id

TELKOMNIKA Telecommun Comput El Control  1157

This manuscript discusses about the uncertainties and disturbances thoroughly. As such, the rest of
the manuscript begins with thorough review on the class of uncertainties and disturbances. The distinguishing
features between match and miss-match uncertainties are discussed in detail. Then, a numerical example on
the match and mismatch handling will be presented to replenish understanding.

2. UNCERTAINTIES AND DISTURBANCES

One of the key reasons for designing a controller is to combat with the adverse effect of uncertainty that
may appear in different forms as disturbances or as other imperfections in the system model. Uncertainty is classified
into two categories namely disturbance signals and dynamic perturbations. Disturbance signal includes sensor noise
(see aircraft noise in [14]), actuator noise [15] and exogenous disturbance such as intermittent wind flow in wind
turbine [16] and gust on aircraft [17]. Dynamic perturbations due to un-modelled dynamics (usually high frequency
dynamics) is known as unstructured uncertainty [18], [19]. Neglected nonlinearities in the system model and
variations in system parameters are known as parametric uncertainty [18], [19]. The unstructured uncertainty
constitutes the lumped dynamic perturbations that may occur in different parts of a system. For linear time invariant
systems, the configuration of unstructured uncertainty can be additive perturbation, inverse additive perturbation (see
the application in wind farms in [20]), input multiplicative perturbation, output multiplicative perturbation, inverse
input multiplicative, inverse output multiplicative perturbation, left coprime factor perturbations and right coprime
factor perturbations [19]. On the other hand, the parametric uncertainties affect the low-frequency range performance
that are normally caused by dynamic perturbations due to variations of certain system parameters. In a dynamical
system model, the uncertain term can be matched with the control input, mismatched with the control input or mixed
matched-mismatched with the control input. Hence, emerge a class of system so-called matched uncertainty,
mismatched uncertainty, and matched-mismatched uncertainty [21], [22].
There is a large volume of published studies describing the method to solve a dynamical system with
uncertainty. Numerous studies in [21]-[23] have attempted to solve matched uncertainty using a sliding mode control
(SMC) and a variable structure control (VSC). A method to solve nonlinear system with matched uncertainty using
Lyapunov redesign are discussed in [24], [25].
Mismatched and mixed matched-mismatched uncertainty are rather hard to handle as they require
matching condition such that the uncertain term matched with the control input. An early matching condition
using a Lyapunov min-max approach can be reviewed in [26]. Choi [27], [28], Lee and Mau [29] proposed SMC
to stabilize linear dynamical system with mismatched uncertainty. For a system with nonlinearity and input matrix
uncertainty, the approach has been proposed by Choi in [30]. To date, very little literature proposed a control
approach for nonlinear system with matched-mismatched uncertainty. Amongst them, Park et al. [31] proposed
VSC for linear system with matched-mismatched uncertainty. In Kamarudin et al. [5], proposed back-stepping
and Lyapunov redesign approach to stabilize nonlinear system with matched-mismatched uncertainty.
To compensate the uncertainty in the design therein, the nominal back-stepping controller is augmented with
nonlinear damping function.
As for linear systems, numerous studies have been revealed. For example, studies in [32] reported the use
of linear matrix inequalities (LMI-based) approach to the analysis and design of closed-loop system under linear
state feedback laws to maximize the disturbance rejection capability. Several studies thus far have linked the use of
LMI-based method to the problem of disturbance tolerance and rejection for a family of linear systems subject to
actuator saturation and 𝐿∞ -disturbances [33], [34], 𝐿2 -disturbance [35]-[37] or 𝐿∞ /𝐿2 -disturbance [38].
More practical treatments on the uncertainties have been discussed by Jamri [39], and overview in power system.
Figure 1 summarizes the uncertainties phenomena and they classification.

Figure 1. Class of uncertainties and disturbances

Brief note on match and miss-match uncertainties (Muhammad Nizam Kamarudin)

1158  ISSN: 1693-6930

3. MISS-MATCH HANDLING
Miss-match (or un-match) uncertainties are rather hard to handle. In certain occasion, system with
miss-match uncertainties needs structural condition and complex mathematical manipulation. Simple
continuous time-invariant system with miss-match exogenous disturbance is shown in (1).

𝑥̇ = 𝐹(𝑥) + 𝐺(𝑥)𝑢 + 𝐻(𝑥)𝓌 (1)

Where 𝐹(∙), 𝐺(∙) and 𝐻(∙) are continuous functions, 𝑢 is the control input, and 𝓌 is the disturbance
input. The conceptual diagram for the system is depicted in Figure 2.

Figure 2. Conceptual diagram for system in (1)

It is clearly shown in both (1) and Figure 2 that 𝓌 is miss-match to 𝑢; and 𝑢 does not affine in 𝑥̇ (𝐹, 𝐻).
As such, one might simply design a stabilizing function 𝑢 under assumption that nominal system in (1) is
stabilizable and the state is available for feedback; that is 𝑌(𝑥) = {𝑥} for output 𝑌(𝑥). However, omitting 𝓌
in the stabilizing function 𝑢 may not realize the stabilization if 𝓌 dominate the function and has no constraint.
Though 𝓌 constraint the closed unit ball (𝓌 ≡ ℬ), the unconstrained control function (𝑢 ≡ 𝒰) unable to
dominate 𝓌. In this case, one might ponder a structural condition for (1) as portrayed in proposition 1.

3.1. Proposition 1
Assume that there exists a continuous function 𝐸(𝑥) that satisfies structural condition 𝐻(𝑥) = 𝐺(𝑥)𝐸(𝑥).
Then the matching condition can be applied to the system (1).

𝑥̇ = 𝐹(𝑥) + 𝐺(𝑥)[𝑢 + 𝐸(𝑥)𝓌] (2)

Through proposition 1, 𝑢 and 𝓌 become affine in 𝐺(𝑥), and 𝓌 is said to be matched with the control
input 𝑢. Thus 𝓌 can be combated easily provided that 𝓌 is available and observable. In case 𝓌 is unobservable,
one might use disturbance estimator provided that the bounded of 𝓌is known (perhaps within a closed unit ball
𝓌 ≡ ℬ). Figure 3 shows affine function (2) where 𝓌 enters through the same input channel as 𝑢.

Figure 3. System in (1) after matching condition

3.1.1. Miss-match versus match uncertainties

To replenish the understanding towards the method, two numerical systems are shown in (3) and (4).
𝑥̇1 = 𝑥2
(3)
𝑥̇ 2 = 𝑢 + 𝑥13 𝓌

𝑥̇1 = 𝑥2 + 𝑥13 𝓌
(4)
𝑥̇ 2 = 𝑢

TELKOMNIKA Telecommun Comput El Control, Vol. 21, No. 5, October 2023: 1156-1162
TELKOMNIKA Telecommun Comput El Control  1159

0 1
Note that both systems in (3) and (4) possess identical dynamic (system matrix [ ]) and input
0 0
𝑇
matrix ([0 1] ). The only difference between the two is the attribute of matrix associated with 𝓌. To this
end, system in (3) satisfies the matching condition because the disturbance can be re-grouped into a function
𝑓(𝑢, 𝓌, 𝑥1 ). Whereas, system in (4) does not satisfy the matching condition. To visualize the issue, let us plug
the numerical dynamics into the system. With simple matrix algebra, one would reach clearer visualization
between match and miss-match appearance in the system at hand. Hence, system (3) becomes:
𝑥̇ 0 1 𝑥1 0
[ 1] = [ ] [ ] + [ ] [𝑢 + 𝑥13 𝓌] (5)
𝑥̇ 2 0 0 𝑥2 1
Whereas system (4) turns into:
𝑥̇ 0 1 𝑥1 0 1
[ 1] = [ ] [ ] + [ ] 𝑢 + [ ] 𝑥13 𝓌 (6)
𝑥̇ 2 0 0 𝑥2 1 0
System (5) portrays a matched uncertainty. Whereas system (6) reveals miss-match uncertainty that
is identical to the priorly defined system in (1). At glance, we can control the 1st state. The system is
controllable if we consider the controllable matrix denotes as:
0 1
𝐶𝑚 ≜ [𝐵 𝐴𝐵 ] = [ ] (7)
1 0
That gives the rank 2 matrix. As the systems are known to be autonomous, the null input matrix results in
unobservable states. As both systems (5) and (6) are controllable, let design the feedback gain 𝐾 that will
produce the control energy 𝑢 = −𝐾𝑥 under the nominal condition (that is 𝓌 = 0). To simplify the process, let
us place the closed loop aigenvalues (𝐴 − 𝐵𝐾) in stable region, for instance 𝑒𝑖𝑔(𝐴 − 𝐵𝐾) ≡ [−1 −2].
The judicious choice of 𝑒𝑖𝑔(𝐴 − 𝐵𝐾) leads to the feedback gain 𝐾 = [2 3]. The history of the states
trajectory is depicted in Figure 4. With the judicious 𝐾-value, both states return to equilibria within 3 seconds
with initial condition 𝑥(0) = [1 1]. Figure 5 shows the control law for system in (6) with matched
disturbance 𝓌 appeared in the system. Despite the disturbance, the system is stabilizable with only slight
distortion in the trajectory, as depicted in Figure 6.

Figure 4. State trajectory for system (6) without Figure 5. Control law u for system (6) when
disturbance 𝓌 with state feedback disturbance 𝓌 occurred

Figure 6. State trajectory for system (6) with disturbance 𝓌 and state feedback

Brief note on match and miss-match uncertainties (Muhammad Nizam Kamarudin)

1160  ISSN: 1693-6930

3.1.2. Example of matching condition

To illustrate the concept of structural matching condition more literally, let consider two-dimensional
nonlinear system of the form.

𝑋̇ = 𝑓(𝑋) + 𝑔(𝑋)𝑦 + ℎ(𝑋)𝜉(𝑋, 𝑡) (8)

𝑦=𝑢 (9)
𝑋
Where [ ] ∈ ℛ 𝑛+1 are the system states with 𝑋 ∈ ℛ 𝑛 , 𝑋 ∈ ℛ is the single control input, 𝜉(𝑋, 𝑡) is the sum of
𝑦
uncertainties and time varying exogenous disturbances, 𝑓(∙), 𝑔(∙) and ℎ(∙) are smooth functions. As compared
with one dimensional system in (2), the control problem for system (8) and (9) become more complicated as the
control command 𝑢 does not directly influence 𝑥1 . In addition, 𝜉(𝑋, 𝑡) is mismatches to 𝑦 and 𝑢. Therefore,
it needs structural matching condition prior to the design steps. By using proposition 1, we introduce a continuous
function 𝑝(𝑋) that satisfies structural condition ℎ(𝑋) = 𝑔(𝑋)𝑃(X). Then the matching condition can be applied to
the system (8), (9). The dynamic equation in subsystem (10) eases the elimination of 𝑝(𝑋)𝜉(𝑋, 𝑡)-term via a virtual
control 𝑦. To this end, the design objective is to eliminate 𝜉(𝑋, 𝑡) by a control law (not to be discussed in this work).

𝑋̇ = 𝑓(𝑋) + 𝑔(𝑋)[𝑢 + 𝑝(𝑋)𝜉(𝑋, 𝑡)] (10)

𝑦=𝑢 (11)

4. CONCLUSION
The necessity of matching condition is emphasized in the existence of unmatched uncertainties
(or mismatched) and exogeneous disturbances. Beforehand, the matching case and unmatching case were
distinguished with example. The outcome showed that the structural matching condition is able to simplify the
systems in order to facilitate control design. In the future, the application of control design will be presented in
conjunction to the matching process discussed in this manuscript.

ACKNOWLEDGEMENTS
Special acknowledge is addressed to the Center for Research and Innovative Management (CRIM),
UTeM for the research fund, and the Faculty of Electrical Engineering, UTeM for the research facility.

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BIOGRAPHIES OF AUTHORS

Muhammad Nizam Kamarudin received the B. Eng (Hons.) Electrical from

the Universiti Teknologi MARA, in 2002, and M.Sc. in Automation and Control from the
University of Newcastle Upon Tyne, United Kingdom in 2007. He received the Doctor of
Philosophy in Electrical Engineering from the Universiti Teknologi Malaysia in 2015. He is
currently with the Universiti Teknikal Malaysia Melaka (UTeM). His research interests
include control systems and hydraulic technology. Before joining UTeM, he worked as a
Technical Engineer at the Magnetron Department of Samsung Electronics Malaysia and
Suwon, South Korea. He can be contacted at email: nizamkamarudin@utem.edu.my.

Sahazati Md Rozali received the Bachelor of Electrical & Electronic

Engineering (Electronic) in 2004 from Universiti Sains Malaysia. Then she pursued her
Master in Electrical Engineering (Control, Automation & Robotics) in Universiti Teknologi
Malaysia on 2006 and Doctor of Philosophy in Electrical Engineering from the same
university in 2014. She is a member of Board of Engineers Malaysia (BEM) and Malaysian
Board of Technologist (MBOT). Her specialization is in control system, modelling and
system identification. She can be contacted at email: sahazati@utem.edu.my.

Mohd H. Hairi received his Bachelor’s degree in Electrical Engineering from

University Sains Malasyia in 2001 and Master of Science degree in Electrical Power
Engineering from Universiti Teknologi Malaysia in 2003. He received his Ph.D degree from
University of Manchester in 2015. He has served as an academic staff at Universiti Teknikal
Malaysia Melaka (UTeM) since 2003 and he is currently a senior lecturer in the Faculty of
Electrical Engineering (FKE), UTeM. He is an expert in power system protection and his
research interests included power system, renewable energy and distributed generation. He
can be contacted at email: hendra@utem.edu.my.

Muhammad I. Zakaria is a lecturer at Universiti Teknologi MARA. He

obtained his B.Eng degree in Mechatronics from International Islamic University Malaysia
in 2010 and his M.Eng and Ph.D degrees in Electrical Engineering from Universiti Teknologi
Malaysia in 2012 and 2019 respectively. His research interests include renewable energy,
photovoltaic system, and control system. He can be contacted at email: iqbal.z@uitm.edu.my.

TELKOMNIKA Telecommun Comput El Control, Vol. 21, No. 5, October 2023: 1156-1162