1496 J. Zhang et al.
ject to random abrupt perturbations due to compo-
nent repairs or failures, sudden environmental changes,
and other unpredictable factors. Over the past few
decades, lots of practical cases have shown that Markov
jump systems (MJSs) are very reasonable for model-
ing abrupt changes [13,14]. Some interesting research
issues of MJSs have been explored such as con-
trol synthesis [16–18] and filtering and state estima-
tion [15,19]. Recently, positive Markov jump systems
(PMJSs), whose subsystems are positive, have been
investigated due to their theoretical challenge and prac-
tical importance [20–24]. Some issues of PMJSs have
also attracted the researchers’ interest. For example,
mean and stochastic stability analysis of PMJSs were
concerned in [20] and [21], respectively. In [22], a
linear programming-based approach for positive fil-
tering of PMJSs was proposed. A detail characteri-
zation of L
1
/
1
-gain performance analysis of PMJSs
can be found in [23]. In [24], L
∞
-gain analysis and
control of PMJSs with time delays were investigated.
The results mentioned above are mainly concerned
with linear PMJSs and general (non-positive) nonlinear
Markov jump systems (NMJSs). In fact, many phys-
ical systems such as epidemiological and biological
models involve variables with nonnegative and non-
linear random changes [ 25]. Therefore, nonlinear posi-
tive Markov jump systems (NPMJSs) are very powerful
for modeling complex dynamics. In the literature, the
positivity analysis and control synthesis of nonlinear
positive systems [26] and nonlinear positive switched
systems [27–29] have been addressed. To the best of
the authors’ knowledge, only few efforts have been
devoted to NPMJSs [25] though there have been many
results on NMJSs (non-positive) [30–34]. In particular,
it seems that the case of complex saturation nonlinear-
ities and time-varying delay has not yet been studied.
This implies that the topic of this paper is full of chal-
lenge. This motivates our investigation.
On the other hand, the saturation control of nonlin-
ear systems has always been one of difficult issues in
the control community. Some control approaches have
been presented to deal with nonlinear saturation con-
trol issues such as fuzzy-scheduling control [35], robust
adaptive control [36], adaptive fuzzy control [37], and
other ones [38,39]. Most of existing saturation control
approaches for nonlinear systems use a linear satura-
tion controller since it is easy to transform the saturation
term into a non-saturation one [40–43]. However, linear
controllers usually contain some errors owing to exter-
nal environmental disturbances and the imprecision in
dynamic systems. Therefore, linear controllers may not
be effective for complex nonlinear systems with satura-
tion and time delay and the robust performance may be
lost in such a case. In other words, the linear controller
may be fragile when dealing with complex nonlinear
systems. Considering these reasons, it is important to
propose a non-fragile controller to guarantee the robust
stability of the closed-loop systems. The non-fragile
controller is to introduce a controller gain uncertainty
term, located in a predetermined admissible range, to
the linear controller. Some interesting results on the
non-fragile controller combine the simplicity of linear
approaches with some robustness properties [44–48].
In this paper, we propose a non-fragile controller
for NPMJSs with actuator saturation and time-varying
delays, where the controller contains multiplicative and
additive perturbations, respectively. First, a positivity
criterion on an auxiliary nonlinear system is addressed
under the sector nonlinearity conditions. By virtue of
the criterion, the positivity of MJSs with the sector
nonlinearity is explored. A novel nonlinear co-positive
Lyapunov functional is constructed for the considered
systems. Using the Lyapunov functional, a set of non-
fragile controllers with auxiliary feedback control gains
are designed using a matrix decomposition technique.
A non-fragile controller with multiplicative perturba-
tion is designed to guarantee the positivity and stabil-
ity with an L
1
-gain performance. A cone set is con-
structed for the attraction domain estimation, and a
design approach to auxiliary controller gains is pre-
sented. Then, the non-fragile control with additive per-
turbation is proposed in terms of a linear programming
approach. The contributions of the paper lie in three
aspects: (i) Non-fragile controllers with multiplicative
and additive perturbations are proposed for NPMJSs,
respectively, (ii) a linear computation method based on
linear programming is presented for dealing with the
corresponding conditions, and (iii) the presented non-
fragile control framework has potential applications in
other issues of hybrid positive systems. The remain-
der of this paper is organized as follows. In Sect. 2,
some preliminaries are introduced. In Sect. 3, the non-
fragile controller design with L
1
-gain is investigated
and the estimation of an attraction domain is addressed.
An illustrative example is given in Sect. 4. Finally, the
paper is concluded in Sect. 5.
Notations Denote by R the set of real numbers,
R
n
(resp. R
n
+
) the n-column real (resp. positive) vector
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