1 Introduction

Some small physical parameters such as time constants, masses, capacitances, etc, increase the order of dynamic systems and introduce the multi-time scales property. Resulting systems can possess simultaneously, slow and fast coupling states increasing the system complexity. The singular perturbation approach [1–4] is a powerful technique for systems order reduction and time scales separation. The method explicit the time scale separation by mean of a small singular perturbation parameter \(\mu \). When \(\mu \) is small enough, the high order system is decomposed into slow and fast subsystems and considered as a singularly perturbed system.

Stability of linear singularly perturbed systems have been extensively studied in past years and a great number of results have been reported in the literature; see, e. g. [5–11], and the references therein. Recently, a great amount of effort has focused on the stability analysis of nonlinear singularly perturbed systems [12–23] where the properties of two lower order slow and fast subsystems are studied by mean of Lyapunov functions to predict the stability properties of the composite system.

In spite of the progression of stability nonlinear singularly perturbed systems analysis, it is not that obvious to apply complex techniques to practical engineering problems. It is still needed to develop the more simple stability technique for general nonlinear singularly perturbed systems. The fuzzy control theory [24, 25] uses collections of linguistic rules in order to model such as systems by considering qualitative aspects of human knowledge and reasoning processes without employing a precise quantitative analysis [26].

Fuzzy set theory has been developed and widely studied in the past two decades. It has been successfully applied in engineering problems due to its capacity of modeling and controlling complex nonlinear systems [27, 28]. The most known methods in the literature designed for synthesizing stability conditions of fuzzy systems are [29–31]: Popov’s stability criterion [32, 33], the circle criterion [34, 35], conicity criterion (extended version of circle criterion) [36, 37], direct Lyapunov’s method [38–40], analysis of system stability in phase space [35, 41], the describing function method [37], methods of stability indices and systems robustness [35, 37], methods based on theory of input-output stability [37, 42], hyperstability theory [43–45] and heuristic methods [37, 46].

Recently, stability of singular fuzzy systems have been investigated [47–55]. Huang [47] proposes a discrete singular T-S (DST-S) model and introduces to stability criteria by non-strict linear matrix inequalities (LMIs) and projection method. Liu et al. [48] synthesizes stability conditions of DST-S systems in term of LMIs and derives stability conditions for feedback controller via the nonlinear matrix inequalities (NMIs). Dong and Yang [49] present a method of evaluating the upper bound of the singular perturbation parameter \(\mu \) for DST-S systems with meeting a prescribed H\(\infty \) performance bound requirement. Xu and Lam [50] propose a necessary and sufficient stability condition for uncertain DST-S systems in terms of a strict linear matrix inequality. Xu et al. [51] considers the problem of robust stability of uncertain DST-S systems with time-varying norm-bounded parameter uncertainties. A sufficient stability condition is proposed in terms of a set of LMIs. Chen et al. [52] treat state feedback robust stabilization problems for DST-S systems with parameter uncertainty, based on a matrix spectral norm approach.

Motivated by the fact that fuzzy sets provide an effective way to describe a nonlinear system, we will investigate, in this paper, the stability problem for T-S fuzzy discrete singular perturbed systems without using conventional Lyapunov function. New sufficient stability conditions, for original and reduced order discrete nonlinear T-S fuzzy models, are developed based on the arrow matrix form and Borne and Gentina criterion.

This paper is organized as follows. In sect. 2, the fuzzy system modeling and decoupling procedure are formulated via the singular perturbation technique. Section 3, stability conditions based on Lyapunov functions are reviewed, and new stability conditions for T-S fuzzy discrete singularly perturbed systems are proposed. In sect. 4, a numerical example is given, and finally, conclusions are presented in sect. 5.

2 Two-time scale singularly perturbed fuzzy model description

Physical processes are very complex in practice and rigorous mathematical models can be very difficult to synthesize, if not impossible. Many of these systems can be expressed in some form of mathematical model locally, or as an aggregation of a set of mathematical models. Here, we consider the Takagi–Sugeno (T-S) model to represent a complex system that includes local analytic nonlinear models \({S_i}\) [56]. The \(i\)th fuzzy inference rule of the fuzzy model is of the following form:

$$\begin{aligned}&{R_i}:{ IF}\,{x_k}\,is\,M_1^i\, \cdots \,\,and\,\,{x_k}\,is\,M_n^i\,\,{ THEN}\nonumber \\&{x_{k + 1}} = {A_i}\left( {.}\right) {x_k},\,\,\ i \in I: = 1,2, \cdots ,m \end{aligned}$$
(1)

where the state vector \(x\left( {kT}\right) \) is noted \({x_k},\,{x_k} \in \mathbb{R }_{}^n,\,kT\) is the discrete time and \(T\) the sampling time such that \( {x_k} = {\left[ {\begin{array}{*{20}{c}}{x_k^{{1^T}}}&{x_k^{{2^T}}} \end{array}} \right] ^T}\). \(x_k^1 \in {\mathbb{R }^{{n_1}}},\,x_k^2 \in {\mathbb{R }^{{n_2}}}\) and \(m\) denotes the number of inference rules and \(M_j^i\) \(( {j = 1,2, \ldots ,n})\) the fuzzy sets. The instantaneous characteristic \(n \times n\) matrix \({A_i}\left( {.} \right) \) of the \(i\)th local model of the studied system is defined by

$$\begin{aligned} {A_i}\left( . \right) = \left[ {\begin{array}{lllllllllllllll} {A_{i,11}^{}}\left( \cdot \right) &{}\,\,\,\,{A_{i,12}^{}}\left( \cdot \right) \\ {A_{i,21}^{}}\left( \cdot \right) &{}\,\,\,\,{A_{i,22}^{}}\left( \cdot \right) \end{array}} \right] \end{aligned}$$
(2)

By using a standard fuzzy inference method -that is, using a singleton fuzzifier, product fuzzy inference and weighted average defuzzifier- the final state of the fuzzy system \(S\) is inferred as follows [31]

$$\begin{aligned} S:{x_{k + 1}}\, = \,\,\sum \limits _{i = 1}^m {{h_i}(x_k)} {A_i}\left( \cdot \right) {x_k} \end{aligned}$$
(3)

with

$$\begin{aligned} h_i\left( {{x_k}} \right) = \frac{{{w_i}\left( {{x_k}} \right) }}{{\sum \limits _{i = 1}^m {{w_i}\left( {{x_k}} \right) } }} \,\,\ and \,\,\ {w_i}\left( {{x_k}} \right) = \prod \limits _{j = 1}^n {M_j^i} \end{aligned}$$
(4)

We assume that \({w_i}\left( {{x_k}} \right) \geqslant 0\) and \(\sum \limits _{i=1}^m {{w_i}\left( {{x_k}} \right) }>0\)   for \(i \in I\). Then, it is easy to see that \({h_i}\left( {{x_k}} \right) \geqslant 0\),   for \(i \in I\) and \(\sum \limits _{i = 1}^m {{h_i}\left( x_k \right) }=1\).

The local system \({S_i}\) is assumed to possess a two-time-scale property, which means that the \(n\) eigenvalues of \({S_i}\) can be separated into \({n_1}\) slow modes and \({n_2}\) stable fast modes related to \(x_k^1\) and \(x_k^2\), respectively. The fast subsystem \(x_k^2,\) assumed to be stable, is active only during a short initial period, after, it is negligible and the system can be described by it slow subsystem \(x_k^1\) [57].

Often, numerical methods for simulation or controller design cannot be applied to large scale systems because of their extensive numerical costs. This motivates model reduction, which is the approximation of the original, large realization by a realization of smaller order. A method that maintains the coordinate system of the original model is based on singular perturbation technique [1, 5, 6]. In most classical and modern control schemes, singular perturbation techniques exploit the two-time-scale nature of the system in order to decompose the design problem into slow and fast modes.

Singularly perturbed systems have the following form [6, 8, 58–60]

$$\begin{aligned} \left[ {\begin{array}{lllllllllllllll} {x_{k + 1}^1} \\ {x_{k + 1}^2}\end{array}} \right] = \left[ {\begin{array}{c@{\quad }c} {I_{n1}^{} + \mu A_{i,11}^ * }&{}{\mu A_{i,12}^ * } \\ {A_{i,21}^ * }&{}{A_{i,22}^ * } \end{array}} \right] \left[ {\begin{array}{lllllllllllllll} {x_k^1} \\ {x_k^2} \end{array}} \right] \end{aligned}$$
(5)

where \(\mu \) is a small positive singular perturbation parameter that indicates separation of the state space variables into slow variables \(x_k^1\) and fast variables \(x_k^2\), and \(\det \left( {{I_{{n_2}}} - A_{i,22}^ * } \right) \ne 0\) [1]. The slow subsystem is defined by neglecting the fast stable dynamics, which is equivalent to replace the second equation of (5) by its steady-state algebraic equation. The fast subsystem, supposed to be stable, is derived by assuming that slow variables are constant during fast transients and \(\mu = 0\).

Described system (5) is dual to system (1) and it is possible to put the system into the singularly perturbed form (5). The relation ship among the system matrices defined in (1) and in (5) are as follows

$$\begin{aligned} \begin{array}{l} A_{i,11}^ * = {\mu ^{ - 1}}\left( {A_{i,11}^{} - {I_{{n_1}}}} \right) , A_{i,12}^ * = {\mu ^{ - 1}}A_{i,12}^{} \\ A_{i,21}^ * = A_{i,21}^{},\qquad \qquad A_{i,22}^ * = A_{i,22}^{} \end{array} \end{aligned}$$
(6)

Applying the decoupling transformation [1, 6, 61, 62] defined by

$$\begin{aligned} \begin{array}{l} \left[ {\begin{array}{lllllllllllllll} {x_{k + 1}^s} \\ {x_{k + 1}^f} \end{array}} \right] = \left[ {\begin{array}{lllllllllllllll} {{I_{{n_1}}} - \mu M_iL_i}&{}\,\,\,{ - \mu M_i} \\ \qquad \quad L_i&{}\,\,\,\quad {{I_{{n_2}}}} \end{array}} \right] \left[ {\begin{array}{lllllllllllllll} {x_{k + 1}^1} \\ {x_{k + 1}^2} \end{array}} \right] \\ \left[ {\begin{array}{lllllllllllllll} {x_{k + 1}^1} \\ {x_{k + 1}^2} \end{array}} \right] = \left[ {\begin{array}{lllllllllllllll} {{I_{{n_1}}}}&{} \qquad \qquad {\mu M_i} \\ { - L_i}&{}\,\,\,\,\,\,{{I_{{n_2}}} - \mu L_iM_i} \end{array}} \right] \left[ {\begin{array}{lllllllllllllll} {x_{k + 1}^s} \\ {x_{k + 1}^f} \end{array}} \right] \end{array} \end{aligned}$$
(7)

the singularly perturbed system (5) can be decoupled into independent slow and fast subsystems [6] as

$$\begin{aligned}&S_i^d:\,\,\left[ {\begin{array}{llllll} {x_{k + 1}^s} \\ {x_{k + 1}^f} \end{array}} \right] = \left[ {\begin{array}{cc} {I_{{n_1}}^{} + \mu A_i^s}&{}0 \\ 0&{}{A_{i,22}^ * } \end{array}} \right] \left[ {\begin{array}{lllllllllllllll} {x_k^s} \\ {x_k^f} \end{array}} \right] \end{aligned}$$
(8)
$$\begin{aligned}&S_i^s:\,\,\,x_{k + 1}^s = (I_{{n_1}}^{} + \mu A_i^s)x_k^s\end{aligned}$$
(9)
$$\begin{aligned}&S_i^f:\,\,\,x_{k + 1}^f = A_{i,22}^ * x_k^f \end{aligned}$$
(10)

with

$$\begin{aligned} A_i^s = A_{i,11}^ * + A_{i,12}^ * {\left( {{I_{{n_2}}} - A_{i,22}^ * } \right) ^{ - 1}}A_{i,21}^ * \end{aligned}$$
(11)

if it exists \(L_i \in \mathbb{R ^{{n_1} \times }}^{{n_2}}\) and \(M_i \in \mathbb{R ^{{n_2} \times }}^{{n_1}}\) matrices satisfying the algebraic equations [6]

$$\begin{aligned}&A_{i,21}^ * + L_i - A_{i,22}^ * L_i + \mu L_i\left[ {A_{i,11}^ * - A_{i,12}^ * L_i} \right] = 0\end{aligned}$$
(12)
$$\begin{aligned}&A_{i,12}^ * + M_i - M_iA_{i,22}^ *+\mu \left[ {A_{i,11}^ * - A_{i,12}^ * L_i} \right] M_i \nonumber \\&\quad -\, \mu ML_iA_{i,12}^ * = 0 \end{aligned}$$
(13)

\({{x}^{s}}\in {\mathbb{R }^{{{n}_{1}}}}\) and \({{x}^{f}}\in {{\mathbb{R }}^{{{n}_{2}}}}\) are, respectively, the slow and the fast subsystems state vectors. Finally, the decoupled discrete nonlinear T-S fuzzy model \(S^d\) of the original system (3), and the corresponding slow \(S^s\) and fast \(S^f\) fuzzy subsystems are respectively given by

$$\begin{aligned}&{S^d}:\left[ {\begin{array}{lllllllllllllll} {x_{k + 1}^s} \\ {x_{k + 1}^f} \end{array}} \right] = \left[ {\begin{array}{cc} {\sum \limits _{i = 1}^m {{h_i}} \left( {I_{{n_1}}^{} + \mu A_i^s} \right) }&{} 0 \\ 0&{}{\sum \limits _{i = 1}^m {{h_i}} A_{i,22}^ * } \end{array}} \right] \left[ {\begin{array}{lllllllllllllll} {x_k^s} \\ {x_k^f} \end{array}} \right] \nonumber \\ \end{aligned}$$
(14)
$$\begin{aligned}&{S^s}:{x_{k + 1}^s}\, = \,\,\sum \limits _{i = 1}^m {{h_i}} \left( {I_{{n_1}}^{} + \mu A_i^s} \right) {x_{k}^s} \end{aligned}$$
(15)
$$\begin{aligned}&{S^f}:{x_{k + 1}^f}\, = \,\,\sum \limits _{i = 1}^m {{h_i}} A_{i,22}^ * {x_{k}^f} \end{aligned}$$
(16)

The main objective of the present paper is to provide conditions ensuring the asymptotic stability of the discrete nonlinear T-S fuzzy system (3). We will show that this corresponds in some case to verify the stability conditions of the slow and fast subsystems (15, 16) synthesized via singular perturbation technique.

3 Stability study

In this section, we recall basic results on stability analysis for T-S fuzzy models based on Lyapunov functions and we formulate the problem. We, then, establish main stability results for the discrete nonlinear original (1, 3) and decoupled (14) T-S fuzzy system.

3.1 Lyapunov functions

Stability analysis of T-S fuzzy systems has been pursued mainly based on Lyapunov stability. Mainly, three different Lyapunov functions, developed in the literature [31], are introduced below.

3.1.1 The common (or global) quadratic Lyapunov functions \(V\left( x \right) = {x^T}Px\) [38, 63]

Theorem 1

[38]: The TS fuzzy system (1), or equivalently (3), is globally exponentially stable if there exists a common positive definite matrix such that the following LMIs are satisfied

$$\begin{aligned} A_i^TP{A_i} - P < 0,\,\, i \in I \end{aligned}$$
(17)

3.1.2 The piecewise quadratic Lyapunov functions \(V\left( x \right) = \sum \limits _{i = 1}^m {{x^T}{P_i}x} \)

Define \(m\) regions in the premise variable space as follows

$$\begin{aligned} {D_i} = \left\{ {x\left| {\,{h_i}\left( x \right) > {h_l}\left( x \right) \quad l \in I,\quad l \ne i} \right. } \right\} ,\quad i \in I \end{aligned}$$
(18)

The T-S fuzzy system (3) can be expressed in each local region as

$$\begin{aligned} {x_{k + i}} = \left( {{A_i} + \varDelta {A_i}\left( h \right) } \right) {x_k},\quad i \in I \end{aligned}$$
(19)

with

$$\begin{aligned}&\varDelta {A_i}\left( h \right) = \sum \limits _{l = 1,l \ne i}^m {{h_l}}\nonumber \\&\varDelta {A_{il}},\,\,\,\varDelta {A_{il}} = {A_l} - {A_i}\\&{\left[ {\varDelta {A_i}\left( h \right) } \right] ^T}\left[ {\varDelta {A_i}\left( h \right) } \right] \leqslant E_{iA}^T{E_{iA}}\nonumber \end{aligned}$$
(20)

In addition, define a set \(\varOmega \) that represents all possible system transitions among regions, that is

$$\begin{aligned} \varOmega : = \left\{ {\left( {i,j} \right) \left| {{x_k} \!\in \! {D_i},\,\,{x_{k + 1}} \in {D_j},\,\,\forall i,j \in I,\,\,i \ne j} \right. } \right\} \end{aligned}$$
(21)

Theorem 2

[64]: The T-S fuzzy system (1), or equivalently (19), is globally exponentially stable if there exists a set of positive-definite matrices \({P_i}\),   \(i \in I\), such that the following LMIs are satisfied

$$\begin{aligned}&\left[ {\begin{array}{cc} {A_i^T{P_i}{A_i} \!-\! {P_i} \!+\! E_{iA}^T{E_{iA}}}&{}{A_i^T{P_i}} \\ {{P_i}{A_i}}&{}{ - \left( {I \!-\! {P_i}} \right) } \end{array}} \right] \!<\! 0,\,\,\, i \!\in \! I \end{aligned}$$
(22)
$$\begin{aligned}&\left[ {\begin{array}{cc} {A_i^T{P_j}{A_i} \!-\! {P_i} \!+\! E_{iA}^T{E_{iA}}}&{}{A_i^T{P_j}} \\ {{P_j}{A_i}}&{}{ - \left( {I \!-\! {P_j}} \right) } \end{array}} \right] \!<\! 0,\,\,\, i,j \!\in \! \varOmega \end{aligned}$$
(23)

3.1.3 The fuzzy (or non-quadratic) Lyapunov functions \(V\left( x \right) = \sum \limits _{i = 1}^m {{h_i}\left( x \right) {x^T}{P_i}x} \) [65, 66]

Theorem 3

[65]: The T-S fuzzy system (1), or equivalently (3), is globally exponentially stable if there exists a set of positive-definite matrices \({P_i}\),  \(i \in I\) such that the following LMIs are satisfied

$$\begin{aligned} A_i^T{P_j}{A_i} - {P_j} < 0, \,\, i\in I,\,\,\ j \in I \end{aligned}$$
(24)

The stability conditions synthesized via the common quadratic Lyapunov functions are very conservative and the introduced approach suffers mainly from few limitations. First, it has been noted that common quadratic Lyapunov functions tend to be conservative, and, might not exist for many complex highly nonlinear systems as shown in [64] and [67]. Second, it appears that a necessary condition, for the existence of this common positive definite matrix, is that all subsystems must be asymptotically stable [38]. Piecewise quadratic Lyapunov functions and fuzzy Lyapunov functions are less conservative but computation cost would be much higher. Vector norms constitute a systematic mean of obtaining comparison systems, which help to overvaluate and analyze nonlinear systems. An adequate choice of the stable overvaluing system may prove the initial system stability. The method is robust and a good choice of the vector norms may allows to obtain conservatism stability conditions [68–72].

In the following, sufficient conditions ensuring asymptotic stability of discrete T-S fuzzy systems (3) with \(m\) nonlinear local models (1) are proposed. The aforementioned conditions are developed for original and reduced order decoupled described systems.

3.2 Proposed stability conditions-main results

Consider the class of systems \({S_i}\) (1) described by the scalar equation

$$\begin{aligned} \tilde{x}_{k + n} + \sum \limits _{j = 1}^n {{a_{i,j}}\left( {\tilde{x}_{k + n - j}}\right) } {\tilde{x}_{k + n - j}} = 0, \,\,\,\ i \in I \end{aligned}$$
(25)

where the corresponding instantaneous characteristic polynomial \({P_{{S_i}}}(.,\lambda )\) is

$$\begin{aligned} {P_{{S_i}}}\left( {\,\,.\,\,,\lambda } \right) { \text{= } }{\lambda ^n} + \sum \limits _{p = 1}^n {{a_{i,p}}\left( . \right) } { }{\lambda ^{n - p}},\,\,\, i \in I \end{aligned}$$
(26)

and define distinct arbitrary constant parameters \({\alpha _j},\,j = 1,2, \cdots ,n-1\).

For \({\alpha _i} \ne {\alpha _j}\),   \(\forall i \ne j\) and \(i \in I\), let us introduce to the following notations

$$\begin{aligned}&{\beta _j}=\prod \limits _{\mathop {k = 1}\limits _{k \ne j} }^{n - 1} {{{(\,{\alpha _j} - \,{\alpha _k}\,)}^{ - 1}}} ,\,\,\, j = 1,2, \ldots ,n - 1\end{aligned}$$
(27)
$$\begin{aligned}&\gamma _j^i(.) = - {P_{{S_i}}}(\,.\,,{\alpha _j}).,\,\,\, j = 1,2, \ldots ,n - 1 \end{aligned}$$
(28)
$$\begin{aligned}&\delta _n^i(.)= - {a_{i,1}}(.) - \sum \limits _{p = 1}^{n - 1} {{\alpha _p}} \end{aligned}$$
(29)

Let \(S\) be a discrete T-S fuzzy system (3), \({S_i}\) a corresponding nonlinear local system of the form (1), \({S_i^s}\) the nonlinear decoupled slow local subsystem (9) and \({S_{}^s}\) a nonlinear decoupled slow fuzzy subsystem (15). By applying the Borne-Gentina practical stability criterion [73–75] to the discrete introduced systems characterized by the Benrejeb arrow form matrix [76–81], we obtain following theorems and corollaries.

Theorem 4

The discrete nonlinear local system \({S_i}\) is asymptotically stable, if there exists constant parameters \({\alpha _i} \in \mathbb{R }\),   \({\alpha _i} \ne {\alpha _j}\,\,\forall i \ne j\), such that

$$\begin{aligned} \left| {{\alpha _i}} \right| < 1\begin{array}{lllllllllllllll} {} \end{array}\begin{array}{lllllllllllllll} {} \end{array}\forall i = 1, \ldots ,n - 1 \end{aligned}$$
(30)

and

$$\begin{aligned} 1 - \left| {\delta _n^i\left( \cdot \right) } \right| - \sum \limits _{j = 1}^{n - 1} {\left| {{\beta _j}} \right| \left| {\gamma _j^i\left( \cdot \right) } \right| } {\left( {1 - \left| {{\alpha _j}} \right| } \right) ^{ - 1}} > 0 \end{aligned}$$
(31)

Proof

(Theorem 4) Let us consider the nonlinear local system \({S_i}\) expressed in the Frobenius form as

$$\begin{aligned} {\tilde{x}_{k + 1}} = A_i^{Fr}\left( \tilde{x}_{n} \right) {\tilde{x}_k} \end{aligned}$$
(32)

with

$$\begin{aligned} {A_i^{Fr}}({\tilde{x}_{n}})\, = \,\left[ {\begin{array}{*{20}{c}} 0&{}\,\,\, \cdots &{}\,\,\,{}&{}\,\,\,0&{}\,\,\,{ - {a_{i,n}}({\tilde{x}_{n}})} \\ 1&{}\,\,\, \ddots &{}\,\,\,{}&{}\,\,\, \vdots &{}\,\,\,{ - {a_{i,n - 1}}({\tilde{x}_{n}})} \\ 0&{}\,\,\, \ddots &{}\,\,\,{}&{}\,\,\,{}&{}\,\,\, \vdots \\ \vdots &{}\,\,\, \ddots &{}\,\,\,{}&{}\,\,\,0&{}\,\,\,{} \\ 0&{} \cdots &{}\,\,\,0&{}\,\,\,1&{}\,\,\,{ - {a_{i,1}}({\tilde{x}_{n}})} \end{array}} \right] \end{aligned}$$
(33)

A change of coordinate defined by

$$\begin{aligned} {y_k} = T{\tilde{x}_k} \end{aligned}$$
(34)

with \({y_k} \in \mathbb{R }_{}^n\) and \(T\) an invertible transformation for \(\forall {\alpha _i}\),  \(i = 1,2, \cdots ,n - 1,\,{\alpha _i} \ne {\alpha _j}\) and \(\forall i \ne j\).

$$\begin{aligned}&T=\left[ {\begin{array}{lllllllllllllll} 0&{}\,\,0&{}\,\, \cdots &{}\,\,0&{}\,\,1 \\ 1&{}\,\,{{\alpha _{n - 1}}}&{}\,\,{\alpha {{_{n - }^2}_1}}&{}\,\, \cdots &{}\,\,{\alpha {{_{n - 1 }^{n - 1}}}} \\ 1&{}\,\,{{\alpha _{n - 2}}}&{}\,\,{\alpha {{_{n - }^2}_2}}&{}\,\, \cdots &{}\,\,{\alpha {{_{n - 2}^{n - 1}}}} \\ \, \vdots &{}\,\, \quad \vdots &{}\,\, \quad \vdots &{}{}&{}\,\, \quad \vdots \\ 1&{}\,\,\,\,\, {{\alpha _1}}&{}\,\,\,\,\,{\alpha _1^2}&{}\,\, \cdots &{}\,\,{\alpha _1^{n - 1}} \end{array}} \right] \end{aligned}$$
(35)
$$\begin{aligned}&\det \left( T \right) = \prod \limits _{\mathop {{1}\le \mathrm{{j}} < \mathrm{{i}}\le \mathrm {n}-{1}}\limits _{i \ne j}} ({{\alpha _i} - {\alpha _j}}) \end{aligned}$$
(36)

leads to the following state space description

$$\begin{aligned} {y_{k + 1}} = {G_i}\left( . \right) {y_k} \end{aligned}$$
(37)

Allowing the synthesis of sufficient stability conditions easy to test, the new instantaneous characteristic matrix \({G_i}\left( . \right) \) is chosen to be in the arrow form [76–81], Appendix 2, as following

$$\begin{aligned} {G_i}(.) = T\,{A_i^{Fr}}(.)\,T_{}^{ - 1}\,\, = \,\left[ {\begin{array}{lllllllllllllll} {\delta _n^i(.)}&{}{{\beta _1}}&{} \cdots &{}{{\beta _{n - 1}}} \\ {\gamma _1^i(.)}&{}{{\alpha _1}}&{}{}&{}{} \\ \vdots &{}{}&{} \ddots &{}{} \\ {\gamma _{n - 1}^i(.)}&{}{}&{}{}&{}{{\alpha _{n - 1}}} \end{array}} \right] \end{aligned}$$
(38)

where \({\beta _i},\,\gamma _j^i\),  \(\delta _n^i\,\) and \({\alpha _i},\,i = 1,2, \ldots ,n - 1\), are defined by the relations (27–29). A pseudo-overvaluing matrix \(M\left( {{G_i}\left( \cdot \right) } \right) \) of the system (37), corresponding to the use of the vector norm (Appendix 1) \(p\left( y \right) \) such that

$$\begin{aligned} p\left( y \right) = {\left[ {\left| {{y_1}} \right| ,\,\left| {{y_2}} \right| ,\,...\,,\,\left| {{y_n}} \right| } \right] ^T} \end{aligned}$$
(39)

\(y = {\left[ {{y_1},\,{y_2},\,...\,,\,{y_n}} \right] ^T}\), for the stability study, can be obtained from the inequality

$$\begin{aligned} p({y_{k + 1}}) \leqslant M\left( {{G_i}\left( \cdot \right) } \right) p({y_k}) \end{aligned}$$
(40)

satisfied for each corresponding component; that leads to the following comparison system

$$\begin{aligned} {z_{k + 1}} = M\left( {{G_i}\left( \cdot \right) } \right) {z_k} \end{aligned}$$
(41)

with

$$\begin{aligned} M\left( {{G_i}\left( . \right) } \right) = \left[ {\begin{array}{lllllllllllllll} {\left| {\delta _n^i\left( \cdot \right) } \right| }&{}{\left| {{\beta _1}} \right| }&{} \cdots &{}{\left| {{\beta _{n - 1}}} \right| } \\ {\left| {\gamma _1^i\left( \cdot \right) } \right| }&{}{\left| {{\alpha _1}} \right| }&{}{}&{}{} \\ \vdots &{}{}&{} \ddots &{}{} \\ {\left| {\gamma _{n - 1}^i\left( \cdot \right) } \right| }&{}{}&{}{}&{}{\left| {{\alpha _{n - 1}}} \right| } \end{array}} \right] \end{aligned}$$
(42)

such as \({z_0} = p\left( {{y_0}} \right) \). If the nonlinearities of the comparison nonlinear system (41) are isolated in one row of \(M\left( {{G_i}\left( \cdot \right) } \right) \), the verification of the Kotelyanski condition (Appendix 1) enables to conclude to the stability of the original system characterized by \({G_i}\left( \cdot \right) \) [74]. It comes the following sufficient asymptotic stability condition of the original system \({S_i}\)

$$\begin{aligned} \left( {{I_n} - M\left( {{G_i}\left( \cdot \right) } \right) } \right) \left( {\begin{array}{lllllllllllllll} 1&{}2&{} \ldots &{}j \\ 1&{}2&{} \ldots &{}j \end{array}} \right) > 0\begin{array}{lllllllllllllll} {} \end{array}j = 1, \ldots ,n \end{aligned}$$
(43)

This ends the proof of Theorem 4. \(\square \)

Theorem 5

The discrete nonlinear decoupled local system \(S_i^d\) (8) is asymptotically stable if there exists \({\alpha _i}\in \mathbb{R }\), \({\alpha _i} \ne {\alpha _j}\)  \(\forall i \ne j\), such that

$$\begin{aligned} \left| {{\alpha _i}} \right| < 1\; \forall i = 1, \ldots ,n - 1 \end{aligned}$$
(44)

and

$$\begin{aligned} 1 - \left| {\delta _{_n}^i\left( \cdot \right) + \sum \limits _{j = {n_1}}^{n - 1} {{\beta _j}\gamma _j^i\left( \cdot \right) {{\left( {1 - {\alpha _j}} \right) }^{ - 1}}} } \right| \nonumber \\ - \sum \limits _{j = 1}^{{n_1} - 1} {\left| {{\beta _j}} \right| \left| {\gamma _j^i\left( \cdot \right) } \right| {{\left( {1 - \left| {{\alpha _j}} \right| } \right) }^{ - 1}}} > 0 \end{aligned}$$
(45)

Proof

(Theorem 5) Note that the satisfaction of the conditions (30), i.e. \(\left| {{\alpha _i}} \right| < 1,\,i = 1, \ldots ,n - 1\), means that the fast system characterized by a diagonal matrix \(\left\{ {{\alpha _i}} \right\} ,\,i = {n_1}, \ldots ,n - 1\) is stable. Conditions \(\left| {{\alpha _i}} \right| < 1,\,i = 1, \ldots ,{n_1} - 1\), are necessary to satisfy the reduced slow subsystem stability. In order to synthesize the stability conditions of the two-time-scale decoupled system \({S_i}\), we, consider the transformed nonlinear system states (38). Resulting \({A_{i,11}},\,{A_{i,12}},\,{A_{i,21}}\) and \({A_{i,22}}\) matrices are then in the form (46) where the matrix \({A_{i,11}}\) is candidate to characterize the slow subsystem of (1) and \({A_{i,22}}\) the fast one.

$$\begin{aligned} {A_{i,11}}&= \left[ {\begin{array}{lllllllllllllll} {\delta _n^i\left( \cdot \right) }&{}{{\beta _1}}&{} \cdots &{}{{\beta _{{n_1} - 1}}} \\ {\gamma _1^i\left( \cdot \right) }&{}{{\alpha _1}}&{}{}&{}{} \\ \vdots &{}{}&{} \ddots &{}{} \\ {\gamma _{{n_1} - 1}^i\left( \cdot \right) }&{}{}&{}{}&{}{{\alpha _{{n_1} - 1}}} \end{array}} \right] \nonumber \\ {A_{i,12}}&= \left[ {\begin{array}{lllllllllllllll} {{\beta _{{n_1}}}}&{} \cdots &{}{{\beta _{n - 1}}\,} \\ 0&{} \cdots &{}0 \\ \vdots &{}{}&{} \vdots \\ 0&{} \cdots &{}0 \end{array}} \right] \\ {A_{i,21}}&= \left[ {\begin{array}{lllllllllllllll} {\gamma _{{n_1}}^i\left( \cdot \right) }&{}0&{}{\,\,\, \cdots }&{}{\,\,0} \\ \vdots &{} \vdots &{}{}&{}{\, \vdots } \\ {\gamma _{n - 1}^i\left( \cdot \right) }&{}0&{} \cdots &{}{\,0} \end{array}\,\,\,\,\,\,} \right] \nonumber \\ {A_{i,22}}&= \left[ {\,\,\begin{array}{lllllllllllllll} {{\alpha _{{n_1}}}}&{}{}&{}0 \\ {}&{} \ddots &{}{} \\ 0&{}{}&{}{{\alpha _{n - 1}}} \end{array}} \right] \nonumber \end{aligned}$$
(46)

Arbitrary constant parameters \({\alpha _i},\,i = {n_1}, \ldots ,n - 1\), are chosen in concordance with the estimation of the dynamics that what we consider physically fast for the studied system. Substituting the relations (46), (6) and (11) into (9) and (10), yields to following discrete slow and fast subsystems, respectively

$$\begin{aligned} x_{k + 1}^s&= F_i^s\left( . \right) x_k^s\end{aligned}$$
(47)
$$\begin{aligned} x_{k + 1}^f&= F_i^fx_k^f \end{aligned}$$
(48)

and then corresponding comparison systems, respectively

$$\begin{aligned} y_{k + 1}^s&= M\left( {F_i^s\left( \cdot \right) } \right) y_k^s\end{aligned}$$
(49)
$$\begin{aligned} y_{k + 1}^f&= M\left( {F_i^f} \right) y_k^f \end{aligned}$$
(50)

where \(F_i^s \in {\mathbb{R }^{{n_1} \times {n_1}}}\) and \(F_i^f \in {\mathbb{R }^{{n_2} \times {n_2}}}\) are given by

$$\begin{aligned}&F_i^s\left( . \right) = \left[ {\begin{array}{lllllllllllllll} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\delta _n^i\left( \cdot \right) \\ + \sum \limits _{j = {n_1}}^{n - 1} {{\beta _j}\gamma _j^i\left( \cdot \right) {{\left( {1 - {\alpha _j}} \right) }^{ - 1}}} &{}{{\,\,\,\,\,\,\,\,\beta _1}}&{} \ldots &{}{{\beta _{{n_1} - 1}}} \\ {\gamma _1^i\left( \cdot \right) }&{}{{\,\,\,\,\,\,\,\,\alpha _1}}&{}{}&{}{} \\ \vdots &{}{}&{} \ddots &{}{} \\ {\gamma _{{n_1} - 1}^i\left( \cdot \right) }&{}{}&{}{}&{}{{\alpha _{{n_1} - 1}}} \end{array}} \right] \end{aligned}$$
(51)
$$\begin{aligned}&F_i^f = \left[ {\begin{array}{lllllllllllllll} {{\alpha _{{n_1}}}}&{}{}&{}{} \\ {}&{} \ddots &{}{} \\ {}&{}{}&{}{{\alpha _{n - 1}}} \end{array}} \right] \end{aligned}$$
(52)

and \(M\left( {F_i^s\left( \cdot \right) } \right) \) and \( M\left( {F_i^f} \right) \) are respectively the pseudo-overvaluing matrices of the slow and fast subsystems (9) and (10), corresponding to the use of the vector norm  (39). By applying the practical Borne-Gentina stability criterion [73–75] to the comparison systems (49) and (50) of (47) and (48), we deduce the stability conditions of the decoupled discrete systems\(S_i^d\) (8). The Theorem 5 is then proved. \(\square \)

Corollary 1

If the discrete nonlinear system \({S_i}\) (1) is asymptotically stable, i.e. the following conditions are satisfied

  1. (i)

    \(\exists \,\varepsilon >0\) and \({\alpha _j} \in \mathbb{R },\,0 < {\alpha _j} < 1, {\alpha _j} \ne {\alpha _k},\,\forall j \ne k,\,j,k = 1, \cdots ,n - 1\) such that

    $$\begin{aligned} \left\{ \begin{array}{ll} \delta _n^i\left( . \right) > 0\\ \gamma _j^i\left( . \right) {\beta _j} > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\forall j = 1,...,n - 1\\ \end{array} \right. \end{aligned}$$
    (53)
  2. (ii)
    $$\begin{aligned}&{\left. {{P_{{S_i}}}\left( {\,.\,,\lambda } \right) } \right| _{\lambda = 1}} \geqslant \varepsilon > 0, \,\,\,\ \mathrm{{i.e.}}\nonumber \\&\qquad \qquad \qquad \qquad \qquad 1 + \sum \limits _{p = 1}^n {{a_{i,p}}\left( . \right) } > 0,\,\,\, i \in I \qquad \end{aligned}$$
    (54)

then, the corresponding decoupled nonlinear system \(S_i^d\)  (8) is asymptotically stable.

Proof

(Corollary 1) By considering conditions (i) of the Corollary 1, and substituting relations (27–29) in (31), the stability condition (31) of the discrete nonlinear local system \({S_i}\) (1) becomes

$$\begin{aligned}&1 + {a_{i,1}}(.) + \sum \limits _{p = 1}^{n - 1} {{\alpha _p}}\nonumber \\&\quad + \sum \limits _{p = 1}^{n - 1} {\frac{1}{{1 - {\alpha _p}}}} {\left( {\frac{{\left( {\lambda - {\alpha _p}} \right) {P_{{S_i}}}\left( {\,.\,,\lambda } \right) }}{{Q\left( \lambda \right) }}} \right) _{\lambda = {\alpha _p}}} > 0 \end{aligned}$$
(55)

with

$$\begin{aligned} Q\left( \lambda \right) = \prod \limits _{p = 1}^{n - 1} {\left( {\lambda - {\alpha _p}} \right) } \end{aligned}$$
(56)

To deduce the stability conditions of the decoupled system \(S_i^d\) (8), let us first observe that

$$\begin{aligned} \frac{{{P_{{S_i}}}\left( {\,.\,,\lambda } \right) }}{{Q\left( \lambda \right) }}&= \lambda + {a_{i,1}}(.) + \sum \limits _{p = 1}^{n - 1} {{\alpha _p}}\nonumber \\&+ \sum \limits _{p = 1}^{n - 1} {\frac{1}{{\lambda - {\alpha _p}}}} {\left( {\frac{{\left( {\lambda - {\alpha _p}} \right) {P_{{S_i}}}\left( {\,.\,,\lambda } \right) }}{{Q\left( \lambda \right) }}} \right) _{\lambda = {\alpha _p}}} \end{aligned}$$
(57)

It, then, follows that the developed stability condition (55) is equivalent to

$$\begin{aligned} {\left. {\frac{{{P_{{S_i}}}\left( {\,.\,,\lambda } \right) }}{{Q\left( \lambda \right) }}} \right| _{\lambda = 1}} > 0 \end{aligned}$$
(58)

or

$$\begin{aligned} {\left. {{P_{{S_i}}}\left( {\,.\,,\lambda } \right) } \right| _{\lambda = 1}} > 0 \end{aligned}$$

which yields

$$\begin{aligned} 1 + \sum \limits _{p = 1}^n {{a_{i,p}}\left( . \right) } > 0,\qquad i \in I \end{aligned}$$
(59)

and constitutes a verification case of the validity of the linear Aizerman conjecture [82, 83]. These conditions, associated to aggregation techniques based on the use of vector norms, have led to stability domains for a class of Lure-Postnikov systems whereas, for example, Popov stability criterion use failed. The proof is easily completed by substituting the conditions (i) in stability condition (45) of the discrete nonlinear decoupled system \(S_i^d\) (8). \(\square \)

Corollary 2

If the discrete nonlinear decoupled system \(S_i^d\) (8) is asymptotically stable, i. e. the following conditions are satisfied

  1. (i)

    \(\exists \,\varepsilon > 0\) and \({\alpha _j} \in \mathbb{R },\,{\alpha _j} \ne {\alpha _k},\,\forall j \ne k\); \(j,k = 1, \ldots ,n - 1\), and \(0 < {\alpha _j} < 1\) \(j = 1, \ldots ,{n_1} - 1\) such that

    $$\begin{aligned} \left\{ \begin{array}{ll} \delta _{_n}^i\left( \cdot \right) + \sum \limits _{j = {n_1}}^{n - 1} {{\beta _j}\gamma _j^i\left( \cdot \right) {{\left( {1 - {\alpha _j}} \right) }^{ - 1}}} > 0\\ \\ \gamma _n^i\left( . \right) {\beta _j} > 0\qquad \forall j = 1,...,{n_1} - 1\end{array}\right. \end{aligned}$$
    (60)
  2. (ii)
    $$\begin{aligned} {\left. {{P_{{S_i}}}\left( {\,.\,,\lambda } \right) } \right| _{\lambda = 1}}&\geqslant \varepsilon > 0\,\,\,\,\,\,\,i.e.\nonumber \\&1 + \sum \limits _{p = 1}^n {{a_{i,p}}\left( . \right) } > 0,\,\,\, i \in I\qquad \end{aligned}$$
    (61)

then, the original discrete nonlinear local system \(S_i\) (1) is asymptotically stable if the following additional conditions are satisfied

$$\begin{aligned} \left\{ \begin{array}{ll} 0 < {\alpha _j} < 1\quad \qquad \forall j = {n_1}, \ldots ,n - 1 \\ \delta _n^i\left( . \right) > 0\\ \gamma _j^i\left( . \right) {\beta _j} > 0\,\qquad \forall j = {n_1},\ldots ,n - 1 \end{array}\right. \end{aligned}$$
(62)

Proof

(Corollary 2) Conditions (i) imply stability condition (ii) as demonstrated in Corollary 1 proof. Indeed if (62) are satisfied, then it is easy to see that stability conditions (30–31) of the original discrete nonlinear system \(S_i\) (1) are verified. \({}\square \)

Theorem 6

The discrete nonlinear T-S fuzzy system \(S\) (3) is asymptotically stable if there exist constant parameters \({\alpha _i} \in \mathbb{R },\,{\alpha _i} \ne {\alpha _j}\) \(\forall i \ne j\), such that \(\forall x \in D\).

$$\begin{aligned} \left| {{\alpha _i}} \right| < 1\;\forall i = 1, \ldots ,n - 1 \end{aligned}$$
(63)

and

$$\begin{aligned} 1 \!-\! \left| {\sum \limits _{i = 1}^m {{h_i}} \delta _n^i\left( \cdot \right) } \right| \! -\! \sum \limits _{j = 1}^{n - 1} {\left| {{\beta _j}} \right| \left| {\sum \limits _{i = 1}^m {{h_i}\gamma _j^i\left( \cdot \right) } } \right| {{\left( {1 \!-\! \left| {{\alpha _j}} \right| } \right) }^{ - 1}}} \!>\! 0\nonumber \\ \end{aligned}$$
(64)

If \(D = {\mathbb{R }^n}\), the stability is global.

Proof

(Theorem 6) Based on the state transformed form of the local nonlinear systems (37), the discrete T-S fuzzy model (3) can be rewritten as

$$\begin{aligned} {y_{k + 1}} = G\left( . \right) {y_k} \end{aligned}$$
(65)

where \(G\left( . \right) \) is given by

$$\begin{aligned} G\left( . \right) = \sum \limits _{i = 1}^m {{h_i}{G_i}\left( . \right) } \end{aligned}$$
(66)

It follows that

$$\begin{aligned} {y_{k + 1}} = \left[ {\begin{array}{lllllllllllllll} {\sum \limits _{i = 1}^m {{h_i}} \delta _n^i\left( \cdot \right) }&{}{{\beta _1}}&{} \cdots &{}{{\beta _{n - 1}}} \\ {\sum \limits _{i = 1}^m {{h_i}} \gamma _1^i\left( \cdot \right) }&{}{{\alpha _1}}&{}{}&{}{} \\ \vdots &{}{}&{} \ddots &{}{} \\ {\sum \limits _{i = 1}^m {{h_i}} \gamma _{n - 1}^i\left( \cdot \right) }&{}{}&{}{}&{}{{\alpha _{n - 1}}} \end{array}} \right] {y_k} \end{aligned}$$
(67)

Now, by introducing the comparison system

$$\begin{aligned} {z_{k + 1}} = M\left( {G\left( \cdot \right) } \right) {z_k} \end{aligned}$$
(68)

where \(M\left( {G\left( \cdot \right) } \right) \) is the pseudo-overvaluing matrix of (3), corresponding to the use of the vector norm (39). By applying the practical Borne-Gentina criterion [73–75] to the comparison system (68), we deduce the stability conditions of the nonlinear discrete T-S fuzzy system (3). This ends the Theorem 6 proof. \(\square \)

Theorem 7

The discrete nonlinear decoupled T-S fuzzy system \(S^d\) (14) is asymptotically stable if there exists \({\alpha _i} \in \mathbb{R },\,{\alpha _i} \ne {\alpha _j}\) \(\forall i \ne j\), such that

$$\begin{aligned} \left| {{\alpha _i}} \right| <1\quad \forall i = 1, \cdots ,n - 1 \end{aligned}$$
(69)

and

$$\begin{aligned}&1 - \left| {\sum \limits _{i = 1}^m {{h_i}} \delta _n^i\left( \cdot \right) + \sum \limits _{j = {n_1}}^{n - 1} {{\beta _j}\sum \limits _{i = 1}^m {{h_i}\gamma _j^i\left( \cdot \right) } {{\left( {1 - {\alpha _j}} \right) }^{ - 1}}} } \right| \nonumber \\&\quad - \sum \limits _{j = 1}^{{n_1} - 1} {\left| {{\beta _j}} \right| \left| {\sum \limits _{i = 1}^m {{h_i}\gamma _j^i\left( \cdot \right) } } \right| {{\left( {1 - \left| {{\alpha _j}} \right| } \right) }^{ - 1}}} > 0 \end{aligned}$$
(70)

Proof

(Theorem 7) By substituting relations (6) and (11) in (15) and (16) where matrices \({A_{i,11}},\,{A_{i,12}},\,{A_{i,21}}\) and \({A_{i,22}}\) are represented in the arrow form (46), we obtain the following slow and fast reduced order discrete T-S fuzzy systems, respectively

$$\begin{aligned} x_{k + 1}^s&= F_{}^s\left( . \right) x_k^s\end{aligned}$$
(71)
$$\begin{aligned} x_{k + 1}^f&= F_{}^fx_k^f \end{aligned}$$
(72)

and then comparison systems, respectively

$$\begin{aligned} y_{k + 1}^s&= M\left( {F_{}^s\left( \cdot \right) }\right) y_k^s\end{aligned}$$
(73)
$$\begin{aligned} y_{k + 1}^f&= M\left( {F_{}^f} \right) y_k^f \end{aligned}$$
(74)

\(F_{}^s\left( \cdot \right) \in {\mathbb{R }^{{n_1} \times {n_1}}}\) and \(F_{}^f \in {\mathbb{R }^{{n_2} \times {n_2}}}\) are respectively given by

$$\begin{aligned}&F_{}^s\!=\! \left[ {\begin{array}{lllllllllllllll} \qquad \qquad \qquad \qquad \sum \limits _{i = 1}^m {{h_i}} \delta _n^i\left( \cdot \right) \\ +\sum \limits _{j = {n_1}}^{n - 1} {{\beta _j}\sum \limits _{i = 1}^m {{h_i}\gamma _j^i\left( \cdot \right) } {{\left( {1 - {\alpha _j}} \right) }^{ - 1}}}&{}{\begin{array}{lllllllllllllll} {} \end{array}\begin{array}{lllllllllllllll} {} \end{array}{\beta _1}}&{\begin{array}{lllllllllllllll} {} \end{array} \ldots \begin{array}{lllllllllllllll} {} \end{array}}&{}{{\beta _{{n_1} - 1}}} \\ {\sum \limits _{i = 1}^m {{h_i}} \gamma _1^i\left( \cdot \right) }&{}{\begin{array} {lllllllllllllll}{} \end{array}\begin{array}{lllllllllllllll} {} \end{array}{\alpha _1}}&{}{}&{}{} \\ \vdots &{}{}&{} \ddots &{}{} \\ {\sum \limits _{i = 1}^m {{h_i}} \gamma _{{n_1} - 1}^i\left( \cdot \right) }&{}{}&{}{}&{}{{\alpha _{{n_1} - 1}}} \end{array}} \right] \end{aligned}$$
(75)
$$\begin{aligned}&F_{}^f = \left[ {\begin{array}{lllllllllllllll} {{\alpha _{{n_1}}}}&{}{}&{}{} \\ {}&{} \ddots &{}{} \\ {}&{}{}&{}{{\alpha _{n - 1}}} \end{array}} \right] \end{aligned}$$
(76)

and \(M\left( {F_{}^s\left( \cdot \right) } \right) \) and \(M\left( {F_{}^f} \right) \) are respectively the pseudo-overvaluing matrices of the slow and fast subsystems  (15) and (16), corresponding to the use of the vector norm  (39). Stability condition for the discrete decoupled system (14) is synthesized by the application of Borne and Gentina stability criterion, that completes the proof. \(\square \)

A generalized form of Corollary 1 and 2 can be developed for original T-S fuzzy system (3) and the decoupled T-S fuzzy system (14) by substituting \({a_{i,j}}(.),\,\delta _n^i\left( . \right) ,\,\gamma _j^i\left( . \right) \) and \({P_{{S_i}}}\left( {\,\,.\,\,,\lambda } \right) \) respectively by \({a^{\prime }{_j}}(.),\,\delta ^{\prime }{_n}(.),\,\gamma ^{\prime }{_j }\left( . \right) \) and \({P^{\prime }_{{S}}}\left( {\,\,.\,\,,\lambda } \right) \) such that

$$\begin{aligned}&{a^{\prime }{_j}}(.) = \sum \limits _{i = 1}^m {{h_i}} {a_{i,j}}(.)\end{aligned}$$
(77)
$$\begin{aligned}&\delta ^{\prime }{_n}(.) = \sum \limits _{i = 1}^m {{h_i}} \delta _n^i(.)\end{aligned}$$
(78)
$$\begin{aligned}&\gamma ^{\prime }{_j }\left( . \right) = \sum \limits _{i = 1}^m {{h_i}} \gamma _j^i\left( . \right) \end{aligned}$$
(79)
$$\begin{aligned}&{P^{\prime }_{{S}}}\left( {\,\,.\,\,,\lambda } \right) = {\lambda ^n} + \sum \limits _{j = 1}^n {{{a^{\prime }{_j}}}\left( . \right) {\lambda ^{n - j}}} \end{aligned}$$
(80)

Corollary 3

If the nonlinear discrete T-S fuzzy system \({S}\)  (3) is asymptotically stable, i.e. the following conditions are satisfied

  1. (i)

    \(\exists \,\varepsilon >0\) and \({\alpha _j} \in \mathbb{R },\,0 < {\alpha _j} < 1, {\alpha _j} \ne {\alpha _k},\,\forall j \ne k;\,\,j,k = 1, \ldots ,n - 1\) such that

    $$\begin{aligned} \left\{ \begin{array}{ll} \delta ^{\prime }{_n}(.) > 0 \\ \gamma ^{\prime }{_j }\left( . \right) {\beta _j} > 0\quad \forall j = 1,...,n - 1\\ \end{array} \right. \end{aligned}$$
    (81)
  2. (ii)
    $$\begin{aligned} {\left. {{{P^{\prime }}_S}\left( {\,.\,\,,\lambda } \right) } \right| _{\lambda = 1}} \geqslant \varepsilon > 0 \end{aligned}$$
    (82)

then, the corresponding decoupled T-S system (14) is asymptotically stable.

Corollary 4

If the nonlinear discrete decoupled T-S fuzzy system (14) is asymptotically stable, i.e. the following conditions are satisfied

  1. (i)

    \(\exists \,\varepsilon > 0\) and \({\alpha _j} \in \mathbb{R },\,{\alpha _j} \ne {\alpha _k},\,\forall j \ne k\); \(j,k = 1, \ldots ,n - 1\), and \(0 < {\alpha _j} < 1\) \(j = 1, \ldots ,{n_1} - 1\) such that

    $$\begin{aligned} \left\{ \begin{array}{ll} \delta ^{\prime }{_n}\left( \cdot \right) + \sum \limits _{j = {n_1}}^{n - 1} {{\beta _j}\gamma ^{\prime }{_j}\left( \cdot \right) {{\left( {1 - {\alpha _j}} \right) }^{ - 1}}} > 0\\ \gamma ^{\prime }{_n}\left( . \right) {\beta _j} > 0\quad \forall j = 1,...,{n_1} - 1\\ \end{array} \right. \end{aligned}$$
    (83)
  2. (ii)
    $$\begin{aligned} {\left. {{P^{\prime }_{{S}}}\left( {\,.\,,\lambda } \right) } \right| _{\lambda = 1}} \geqslant \varepsilon&> 0\,\,\,\,\,\,\,\mathrm{{i.e.}}\nonumber \\&1 + \sum \limits _{p = 1}^n {{a^{\prime }{_p}}\left( . \right) } > 0 \end{aligned}$$
    (84)

then, the original discrete nonlinear T-S fuzzy system (3) is asymptotically stable if the following additional conditions are satisfied

$$\begin{aligned} \left\{ \begin{array}{ll} 0 < {\alpha _j} < 1\qquad \qquad \qquad \qquad \forall j = {n_1}, \ldots ,n - 1\\ \delta ^{\prime }{_n}\left( . \right) > 0 \\ \gamma ^{\prime }{_j}\left( . \right) {\beta _j} > 0\qquad \qquad \qquad \quad \forall j = {n_1},\ldots ,n - 1 \\ \end{array} \right. \end{aligned}$$
(85)

4 Example: case of third order system

Consider a T-S fuzzy model based system such that the consequence of the rule \({R_i}\) is in the form

$$\begin{aligned}&{x_{k + 1}} = {A_i(.)}{x_k},\,\,\ i = 1,2\end{aligned}$$
(86)
$$\begin{aligned}&{A_i}\left( . \right) \,\, \!=\! \,\,\left[ {\begin{array}{llllllllllllll} 0&{}{\,\,\,\,\,\,\,\,0}&{}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ - {{1,19.10}^{ - 6}}{f_i}\left( . \right) } \\ 1&{}{\,\,\,\,\,\,\,\,0}&{}{\,\,\,\, - 0,13 + {{0,23.10}^{ - 1}}{f_i}\left( . \right) } \\ 0&{}{\,\,\,\,\,\,\,\,1}&{}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ 1,13 - 1,92{f_i}\left( . \right) } \end{array}} \right] ,\quad i \!=\! 1,2\nonumber \\ \end{aligned}$$
(87)

The local systems (86) with the characteristic matrix \({G_i}\left( . \right) \) and the synthesized T-S fuzzy system with \(G\left( . \right) \) can be, respectively, expressed in the arrow form as following

$$\begin{aligned}&{G_i}\left( . \right) = \left[ {\begin{array}{lcc} {\,0,14 - 0,19{f_i}\left( .\right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}&{}{1,20}&{}{- 1,20} \\ {{{0,69.10}^{ - 1}} - 0,14{f_i}\left( .\right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,}&{}{0,90}&{}0 \\ { - {{0,32.10}^{ - 2}} - {{0,37.10}^{ - 3}}{f_i}\left( .\right) }&{}0&{}{0,10} \end{array}} \right] \nonumber \\&\qquad \qquad \quad i = 1,2\nonumber \\ \end{aligned}$$
(88)
$$\begin{aligned}&G\left( . \right) =\nonumber \\&\left[ {\begin{array}{lcc} {0,14 - 0,038{f_1}\left( .\right) \, - 0,152{f_2}\left( .\right) }&{}{1,20}&{}{- 1,20} \\ {{{0,69.10}^{ - 1}} - 0,028{f_1}\left( . \right) \, -0,112{f_2}\left( .\right) }&{}{0,90}&{}{0} \\ {\begin{array}{l} - {{0,32.10}^{ - 2}} - {0}{,74}{{.10}^{ -4}}{f_1}\left( .\right) \\ \qquad -{0}{,296}{{.10}^{ - 4}}{f_2}\left( .\right) \end{array}}&0&{0,10} \end{array}} \right] \end{aligned}$$
(89)

for \({\alpha _1} = 0.9\) and \({\alpha _2} = 0.1\) satisfying (30), \(h1=0.2\) , \(h2=0.8\) and \(\mu = 0.1\). The decoupled slow and fast subsystems for the local nonlinear systems (86) are given respectively by

$$\begin{aligned}&F_i^s\left( . \right) = \left[ {\begin{array}{lllllllllllllll} {0,14 - 0,19{f_i}\left( . \right) \,\,\,\,\,\,\,\,\,\,}&{}{1,20} \\ {{{0,69.10}^{ - 1}} - 0,14{f_i}\left( . \right) }&{}{0,90} \end{array}} \right] \,\,\,\, i = 1,2\\&F_i^f = 0,10\nonumber \end{aligned}$$
(90)

and for the T-S fuzzy system (89) respectively by

$$\begin{aligned}&F_{}^s\left( . \right) = \left[ {\begin{array}{llllllllllllll} {0,14 - 0,038{f_1}\left( . \right) \, - 0,152{f_2}\left( . \right) \,\,\,\,\,\,\,\,\,\,}&{}{1,20} \\ {{{0,69.10}^{ - 1}} - 0,028{f_1}\left( . \right) \, - 0,112{f_2}\left( . \right) }&{}{0,90} \end{array}} \right] \nonumber \\ \\&F_{}^f = 0,10 \nonumber \end{aligned}$$
(91)

In the following, we determine the stability domains of original and decoupled described systems. For the chosen \({\alpha _1}\) and \({\alpha _2}\), synthesized stability condition of the discrete T-S fuzzy system (89) deduced from Theorem 6, is the following

$$\begin{aligned}&1 - \left| {0,14 - 0,038{f_1}\, - 0,152{f_2}} \right| \nonumber \\&- 12\left| {{{0,69.10}^{ - 1}} - 0,028{f_1}\, - 0,112{f_2}} \right| \nonumber \\&- 1.33 \left| { - {{0,32.10}^{ - 2}} - { 0}{,74}{{.10}^{ - 4}}{f_1} - {0}{,296}{{.10}^{ - 4}}{f_2}} \right| > 0 \nonumber \\ \end{aligned}$$
(92)

Using condition (92), system (89) is stable if nonlinear functions \({f_1}\left( . \right) \) and \({f_2}\left( . \right) \) are, respectively, within the following limits, given in Table 1. Furthermore, applying Theorem 4 to the nonlinear local system (86) yields

$$\begin{aligned} - 0.0148 < {f_i}\left( . \right) < 1.0498 \,\,\,\ i = 1,2 \end{aligned}$$
(93)

Now, for the synthesized decoupled discrete T-S fuzzy system (91), sufficient stability condition issued from Theorem 7, is given by

$$\begin{aligned}&1 - \left| {0,14 - 0,038{f_1}\left( . \right) \, - 0,152{f_2}\left( . \right) } \right| \nonumber \\&\qquad - 12\left| {{{0,69.10}^{ - 1}} - 0,028{f_1}\left( . \right) \, - 0,112{f_2}\left( . \right) } \right| > 0\nonumber \\ \end{aligned}$$
(94)

Deriving additional conditions on \({f_1}\left( . \right) \) and \({f_2}\left( . \right) \) for the existence of a solution to stability condition (94), results of Table 2 are obtained. Moreover, according to Theorem 5, the nonlinear local systems (90) is stable for

$$\begin{aligned} - 0.0171 < {f_i}\left( . \right) < 1.0524 \,\,\,\ i = 1,2 \end{aligned}$$
(95)
Table 1 Stability domain of the original T-S fuzzy system (89)
Full size table
Table 2 Stability domain of the decoupled T-S fuzzy system (91)
Full size table

Figure 1 illustrates the stability domains \({D_1},\,{D_2},\,{D_3}\) and \({D_4}\) associated respectively to the original discrete T-S fuzzy system (77), the decoupled T-S fuzzy system (91), the nonlinear local model (86) and the decoupled nonlinear local model (90). As shown, the stability domain of the decoupled systems (90) and (91)are, respectively, very close to the original ones (86) and  (89). Furthermore, one can see that the stability conditions (30–31) and (44–45) of local systems are conservative and induce smaller stability domains. Discrete T-S fuzzy and local models have the common restricted stability domain \({D_5} = {D_1} \cap {D_2} \cap {D_3} \cap {D_4}\). \({D_5}\) is smaller than the common estimated stability region of local systems; the stability of each local model does not ensure the stability of the global system.

Fig. 1
figure 1

Stability domains

Full size image

5 Conclusion

In this paper, we have investigated the stability problem of singular T-S fuzzy systems under the discrete-time framework. By using the arrow matrix form and Borne and Gentina criterion, sufficient stability conditions for of the reduced order decoupled T-S fuzzy system, as well as the original T-S fuzzy system are derived. Supplementary stability conditions are synthesized to ensure a common stability domain for the original and the decoupled T-S fuzzy system. In the simulation, an illustrative example demonstrated that obtained results are less conservative than existing ones.