1. Introduction
The traditional T-S fuzzy system has been studied and applied widely by many researchers due to its advantages in structure and theory. In fact, the T-S fuzzy system is a convex combination of linear systems, and is also a direct generalization of state-space model in terms of LPV theory. In theory, there is a connection between T-S fuzzy systems LPV or qLPV systems and they can transform into each other [
1]. In [
2], the T-S fuzzy system was proven to be equivalent to the LPV system. For the the modeling problem, it presented two methods for the automated generation of LPV and T-S systems. The references [
3] proposed a tensor product (TP) model transformation-based framework requiring minimal human intuition to numerically reconstruct LPV and T-S fuzzy model-based linear parameter varying and quasi-linear parameter varying representations of state-space model. The problem “How to Vary the Input Space of a T-S Fuzzy Model” was solved in [
4].
When dealing with a complex nonlinear system, we often use T-S fuzzy system to approximate it, because this method has high precision. Markov jump is a commonly used tool for describing system randomness. Many complex nonlinear systems Markov jump [
5,
6] are given and investigated under the frame of T-S fuzzy systems. A large number of Markov jump systems are described and investigated applying the fuzzy systems [
7,
8,
9,
10,
11,
12]. The T-S fuzzy system theory has been applied to investigate semi-Markov jump model reliable mixed
/passive control problem [
13]. In [
14], Shen used an event-triggered communication scheme to stabilize the closed-loop system with a prescribed
performance level. Mode-dependent non-fragile observer-based controller [
15] is designed for fractional-order T-S fuzzy systems with Markovian jump via non-PDC scheme. The reference [
16] examines the stochastic control problem of continuous-time fuzzy systems with Markov jump. On the other hand, many T-S fuzzy systems are investigated in view of Markov jump. For the resilient estimation problem, the T-S fuzzy system theory is generalized to Markov systems under a dynamic event-triggered scheme [
17]. Many other continuous-time generalized T-S fuzzy systems with Markov jump are adequately analyzed via different methods [
18,
19,
20,
21].
The positive stochastic stabilization issue is investigated for discrete-time T-S fuzzy Markov jump nonlinear systems in [
22]. In the point of partial differential T-S fuzzy systems, Ji and Zhang [
23] are concerned with a finite-dimensional guaranteed cost sampled-data fuzzy control problem. Song [
24] investigated the problem of state estimation for partial differential T-S fuzzy Markov jump delayed neural networks and proposed a memory-based control scheme that contains a constant signal transmission delay.
In practice, the T-S fuzzy system with varying parameters is worth researching for many time-varying cases. The idea behind this paper comes from references [
25,
26,
27,
28,
29] about T-S fuzzy system, LPV system with Markov jump.
where
is the premise variable,
,
,
, is a fuzzy or linguistic term, which can be quantified by membership functions. The given
I and
J are positive integers which specify the number of fuzzy rules and premise variables, respectively.
is the system state,
is the measurement output and
is the control input signal. The matrix functions
,
,
and
depend on the time-varying parameter
. The time-varying parameter
is independent from the system state variables and can be measured on-line. For the convenience of the following expression, define
and abbreviate
,
and
as
,
and
. The parameter subset
is a bounded closed set and contains
vertices at
. The parameter
takes discrete values in a finite state space
which is a right continuous homogeneous Markov process and the jumping of
satisfies the transition probability matrix
determined by
where
,
,
denotes the transition rate from mode
s to
ℓ at time
if
and
.
Applying the “algebraic product” operation to quantify the linguistic term “and”, the “product” implication method to infer the local output according to each fuzzy linguistic rule and the center-average defuzzification method, the fuzzy parameter varying system with Markov jump described in the linguistic form (
1) could be represented by the following analytical formula:
where
Under the frame of LPV systems, assume that system matrices of (
3) are affine on
such that when
the system matrices are affine dependent on
as
The system (
3) can describe T-S fuzzy system, LPV system and Markov switch system. Specifically, when the system (
3) contains only one fuzzy rule, it degenerates into an LPV (to be precise, it is qLPV) system with Markov jump [
30,
31]
when the system matrices of (
3) are not dependent on
, it degenerates into a T-S fuzzy system with Markov jump [
14]
when the system (
3) ignores the time-varying parameter and Markov jump, it degenerates into a conventional T-S fuzzy system [
32,
33]
Therefore, the system (
3) merges several kinds of common ones. Under the frame of T-S fuzzy system, it is also a universal approximator with arbitrary precision for any nonlinear system [
34]. Differently, the system (
3) will contain less rules than the conventional T-S fuzzy system when it is used to deal with nonlinear time-varying systems. This makes it convenient to obtain stability conditions and design controllers. The system (
3) combined in convex hull is more compendious to model a nonlinear system than LPV system. On the other hand, the system (
3) can model more complex nonlinear systems due to consideration of Markov jump. Each of its subsystems is a Markov jump system describing randomness. It is a novel nonlinear time-varying fuzzy system worth studying.
The notations used in the paper are rather standard
Table 1.
stands for the set of real numbers.
is the set of real
matrices. We use
to denote real symmetric
matrices, and
for positive-definite matrices. The
n order identity matrix is denoted by
. In some matrix, the sign “★” means symmetry element.
means
P is a symmetric positive definite matrix.
and
are the inverse and the transposition of matrix
P, respectively. The parameter
ℓ present the system mode, for example,
. The sign “
” is defined as
. For any matrices
and
,
and
denotes the matrices
and
. For convenience of expression and proof, the notation
is introduced
Due to
,
For the set of membership functions, we define
,
with
,
for convenience of presentation.
In this paper, we also study the system (
3) with external disturbance and give the
gain schedule controllers design methods. The system (
3) with external disturbance is
where
is exogenous input. The superscripts
,
and
are just to distinguish different system matrices. The superscript
ℓ is the system mode. The definition subscript
is the same as (
7). Other nations are the same as them in the system (
3).
In the paper, we investigate a novel system, which merges the T-S fuzzy parameter varying system and Markov jump. We obtain its stable conditions and gain-scheduling controller design method. Applying Lyapunov function depending on the operation mode and full block S-procedure Lemma, we obtain stochastic stabilization conditions. It inherits the advantages of traditional T-S fuzzy systems in handling nonlinear objects. This study expands the scope of application of T-S fuzzy system theory.
The remainder of this article is organized as follows: In
Section 3, the stability analysis of open-loop fuzzy parameter varying system with Markov jump is provided based on a Lyapunov function that depends on the operating mode. In
Section 4, state feedback fuzzy gain schedule controllers are designed using the full block S-procedure lemma and Lyapunov stability theory. At the end of
Section 4, the
controller will be provided in the form of corollary. A numerical example is utilized to validate our the controller synthesis conditions in
Section 5.
Section 6 concludes the article.
2. Preliminary
In the remainder of this article, the full block S-procedure lemma will be extensively and continually examined. It plays an important role in the proof of our main results. To improve readability of this article, this lemma is introduced as below.
Lemma 1 (Full Block S-procedure [
35])
. For a given linear functional transformation (LFT) expressionwith and , the inequality is equivalent to the existence of a full block multiplier satisfyingfor all . This lemma has been used in robust control researches [
36,
37,
38] to convert the inequality containing uncertainty matrix
and the nominal part of the system decomposes into two separate inequalities. According to Lemma 1, the uncertain
can be separated from the original matrix inequality
. However,
is not simply treated as uncertain in the paper, but a time-varying parameter measured on line. Note that the second inequality contains quadratic terms of
with indefinite sign. It is reasonable to believe that the restriction
could render some conservatism. Nevertheless, the comparison examples in reference [
27] have shown that the proposed relaxation based on full block S-procedure is much better than existing relaxation methods. In our future research, we would like to remove the requirement of
for further improvement of the full block S-procedure relaxation. Generally speaking, infinite number of inequalities should be checked to ensure this inequality over the set
. Nevertheless for a convex polytope
, a condition with finite LMIs can be derived by enforcing the sub-matrix
.
Remark 1. In Lemma 1, let and , then can be expressed as a LFT, namely,where . In
Section 4, the parameter
in Remark 1 will be replaced by some other matrix functions depending on time-varying parameters or premise variables in the fuzzy rules. Consequently, the state feedback fuzzy gain schedule controller will be derived by using the full block S-procedure lemma.
Lemma 2 (Bound Real Lemma [
13]).
For a system G, the expression form of state space iswhere x and w are the state and external disturbance. The system matrices are A, B, C and D, then the following two conditions are equal,1. The systems is exponentially stable and .
2. There is a positive definite matrix P satisfying Definition 1 ([
15])
. The system (9) is said to robustly stochastically stable if for any finite initial condition and , one has as . 4. Full States Feedback Fuzzy Gain Schedule Controller Design
The full states feedback controller is the most direct control method in control theory when all system states are measurable. Inspired by T-S fuzzy parallel compensation control method and gain schedule control of LPV theory, full states feedback fuzzy gain schedule controller is investigated. However, the quadratic terms of normalized fuzzy weighting functions, e.g.,
, present a great challenge in the process of recasting the controller synthesis condition into LMIs conditions. Traditional methods admit every sub-item is negative shielding membership functions. To solve the problem, many various relaxation techniques have been studied in [
26,
27]. More difficultly, the quadratic terms
include not only parameters dependent on fuzzy weighting functions but also varying parameters
. In this section, we apply twice full block S-procedure lemma to separate
H and
, respectively.
According to the idea of PDC controller design [
40], we state a traditional result based on common Lyapunov function to design full states feedback gain-schedule fuzzy controller. Since the proof is very simple, we directly give the result.
Theorem 2. If there exist and matrix functions satisfying the following matrix inequalitiesfor any , then the system (3) is stabilized by the state feedback fuzzy gain schedule controller defined in (20)where, Remark 6. The inequality (19) in Theorem 2 is not linear about unknown variable matrices P and K. However, it can be converted to a LMI of . Remark 7. Though the inequality (19) can converted to a LMI, the quadratic term of θ needs to be dealt with via the grid method of LPV theory. It will also require a lot conservatism [41]. So it is a challenge for fuzzy parameter varying system with Markov jump. Remark 8. In Theorem 2, the inequality (19) is conservatively simplified from The inequality (21) is equal towhere . The inequality (19) is equal to every entry of block core matrix. The simplification from (22) to (19) has some conservatism. The conservatism of such a relaxation procedure can be revealed by a simple example: The inequality clearly holds for any non-negative satisfying , but not every entry of core matrix is negative. In the following result, we will overcome some conservatism as far as we can based on Lyapunov function dependent on the system mode and full block S-procedure lemma. A relaxation technique proposed in this section can be used in effectively solving controller synthesis problem of fuzzy parameter varying system with Markov jump.
Theorem 3. If there exist positive definite matrices , some constant matrices , , and , and two symmetric matrices , with appropriate dimensions satisfying, for any ,where , ,with , , , then the closed-loop fuzzy parameter varying system with Markov jump (3) is stabilized by the state-feedback fuzzy gain schedule controller in the form ofwhere . Proof. Consider closed-loop system (
3) with the full states feedback fuzzy gain schedule controller
where
. Because the membership functions satisfy
, then the system (
26) is rewritten as
Take the Lyapunov function candidate dependent on system mode
, where
. Directly computing the derivative with respect to the time along the trajectory of the system (
27), the following stability condition can be obtained
Applying Schurer’s lemma, the inequality (
28) is equal to
It is easy to see that the inequality has the following decomposition regard
H as argument
where
is a matrix respecting to
and system mode
ℓ,
According to the full block S-procedure lemma, the inequality (
30) is guaranteed if there is a full-block multiplier
satisfying
where
Notice that the inequality (
31) depends on the parameter
and it is observed that
has the following decomposition
Then the inequality (
31) can be rewritten as
It is in the form of
. Therefore, we can apply the full block S-procedure lemma again for the inequality (
33). That is to say that the existence of a full-block multiplier
satisfies
Because of the negative definitive
and
, it is adequate to verify the inequalities (32) and (35) at vertices
and
, which are the same as LMIs (24) and (25). Therefore it follows from the Lyapunov stability theory that the system (
27) or (
3) are stable at the origin, which concludes the proof. □
Remark 9. The full block S-procedure lemma is used twice in the proof Theorem 3. To make it easier for the reader to understand, we omit some complex calculations. In the first time for the inequality (30), the constant matrices and are given as the same idea as Remark 1. In the second time for the inequality (34), the constant matrices are a little different from the first time because needs to be divided into and . Remark 10. The inequality (23) plays an important role in Theorem 3. It contains that the key matrix may be not negative definite because of . If we take the matrix , Theorem 3 will degenerate to the result based on the common quadratic Lyapunov function. Remark 11. The core matrix in Theorem 3 contains the old information of quadratic Lyapunov function method, for example, . The terms , and are the new information from the Lyapunov function depending on the operating mode. They have contributed to reducing conservativeness for our method.
Remark 12. Notice that the inequality (24) contains the quadratic terms of H. Honestly speaking, infinite number of matrices inequalities should be held to ensure (32) even if the space containing H is convex. However, the inequality (24) will be convex with respect to H with enforcing the matrix . Therefore, if the space containing H is a convex polytope containing zero, the inequality (32) only needs to be checked at the vertices of H, that is the inequality (24). This would lead researchers to check the finite number of linear matrix inequalities. A similar situation also occurs in inequality (25).
Remark 13. The controller design method in Theorem 3 is effective for system matrices uncertainty problem regarding θ as uncertainty. The controller will be independent of θ and be a fuzzy gain-schedule controller with respect to the membership function . It only needs the matrices in Theorem 3.
In the actual system, the external disturbance is inevitable and is worth studying in depth. Fortunately, the design method in Theorem 3 is also used to design fuzzy gain-schedule
controllers for the system (
9). The method of Theorem 3 can be transplanted to the system (
9) in parallel applying Lemma 2. We give the result directly without the proof because it is very similar to the proof of Theorem 3.
Corollary 1. For a given , if there exist a positive definite matrix , some constant matrices and , and two full block multipliers , satisfying for any , ,where , the signal ⊗ represents the Kronecker product, is defined asthen the closed-loop fuzzy parameter varying system (3) with Markov jump is stabilized by the fuzzy gain schedule controller in the form ofwith , which renders the gain of the closed-loop system less than μ for any disturbance . 5. Simulation
In this section, a numerical example is provided to illustrate the advantages and efficiency of the controller design methods. It demonstrates the usage of Theorem 3 and Corollary 1. The full state feedback fuzzy gain schedule controllers will be synthesized. We are going to compare our design methods with the conventional T-S fuzzy model based the numerical example.
For simplicity, we assume that there are only two fuzzy rules in the rule base and the local time-varying system is of second-order and has two system modes
Figure 1, i.e.,
,
. Therefore, the analytical formula of the FPV system to be controlled is described as below
where
is the system state,
y is performance output.
is the disturbance signal satisfying
showed in
Figure 2. For convenience, only one state variable, i.e.,
is utilized as the scheduling variable and the membership functions (seeing
Figure 3) are selected as
. For simulation purposes, the varying parameter
is chosen to be a sinusoidal function with respect to time
t, i.e.,
. The system parameter matrices for the two local linear varying systems with Markov jump are listed in
Appendix A and set two parameters
in system matrices to compare conservatism with other design methods. The transfer matrix is characterized by
Firstly, we compare the design method Theorem 3 with the T-S method with less conservatism provided in [
26]. The method of Theorem 3 is simpler in algorithm complexity. When
are integers and vary from 0 to 10, it is found that the system (
39) is stabilized by Theorem 3 for all
but the method provided in [
26] not (see
Figure 4). It implies that Theorem 3 we provide has less conservatism than reference [
26]. In addition, Corollary 1 is used to obtain optimized
gain compared with the method provided in [
26]. In
Table 2, we can find that the optimal
obtained by Theorem 3 is less than the references [
13,
26]. Specially, we simulate the system (
39) when
and
by the LMI toolbox and Simulink in MATLAB(2018a) to solve the inequalities (
36)–(38) in Corollary 3 (or (
23)–(25) in Theorem 3) and obtain the gain matrices of full state feedback
fuzzy gain schedule controllers,
where
It can be observed from these simulations that the designed full state feedback fuzzy gain schedule controller keeps the closed-loop system asymptotically stable (see
Figure 5) and succeeds in attenuating the disturbance (see
Figure 6).