Fuzzy-model-based robust control of Markov jump nonlinear systems with incomplete transition probabilities and uncertain packet dropouts

Interval type-2 fuzzy Markov jump systems (IT2FMJSs) have received much attention because they can well describe complex nonlinear systems with uncertainties and stochastic system mode switching. However, the transition probabilities of fuzzy MJSs (FMJSs) have been assumed to be completely known, limiting real-world applications of existing results. Different from the previous studies, transition probabilities between system modes switching are partially unknown, and packet dropouts of data transmission are uncertain in this study. Compared with the previous studies, the main advantages of this work are as follows: (1) To analyze stochastic stability and reduce conservatism of existing approaches, a novel Lyapunov function that depends on both system mode and fuzzy basis function is constructed; (2) the existence of a mode-dependent and fuzzy-basis-dependent state feedback controller is revealed; (3) stochastic stability of closed-loop system with a desired H ∞ performance is established, and the problem of incomplete transition probabilities and uncertain packet dropouts has been completely addressed. An illustrative example of a robot arm is used to demonstrate the effectiveness and practicality of the proposed control strategy. By virtue of the proposed strategy, the effects of incomplete transition probabilities and uncertain packet dropouts on IT2FMJSs have been completely alleviated.


INTRODUCTION
Practical engineering systems invariably contain nonlinearities.Because of the capability of approximating nonlinear systems, Takagi-Sugeno (T-S) fuzzy models have been widely used in control systems [1][2][3][4].In general, by blending all linear subsystems through a set of IF-THEN rules, the T-S fuzzy model can describe any smooth nonlinear system [4].Many fruitful results on stability analysis, control synthesis, and filtering synthesis of fuzzy systems have been reported [5][6][7][8][9][10].By considering nonlinear terms and uncertain parameters, the interval type-2 (IT2) technique has been introduced into T-S fuzzy systems to improve their approximation capabilities [11].By considering uncertain information and upper and lower membership functions (ULMFs), a stable IT2 fuzzy controller has been designed to carry out a control process.Some typical control methods based on IT2 fuzzy systems (IT2FSs) have been proposed [12].Moreover, studies on fuzzy systems have been witnessed in other fields, such as sensor networks [13], switched systems [14], and multi-agent systems [15].These latest research results make hybrid systems a mainstream trend dealing with the complexity of dynamic systems.However, many complex dynamic systems are not sufficiently expressed by IT2FSs, especially systems with stochastically varying parameters subject to the Markov process.IT2 fuzzy Markov jump systems (IT2FMJSs) can be adopted in this paper.
As a branch of hybrid systems, IT2FMJSs have received great attention because they can not only represent systems with Markovian stochastic variations such as stochastic characteristics of dynamic systems themselves, stochastic failures or repairs of components, and sudden environmental disturbances but also address the nonlinearity and uncertainty of complex systems [16].IT2FMJSs can combine the advantages of IT2FSs and Markov jump systems (MJSs), which can better describe complex nonlinear systems with uncertainties and system mode switching.Over the past decade, many favorable results of fuzzy MJSs (FMJSs) have been reported, including stability [17,18], stabilization [19], and H ∞ control [20,21].However, transition probabilities that govern the dynamic behavior of MJSs have been usually assumed to be completely known.To a great extent, this assumption has limited wide spread applications of MJSs because it is difficult to obtain complete description of transition probability [22].In practical systems, MJSs with incomplete transition probability are more common and general than those with completely known transition probability.This has more necessary research significance from the control perspective.Many practical systems with unpredictable structural variations can be described by MJSs with generally incomplete transition information [23].In [24], the H ∞ control problem of MJSs considering time-varying delay and incomplete transition probability has been addressed.Furthermore, when sensor failures are considered, a dissipative filter of FMJSs with incomplete transition probability is proposed in [25].Moreover, in [26][27][28], other control and filtering methods of FMJSs with complete transition probability have been presented, but these results rarely involve the control problem of incomplete transition probability.
In addition, because FMJSs is one kind of networked control systems (NCSs), it is necessary to study the problem of data packet dropout in NCSs.In recent years, some solutions to packet dropouts have been presented [29,30].In [31], a stochastic process is adopted to describe the packet dropout process.By modeling the packet dropout process using stochastic variables that satisfy the Bernoulli distribution, the H ∞ fuzzy control problem of FMJSs has been addressed in [29,32].However, the aforementioned research results assume almost all that the expectation of packet dropouts is certain.On the contrary, the expectation of packet dropouts is uncertain in practical systems.Regarding the aforementioned issues, few attempts have been made on the control problem of FMJSs with uncertain packet dropout.
Different from existing approaches, this paper focuses on controller design of IT2FMJSs with incomplete transition probabilities and uncertain packet dropouts.The main contributions of this work are as follows: (1) The proposed IT2FMJSs with incomplete transition probabilities are more general than existing FMJSs and the difficulty of incomplete transition probability has been effectively addressed.(2) A mode-dependent and fuzzy-basis-dependent state control strategy is proposed to simultaneously address the problem of incomplete transition probabilities and uncertain packet dropouts.(3) The sufficient conditions that guarantee the resulting closed-loop systems are stochastically stable with the desired H ∞ performance are obtained and the conservatism of existing control strategies is significantly reduced.

Notation
P > 0 (P < 0) indicates that P is a positive (negative) definite matrix." * " denotes an ellipsis for the parts introduced by symmetry.I n and 0 are n × n identity matrix and zero matrix with appropriate dimension, respectively.diag{...} denotes a block-diagonal matrix.Pr {A|B} denotes the condition probability of event A conditional on B. E {a} and E {a|b} denote the expectation of a and expectation of a conditional on b, respectively.Moreover, ‖•‖ E2 denotes expected l 2 [0, ∞) norm.

Physical plant
By considering the discrete-time IT2FMJS with incomplete transition probabilities and uncertain packet dropouts, the IT2FS of a physical plant can be expressed as follows: and H i,c k are known real-valued matrix functions with respect to c k .The firing interval of the ith rule can be depicted by where  i ( c k (k)) and λi ( c k (k)) denote the lower and upper fuzzy basis functions (LUFBFs) for the ith rule, and Hence, the fuzzy basis function can be expressed as [33,34] where  i ( c k (k)) denotes fuzzy basis functions for the ith rule  i ( c k (k)) and  i ( c k (k)) are the lower and upper weighting coefficient functions (LUWCFs) that can capture the variation of uncertain parameters.For simplicity, By leveraging the concept of singleton fuzzification, product inference, and center-average defuzzification, the global model of discrete-time IT2FMJS can be written as

Partially unknown transition probability
The stochastic process {c k , k ≥ 0} is modelled by a discrete-time Markov chain, and the transition probability between mode switching is given by where ∀m, n ∈ M,  mn ≥ 0, and ∑  n=1  mn = 1.Moreover, the transition probability matrix (TPM) is defined as where "?" denotes inaccessible elements.To facilitate analysis, ∀m ∈ M, M (m) ∈ N + denotes the bth known element in the mth row of the matrix  .Besides, sum of known elements and sum of unknown elements, respectively, are denoted by

Uncertain packet dropout
Since packet dropouts occur in communication links between sensors and the controller as well as between the controller and actuators, two stochastic processes are adopted to describe the packet dropout, as follows: where x d (k) and u d (k) are the input and output of the controller due to packet dropouts, respectively.The terms (k) and (k) are individually independent Bernoulli processes.
when the data transmission of the links succeeds (or fails).Similarly, (k) holds.By defining two independent processes can be combined into one [31,35].
and (k) = 0, otherwise.Moreover, by considering uncertain packet dropout, the following equation is obtained: where  + Δ denotes the success packet rate, and ε and Δ are the nominal expectation and norm-bounded uncertainty of the success packet rate, respectively.By defining where  ∈ R ≜ {1, 2, … , r}, E  a,c k is the fuzzy set.The firing interval of the th rule can be depicted by where   ( c k (k)) and κ ( c k (k)) denote LUFBFs for the th rule, and a,c k .Hence, the fuzzy basis function can be expressed as where   ( c k (k)) denotes fuzzy basis functions for the th rule and   ( c k (k)) and   ( c k (k)) are LUWCFs that can cap-ture the variation of uncertain parameters.For simplicity,   ( c k (k)) is denoted by  ,c k .Thus, ( 12) can be written as

Closed-loop system
By substituting ( 16) into ( 5), the resulting closed-loop system can be expressed as where To facilitate further concerning, the following definition about the stability and performance of IT2FMJSs is introduced.

Definition 1.
(see [36]) The closed-loop system (17) with For the system (5), the TPM is incomplete and the packet dropout is uncertain.By addressing the aforementioned problems, a fuzzy H ∞ controller with a prescribed scalar  > 0 in the form of ( 12) is designed for any  mn satisfying (8).The following two observations can be made: (a) In the sense of ( 18), the closed-loop IT2FMJS given by ( 17) is stochas-

Stochastic stability analysis
Concerning the stochastic stability of the closed-loop system (17), the sufficient conditions that guarantee it is stochastically stable with a prescribed H ∞ performance index are now stated. where Proof 1: By defining c k = m, c k+1 = n, the following mode-dependent and fuzzy-basis-dependent Lyapunov function is constructed: The stochastic stability of the closed-loop system (17) without disturbance input, that is, (k) ≡ 0 (k = 1, 2, ...), is now analyzed.
Following the above derivation, ( 22) can be written as where From (19), the following inequalities are obtained: It can be seen from ( 24) that By combining (23) with (25), it is easy to obtain: where (19) for any l ∈ R, m ∈ M, it is easy to obtain ℵ > 0 for any l ∈ R, m ∈ M. Therefore, E {ΔV(k)} < 0, which proves that the IT2FMJS given by (17) with (k) ≡ 0 is stochastically stable.
Remark 1.In Theorem 1, the fuzzy-basis-dependent and mode-dependent Lyapunov function can reduce conservatism in stochastic stability.

Controller design
Based on Theorem 1, sufficient conditions on the existence of the desired state-feedback controller (12) ensure that the closed-loop system ( 17) is stochastically stable with the optimized H ∞ performance.

Theorem 2. By considering the closed-loop IT2FMJS (17) with incomplete transition probabilities and uncertain packet dropouts, for given
UK , it is stochastically stable with a prescribed H ∞ performance index  if there exist positive definite matrices X The controller gains of the desired state-feedback (12) are given by Proof.For any i ∈ R and m ∈ M, according to X = N T m + N m − X i,m , and noting X i,m > 0, we have According to (36) and (38), it is not difficult to obtain Applying (39) to ( 35) yields where By performing congruence transformation to ( 35) by diag{N −1 m , I, I, … , I}, the following inequality can be obtained: where UK , (41) can be rewritten as where By performing congruence transformation to Ψ l,m in (35) by diag } , the following inequality can be obtained: On the other hand, for any i, , l ∈ R, m ∈ M, we have By Schur complement, (43) can be rewritten as According to ( 36)-( 43), (47) can be expanded as follows: where According to (45) and ( 46), (48) can be written as ] ( Furthermore, according to Schur complement, (50) can be written as It can be seen from ( 35)-( 51) that ( 51) is extrapolated from (35); that is, ( 35) is a sufficient condition of (51).
Hence, the developed criteria based on Theorem 2 are more conservative than those of Theorem 1.By comparing (51) and ( 19)-( 21), it can be seen that ( 51) is a sufficient condition that ensures  T i,m  i,m −  i,m + Ψ il,m < 0. Furthermore, in the mean square sense, stochastic stability of the closed-loop system and the desired H ∞ performance are ensured.This completes the proof.Remark 2. It is observed that the computational complexity of Theorem 2 depends on the number of unknown elements of TPM, system modes, and fuzzy rules.Hence, the number of unknown elements of TPM, the number of system modes, and the number of fuzzy rules should be appropriately chosen.

AN ILLUSTRATIVE EXAMPLE
In this section, a practical example is used to demonstrate the effectiveness and practicality of the proposed control approach.A single-link robotic arm system with Markov jump modes and uncertain packet dropouts employed in [31] is considered.The dynamic equation of the robotic system is given by where (t), u(t), and (t) are the angle position of the robot arm, control force, and the disturbance input belonging to l 2 [0, ∞), respectively.The terms g, L, and  denote gravitational acceleration, length of the arm, and mass of the load, respectively.The inertial moment and coefficient of viscous friction are represented by  and Q, respectively.First, let x 1 (t) = (t), x 2 (t) = .(t).In the system, two fuzzy rules and three system modes are set.The terms  and  have three values, that is, 2, and  3 = 1.6 of which variations are subject to a Markov process.Other parameters are set to g = 9.81, L = 0.5, and The sampling interval is set to T=0.1.Here, the fuzzy basis functions are designed by where  denotes parameter uncertainties,  ∈ [0.01, 0.1]∕.Hence, when r = 2, real fuzzy basis functions can be determined as follows: where LUWCFs are denoted by  1 i (x 1 (k)) = cos 2 (x 1 (k)) and Hence, the partition of fuzzy rules can be based on x 1 (k) with respect to 0 rad and ± rad.Simultaneously, the fuzzy basis functions of the controller are designed by Thus, when r = 2, real fuzzy basis functions can be determined as follows: where LUWCFs are denoted by Next, the parameters of the IT2FMJS approximating the robotic arm system are given by for i = 1, 2 and s = 1, 2, 3. Besides, the TPM with partially unknown elements is given by ? 0.5 ?0.2 0.5 0.3 0.6 ??
Figure 1 shows the state response of the open-loop system (u(k) ≡ 0 in (17)).Evolution of the system mode under the condition of (55) and the packet dropout sequence with nominal expectation  and uncertainty  are shown in Figures 2 and 3, respectively.It can be seen from Figure 1 that the open-loop system is unstable.Comparative studies of the optimal H ∞ performance indices  min for different packet dropouts under the condition of (55) are given in Table 1.As  increases, the optimal H ∞ performance   index  min becomes smaller, which means that more reliable communication conditions and smaller value of  min lead to better H ∞ performance.Furthermore, for a fixed , Table 1 shows that  min increases as  grows, which makes the H ∞ performance worse.This illustrates that nominal expectation and uncertainties of the packet dropout have great effects on the desired performance.
The initial state values are set to x 0 = [0.25− 1].To test H ∞ performance, zero initial condition with disturbance input is assumed, where 4 State response of the closed-loop system with success data packet probability [0.9 − 0.1, 0.9 + 0.1].

𝜉(k)
When  = 0.9 and  = 0.1, the closed-loop state responses of x 1 and x 2 converge to zero, as shown in Figure 4, which shows that the designed controller is efficient against incomplete transition probability and uncertain packet dropout.
To evaluate the  under the same condition are almost identical and are all smaller than the optimal value of  min = 0.6662 in Table 1.This shows that variations of stochastic sequences have little effect on the system performance.Besides, actual H ∞ performance indices about two packet dropout sequences corresponding to  = 0.1 and  = 0 are  1 = 0.1078 and  2 = 0.1064, respectively.By employing Theorem 2, the controller gains (0.9, 0.1) and (0.9, 0) are obtained for  = 0.1 and  = 0, respectively.If the controller gain (0.9, 0) ( = 0) is adopted to test the packet dropout sequence with uncertainty  = 0.1, actual H ∞ performance indices can be calculated as  3 = 0.1075 and the resulting state responses of x1 and x2 are obtained in Figure 4.It can be seen from  3 >  2 that uncertainties of packet dropout need to be considered in the controller design.Simultaneously, by comparing x 1 , x 2 with x1 , x2 , the controller with uncertain packet dropout is much better than that without uncertain packet dropout in terms of overshoot and convergence time.
To investigate the effect of incomplete TPM on H ∞ performance of the system, values of known elements in the rows with unknown elements are increased, that is, sum of unknown elements in the same rows is reduced.Compared with (55), the TPM after enlarging values of known elements is given by where boldface denotes enlarged element.Under the condition of (57), the comparison of optimal H ∞ performance indices  min for different  with different  is obtained in Table 2.By comparing Table 1 with Table 2, γmin with Υ is smaller than  min with  when  changes from 0.7 to 0.9 and  changes from 0 to 0.1.This indicates that the increase of known degree of description information in the TPM will improve the system performance.
To further test the effect of the known degree of description information in the TPM on the H ∞ performance,  12 and  31 in  and Υ change from 0.2 from 0.8 to evaluate minimum H ∞ performance indices  min , as shown in Figure 6.The main trend of  min decreases with increase of  12 and  31 .It can be seen that more complete the description information of the TPM is, better H ∞ performance is, and smaller the value of  min is.

2 FIGURE 1
FIGURE 1 State response of the open-loop system.

FIGURE 2
FIGURE 2 Evolution of system mode.

FIGURE 3
FIGURE 3 Sequence of packet dropout.
under zero-initial condition, and  = 0.9,  = 0.1 with  are shown in Figure 5.It is observed that values of the actual H ∞ performance indices  for 100 different Markov jump sequences

FIGURE 5
FIGURE 5 Actual H ∞ performance indices  for 100 different Markov jump sequences.
Engineering with Nanyang Technological University, Singapore, from 1992 to 2020.He has authored six books, 21 book chapters, and more than 500 refereed journal and conference papers achieving a total citation count of more than 14000 and an h-index of 57 in Google citations.He was the recipient of the Web of Science Top 1% Best Cited Paper and the Elsevier Top 20 Best Cited Paper Award in 2007 and 2008, respectively.He is currently the Editor-in-Chief of two international journals, namely, Transactions on Machine Learning and Artificial Intelligence and the International Journal of Electrical and Electronic Engineering and Telecommunications.He is an Associate Editor for the IEEE Transactions on Systems, Man and Cybernetics-Systems, IEEE Transactions on Fuzzy Systems, Asian Journal of Control, Sensors, ETRI Journal, and International Journal of Modeling, Simulation, and Scientific Computing and an Editorial Board Member of the AI, Computer Science and Robotics Technology Journal at IntechOpen.Zhijian Hu received the PhD degree in control science and engineering from Harbin Institute of Technology, Harbin, China, in 2022.From 2019 to 2020, he was a Joint PhD Student with the Department of Electronics, Carleton University, Ottawa, Canada.He is currently a research fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore.His research interests include model predictive control, robust control, fuzzy control, and their applications in power systems and robotics.How to cite this article: Z. Xu, S. Shi, M. J. Er, and Z. Hu, Fuzzy-model-based robust control of Markov jump nonlinear systems with incomplete transition probabilities and uncertain packet dropouts, Asian J Control 25 (2023), 4201-4214, DOI 10.1002/asjc.3188

Theorem 1 .
By considering the problem of incomplete transition probabilities and uncertain packet dropouts, for a guaranteed H ∞ performance index  > 0,  > 0, Δ > 0,  > 0, the discrete-time IT2FMJS (17) is stochastically stable for any uncertain packet dropout rate satisfying − ≤ Δ ≤ , if there exists positive definite matrices P i,m , i ∈ R, m ∈ M, and symmetric matrices Ψ il,m , i, , l ∈ R, such that

TABLE 1
Optimal H ∞ performance indices  min for different  and  under the condition of (55).Infeasible Infeasible Infeasible Infeasible 0.3923 0.10 Infeasible Infeasible Infeasible Infeasible 0.6662

TABLE 2
Optimal H ∞ performance indexes  min for different  and  under the condition of (57).Infeasible Infeasible Infeasible 0.5928 0.3185 0.08 Infeasible Infeasible Infeasible Infeasible 0.3858 0.10 Infeasible Infeasible Infeasible Infeasible 0.6281 FIGURE 6 Minimum H ∞ performance indices  min for different incomplete TPMs.