Robust Mixed Performance Control of Uncertain T-S Fuzzy Systems with Interval Time-Varying Delay by Sampled-Data Input

In this paper, a sampled-data parallel distributed compensator (PDC) is proposed to guarantee mixed H2/H∞ performance of uncertain T-S fuzzy systems with interval time-varying delay and linear fractional perturbations. A full matrix formulation approach is developed to present our main results in LMI conditions. To achieve better results, new inequality and Lyapunov-Krasovskii functional are developed to improve the conservativeness of the proposed results. Finally, some numerical examples are illustrated to show the use of our main results. In this paper, interval time-varying delay and interval sampling are considered instead of constant delay and periodic sampling in published literatures.


I. INTRODUCTION
Time-delay phenomena are often encountered in various practical systems; such as aircraft stabilization, biology and medicine engineering, chemical engineering systems, control of epidemics, distributed networks, inferred grinding model, manual control, mechanical operation; microwave oscillators, models of lasers, neural networks, nuclear reactors, population dynamic models, rolling mills, ship stabilization, and systems with lossless transmission lines. On the other hand, time delay is often the source of instability and generation of oscillation in many physical systems. Hence, the stability issues of T-S fuzzy systems with time delays have been investigated in recent years [1]- [4]. It is interesting to note that the models of practical systems are always containing several nonlinear properties. Hence the Takagi-Sugeno (T-S) fuzzy system models [5]- [6] were introduced to approximate these nonlinear elements in many physical examples. T-S fuzzy system is a useful tool to solve the control design problems in many nonlinear practical applications for dynamic systems; such as guidance and mooring control in autonomous surface vehicle; nonfragile control of permanent-magnet synchronous motor; stabilization of inverted pendulum and motor drive control; predictive and dissipative control of neural networks; predictive control for a diesel engine; performance control of truck-trailer model; dissipative control of wind turbine model; sampled-data control of reaction-diffusion neural networks, vehicle suspension systems, and wind energy conversion systems [7]- [18]. This approach provides a connection between the linear control theory and the fuzzy concept. It is also interesting to note that interval time-varying delay in [10]- [12], [19] is more suitable to describe the transportation delay than constant delay in [20]- [22]. In recent years, stability and performance for T-S fuzzy systems with time delays were investigated by Lyapunov theory and LMI (Linear Matrix Inequality) approach in [10]- [23].
Sampled-data state feedback input is a useful approach to implement some complicate control schemes; such as parallel distributed compensator (PDC) in T-S fuzzy system [20]- [23] and switching control in switched system [24]. Parallel distributed compensator and switching control are always used to enrich the performance for T-S fuzzy systems and switched systems, but PDC and switching control are difficult to implement by analog devices. Hence sampled-data state feedback control input is an available consideration to implement PDC and switching control for systems under consideration. Suppose the control input is calculated by a digital device (computer or chip), then the feedback value will be remained until the next sampling instant to reflect the sampled value [16]- [18], [20]- [24]. The allowable upper bound for fixed sampling period > 0 will be an important issue to guarantee the performance of systems under consideration. To implement the distant from state feedback control, networked control technology was provided to finish the goal in the recent years. Aperiodic sampling concept in [20] is a more practical application, but only pointwise sampling period can be guaranteed the performance of systems under consideration. Since the congestion for transmission in network or signal processing of sampler, the actual information transmits to actuator will produce in different sampling periods [19], [22]- [23]. Hence interval sampling period is more suitable for practical implementation in sampled-data control systems than constant and pointwise sampling periods [13]- [14], [16]- [19], [22]- [23], [25]. In this paper, we propose a novel inequality and new Lyapunov-Krasovskii functional to guarantee the mixed performance and design the robust PDC state feedback sampled-data control input with interval sampling period.
In the past, the ∞ performance of systems under consideration was used to minimize the effect of regulated output with respect to disturbance input and guarantee that the closed-loop system is stable [4], [7], [9], [12], [20]- [21], [25]- [27]. On the other hand, the 2 performance of systems was applied to minimize the dynamics with respect to initial condition of system under consideration and zero disturbance. Hence the system with mixed performance has been an interesting research topic in recent years [20], [26]- [27]. In this paper, the mixed 2 / ∞ performance scheme is proposed to minimize upper bound of ∞ performance with respect to 2 measure. Linear fractional perturbation is a general presentation about systems with some uncertain elements or nonlinearities [28]- [29].
In this paper, we use LMI optimization approach in [30] to guarantee the mixed 2 / ∞ performace and design the sampled-data PDC. The main contributions of this paper can be highlighted as follows: • In this paper, the optimal ∞ performance for uncertain T-S fuzzy system with interval time-varying delay and linear fractional perturbations is achieved by sampled-data PDC. The 2 measure can be provided to guarantee the upper bound in response for regulated output of system under consideration.

•
To overcome the difficulty about the multiplication and combination of matrices, the full matrix formulation approach is developed in this paper. With the proposed approach, our results can be shown in LMI optimization formulation which can be solved by LMI toolbox of Matlab directly. For more complex system under consideration or other inequalities used, our developed approach is also a good tool for further analysis.

•
An upper bound about the sampling period can be evaluated instead of pointwise values in our past results [20]. Interval time-varying delay is considered instead of constant delay in [20]- [22] for the uncertain T-S fuzzy system under consideration. The proposed LMI conditions are easier to solve than the proposed ones in [20]. Notations: For a matrix A, we denote the transpose by , symmetric positive (negative) definite by > 0 ( < 0). ≤ means that matrix − is symmetric positive semidefinite.
( ) = + , and 0 denote the identity matrix and zero matrix with appropriate dimension, respectively,
The delay-dependent LMI optimization results are developed to guarantee the asymptotic stability and mixed performance by the design of sampled-data PDC in (4b).

Remark 1:
In this paper, the sampling period = +1 − ≤ can be allowed in an interval. It is more efficient than periodic sampling and easy to implement in the real world. The delay 0 < ℎ ≤ ℎ( ) ≤ ℎ is also assumed varying in a given interval. It is more practical than constant delay. This term ( ) ( ) used in Lyapunov-Krasovskii functional (7) will be a flexible choice. The possible dynamics of system can be included into the time derivative of the functional in (7). Since the high dimensional matrix operations, a full matrix formulation approach is developed to present the multiplication and combination of matrices ( , ). The inequalities in Lemmas 1 and 2 are simultaneous applied in derivations for positive definitive matrices 3 and 5 for interval time-varying delay ℎ( ), 4 and 6 for sampling period , respectively. It is interesting to note that the inequality in Lemma 1 is less conservative than Wirtinger-based one [20]. The major advantage of Lemma 2 is that the delayed state terms ( − ℎ( )) and ( − ( )) can be included into the derivation for the main LMI conditions. More efficient results can be proposed for interval time-varying delay and sampling period.

Remark 2:
If the upper bound of variation of interval time-varying ℎ is larger than 1 or unknown, the proposed results in Theorem 1 of this paper are also valid by selecting the matrix ̂4 = 0. [20], some given pointwise sampling intervals will be proposed to improve the fixed sampling. In this paper, the proposed approach will allow that the sampling period belongs to an estimated interval [0, ] . The interval time-varying delay is considered in this paper instead of constant delay in [20]. It will be more practical and flexible than the our published results in [20].

Remark 4:
The larger values for sampling interval , lower bound ℎ , upper bound ℎ , interval ℎ − ℎ of interval time-varying delay are better to provide flexibility and less conservativeness for our proposed results. The smaller value for will provide better at disturbance attenuation for system under consideration.

Remark 5:
In Theorem 1, we use LMI conditions in (6a)-(6c) to find the feasible solution for a known positive parameter ̅ = 2 . The 2 measure of the system under consideration can be calculated from our proposed result in (6d). Smaller values of and will imply better disturbance attenuation and 2 measure, respectively.

III. ILLUSTRATIVE EXAMPLES
This section includes three examples to demonstrate the use and main contribution of the proposed results. Some comparisons have been made in Examples 1-2. Example 3 is a practical nonlinear system which has been designed by our developed approach to achieve mixed performance for its corresponding T-S fuzzy system. Some simulation diagrams have been provided to show the efficiency of proposed results.
In this simulation, the selection of fuzzy rule is setting on center and boundary points of displacement variable ( ). The number of fuzzy rule is 2. If we would like to make better approximation for original nonlinear system in (15) (19) is illustrate in Figure 8. In general, larger number mr ruzzy rule will make mmre apprmximate tm mriginal nmnlinear system un er cmnsi eratimn. But it may cause that LMI mptimizatimn prmblem mr esign scheme rmr sample -ata PDC is inreasible.

IV. CONCLUSION
In this paper, a sampled-data PDC has been proposed to guarantee mixed 2 / ∞ performance of uncertain T-S fuzzy systems with interval time-varying delay and linear fractional perturbations. Full matrix formulation approach has been investigated to improve the conservativeness of proposed results. New inequality and Lyapunov-Krasovskii functional have been developed to guarantee the efficiency of the proposed results in this paper. Finally, some numerical examples have been illustrated to show the main results. In this paper, we consider interval time-varying delay and interval sampling period instead of constant delay and pointwise sampling period in [20], respectively. Furthermore, some interesting research topics for T-S fuzzy systems can be investigated, e.g., asynchronous non-PDC controller and quantizer [32]- [33], dissipativity and dissipative control [15]- [16], event-triggered control [33]- [34], finite-time control [27]- [32], nonfragile control [14], passivity and passive control [25]. All these would constitute our future research work.