Sampled-Data Control of Nonlinear Systems with Quantization

This paper is concerned with sampled-data control problem for a class of nonlinear systems with input quantization.The nonlinear system is converted into a linear-like system with modeling error by using partition of unity method. A time-dependent Lyapunov functional is introduced to capture the characteristic of nonlinear sampled-data systems and the exponential stability conditions are derived by the use of inequality techniques. The desired sampled-data controller is then synthesized. An example is provided to illustrate the effectiveness and benefits of the proposed scheme.


Introduction
In modern control systems, such as computer-based control systems, the continuous-time plant is usually controlled by a discrete-time controller with sample and hold devices.Such control systems are referred to as sampled-data systems.The analysis and synthesis of sampled-data systems have been a research focus for nearly three decades and many results have been reported in the literature; see, for example, [1][2][3][4][5] and the references therein.Among these references, lifting technique method and impulse model method are two main approaches.In the past years, some papers have considered the modeling of continuous-time systems with sampledcontrol in the form of continuous-time systems with delayed control input, which was introduced in [6].Compared with the other two approaches, this input delay approach can be applied to systems with nonuniform uncertain sampling and can cope with the system with parameter uncertainties.Many results related to this approach have been reported in the literature.For example, stabilization and stability analysis were investigated in [7][8][9][10][11][12][13], and filtering problems were addressed in [14,15].Improvements were provided in [16][17][18][19][20], where new time-dependent Lyapunov-Krasovskii functionals (LKFs) were defined to capture the characteristic of sampleddata systems.
The aforementioned results about sampled-data systems are mainly on linear systems or linear systems with nonlinear terms because the nonlinear dynamics are extremely difficult to deal with.Most of the existing results for nonlinear sampled-data systems are based on certain T-S fuzzy models.The T-S fuzzy model-based technique is an efficient approach for taking advantage of modern linear sampled-data control theory to nonlinear control.In [21][22][23][24][25][26], based on input delay approach, the considered systems were simply treated as ordinary continuous-time systems with a bounded fastvarying delay.The work in [27,28] has improved the results by applying an improved Lyapunov functional, which captures the characteristic of fuzzy sampled-data systems.It should be pointed out that the fuzzy controller design is based on the assumption that the fuzzy model exactly matches the plant and the modeling error may be neglected despite that the existence of modeling error may cause the instability of the system [29].
On the other hand, partition of unity is an important concept in differential geometry and is close to a group of open covering sets.The partition of unity method has proved that certain finite linear combinations of partition of unity have the ability to approximate continuous functions at any arbitrary precision on a compact region of Euclid space, which was first introduced in [30].When a nonlinear system is well approximated by using partition of unity, the remaining control problems become easier [31].Compared with T-S fuzzy approach, the main difference lies in that the former is more flexible in the modeling of nonlinear systems.In addition, the fuzzy method ignores the modeling error, while partition of unity method yields an equivalent
Now, we rewrite system (1) into a mathematically equivalent linear-like system with modeling error using Lemma 2.
where  and  are positive real constants.
For sampled-data control, only discrete measurements of () are available for control purposes, and the control signal is assumed to be generated by using a zero-order-hold (ZOH) function with a sequence of hold times Moreover, the sampling is not required to be periodic, and the only assumption is that the distance between any two consecutive sampling instants belongs to an interval.Specifically, it is assumed that For all  ≥ 0, where ℎ 2 ≥ ℎ 1 > 0 represents the upper and lower bounds of sampling interval, respectively.
In this paper, we propose the following state feedback controller: It is assumed that the output signals of controller (11) are passed via a quantizer and the quantizer is denoted as (⋅) = [ 1 (⋅),  2 (⋅), . . .,   (⋅)]  , which is assumed to be symmetric; that is, (−]) = −(]).The set of quantized levels is described by Theoretically, many different forms are possible for a quantizer.There is always a need to find a compromise between the performance and simplicity for real-world sampled-data control applications.In this paper, we are interested in logarithmic static and time-invariant quantizer, which is relatively simple and suitable for sampled-data systems.According to [40,41], a quantizer is called logarithmic if the set of quantized levels is characterized by For the logarithmic quantizer, the associated quantizer (⋅) is defined as follows: where Then, considering the above quantization behavior, we obtain the following quantized sampled-data controller: where Substituting the quantized controller ( 16) into (6), we obtain the following closed-loop system: where The goal of this paper is to design quantized sampleddata controller (16) to make the closed-loop system (17) be exponentially stable; that is, there exist two constants  > 0 and  > 0 such that Throughout this paper, the following lemma will be used.
Lemma 5. Considering system (17), the following inequality holds: where Mathematical Thus, Applying the Gronwall-Bellman lemma, we can conclude that (19) holds.This completes the proof.

Main Results
In this section, the problem of quantized sampled-data control will be studied for nonlinear systems by using partition of unity.Some results are provided as follows.Theorem 6.Consider system (1) satisfying Assumptions 1 and 4, for given scalar  > 0, if there exist matrices  > 0,  > 0, [ ] and constants  1 > 0,  2 > 0,  3 > 0, and  4 > 0 such that, for any ,  ∈ R, the following two equations hold: where Then, system (1) is exponentially stable.
Calculating the derivative of () along the trajectories of (17), we obtain Based on Schur complement, it can be found that for any appropriately dimensioned matrix   , ,  ∈ R, where From ( 31), we have It is noted that, for any appropriately dimensioned matrices  1 and  2 , the following equation holds: On the other hand, for any arbitrary  1 > 0,  2 > 0,  3 > 0, and  4 > 0, we can get from (8) that Adding the right side of (9) into V(), we obtain from ( 30), (33), and (35) From ( 24) and ( 25), it is easy to get Then, we obtain from ( 36) and (37) that Thus, it follows that for  ∈ [  ,  +1 ) Applying Lemma 5 and (39), we have for  ∈ [  ,  +1 ) From ( 40), we get Then, system (17) is exponentially stable.This completes the proof.Remark 7. Theorem 6 provides a stability criterion for system (1).Inspired by [19,20], the time-dependent Lyapunov functional is adopted in the proof of Theorem 6, which can capture the characteristic of sampled-data systems and reduce the conservatism.In addition, unlike most of the existing works, the obtained results depend not only on the upper, but also on the lower bound of the variable sampling pattern, which can fully adopt the actual sampling pattern and reduce conservatism.Now, based on Theorem 6, we focus on the quantized sampled-data controller design for system (17).The following theorem presents a sufficient condition of the existence of the desired controller.] and constants  1 > 0,  2 > 0,  3 > 0,  4 > 0, and  5 > 0 such that, for any ,  ∈ R, (42) and ( 43) hold: where Ĝ . ( Before and after multiplying (24) by diag{ Ĝ, Ĝ, Ĝ, Ĝ} and diag{ Ĝ , Ĝ , Ĝ , Ĝ }, we get where Using Schur complement, we obtain where From lemma, for any  5 > 0, Using Schur complement again, we get (42).Before and after multiplying (25) by diag{ Ĝ, Ĝ, Ĝ, Ĝ, Ĝ} and diag{ Ĝ , Ĝ , Ĝ , Ĝ , Ĝ }, by using the above similar method, we can also get (43).This completes the proof.Remark 9. Theorem 8 provides a design method of the sampled-data controller with quantization for nonlinear systems.The conditions in Theorem 8 are formulated in LMIs and thus can be effectively solved by using LMI toolbox in MATLAB.
In this example, we choose the initial condition (0) = [1 0.5]  , the minimum sampling period ℎ 1 = 0.01, the maximum sampling period ℎ 2 = 0.03, and the convergence rate  = 0.2.In addition, the parameter for the quantizer is assumed to be  = 0.9.By solving the LMIs in Theorem 8, we can get the controller gains from (45) that The model in (52) has been studied by T-S fuzzy systems (T-S) method in [42].To illustrate the effectiveness of partition of unity (PoU) method, the comparison of simulation results is shown in Figure 1.It can be shown from Figure 1 that the method used in our paper gives faster convergence rate.The reason is that the fuzzy method ignores the modeling error, while partition of unity method yields an equivalent model with modeling error, better representing the characteristics of the origin nonlinear system.Therefore, the results will hold if the modeling error exists by using fuzzy control method.

Conclusion
In this paper, the problem of quantized sampled-data control for a class of nonlinear systems has been investigated.Based on partition of unity method, the system has been converted into a linear-like system with modeling error.By introducing a Lyapunov functional, sufficient conditions for the existence of desired controller have been derived.An example has been presented to show the effectiveness and benefits of proposed method.

Figure 1 :
Figure 1: State response using T-S fuzzy and partition of unity methods, respectively.