STABILIZATION OF SYSTEMS WITH SECTOR BOUNDED NONLINEARITY BY A SAWTOOTH SAMPLED-DATA FEEDBACK

The paper considers a nonlinear Lur’e type system with a sector bounded nonlinearity. The zero equilibrium of the system may be unstable, so it is stabilized by a periodically sampled feedback signal. Such stabilization problems were previously explored by a number of researches with the help of the zero-order hold (ZOH) control that is kept constant between successive sampling times. The main disadvantage of this method is that the time delay introduced by ZOH has a destabilizing impact on the closed feedback system, especially in the case when the sampling frequency is sufﬁciently low and the feedback gain is high. To reduce this effect it is proposed to modify the form of the stabilizing signal. In this paper the reverse sawtooth control is introduced instead of ZOH. The stability criterion is obtained in the form of a feasibility problem for some linear matrix inequalities (LMI). A numerical example demonstrates how the new stabilization method allows to reduce the sampling frequency required for stabilization.


Introduction
Stability problems for a continuous-time system under a sampled-data feedback attracted much attention in the last decade. This study was largely motivated by appli-cations to networked control systems (see, e. g., [Hespanha et al., 2007]). Sampled-data stabilization of a sector bounded nonlinear system was treated in [Seifullaev and Fradkov, 2015a;Seifullaev and Fradkov, 2015b;Seifullaev and Fradkov, 2015c;Seifullaev and Fradkov, 2016;Zhang et al., 2017;Bryntseva and Fradkov, 2018].
For networked control systems it is desired that the sampling rate of a stabilizing feedback signal should be sufficiently low. As for the intersample control, most of researchers employ the zero-order hold (ZOH) scheme, when the control function is kept constant throughout the sampling interval. It is well known that a system with a ZOH feedback can be rewritten as a system with a linearly increasing time-varying delay (see, e. g., [Fridman, 2010]), and this delay is the greater, the greater the length of the sampling interval. Such long delays have a destabilizing effect on the system's behavior. It looks reasonable to replace ZOH for something more sophisticated, which allows to reduce the negative influence of the delay. A reverse sawtooth picewise-linear control was introduced to this end. This paper continues a series of works [Churilov, 2018;Churilov, 2019a;Churilov, 2019b] devoted to an application of the absolute stability theory to sampled-data stabilization. For the stability analysis of the obtained hybrid system we use the Gelig's averaging method [Gelig, 1982;Gelig and Churilov, 1993b;Gelig and Churilov, 1998] reformulated in terms of linear matrix inequalities [Boyd et al., 1994]. Besides the averaging, we employ such methods as S-procedure for multiple quadratic forms [Fradkov and Yakubovich, 1979;Yakubovich, 1992] and integral quadratic constraints (see, e. g., [Yakubovich, 1968;Yakubovich, 1988;Megretski and Rantzer, 1997;Yakubovich, 2002]).
The paper is organized as follows. First the system equations are given. Then we present quadratic and integral quadratic constraints that will be needed to prove the main theorem. Further, the main result of the paper is formulated and proven. Finally, we provide an illustrative numerical example. The upper bounds of the sampling period obtained by the theorem of this paper are compared with the similar bounds for ZOH stabilization.
Consider a system with a uniform sampling with the period T . The feedback control function u(·) is defined as n = 0, 1, . . .. The function u(t) is a reverse sawtooth wave that linearly ramps downwards (for σ(nT ) > 0) or upwards (for σ(nT ) < 0) until it reaches zero (see Figure 1(b)). We are interested in obtaining effectively verifiable conditions for asymptotic to zero of the solutions of system (1)-(4).

The Idea of Averaging
The pulse averaging technique that we use here originates from the principle of equivalent areas (PEA) [Andeen, 1960a;Andeen, 1960b]. For sector bounded constraints it was firstly obtained in [Gelig, 1982] and the refined in [Gelig and Churilov, 1993a;Gelig and Churilov, 1998]. Following the style of that time, the mathematical statements in [Gelig and Churilov, 1998] were formulated in terms of frequency-domain inequalities, however they can be techically reformulated as LMI by using the celebrated Kalman-Yakubovich-Popov (KYP) lemma (see, e. g., [Popov, 1973;Boyd et al., 1994;Yakubovich et al., 2004]).
Following [Gelig, 1982], compute the average of the nth control pulse Let the function w(t) be the integrated error of replacing u(t) for its averages: for any n 0. From (5) it follows that the function w(t) is continuous and turns to zero at all the sampling instants nT . The main mathematical technique used in [Gelig and Churilov, 1993a;Gelig and Churilov, 1998] was the construction of quadratic Lyapunov fuctions that depend on x(t) and w(t). The functions x(t), w(t) and the sequence v n were linked by quadratic and integral quadratic constraints. When the sampling frequency tends to infinity, the Gelig's criteria reduce to classical criteria of absolute stability, such as circle or Popov.
The physical aspects of PEA were discussed in [Churilov, 2019a].
Notice that some distant similarity in technique can be found in later works [Briat and Seuret, 2012;Seuret, 2012;Briat, 2013], where loop functionals were introduced and explored. These functionals also become zero at sampling times. However, looped functionals were not related to averaging and were usually added to Lyapunov-Krasovskii functionals.

Quadratic and Integral Quadratic Constraints
For brevity, denote t n = nT , n = 0, 1, . . .. Besides the sectoral bound (3), we will need a number of auxiliary constraints. Following the averaging scheme described in the previous section, from (4), (5) we get v n = 1 2 σ(t n ).
Define a piecewise constant function v(t) = v n , t n t < t n+1 .
By a straightforward calculation, from (6) we have is valid. By a direct calculation we get tn+1 tn w(t) dt = 1 6 v n T 2 tn+1 tn w(t) 2 dt = 1 30 v 2 n T 3 .
If we introduce a vector column then from (14), (15) we obtain where
6 Necessary Conditions for the Fulfillment of the Main Statement Preposition 1. Let conditions of Theorem 1 be satisfied. Then the matrix A µ = A + µB 0 C 0 + 1 2 BK is Hurwitz stable for any number µ, µ 1 µ µ 2 .

Numerical Example
Consider the system (see [Seifullaev and Fradkov, 2015c])ẋ Here σ 0 (t) = x 2 (t), ϕ 0 (σ 0 ) = sin σ 0 , and we can take µ 1 = −0.2173, µ 2 = 1. The control function u(t) is defined by (4), where σ(t) = kx 2 (t) and k is a scalar feedback gain. Thus system (29) can be written in the form of (1)-(4) with Feasibility of inequalities (21) of Theorem 1 was explored with the help of YALMIP software package for interface and SeDuMi solver for semidefinite programming [Löfberg, 2004;Sturm, 1999]. System (29) was previously studied in [Seifullaev and Fradkov, 2015c] with the use of ZOH control The results of computer experiments are consolidated in Table 1. The computer experiment was produced for the values k = 2, 3, 4, 5, 10 (see column 1). The maximal values of T computed with the help of Theorem 1 are given in column 2. Column 3 contains the maximal values of T obtained by the Lyapunov-Krasovskii method for the case of ZOH control in [Seifullaev and Fradkov, 2015c] (the value k = 4 was not considered). For k = 2, 3 Theorem 1 gives no feasible values, but for k = 5, 10 the maximum of T obtained from Theorem 1 is greater than that was found in [Seifullaev and Fradkov, 2015c].
The comparison of the reverse sawtooth (RS) control and the ZOH control is presented in Table 2. The maximal values of T are obtained by a direct computer simulation under the T -periodic sampling (i. e., they are irrelevant to any stability criteria). It is seen that for the same feedback gain k the RS control provides a sampling period T approximately 1.9 times greater than that of ZOH control. Some transients for system (29) with k = 5 and the T -periodic sampling are shown in Figures 2,3. With the increase of T the zero equilibrium turns from a stable node to a stable focus. With the further increase of T this focus loses stability.  [Seifullaev and Fradkov, 2015c] for ZOH feedback k T T (Theorem 1) (Seifullaev, Fradkov, 2015)

Acknowledgement
The work was supported by the Russian Foundation for Basic Research (Grant 17-01-00102-a) and by the Government of Russian Federation (Grant 08-08).

Conclusion
The paper introduces a new control function based on the reverse sawtooth signal wave, which can significantly increase the sampling period compared with the zero-order hold. Based on the Gelig-Yakubovich mathematical technique, the paper suggests an easily verifiable stability criterion formulated in terms of linear matrix inequalities. Simulation shows its reasonable conservatism and an agreement with the previously obtained results.