Finite-Time Boundedness and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.3443pt" id="M1" height="15.7437pt" version="1.1" viewBox="-0.0657574 -11.3994 26.6531 15.7437" width="26.6531pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.012,0,0,-0.012,13.299,4.134)"><path id="g50-30" d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z"/></g></svg> Control for Affine Switched Systems

Han, Lu; Qiu, Cunyong; Jiang, Lin

doi:https://doi.org/10.1155/2018/2309395

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 2309395 | https://doi.org/10.1155/2018/2309395

Finite-Time Boundedness and Control for Affine Switched Systems

Lu Han,¹Cunyong Qiu,¹and Lin Jiang¹

Academic Editor: Guangming Xie

Received06 Oct 2017

Revised05 May 2018

Accepted30 May 2018

Published18 Jul 2018

Abstract

For affine switched systems, the existence of multiple equilibria is related to subsystems owing to the affine terms, which makes asymptotic and finite-time stability analysis nontrivial. In this paper, the problems of finite-time boundedness (FTB) analysis and stabilization are addressed for affine switched systems, and several definitions and sufficient conditions are proposed to study FTB and performance. At first, the definition of FTB for affine switched systems is improved concerning the affine terms and multiple equilibria. Based on the FTB definition, sufficient conditions ensuring finite-time boundedness for affine switched systems under a prespecified state boundary are given. Then the results are extended to solve finite-time boundedness problem, in which the controllers are designed to guarantee the finite-time boundedness of affine switched system with performance. In our investigation, average dwell-time approach is employed to study the time-dependent constrained switching case. Finally, several numerical examples are given to illustrate the effectiveness of the proposed results.

1. Introduction

Switched systems are distinctive subclass of hybrid systems. They are composed of a family of continuous-time or discrete-time subsystems with a criterion that rules the switching among them. This switching rule can be classified as time-dependent, state-dependent, or time-state-dependent [1]. Since many physical processes possess switching nature, and many real-world applications resort to switching strategy to improve the control performance, the theory and application of switched systems have received a great attention during the recent decades. For more details on the recent results about the basic problems in stability and stabilization for switched systems, readers are referred to surveys [2–4] and books [1, 5] and the references cited therein.

The issue of stability analysis and stabilization is an important topic for switched dynamical systems [6–10]. Finding sufficient conditions ensuring the Lyapunov asymptotic stability dealing with infinite time interval has been the major concern for switched systems. Numerous published results discussed the asymptotic stability analysis and stabilization employing different variations of Lyapunov function [7, 11, 12]. Average dwell-time approach [13, 14] and Lie-algebraic condition technique [15, 16] are effective tools for analysis of switched systems. On the contrary, the finite-time behavior of dynamical systems is also of interest in many practical applications. It concerns that the states do not exceed a certain bound during a fixed time interval, e.g., to avoid saturations or excitation. The theory of finite-time stability (FTS) and finite-time boundedness (FTB) focuses on the transient response of dynamical systems over a finite-time interval, while asymptotic behavior is for infinite time. In the survey of recent development of this innovative theory, some necessary and sufficient conditions for finite-time stability and stabilization of continuous-time systems or discrete-time systems have been provided in [17, 18]. Based upon it, necessary and sufficient conditions for finite-time stability of systems with impulsive effects were obtained in [19, 20]. The authors [21, 22] applied FTS/FTB conceptions to switched systems and compared the conservativeness among different conditions. In [23], the mixed /finite-time stability control problem was discussed. For quadratic input-output finite-time stability with an bound, [23] provided a necessary and sufficient condition. Then the method was extended to robust controller and filter design for switched system with exogenous noise [24, 25]. It should be noted that finite-time stability and Lyapunov asymptotic stability are independent concepts: a Lyapunov asymptotic stable system may not fulfill FTS/FTB criteria since the transient response of a system may exceed the bound, and vice versa [26]. In many practical applications, switching is likely to occur in some short-time intervals, whereas for remaining long time no switching occurs. Since Lyapunov stability concerns with infinite time, it may not be influenced by such short-time switching. However, the boundedness of state may be affected by the switching. Hence, FTB criteria are needed to be considered for designing controller and switching laws during such applications.

Most of the existing literatures on stability issues of switched systems are based on the premise that all subsystems share a common equilibrium (typically the origin). On the other hand, for affine switched system, subsystems have different equilibria, so complex and interesting phenomena emerge. Almost all the practical hybrid systems can be modeled as affine switched systems. Many results like [27–30] analyzed interesting behaviors similar to those of asymptotically stable systems near an equilibrium for affine switched systems and depicted their real-world applications. Many extensions of the conventional stability concepts have been obtained for affine switched systems. S-Procedure method with the extensional state vector has been proposed in [31, 32] to analyze the asymptotic stability for continuous affine switched system. The relative results were extended to discrete affine switched systems in [33]. In [34], a method for designing switching rules driving the state of affine switched system to a desired equilibrium was investigated. Almost all the existing literatures on stability analysis of affine switched systems focused on the asymptotic stability. However, the boundedness of state for affine switched systems under constrained dwell-time switching is also of significant interest for affine switched systems. In FTB analysis, we also need to deal with affine terms leading to multiple equilibria for affine switched systems, but the investigation of this problem lacks researchers’ interest previously. Potential of affine switched systems theory and importance of finite-time transient behavior from the perspective of real-world applications are the major motivations for this investigation presented in this paper.

The main objective in this paper is to find sufficient conditions ensuring the FTB of affine switched systems by switching signal and feedback controllers design and to drive the state of affine switched system to the prescribed neighborhood of a desired equilibrium during a finite-time interval. Taking into account the influence of affine terms on FTB for affine switched system, we propose an innovative FTB concept. Based on this definition, sufficient conditions ensuring the affine switched system finite-time bounded are proposed. Specifically, with the prespecified state boundary, average dwell time and state-feedback controllers for each subsystem are determined to guarantee the finite-time boundedness. The paper [22] points out that the more information about switching signal we know, the less conservative results can be derived. We extend this idea to switched affine systems to further reduce the conservatism. Classifying subsystems into asymptotically stable and unstable systems, we get the less conservative results of finite-time boundedness for affine switched system with the help of additional information of switching signal. Then, results are extended to solve the FTB problem for controller design.

The rest of this paper is organized as follows. In Section 2, definitions of finite-time boundedness and finite-time boundedness for affine switched system are revisited. Based on these definitions, finite-time boundedness analysis and finite-time stabilization are presented in Section 3. Then in presence of exogenous signals, finite-time boundedness and the controllers design are investigated in Section 4. In Section 5, several numerical examples are presented to validate the proposed results. Conclusions are given in Section 6.

2. Preliminaries and Problem Formulation

For our investigation, we consider continuous-time affine switched system described aswhere is the system state, is the control input, is the measurement output, , , and are system matrices with appropriate dimensions, constants are affine terms, and is switching signal. For notational simplicity, we use in place of .

Matrix variables , , and give rise to an equilibrium (stable or unstable) for each subsystem; assuming all to be nonsingular, we consider a given reference as the required equilibrium for the whole system, referred to as switched equilibrium. Without loss of generality, it is assumed that the desirable equilibrium is different from all the equilibria of subsystems. Now although the asymptotic stability of affine switched system may be achieved by other types of switching strategy such as min-switching and sliding method, the state will not exactly converge to under dwell-time constrained switching. The reason is that there always exist time interval (dwell time is always greater than zero) in which state must diverge from . In our FTB investigation, we provide solution for boundedness of error state under dwell-time switching, which depicts the importance and innovation of our approach.

Here first we will extend the FTS and FTB concepts for affine switched systems keeping in view prescribed equilibrium . In absence of control input, system (1) can be stated as

Definition 1. Autonomous affine switched system (2) is said to be finite-time bounded with respect to if the following inequalities hold:where , , , , , , and .

Remark 2. Given equilibrium and system (2), its tracking error system can be written aswhere , . According to Definition 1, we can conclude that affine switched system (2) is finite-time bounded with respect to if whenever and . The FTB criteria of affine switched systems ensure the state tracking the desired equilibrium within the boundary . In other words, it guarantees the error state tracking the origin in finite-time interval. Therefore, our study about FTB of affine switched systems can be turned into analyzing its corresponding tracking error system. Moreover, it is worth noting that FTB theory for general switched systems is related to initial state [35, 36]; whereas for affine switched systems, we are concerned with as well as the desired equilibrium and affine terms . Thus, in the Definition 1, the premise constraint conditions are extended to both initial state and to analyze the FTB of affine switched systems, where is related to the desired equilibrium and affine terms .

Remark 3. With the state-feedback controller , , affine switched system (1) can be rewritten into the following closed-loop system:where and the FTB analysis method can be used directly. Similar to the significant impact of switching laws on asymptotic stability, the switching signals affect the finite-time boundedness of affine switched systems property significantly. Therefore, both switching signals and robust controllers should be designed during the FTB analysis of affine switched systems.

On the other hand, external disturbances are inevitable to dynamical systems. We can state affine switched system with time-varying disturbance as is assumed to be energy-bounded and hence for some scalar it satisfies the inequality . For simplifying FTB analysis, following Definition 1 we can transform affine switched system (6) to its error tracking switched system aswhere , , is the desirable reference point, is the controlled output, and the switched equilibrium of system is moved to the origin accordingly. Considering state-feedback controller , we derive the following closed-loop switched system:where , . Now we are able to state the following definition.

Definition 4. For affine switched system (7), considering state-feedback controller and H_∞ performance index , if the following two conditions are satisfied:(1)the closed-loop error tracking switched system (8) is finite-time bounded;(2)under zero-initial condition, the controlled output satisfies the inequalitywhere , , then is called ‘finite-time controller’.

Assuming , system (7) is expressed as

Now Definition 4 can be reduced to the following form.

Switched system (10) is said to be H_∞ finite-time bounded with performance index , if(1)the error tracking switched system (10) is FTB;(2)under zero-initial condition, the controlled output satisfies

Based upon the above preliminaries we will focus on how to find sufficient conditions to ensure the finite-time boundedness of affine switched systems and address the analysis and synthesis of piecewise linear state-feedback controllers resorting to LMI-based algorithms. The main problems we concern in this paper can be stated as follows.

Problem 5 (finite-time boundedness for affine switched systems). Given affine switched system (2), find sufficient conditions ensuring the finite-time boundedness with respect to .

Problem 6 (state-feedback stabilization under FTB). Given affine switched system (1), find set of static state-feedback controllers to ensure that the closed-loop system (5) is finite-time bounded with respect to .

Problem 7 ( performance and controller design). Given affine switched system (8), analyze the performance and design set of controllers defined in Definition 4 to ensure the finite-time boundedness with respect to and reduce the effect of the exogenous signal and on the controlled output to a prescribed level .

3. Finite-Time Boundedness and State-Feedback Stabilization

In this section, Problems 5 and 6 are taken into consideration. Our main aim is to find sufficient conditions and state-feedback controllers to ensure the finite-time boundedness of affine switched system in the form of (2). For a finite-time interval , we consider finite switchings . Each subsystem has an (stable or unstable) equilibrium point . Regarding reference point as an equilibrium point for the whole system called switched equilibrium and taking into account average dwell time, we will derive sufficient conditions ensuring finite-time boundedness.

Theorem 8. Affine switched system (2) is finite-time bounded with respect to , if there exist positive definite matrices , scalars , , such that

Proof. Consider the error tracking switched system (4), let , . We choose piecewise Lyapunov function . From condition (12b) we haveEmploying (12a) we deriveLet , be the switching instant of switched system. For overall system we can write . Now from inequality (14),where , denotes the finite-time interval. Accordingly, the Lyapunov inequality in single step satisfiesSuppose system switches from mode to at some instant ; then from condition (12a),It is evident that and following (16) iteratively we can derive easily thatApplying (15) and (18) we deduceOn the other hand, from condition (12a), we haveUsing the fact that , in order to ensure the finite-time boundedness of switched system (4), i.e., , the following condition should be satisfied:which can be rewritten as condition (12c). Therefore, we get and we conclude that the affine switched system (2) is finite-time bounded which completes the proof.

Remark 9. When other parameters are fixed, condition (12c) can be described by average dwell time as [37]where . In other words, the average dwell time should be chosen large enough to ensure that inequality (22) is satisfied, which is necessary to guarantee the finite-time boundedness of affine switched system (2). Moreover, assuming , from (12a) and (19) we deduceWhen , and the term on the right side of (23) will become infinite, which explains that the affine switched system (2) is not ultimately bounded, which illustrates FTB and ultimately boundedness are independent concepts.

Remark 10. Once the state bound is not ascertained, the minimum value is of interest, which can be found through optimization problem subject to (12a) and (12b). If we fix the parameter and let , , the optimization problem becomesThen can be derived with the optimized value .

It is evident that smaller value of gives rise to less conservative FTB conditions. In Theorem 8, the parameter indicates the asymptotic stability property of each subsystem. It is well known that when in condition (12b), this condition can be regarded as Lyapunov function condition which ensures each subsystem to be asymptotic stable; whereas when , the condition that must be negative is relaxed in FTB sense, and just should be no greater than to guarantee the boundedness of state in finite-time interval . The parameter in condition (12b) covers both the asymptotic stable and unstable subsystems. Now let subsystems be asymptotic stable and are unstable, and , denote the total activation time for stable and unstable subsystems during . Then the less conservative results about FTB of affine switched system can be obtained in the following corollary.

Corollary 11. Switched system (2) is finite-time bounded (FTB) with respect to , if there exist a set of positive definite symmetric matrices , , scalars , , and such that the following conditions are satisfied:

Proof. Consider the error tracking switched system (4), let , , ; we choose piecewise Lyapunov function .
From condition (25b), we getLet , be the switching instant of switched system, and . From inequalities (25a) and (26), we havewhere . Following (27) iteratively By the same proof line in Theorem 8, we know that in order to ensure the finite-time boundedness of switched system (4), i.e., , the following condition should be satisfied:Since , we havewhich can be rewritten as (25c). Hence, and proof is complete.

Remark 12. Similar to the optimization problem of state bound described in Remark 10, the optimal value can be found according to subject to (25a) and (25b). We fix the parameter and let , , the optimization problem becomesThen the minimum can be derived with the optimized value . Since , comparing the value of the optimal state bound in Theorem 8 and Corollary 11, we know that, by classifying subsystems into asymptotically stable and unstable, the FTB conditions derived in Corollary 11 are less conservative than that in Theorem 8.

Constituting state-feedback controller of the form , affine switched system (1) can be transformed into the closed-loop form of (5) and Definition 1 of FTB can be used directly. Now we will consider problem-2 to provide sufficient conditions for finite-time state-feedback stabilization.

Theorem 13. For affine switched system (1) holding Definition 1, if there exist state-feedback controllers , positive definite matrices , matrices , and scalars , such thatthen closed-loop system (5) is FTB with respect to with .

Proof. Assume is the switched equilibrium point of affine switched system (1). Applying coordinate transformation we can get its corresponding error tracking switched system aswhere and . Then under the state-feedback controllers , and the closed-loop error system can be written aswhere . From Remark 2, we know that FTB analysis and finite-time control can be realized employing tracking error system. Hence, we consider the closed-loop error system (34) here to design the controllers stabilizing the system (1) in finite-time interval.
Let , , we choose piecewise Lyapunov function for each subsystem; then the derivative of along the solution of system (34) is described asLetting , pre- and postmultiplying (32b) by we getDue to condition (32a) and Schur’s complement formula [38], we deduceNow using (36), from (35) we can deriveBy the same proof guidelines of Theorem 8, FTB conditions (32a) and (32c) of closed-loop error system (34) can be derived. Accordingly we get , which proves that the affine switched system (1) is finite-time bounded under state-feedback controllers .

4. Performance Analysis and Controller Design of Affine Switched Systems

Based upon FTB investigation of previous section, our main aim now is to design a set of controllers to solve Problem 7. As stated in Remark 2, finite-time control can be realized through tracking error system, and this will be the main focus in this section. For the sake of simplicity, we firstly consider the autonomous error switched system in the form of (10) assuming that , and the corresponding theorem is stated as follows; then we will show how to remove the assumption and extend the results to the general affine switched system with exogenous signal input.

Theorem 14. Given autonomous robust switched system (10), if there exist positive definite matrices , scalars , , and such thatthen this system is finite-time bounded with H_∞ performance with respect to .

Proof. Let , , and we opt Lyapunov function andSince , condition (39b) implies thatFrom (40) and (41), and together with condition (39a), we getLet , be the switching instants, and , where is the indication function for activated subsystem. From inequality (42), we havewhere . Accordingly, the Lyapunov inequality in single step satisfiesLet system switch from mode to at instant ; then condition (39a) implies thatNoting that , following relation (44) iteratively, we can deriveApplying (43) and (46), the following inequality is obtained:Using the fact , (47) can be rewritten asOn the other hand, from condition (39a), we haveSince we conclude that in order to ensure FTB for system (10) such that , the following condition should be satisfied:which can be rewritten as condition (39c). Hence, FTB analysis for system (10) is completed.
Considering , from (39a) and (39b) we deduceIntegrating both sides of (51) and through iterations, we can deducewhere . Then following the proof line of Theorem 8, we getSince the zero-initial condition, we have ; thus,Setting , , the H_∞ performance condition (11) is satisfied which completes the proof.

Remark 15. The parameter is performance index and its minimum value is often of interest from practical viewpoint; hence, we can state the optimization problem as

Similarly, fulfilling FTB criteria, minimum value of state bound is also desired, which can be found as the optimization problem: subject to (39a) and (39b). If we fix the parameter and let , , then we can state optimization problem asand is derived with the optimized value of . We can adopt a convex combination of and as , and a more general convex optimization problem can be stated as

Now we will extend the results to design the controllers, ensuring FTB of the closed-loop affine switched system (8). Different equilibria for subsystems exist because of the affine terms , and hence stability analysis and control are not trivial. To solve this problem, a few results are available proposing extended state space method in [12, 31]. However, this approach seems conservative for system synthesis because the eigenvalues of the extended state matrices related to the affine terms are not exactly the same as for the original state matrices . For state-dependent affine switched system, S-procedure method can be used to reduce the conservatism. However, for time-dependent affine switched systems, there are only few effective results. In our investigation, we redefine exogenous signal as

Hence, the closed-loop switched system (8) can be rewritten aswhere , . Since in the framework holds, the proposed extension of the disturbance input is reasonable and we can design the controllers of the equivalent closed-loop error switched system (59) to ensure the finite-time boundedness for original affine switched system (8).

Theorem 16. The closed-loop switched system (59) is FTB with H_∞ performance regarding , if there exist constant state-feedback controller , positive definite matrices , matrices , scalars , such that where .

Proof. Let and , . Defining as before, for system (59) we can state thatAssuming , pre- and postmultiplying (60b) by ,Using Schur lemma, (62) can be rewritten asSince , we can getwhich implies thatEmploying (60a) and using Schur complement formula,Following the proof guidelines of Theorem 14, condition (60c) which guarantees the FTB of robust affine switched system can be obtained.
Now we need to prove condition (9) for performance under zero-initial conditions. From (60b),Applying integration and iterations, and setting under zero-initial conditions, we getThen setting , , we obtain that which illustrates that condition (9) is satisfied. We conclude that the affine switched system (59), and hence closed-loop affine switched system (8), is FTB with H_∞ performance .

Remark 17. Unlike the normal switched system, the existence of multiple equilibria for affine switched systems is related to subsystems owing to the affine terms , which makes asymptotic and finite-time stability analysis nontrivial. As for the finite-time controller design, concerning with both the external disturbance and the affine terms , we redefined the conception of controller for affine switched system in Definition 4, based on which the results in this section are obtained. It is worth noting that the performance of affine switched system reduces to normal performance when assuming .

5. Numerical Examples

Example 1. Consider the affine switched system (2) with two modes of operation: is unstable, is Hurwitz stable, and eigenvalues , . Assuming desired reference , error tracking switched system will be

Evidently . Let , , , , , and . From the FTB condition (24), we get the average dwell time to ensure the finite-time boundedness with respect to , so that the switching signal can be chosen as a periodical signal with , which implies that and , during the finite-time interval . Given the initial error state , the conditions and are separately satisfied, then the error state trajectory of error affine switched system and the value of under the switching signal are shown in Figure 1.

It is easy to see in Figure 1 that subject system is FTB with conditions (12a), (12b), and (12c) satisfied. Moreover, assuming and using optimization process (24) and (31), the optimal value , , can be obtained. Then substituting into (12c) and (25c) separately, we getwhere , denote the minimum bound of state derived by Theorem 8 and Corollary 11. It is obvious that Corollary 11 is less conservative than Theorem 8 since .

Example 2. Keeping in view autonomous error switched system (10), we consider this system:with disturbance signal:which satisfies. Let , , , , and . From FTB condition (39c), we get the average dwell time to ensure FTB with respect to . Then for the finite-time performance, we should haveThe simulation results with initial error state are shown in Figure 2.

We observe in Figure 2 that the system is FTB, and the performance satisfies

Thus, according to Definition 4, the autonomous robust error switched system can be regarded as finite-time bounded. Moreover, using optimization procedure (56) we get , , . Putting in (39c), we get .

Example 3. Consider the affine error switched system (7) with two modes of operation:which implies that and .

The objective in this example is to design a set of robust controllers ensuring finite-time boundedness of closed-loop error switched system with respect to , where , , , , , , and . Setting switching signal as a periodical signal with , based on Theorem 16, we calculate

Then we can get the set of controllers for each subsystem asSubstitute controller gains into system (8), the closed-loop error switched system can be written asState responses under state-feedback controllers and switching signal are shown in Figure 3. We can observe that the closed-loop system is FTB, and the performance satisfiesThus, according to Definition 4, the given affine switched system can be regarded as finite-time bounded under designed controller gains.

6. Conclusion

In this paper, the problem of finite-time boundedness and finite-time control for affine switched systems has been investigated. Several definitions and sufficient conditions for FTB and performance are proposed. Based on the average dwell-time method, the FTB conditions of affine switched linear system with known state boundary are derived first in this investigation. To reduce the conservatism of FTB conditions, by classifying subsystems into asymptotically stable and unstable systems, we get the improved FTB conditions for affine switched system presented in Corollary 11. The conservatism of conditions under the two situations is compared. Then applying the finite-time boundedness analysis results, finite-time performance is discussed. Finite-time controllers are designed to ensure the corresponding closed-loop switched system FTB with performance. Numerical examples are finally provided to validate our theoretical results. Many real-world systems concern with finite-time and transient behavior; meanwhile, many engineering applications can be modeled as affine switched systems. Therefore, our theoretical results about finite-time boundedness of affine switched systems are supposed to have great potential in the application of practical switched systems. Furthermore, the proposed results in this paper can be extended to the nonlinear affine switched systems which will be considered in future work.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 51607151), Scientific Research Startup/Planning Project of Southwest Petroleum University (no. 2014QHZ027), and Young Scholars Development Fund of Southwest Petroleum University (no. 201599010110).

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Copyright © 2018 Lu Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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