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Zhen Ye 1 and Qing-Guo Wang 1 and Chong Lin 2 and Chang Chieh Hang 1 and Andrey E. Barabanov 3
Recommended by Bijoy K. Ghosh
1, Department of Electrical and Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260, Singapore
2, Institute of Complexity Science College of Automation Engineering, Qingdao University, Qingdao 266071, China
3, Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskij pr. 28, Petrodvoretz, St. Petersburg 198504, Russia
Received 18 October 2010; Revised 10 January 2011; Accepted 21 February 2011
1. Introduction
Relay feedback forms one important class of nonlinear systems and can cause complex nonlinear behaviors. Its early analysis can be traced to 1950-1960s, and afterwards, two basic approaches, the time domain approach [1, 2] and the frequency domain approach [3], emerged. Although they are almost identical, the frequency approach is more popular because of its ease of manipulation. However, as a general method for relay analysis, such a frequency approach also has some limits in itself. Firstly, only necessary conditions on the existence and stability of possible limit cycles are obtained. Secondly, the conditions are usually expressed as a summation of infinite items. It is certainly desirable to find both sufficient and necessary conditions of the existence and stability of limit cycles for relay feedback systems, as well as to give such conditions explicitly in terms of system parameters without any requirement on numerical computation.
In our recently published paper [4], we presented a complete relay analysis for a class of servo plants, G(s)=K/[s(s+a)] , (a>0 ), including the uniqueness of solutions, existence, and stability of limit cycles, and its amplitude and period, using a similar analysis to [5]. Naturally, it is desirable to investigate whether such an analysis can be extended to double integral plants, that is, the case of a=0 . Most people believe that double integral plants have no limit cycle under relay feedback [1, 3], but our analysis shows that this is not true. Actually, a limit cycle for double integral case is observable, which completely depends on the system initial conditions. Moreover, we also find that the analysis for the case of a>0 in [4] is not applicable to the case of a=0 . This is because for a>0 , the derivative of system output is stable and bounded as G[variant prime](s)=K/(s+a) . While for a=0 , G[variant prime](s)=K/s and the derivative of system output becomes unstable and not bounded anymore. Hence, the extension is not straightforward and the results for double integral plants are quite different from our previous ones.
This paper is organized as follows. Section 2 gives the main result followed by the proofs and a few remarks, where more details and explanations are given. Conclusions are drawn in Section 3.
2. The Result
Consider a double integral plant [figure omitted; refer to PDF] whose state-space representation in the controllable canonical form is given by [figure omitted; refer to PDF] [figure omitted; refer to PDF] where x(t)=[x1 (t),x2 (t)]T ∈...2 , y(t),u(t)∈... are the state, output, and input of the system, respectively. The plant is under the relay feedback control: [figure omitted; refer to PDF] where u+ and u- are the relay amplitudes, e(t)=-y(t) , [straight epsilon]+ and [straight epsilon]- are the relay hysteresis with [straight epsilon]- ≤[straight epsilon]+ . We assume u+ ≠u- since otherwise (4) becomes a constant but no longer a relay control. The relay control is depicted in Figure 1. The initial function for t=t0 is [figure omitted; refer to PDF] where t0 is the initial time and U:={u- ,u+ } . We call (1)-(5) a relay feedback system (RFS) which is depicted in Figure 2, where C(s) represents the controller.
Figure 1: Relay function.
[figure omitted; refer to PDF]
Figure 2: Relay feedback system.
[figure omitted; refer to PDF]
If the RFS generates a limit cycle, let T+ and A+ be the half period and the extreme value corresponding to u(t)=u+ , respectively, and T- and A- be the half period and the extreme value corresponding to u(t)=u- , respectively, as shown in Figure 3. Tr (resp., Td ) is the rising (resp., decreasing) period between the switching time, ti (resp., ti+1 ), when the limit cycle trajectory traverses ...AE;+ (resp., ...AE;- ) and the peak (resp., valley) time.
Figure 3: Limit cycles.
[figure omitted; refer to PDF]
We are now in a position to state the sufficient and necessary conditions for the existence of solutions, the existence and stability of limit cycles, and the amplitudes and periods of limit cycles.
Theorem 1.
Consider the RFS for double integral plant G(s) in (1).
(i) A unique solution exists for any initial condition.
(ii) A limit cycle exists if and only if Ku+ >0>Ku- , [varepsilon]+ =[varepsilon]- =[varepsilon] and any of the following holds:
(a) x1 (t0 )≠-[straight epsilon]/K ;
(b) x1 (t0 )=-[straight epsilon]/K but x[variant prime](t0 )≠0 .
If this is the case, the limit cycle is formed after the first switch, which is unique with two switchings per period.
(iii): If a limit cycle exists, it is globally marginally stable.
(iv) If a limit cycle exists, its amplitude and period are described by [figure omitted; refer to PDF]
Proof.
For a RFS, the input u(t) is a piecewise constant function. Without loss of generality, suppose [figure omitted; refer to PDF] as shown in Figure 4, where [figure omitted; refer to PDF] Define the switching planes: [figure omitted; refer to PDF] If the trajectory of x(t) traverses ...AE;+ (resp., ...AE;- ), that is, -Cx(ti )=[straight epsilon]+ (resp., -Cx(ti )=[straight epsilon]- ) at some instant t=ti >t0 with -Cx(ti- )<[straight epsilon]+ (resp., -Cx(ti- )>[straight epsilon]- ) and -Cx(ti+ )>[straight epsilon]+ (resp., -Cx(ti+ )<[straight epsilon]- ), then the instant t=ti is called a switching time. In particular, ti denotes the switching time when the i th switching takes place.
The state response of (2) to u(t) in (7) is given by [figure omitted; refer to PDF] where t∈[t0 ,∞) , t0 <t1 <... . It is easy to show that both x1 (t) and x2 (t) are continuous at switching time ti ,i=1,2,... , as t-ti =0 for t=ti . However, x...2 (t1- )=lim t[arrow right]t1-x...2 (t)=u- , while x...2 (t1+ )=lim t[arrow right]t1+x...2 (t)=x...2 (t1 )=u- +(u+ -u- )=u+ indicating x...2 (t) is not continuous at t=t1 due to u+ ≠u- by our assumption. Straightforwardly, this discontinuity persists for the general case of t=ti .
If considering the time between two consecutive switchings only, that is, t∈[ti ,ti+1 ) , (10) is simplified as [figure omitted; refer to PDF] where i=0,1,2,... , and [figure omitted; refer to PDF] Since e(t)=-y(t)=-Kx1 (t) and e[variant prime] (t)=-y[variant prime] (t)=-Kx2 (t) , it yields [figure omitted; refer to PDF] [figure omitted; refer to PDF]
It can be seen from (13) that e(ti ) , e[variant prime](ti ) and Kμ will affect behavior of e(t) and thus determine whether or not e(t) will reach the switching level of [varepsilon]+ or [straight epsilon]- . Without loss of generality, suppose the initial condition e(t0 )≤[straight epsilon]+ , thus u(t0 )=u- . To see how e(t) evolves with time, the following four cases are considered which are mutually exclusive and cover all possible cases.
Case 1.
e[variant prime] (t0 )≤0 and Ku- ≥0 . For t∈[t0 ,t1 ) , e[variant prime] (t)=e[variant prime] (t0 )-(t-t0 )Ku- ≤0 , which implies e(t)≤e(t0 )≤[straight epsilon]+ so that x(t) never traverses ...AE;+ for all t>t0 . The trajectory of x(t) is governed by (11) with ti =t0 and μ=u- .
Case 2.
e[variant prime] (t0 )≤0 and Ku- <0 . For t∈[t0 ,t1 ) , it follows from (14) that e[variant prime][variant prime] (t)=-Ku- >0 , which implies there always exists some t1 >t0 such that e(t1 )=[straight epsilon]+ and e[variant prime] (t1 )>0 , that is, x(t) traverses ...AE;+ at t=t1 . For t∈[t1 ,t2 ) , it follows from (14) that e[variant prime] (t)=e[variant prime] (t1 )-(t-ti )Ku+ >0 if Ku+ ≤0 , which implies e(t)>e(t1 )=[straight epsilon]+ ≥[straight epsilon]- so that x(t) never traverses ...AE;- for all t>t1 . The trajectory of x(t) is governed by (13) and (14) with ti =t1 and μ=u+ . If Ku+ >0 , it follows from (14) that e[variant prime][variant prime] (t)=-Ku+ <0 , which implies there always exists some t2 >t1 with e(t2 )=[straight epsilon]- and e[variant prime] (t2 )<0 so that x(t) traverses ...AE;- at t=t2 . Afterwards, with the same analysis as above, it is straightforward to verify that x(t) will traverse ...AE;+ and ...AE;- alternatively and consecutively.
Case 3.
e[variant prime] (t0 )>0 and Ku- <0 . For t∈[t0 ,t1 ) , it follows from (14) that e[variant prime] (t)=e[variant prime] (ti )-(t-ti )Ku- >0 , which implies there always exists t1 >t0 such that e(t1 )=[varepsilon]+ and e[variant prime](t1 )>0 , that is, x(t) will traverse ...AE;+ at t=t1 . From t=t1 onwards, following the same analysis in Case 2, if Ku+ ≤0 , x(t) never traverses ...AE;- for all t>t1 , the trajectory of x(t) is governed by (13) and (14) with ti =t1 and μ=u+ . Otherwise, if Ku+ >0 , x(t) will traverse ...AE;- and ...AE;+ alternatively and consecutively.
Case 4.
e[variant prime] (t0 )>0 and Ku- ≥0 . For t∈[t0 ,t1 ) , it follows from (14) that e[variant prime][variant prime] (t)=-Ku- ≤0 . If e[variant prime][variant prime] (t)<0 , there always exists t0m such that e(t)≤e(t0m ) for all t≥t0 . If e(t0m )≤[straight epsilon]+ , then e(t)≤e(tm )≤[straight epsilon]+ so that x(t) never traverses ...AE;+ for t≥t0 . The trajectory of x(t) is governed by (13) and (14) with ti =t0 and μ=u- . On the contrary, if e(tm )>[varepsilon]+ , there always exists t1 satisfying t0 <t1 <t0m such that e(t1 )=[straight epsilon]+ and e[variant prime] (t1 )>0 , that is, x(t) traverses ...AE;+ at t=t1 . If e[variant prime][variant prime] (t)=0 , then e[variant prime] (t)=e[variant prime] (t0 )>0 , which also implies x(t) traverses ...AE;+ at some t=t1 . For t∈[t1 ,t2 ) , it follows from (14) that e[variant prime] (t)=e[variant prime] (t1 )-(t-ti )Ku+ >0 if Ku+ ≤0 , which implies e(t)>e(t1 )=[straight epsilon]+ ≥[straight epsilon]- so that x(t) never traverses ...AE;- for all t>t1 . The trajectory of x(t) is governed by (13) and (14) with ti =t1 and μ=u+ . If Ku+ >0 , it follows from (14) that e[variant prime][variant prime] (t)=-Ku+ <0 , which implies there always exists some t2 >t1 such that e(t2 )=[straight epsilon]- and e[variant prime] (t2 )<0 , that is, x(t) traverses ...AE;- at t=t2 . For t∈[t2 ,t3 ) , it follows from (14) that e[variant prime] (t)=e[variant prime] (t2 )-Ku- <0 , which implies x(t) never traverses ...AE;+ again for all t>t2 . The trajectory of x(t) is governed by (14) with ti =t2 and μ=u- .
In view of the above analysis, the solution to (2) always exists. This completes (i).
We can also see from the analysis in (i) that consecutive switchings between ...AE;+ and ...AE;- take place if and only if Ku+ >0>Ku- . Under this condition, let ti be the switching instant of x(t) on the plane ...AE;+ , that is, e(ti )=[straight epsilon]+ and e(t)<[straight epsilon]+ , u(t)=u- for sufficiently close t<ti . This implies e[variant prime] (ti )≥0 . The case e[variant prime] (ti )=0 can occur only when [straight epsilon]- =[straight epsilon]+ and then ([straight epsilon]+ ,0) is a fixed point of x(t) . But a fixed point is unique. Hence, no limit cycle exists in this case.Assume e[variant prime] (ti )>0 , then u(t) switches from u- to u+ at t=ti . In Case 2, we have shown that e[variant prime] (t) decreases and becomes negative until x(t) traverses ...AE;- at t=ti+1 . Consider the Poincare mapping [figure omitted; refer to PDF] The function [varphi]+ is defined on the ray [0,+∞) and takes values in the ray (-∞,0] . Let χ=e[variant prime] (ti )>0 and τ=ti+1 -ti . By definition, e(ti )=[straight epsilon]+ and e(ti+1 )=[straight epsilon]- . Since both e(t) and e[variant prime] (t) are continuous at t=ti+1 , it follows from (13) and (14), respectively, by taking t=ti+1 that [figure omitted; refer to PDF] [figure omitted; refer to PDF] It follows from (17) that [figure omitted; refer to PDF] Substituting (18) into (16) yields [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] If [straight epsilon]+ >[straight epsilon]- , ">" holds in (20) so that the Poincare mapping (15) is not a contraction and no limit cycles exist. If [straight epsilon]- =[straight epsilon]+ =[straight epsilon] , "= " holds in (20), which implies a limit cycle is formed after the first switching. As e[variant prime] (ti )>0 , x[variant prime] (t1 )≠0 must be satisfied so that (a) and (b) can be easily derived with the help of (11). This completes (ii).
Once a limit cycle is formed, if e[variant prime] (ti ) is changed due to some disturbance, x(t) cannot remain at the original limit cycle, but form a new one with different amplitude and period, which means the limit cycle is marginally stable. Since the initial condition x(t0 ) is arbitrarily chosen, the limit cycle is globally marginally stable. This completes (iii).
It has been shown in (ii) that for Ku+ >0>Ku- , a limit cycle always exists after the first switching. For t∈[t1 ,t2 ) , it follows from (14) and with the help of (20) that [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Let e[variant prime] (t)=e[variant prime] (t1 )-(t-t1 )Ku+ =0 , then t-t1 =e[variant prime] (t1 )/(Ku+ ) . Substituting it into (13) yields [figure omitted; refer to PDF] In the similar way, we have [figure omitted; refer to PDF] [figure omitted; refer to PDF] For t∈[t0 ,t1 ) , it follows from (13) that [figure omitted; refer to PDF] which gives [figure omitted; refer to PDF] where negative root is ignored since e[variant prime] (t1 )>0 . Substituting (27) into (14) yields [figure omitted; refer to PDF] Substituting (28) into (22)-(24), respectively, yields [figure omitted; refer to PDF] which completes (iv).
Figure 4: A piecewise constant input.
[figure omitted; refer to PDF]
Remark 1.
In our previous relay analysis for a class of servo plant as G(s)=K/(s(s+a)) , (a>0 ), we proved the existence of a limit cycle by showing the Poincare map [varphi]+ is a strict contraction, that is, |[varphi]+[variant prime] (χ)|<1 for all χ∈(0,+∞) . Here for the case of a=0 , differentiating both sides of (16) yields [figure omitted; refer to PDF] Differentiating both sides of (16) and using the help of (20) yield [figure omitted; refer to PDF] If [straight epsilon]+ ≠[straight epsilon]- , |[varphi]+[variant prime] (χ)|<1 for all χ∈(0,+∞) , but no limit cycle exists in this case as we show. It seems to be contradiction there and the reason is that for a=0 , [varphi]+ (χ) is not bounded any more. This also shows the previous results for a>0 cannot be simply extended to the double integral case here by just letting a[arrow right]0 .
Remark 2.
The limit cycle forms after the first switching but is marginally stable. This means the trajectory will not remain at the original limit cycle even if a very small disturbance is introduced. Nevertheless, it will form a new limit cycle with different amplitude and period immediately after the next switching.
Remark 3.
The limit cycle is a parabolic curve with the symmetry of x[variant prime] (ti )=-x[variant prime] (ti+1 ) , where ti is the switching time for i=1,2,... . Its amplitude and period are only decided by the initial conditions, which is also consistent with the result from [6].
Example 1.
Consider a double integral plant G(s)=1/s2 under the relay feedback with [straight epsilon]=1 , u+ =1.0 and u- =-0.8 . The initial conditions are assumed to be x1 (t0 )=0.5 and x2 (t0 )=1 . The simulated response of the system is shown in Figure 5. By Theorem 1(iv), the amplitude and period of the limit cycle occurred are also calculated. The comparison of the results from the theory and the simulation as shown in Table 1, where one can see the error, is quite minor.
Table 1: Limit cycle characteristics of Example 1.
Method | A+ | A- | T+ | T- |
By Theorem 1 | 1.7000 | 2.1250 | 3.6878 | 4.6098 |
By simulation | 1.7002 | 2.1255 | 3.6879 | 4.6100 |
| ||||
Error | -0.0118% | -0.0235% | -0.0027% | -0.0043% |
Figure 5: A piecewise constant input.
[figure omitted; refer to PDF]
3. Conclusion
In this paper, a double integral plants under relay feedback is addressed. Complete results have been established on uniqueness of solutions, existence, and stability of limit cycles and its amplitude and period. Comparing with the plant G(s)=K/[s(s+a)] , a>0 in [4], analytical expressions for the amplitude and period of limit cycles with terms of system parameters are available, but the conditions on the existence of limit cycles are quite different. Reasons for such differences as well as some explanations are provided.
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[2] J. K-C. Chung, D. P. Atherton, "The determination of periodic modesin relay systems using the state space approach," International Journal of Control , vol. 4, pp. 105-126, 1966.
[3] J. A. Tsypkin Theorie der relais systeme der automatischen regelung , R. Oldenbourg, Munich, Germany, 1958.
[4] Z. Ye, Q. G. Wang, C. Lin, C. C. Hang, A. E. Barabanov, "Relay feedback analysis for a class of servo plants," Journal of Mathematical Analysis and Applications , vol. 334, no. 1, pp. 28-42, 2007.
[5] C. Lin, Q. G. Wang, T. H. Lee, "Relay feedback: a complete analysis for first-order systems," Industrial and Engineering Chemistry Research , vol. 43, no. 26, pp. 8400-8402, 2004.
[6] K. H. Johansson, A. Rantzer, K. J. Astrom, "Fast switches in relay feedback systems," Automatica , vol. 35, no. 4, pp. 539-552, 1999.
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Copyright © 2011 Zhen Ye et al. Zhen Ye 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.
Abstract
Double integral plants under relay feedback are studied. Complete results on the uniqueness of solutions, existence, and stability of the limit cycles are established using the point transformation method. Analytical expressions are also given for determining the amplitude and period of a limit cycle from the plant parameters.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer