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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 735623, 10 pages
Limit 2-Cycles for a Discrete-Time Bang-Bang Control Model
1Department of Mathematics, Yanbian University, Yanji 133002, China
2Department of Mathematics, Tsing Hua University, Taiwan 30043, Taiwan
Received 3 August 2012; Revised 19 September 2012; Accepted 24 September 2012
Academic Editor: Raghib Abu-Saris
Copyright © 2012 Chengmin Hou and Sui Sun Cheng. 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.
A discrete-time periodic model with bang-bang feedback control is investigated. It is shown that each solution tends to one of four different types of limit 2-cycles. Furthermore, the accompanying initial regions for each type of solutions can be determined. When a threshold parameter is introduced in the bang-bang function, our results form a complete bifurcation analysis of our control model. Hence, our model can be used in the design of a control system where the state variable fluctuates between two state values with decaying perturbation.
Discrete-time control systems of the form with and , are of great importance in engineering (see, e.g., any text books on discrete-time signals and systems).
Indeed, such a system consists of a linear part which is easily produced by design and a nonlinear part which allows nonlinear feedback controls of the form commonly seen in engineering designs.
In a commonly seen situation, and belong to , while takes on two fixed values (on-off values) depending on whether the state variable is above or below a certain value (as commonly seen in thermostat control). In some cases, it is desirable to see that the state value fluctuates between two fixed values with decaying perturbations as time goes by (an example will be provided at the end of this paper). Here, the important question is whether we can design such a control system that fulfils our objectives.
In this note, we will show that a very simple feedback system of the form can achieve such a goal provided that:
(i) we take to be -periodic sequences with ,
(ii) while the control function is taken to be the step (activation) or bang bang function  defined by where may be regarded as a threshold parameter.
Remarks: (i) Note that in case , our function is reduced to the well-known Heaviside function These bang bang controllers are indeed used in daily control mechanisms; for example, a water heater that maintains desired temperature by turning the applied power on and off based on temperature feedback is an example application.
(ii) As for the sequences and , we have assumed that they are periodic with a prime period . We could have considered more general periodic sequences since a large number of environmental parameters are generated in periodic manners, and such structural nature should be reflected in the choice of our sequences. However, in the early stage of our study, it is quite reasonable to assume that they have a common prime period (instead of various prime periods).
(iii) Finally, we have selected . A simple reason is that without the feedback control and forcing sequence , our system is a stable one and which can easily be realized in practice.
(iv) Equation (1.3) has a second-order delay in the open loop part and a first-order delay in the control function. It may equally be well to choose a system that has a first-order delay in the open loop part and a second-order delay in the control function. Such a model will be handled in another paper.
(v) The simple prototype studied here is representative of a much wider class of discrete-time periodic systems with piecewise constant feedback controls [2–10], and hence we hope that our results will lead to much more general ones for complex systems involving such discontinuous controls.
Clearly, given any initial state value pair in , we can generate through (1.3) a unique real sequence . Such a (state) sequence is called a solution of (1.3) originated from . What is interesting is that by elementary analysis, we can show that for any value of the threshold parameter , there are at most four possible types of limiting behaviors for solutions of (1.3), and we can determine exactly the range of the parameter values and the exact “initial region” from which each type of solutions originates from (see the concluding section for more details).
To this end, we first note that by the transformation , (1.3) is equivalent to where . Next, by means of the identification and for , we note further that (1.6) is equivalent to the following two-dimensional autonomous dynamical system: which is a special case of the system (1.1). By such a transformation, we are then considering the subsequences and consisting of even and odd terms of the solution sequence of (1.6). Therefore, all the asymptotic properties of (1.6) can be obtained from those of (1.7).
To study the asymptotic properties of (1.7), we first note that its solution is of the form where is now a point in the real plane. By considering all possible initial data pairs , we will be able to show that every solution of (1.7) tends to one of four vectors. To describe these four vectors, we set Then, the four vectors are and since , we see that and and hence they form the corners of a rectangle.
Depending on the relative location of the origin with respect to this rectangle, we may then distinguish eleven exhaustive (but not mutually distinct; see Section 3 in the following) cases:(i), (ii), (iii), (iv); (v), (vi); (vii), (viii); (ix), (x); (xi).
For each case, we intend to show that solutions of (1.7) originated from different parts of the plane will tend to one of the four vectors in (1.9). To facilitate description of the various parts of the plane, we introduce the following notations:
2. Main Results
Cases (i), (ii), (iii), and (iv)
First of all, the first four cases , and are relatively easy. Indeed, suppose . Let be a solution of (1.7). By (1.7), . Then, Therefore, there exists an such that for all . By (1.7) again, and for . Then, imply In summary, suppose and suppose , then the solution originated from will tend to . We record this result as the first data row in Table 1.
By symmetric arguments, the second data row is also correct. To see the validity of the third data row, we first note that if and only if or . If holds, then by (1.7), for all . Hence, . Therefore, there exists an such that for . Thus, for . Then . Therefore, there exists an such that for all . Then, by (1.7) again, for all , and hence . The case where is similarly proved. Finally, the fourth data row is established by arguments symmetric to those for the third row.
Next, we assume that . Since for , then for . We see that , and for . Therefore, for . Furthermore, if , then , and if , then . We need to consider three cases: (i) , (ii) , and (iii) . By arguments similar to those used in the derivation of Table 1, we may derive Table 2.
For instance, suppose . Let . Then, by (1.7), we have , and by induction, we may easily see that for all . Thus, , and hence . As another example, let , then for some . By (1.7) and induction, we may easily see that . Our conclusion comes from the previous case. As a further example, let , where , then by (1.7) and induction, we may easily see that . Our conclusion now follows from the fourth data row.
This case is a dual of the Case (v). Indeed, assume that . Then, and for . Thus, for . Furthermore, if , then , and if , then . We need to consider three cases: (i) , (ii) , and (iii) . By arguments similar to those in the previous case, we may obtain the asymptotic behaviors of (1.7) summarized in Table 3.
We remark that the different Cases (i)–(xi) discussed above may not be mutually distinct. For instance, the conditions and are not mutually exclusive. However, the corresponding conclusions in Table 1 show that they are compatible (and hence should not cause any problem).
Next, we turn our attention to our original equation (1.3). By , we may see that Therefore, the results in the previous section for the system (1.7) can easily be translated into results for (1.3). For instance, by Table 1, we may see that when , a solution with will satisfy As another example, the condition is equivalent to . Let be a solution of (1.3) with . Then, by Table 2, we may see that
By arguments similar to those just described, the corresponding asymptotic behaviors of solutions of (1.3) can be summarized as follow:(i)if , then (ii)if , then (iii)if , then (iv)if , then (v)if , then
We remark that the precise initial regions of each type of solutions in the above statements can be inferred from our previous tables. Such repetitions, however, need not to be spelled out in detail for obvious reasons. Instead, based on the statements made above, it is more important to point out that our original motivation can be fulfilled.(i)Equation (1.3) possesses exactly four -periodic solutions with . Every other solution tends to one of these four solutions "according to the information given in the previous section."
As an example, consider a plant which is supposed to produce a type of products with capacity , where now denotes economic stages. Suppose that the stages reflect booms and busts experienced by an economy characterized by alternating periods of economic growth and contraction. Then, during busts, the plant should be managed in a fashion so as to produce at low capacity and during booms at high capacity. Suppose that it is estimated that unit capacity is demanded during busts and unit capacity during booms. Then, an automated plant of the form (1.7) may be built to fit the estimated demands: where the “structural” parameters are chosen since they, as may be checked easily, guarantee that the capacities and will tend to and , respectively. In fact, for all , by (1.7), we have , and for. Thus, . Therefore, there is such that for. Then, We get and .
Project supported by the National Natural Science Foundation of China (11161049). Mathematics Subject Classifications: 39A11, 39A23, 92B20.
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