Some Problems of Feedback Control Strategies and It’s Treatment

This paper extends and improves the feedback control strategies. In detailed, the ordinary feedback, dislocated feedback, speed feedback and enhancing feedback control for a several dynamical systems are discussed here. It is noticed that there some problems by these strategies. For this reason, this Letter proposes a novel approach for treating these problems. The results obtained in this paper show that the strategies with positive feedback coefficients can be controlled in two cases and failed in another two cases. Theoretical and numerical simulations are given to illustrate and verify the results.


Introduction
Chaos control, an important topic in nonlinear dynamical science (Dou, Sun, Duan, & Lü, 2009;Tao & Yang, 2008;Yang, Tao & Wang, 2010) and is one of the main features of chaos applied in practical engineering (Yassen, 2003;Zhu, 2009;Zhu & Chen, 2008).and its play a very important role in the study of dynamical systems and has great significance in the application of chaos.Since Ott et al,1990, first introduced in the notation of chaos control (AL-Azzawi, 2012;Pang & Liu 2011).Various kinds of control schemes and techniques such as OGY method, time-delay feedback, Lyapunov method, impulsive control, sliding method control, differential geometric,  H control, adaptive control, chaos suppression method, and so on have been successfully applied to achieve chaos control (Aziz & AL-Azzawi 2016;Tao, Yang, Luo, Xiong & Hu 2005).Among them, the feedback control is especially attractive and has been commonly applied to practical implementation due to its simplicity in configuration and implementation (Tao & Yang, 2008).Generally speaking, there are two main approaches for controlling chaos: feedback control and non-feedback control.The feedback control approach offers many advantages such as robustness and computational complexity over the non-feedback control method (Yang, Tao, & Wang, 2010;Zhu & Chen 2008).
On the other hand, there are some problems with this approach and one of these problems is that we get a negative feedback coefficient and the second problem is that when the feedback coefficient vanishes, Consequently, these strategies are failing.However, in a feedback control strategies the necessary condition for suppressing the dynamical system is that the feedback coefficient must be positive.Most of the previous work on chaos control was mostly focused on classical dynamical systems under this condition such as the Liu system (Dou, Sun, Duan, & Lü, 2009;Zhu & Chen 2008), L̈ system (Pang & Liu 2011), Unified system (Tao, Yang, Luo, Xiong, & Hu 2005), Lorenz system (Zhu, 2010), Chen system (Yan, 2005), and et al.But in works (Aziz, & AL-Azzawi 2015;Zhu, 2010) founded some cases it can't be controlled, although with positive feedback coefficients.This paper answer this equation and it focused on these problems and we suggest a new method which includes four cases: two cases have at least two positive feedback coefficients and other cases have only one positive feedback coefficient.Finally, we found the system can be controlled if there is an intersection between these coefficients and they can't be controlled if it there is not for the first two cases.
Briefly, this study presents three fundamental questions.First, when can we get a positive feedback coefficient and can suppress a system?Second, when can we get a positive feedback coefficient and can't suppress the system?And third, how can we distinguish between these two cases?This paper begins with the suggestion of a new method that will answer these questions.

Problem Formulation and Our Methodology Using Feedback Control Strategies
In this section, we describe the problem formulation for the chaos control for dynamical systems and our methodology using feedback control strategies.
Let us consider the dynamical system in the following form:
As we know, there are four standard kinds of feedback control techniques: ordinary, dislocated, speed and enhancing feedback control.and the definition of each kind as: Definition 1 Ordinary feedback control The system's variable is often multiplied by a coefficient as the feedback gain, and the feedback gain is added to the right -hand of the corresponding equation (Aziz & AL-Azzawi, 2015;Zhuang, 2012;Zhuang & Chai, 2012).
Definition 2 Dislocated feedback control If a system variable multiplied by a coefficient and its added to the right -hand of another equation, this strategy is called a dislocated feedback control (Zhuang & Chai, 2012;Pang & Liu 2011;Tao & Yang, 2008;Zhu, 2010).

Definition 3 Speed feedback control
The independent variable of a system function is often multiplied by a coefficient as the feedback gain, so the method is called displacement feedback control.Similarly, if the derivative of an independent variable is multiplied by a coefficient as the feedback gain (Tao, Yang, Luo, Xiong & Hu 2005;Zhuang, 2012;Yan, 2005;Zhu & Chen 2008).

Definition 4 Enhancing feedback control
It is difficult for a complex system to be controlled by only one feedback variable, and in such cases the feedback gain is always very large.So we consider using multiple variables multiplied by a proper coefficient as the feedback gain.This method is called enhanced feedback control (Aziz, & AL-Azzawi, 2015;Zhuang, 2012;Dou, Sun, Duan, & Lü, 2009;Zhu, & Chen, 2008;Pang & Liu 2011).
Obviously, from the above definitions, the three first feedback control strategies take just a single controller while in enhancing feedback control includes more than one feedback controller.Consequently, the controller  can represent for each kind as: where k is called a feedback coefficient, and  > 0. when substitute one of these formulations in Eq. 2 and in order to find the feedback coefficients, we can use the following formula: ̇=  +  (4) i.e. deals with linear terms only of this strategy, but with other methods such as nonlinear feedback method, adaptive and active techniques etc. deals with linear and nonlinear terms.
At some time, by this strategy we got more than one positive feedback coefficient based on Routh-Hurwitz method and k is interval, in order to select a suitable feedback coefficient we used the following formulation (5) According to above these strategies, the system can control it if we have a positive feedback coefficient.This condition is necessary and sufficient for the controlled hyperchaotic systems.But in some cases, we establish that the system can't be controlled, although satisfied this condition.In order to overcome this problem and this weakness, we improve and extend these strategies by the following a novel approach.
Novel approach: The control problem by applying feedback control strategies for dynamical systems has posed four cases with a positive feedback coefficient, these cases as the following: 2. No control if it has more than one positive feedback coefficients and there are not an intersection between them.
3. Control it if it has only one positive feedback coefficient and satisfied all Routh-Hurwitz conditions.4. No control if it has only one positive feedback coefficient and it does not satisfy any one of Routh-Hurwitz conditions.

Applications
In this section, we take several four-dimensional hyperchaotic systems, for example to show how to use the results obtained in this paper to analyze the controlling a class of hyperchaotic systems.
Therefore, we find two positive feedback coefficients by this scheme.But, which feedback coefficient that effective in system?After testing, we noticed that both of a positive coefficients can't be effective and active on the system (6), Fig. 1 with  = 0.3 and Fig. 2 with  = 12.5 explain this result numerically.For this reason, we use our new approach to knowledge the cause of this problem.
Let us the two positive feedback coefficients as  1 < 0.3571 ,  2 > 12.1139.Now, based on formulation (5) and a novel approach then  =  1 ∩  2 = (0, 0.3571) ∩ (12.1139, ∞) = ∅ Hence, there is no intersection between them, according to a second case of a new approach.So, this strategy fails to control the system (6).Thus the proof is complete.
Remark1.We can apply theoretical and numerical methods for of a positive feedback coefficient.
Proof.According to the formulation (4), system (6) with this control is then the characteristic equation is According to the Routh-Hurwitz method, system (10) has only one positive feedback coefficient as  < 1.To test this coefficient theoretically, let substitute the value of feedback coefficients  = 1 (critical value) in above Equation we get the following Equation as 12) and the roots of Equation ( 12) are λ 1 = 0 , λ 2,3 = −5 ± 15.3297 and λ 4 = − 8 3 ⁄ , Consequently, the roots of Eq (12) don't contain a simple pair imaginary, So, we don't get a Hopf bifurcation, therefore, the feedback coefficients is not effective in a system (6).Also Fig. 3 justifies this result numerical with positive feedback coefficient  = 0.5.In order to discover the main causes for this problem, we used new approach.Obviously, one condition of Routh-Hurwitz method is not satisfied (third condition of Routh-Hurwitz), Therefore, this strategy is failing to control system (6) according to the four case of the new approach, the proof is complete.
Remark 2. In the context of ordinary differential equations ODEs the word "bifurcation" has come to mean any marked change in the structure of the orbits of a system (usually nonlinear) as a parameter passes through a critical value [1], Bifurcation refers to qualitative changes in the solution structure of dynamical systems with slight variation in system parameters as well as one conditions of bifurcation that dynamical system has a simple pair of pure imaginary eigenvalues and no other eigenvalues with zero real parts.
Proof.System (6) with this control can be reformulated in the following form: then the characteristic equation is Based on Routh-Hurwitz method, we Obvious ,  > 0 , from the condition  >  , we get inequality form 100 − 2800 > 0 imply that  > 28 and Fig. 4 show the convergence to equilibrium point with  = 29.Consequently, this strategy is success to control the system (6).Therefore, satisfied the three case of a new approach, the proof is complete.
In the light of this study, the necessary and sufficient condition for the control hyperchaotic systems to be asymptotically stable is we get a positive feedback coefficient.But in theorem 1 and theorem 2 we get a positive feedback coefficient and can't control it.And a new approach given causes for each case.These cases are one problem of feedback control strategies.
In adding, there are other problems when that vanished the feedback coefficient as the following dislocated feedback controller: Consequently, this strategy has failed to control the system (6).According to the above discussion, we found some problems and weakness of feedback control strategies.In table 1, we can briefly describe all cases that we have a positive feedback coefficient via these strategies.
In order to focus on the goal and the main idea of this search will be shortened and we are going to reduce some mathematical steps.
In addition, if we make simple changes into above control  = , 1 , 0,0,0- i.e.  1 = −,then the system (18) can't asymptotically converge to the unstable equilibrium although we have one a positive feedback s.t. < 10 and the characteristic equation is because one condition of Routh-Hurwitz is not satisfied.Consequently, this a positive feedback coefficient is not active and the system can't be controlled by this strategy based on four case of a novel approach ,The proof is now complete.
Similarly, if we make another simple change in to above control based on the speed feedback control  = , 1 , 0,0,0- ,i.e. 1 = −̇ ,then the system (18) can't be controlled, although we have two positive feedbacks such that  1 < 2.2 and  2 > 2.2355 .where the characteristic equation is and according to the novel approach, second case, there are no intersection between these two positive feedback coefficients Consequently, the system can't be controlled by this strategy.
Also, if we take the control based on enhancing feedback as  = , 1 , 0,  3 , 0- i.e.  1 = − and  3 = −, then the system (18) can be control with single a positive feedback s.t. < 270.1778 according to the third case of a novel approach
On the other hand, if design control based on dislocated feedback control as  = , 1 , 0,0,0- , i.e.  1 = −, then the zero solution of the hyperchaotic system(23) can't be control when  1 < 10 and  2 > 10.where the characteristic equation is The cause in this case, by basing novel approach, second case, we find there are no intersection between these two positive feedbacks  =  1 ∩  2 = (0,10) ∩ (10, ∞)  =  1 ∩  2 ≠ ∅ Therefore the system (23) can't be controlled by this strategy.The proof is now complete.
In addition, if we make a simple change in the dislocated feedback control such that  = ,0,  2 , 0,0- i.e.  2 = − ,then the system (23) can't asymptotically converge to the unstable equilibrium although we have one a positive feedback s.t. < 10.6.and the characteristic equation is because one condition of Routh-Hurwitz is not satisfied.Consequently, this a positive feedback coefficient is not active on a system, So this strategy failed to control this system based on four case of a novel approach, The proof is now complete.
Finally, if design control based on enhancing feedback control as  = −,, , 0, - , then system ( 23) can be control it with three positive feedback ( 1 > 7.8576), ( 2 > 15.4484) and ( 3 > 0.2668), where the characteristic equation is And the active value of positive feedback calculated based on formulation (4) as Hence, there are intersection between three positive feedback coefficients, So, this strategy satisfy fist case of a novel approach.Therefore the system can be controlled by this strategy.The proof is now complete.
Remark 3. Most of previous works used the following formulation  = *  + to find the a positive, active feedback coefficient when we get more than one positive, but this formulation failed in some time and theorem 5 with control  = ,−, 0,0,0- is example, in this case, So prefer using the formulation (5) always.

Conclusions
The results obtained in this paper show that the strategies with positive feedback coefficients can be controlled in two cases and failed in another two cases, which are performed in the above theorems.
Also we consider the weakness of feedback control strategies, when we get positive feedback coefficients and can't suppress the system.This is a try to understand this problem, when it happen and what is that reason?So, we suggest a new method which includes answers to all these questions.Explain this method, theoretical, numerical and justify the results.Finally, we can apply this method for another chaotic and hyper chaotic systems to check the control of these systems.