Exact eigenvalue assignment of linear scalar systems with single delay using Lambert W function
Huang-Nan Huang Chew Chun Yong



1]Huang-Nan Huang
2]Chew Chun Yong

[1]Department of Applied Mathematics, Tunghai University, Taichung, Taiwan (E-mail: nhuang@thu.edu.tw)
[2]Department of Mathematical & Actuarial Sciences, University Tunku Abdul Rahman, Bandar Sungai Long, Selangor, Malaysia (E-mail: chewcy@utar.edu.my)




Exact eigenvalue assignment of linear scalar systems with single delay using Lambert W function
    [
    March 15, 2018
===============================================================================================


Eigenvalue assignment problem of a linear scalar system with a single discrete delay is analytically and exactly solved. The existence condition of the desired eigenvalue is established when the current and delay states are present in the feedback loop. Design of the feedback controller is then followed. Furthermore, eigenvalue assignment for the input-delay system is also obtained as well. Numerical examples illustrate the procedure of assigning the desired eigenvalue.

1em 2010 AMS Classification:
39B82, 93B55
1em Keywords and phrases:
Lambert W, time-delay, eigenvalue assignment








§ INTRODUCTION

During recent decades, the stabilization and control of linear systems with delays are extensively studied, for example, the spectrum (eigenvalues) assignment for linear delay systems in 1978 by Olbrot <cit.>. Recently, an approach for the solution of linear time-delay system is based on the Lambert W function proposed by Asl and Ulsoy <cit.>. Hence the robust stability as well as related topics to design the feedback controller are well established <cit.> and reference therein. A good introduction about the Lambert W function is given by Coreless et. al. <cit.> and this function possesses many applications within these two decades.

Eigenvalue assignment for delay systems with single delay via Lambert W function is first developed by Yi et. al. <cit.> to assign the rightmost eigenvalue of the delay system to a predefined (desired) location in order to stabilize the system, but unfortunately only a real or real part of the rightmost eigenvalue can be assigned for the scalar case. Alternatively, Shinozaki <cit.> discussed the assignment of the complex eigenvalue to the largest one of a scalar single delay system with the output of a complex feedback gain which is not realistic. Later an analytic eigenvalue assignment method is proposed for scalar and some special delay systems <cit.>. All these studies design the controller by feedback only the current state and no condition is drawn on the value of desire eigenvalue such that the feedback controller always exists. On the other hand, although a more general time-delay system can be analyzed by using matrix Lambert W function, but the rightmost eigenvalue can be computed not by using the principal branch which contradicts the main proposition of this method <cit.>.
 
Avoiding using the matrix Lambert W function, only a scalar system with a single delay is considered in this paper. It mainly focuses on deriving the existence condition of the feedback controller related to assign the rightmost eigenvalue of the system to the desired value. The formula to compute feedback gains for the current and delay states is then obtained. Furthermore, an input-delay type system is also considered. Two examples are provided to demonstrate the associated idea. The result consolidate further studies on the eigenvalue assignment of linear systems with multiple delays via the Lambert W function approach.





§ LAMBERT W FUNCTIONS




The Lambert W function is defined as a complex multivalued function which has infinite number of branches, W_k(x), where k=0, ± 1,  
± 2,⋯,±∞,
(regard W_∞ and W_-∞ as fixed mappings), such that


    W_k(z) e^W_k(z) =z,   z∈ℂ.


For any x∈ℝ, when -1/e≤ z=x < 0, the principal branch W_0(x) satisfies W_0(x)≥-1 and the -1 branch  W_-1(x) satisfies W_-1(x)≤ 
-1. 
By partitioning the z-plane with horizontal boundaries z=(2k+1)π+i 0  for k∈ℤ (where i=√(-1)), the ranges of branches of W_k(z) are images of the z between branch cuts in the z-plane. W_0 has a branch cut linking to W_1 and W_-1 which is defined as BC={x+i 0| x∈(-∞,-1/e)}, i.e., W_0(BC) is the boundary between ranges of W_0 and W_1 and so as W_-1(BC) for W_-1 and W_0. The range of branches as well as its real counterpart of this function are shown in Figure <ref>. 




An important property of the Lambert W function is given by <cit.>

	
	Let z∈ℂ. Then  W_k(z) = W_-k(z̅)
		and max_k=0, ±1,±2,⋯,±∞[ W_k(z)]=[ W_0(z)].




§ MAIN RESULT


We here consider a scalar delay system with an exogenous input from environment: 

    ẋ (t) =a x(t)+ a_1d x(t-h)+b u(t),   h>0,
        x(0)  = x_0,      x( τ ) =ϕ(τ),   -h≤τ<0,
   
where a, a_1d, b≠ 0, x_0, h∈ℝ, and ϕ is the initial function
to specify  x(τ), τ∈[-h,0).

Suppose a proportional control is proposed to stabilize the system with feedback of current and delay states:


    u(t)=k x(t) +k_1d x(t-h)


with two parameters k,  k_1d∈ℝ to be designed.
The closed-loop system is then described by

    ẋ =(a+bk) x(t)+ (a_1d+b k_1d) x(t-h)  ≜α x(t) + β x(t-h)

where α=a+b k and β=a_1d+b k_1d which are both real numbers. The characteristic equation of the closed-loop given by <cit.>:


    s-α=β e^-s h,
        
        s-(a+bk) = (a_1d+b k_1d) e^-s h


whose roots (also known as eigenvalue of the system) are
expressed by using Lambert W function


    s_k=α+1/hW_k(β h e^-α h),
          k=0, ± 1, ± 2,…,±∞.


From Lemma <ref>, the system
(<ref>)
is stable iff the real part of the rightmost eigenvalue, s_0, is negative.


The control design about a system with only input delay Whose equation is given by

    ẋ =a x(t) + b u(t-h),   x(0)  = x_0,  h>0,

is much more a demanding challenge since the corresponding input-output operator is not a compact one. 
Applying the state feedback controller
u(t) =k x(t), then the associated closed-loop system becomes
ẋ = a x(t)+b k x(t-h)
which is also of the form (<ref>) with α=a and β = b k.
Thus, we only need to focus on the system (<ref>).


How to assign the rightmost  eigenvalue s_0 of the closed-loop system to a desired location?  
Suppose the desired location is denoted by S_0,des∈ℂ (with positive imaginary part) but not a real number, we want to calculate real parameters k and k_1d (i.e., adjust the values of α and β) such that s_0=S_0,des.  As discussed in <cit.>,
S̅_0,des must be the eigenvalue s_-1 of the system belonged to the range of W_-1.
Let S_0,des=u+i v and substituting s in (<ref>) leads to
u+i v-α = β e^-(u+i v)h, or equivalently,


    u-α  = β e^u hcos v h, 
    
            v =-β e^u hsin v h.


Combine to yield (u-α)h=-v h  v h, i.e., (S_0,des-α)h = (u-α) h+ i v h ∈ W_0(BC), i.e., S_0,des-α lies in the boundary between the ranges of W_0 and W_1. From (<ref>) we have β =- v e^-u h v h (note v≠ 0) and α=u+v  v h and then the associated control feedback gains are given by


    k =u+v  v h -a/b, 
       k_1d = -v e^-u h v h+a_1d/b.



What happens when one of the states in (<ref>) is not included in the feedback? Suppose the current state is not feedback (k=0), the rightmost eigenvalue is assignable if (S_0,des-a) h∈ W_0(BC), or equivalently, a=u+v  v h, and the delay state feedback gain is then given by


    k_1d= (S_0,des-a) e^S_0,desh-a_1d/b.


Furthermore, suppose the delay state is not used (k_1d=0), we obtain S_0,des-a_1de^-S_0,desh =  a+bk∈ℝ from (<ref>), i.e., S_0,des must satisfy the condition a_1d+v e^u h vh=0. The current state feedback gain is then described by


    k= S_0,des-a -a_1d e^-S_0,desh/b.





On the other hand, if S_0,des∈ℝ, then (<ref>) becomes
S_0,des = α + 1/h W_0(β h e^-α h), i.e.,
S_0,des≥α -1/h while
β h e^-α h≥ -1/e, due to the range of W_0, lies in the right-hand side of the vertical line u=-1. Unless the system is an input-delay system, we always can find an α (or k) such that α≤ S_0,des+1/h and β=(S_0,des-α) e^S_0,desh with corresponding k=(α-a)/b and k_1d=(β-a_1d)/b, i.e.,


    k≤S_0,des-a+1/h/b, 
     k_1d = [S_0,des-(a+bk)]e^S_0,desh-a_1d/b≥e^S_0,desh/hb.


Suppose the current state is not present in the feedback (k=0), the rightmost eigenvalue is assignable if S_0,des≥ a-1/h, and the delay state feedback gain is still given by (<ref>).
Similarly, when the delay state is not used (k_1d=0), we have S_0,des-a_1de^-S_0,desh =a +b k which is always achievable by the state feedback gain from (<ref>).



An input delay system (<ref>)is assignable to any complex S_0,des if
(S_0,des-a)h∈ W_0(BC), i.e., (S_0,des-a)h e^(S_0,des-a)h =z for some real number z<-1/e. Then the associated real feedback gain for the controller u=k x(t) is determined by


    k=S_0,des-a/be^ S_0,desh.


Or when S_0,des is real, it must satisfy S_0,des≥ a-1/h and the feedback gains is still given by (<ref>) which is the same as the result presented in <cit.>.


From the above derivation, the following result is asserted:

Suppose the system (<ref>) in not an input-delay system, the following statements hold:

    
  (i) For a given S_0,des=u+i v∈ℂ, the rightmost eigenvalue of the closed-loop system (<ref>) can be assignable to any desired location S_0,des via the controller (<ref>) with both current- and delay-state feedback gains defined by (<ref>).

    Furthermore, if the current or delay state is not included in the feedback loop, the S_0,des must satisfy the condition a = u+v  v h or a_1d=-v e^u h vh such that the associated gain is described by (<ref>) or (<ref>), respectively.
   
    
  (ii)
    For a given S_0,des∈ℝ, the rightmost eigenvalue of the system (<ref>) can be assignable to any desired location S_0,des via the controller (<ref>) with feedback gains defined by (<ref>). Furthermore, if the current or delay state is not included in the feedback loop, the S_0,des must satisfy the condition S_0,des≥ a-1/h or no constraint such that the associated gain is still described by (<ref>) or (<ref>), respectively. 


 For an input-delay system (<ref>), if S_0,des-α belongs to the upper boundary on the range of W_0 or [-1,-∞), a real feedback gain k through (<ref>) is obtained. 



 
Consider the system (<ref>) with the given data set a=1, a_1d=-1, b=1, and h=1. Determine the values of k and k_1d such that the rightmost closed-loop eigenvalue s_0 is located at S_0,des=−0.092484+1.9973i, −0.60502+1.7882i, or −1, respectively.

By Theorem <ref> this system is eigenvalue-assignable to three desired eigenvalues with the feedback gains given by (<ref>) for the first two or (<ref>) for the third one. Table <ref> shows the computational result for these parameters. And the corresponding variation of characteristic roots of the closed-loop system before and after the eigenvalue assignment are also shown in Table <ref> by using the method proposed in <cit.>. For the third eigenvalue, the closed-loop system becomes a non-delay system which has only one eigenvalue -1 as expected.
        




        
 





§ CONCLUSION

The sufficient condition on the solving the eigenvalue assignment of a linear scalar systems with single delay is analytically developed. The feedback controller is designed by using both the current and delay states. Furthermore, similar sufficient condition for input-delay systems is also depicted. The gains to associate feedback states are computed accordingly such that the closed-loop system behaves the same as expected. Examples are presented for illustrate purpose. This result provides more flexibility in real-world applications than previous studies.



 §.§ Acknowledgement

This work was partially supported under the grant No. MOST 106-
2115-M-029-004.

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