# Feedback Stabilization of First Order Neutral Delay Systems Using the Lambert W Function
## 4 Examples
Figures 2b and 3b show the rightmost spectrum and time response, respectively of the NDDS with the state feedback controller. Applying the QPmR to the NDDS with a feedback system, the four rightmost roots were determined to be \(-0.2652\), \(-0.5355\), \(-0.9979\) + \(\mathrm{i}35.9105\), and \(-0.9979\) - \(\mathrm{i}35.9105\), which indicates that all the roots of its characteristic equation are located to the left of the imaginary axis. The characteristic equation of Equation (35), with the state feedback controller, becomes \(\lambda-0.5\lambda e^{-0.7\lambda}+0.0768e^{-1.2\lambda}=0\), and the condition described by Equation (26) gives a value of \(-0.0645668\). Based on Theorem 4, the NDDS with the above state feedback controller has two negative real roots. It should be noted that the calculated root \(\lambda_{m}\) equals one of the rightmost roots produced by the QPmR. Therefore, we can assign a pole of the NDDS by adjusting the controller gains. **Example 5**. : _Consider the following single-species population growth with time delay and feedback control [30]:_  
\[\dot{N}(t)=rN(t)\Big{(}1-\frac{1}{K}\Big{(}N(t-\tau)+c\dot{N}(t-\tau)\Big{)}-q \Big{(}N(t-\tau),\dot{N}(t-\tau)\Big{)}, \tag{37}\]  
_where \(N(t)\) is the population, \(r\) is the intrinsic growth rate, \(\tau\) is the recovering time, \(c\) is the growth rate associated with the growth rate at time \(t-\tau\), and \(K\) is the environmental carrying capacity. Parameters \(r,\tau,c,K\) are positive. In Equation (37), if \(q(\cdot)=0\), it is sometimes called a dynamic food-limited population model [31]._  
Equation (37) can be rewritten as:  
\[2-W(-ae^{-1})+ae^{W(-ae^{-1})-1}+b\tau_{1}=3.29266.\]0  
Let \(z(t)=1-\frac{1}{K}N(t)\), then \(N(t)=K(1-z(t))\), and  
\[2-W(-ae^{-1})+ae^{W(-ae^{-1})-1}+b\tau_{1}=3.29266.\]1  
\[2-W(-ae^{-1})+ae^{W(-ae^{-1})-1}+b\tau_{1}=3.29266.\]2 System response (**a**) without state feedback, and (**b**) with a state feedback controller for \(k_{p}=1.23336\) and \(k_{d}=-2.5\). The chosen state feedback controller gain exponentially stabilizes the NDDS. The variational equation of Equation (39) about the trivial solution, \(z(t)=0\), takes the form:  
\[2-W(-ae^{-1})+ae^{W(-ae^{-1})-1}+b\tau_{1}=3.29266.\]3  
Compare Equation (40) with Equation (4), we see in this case, \(a=rc\), \(b=r\), \(\tau_{1}=\tau_{2}=\tau\). We assume that the feedback controller is in the following form:  
\[2-W(-ae^{-1})+ae^{W(-ae^{-1})-1}+b\tau_{1}=3.29266.\]4  
A dynamic bacteria population model which is represented by an autonomous system of Equation (37), was introduced and discussed in [31]. According to [24], if \(r,c,\tau\) are positive constants, then every non-oscillation solution of the autonomous system of Equation (37) tends to zero as \(t\rightarrow\infty\). However, the real root is not guaranteed to be the rightmost pole. Table 1 shows the five rightmost poles as a function of \(r,c\), and assuming \(K=1\) and \(\tau=0.1\). The odd columns are without feedback control, and the even columns are with feedback control. In the second and third cases, without feedback control, some complex-conjugated roots are found on the right-hand side of the complex plane and they make the system unstable. With a state feedback controller designed by Theorem 6, all roots are located on the left-hand side of the complex plane. Based on Theorem 4, Equation (40) with the state feedback controller has two negative real roots, and these roots are the rightmost. The simulation results show that the designed state feedback controller makes the NDDS exponentially stable.