In this section, we design a nonlinear controller to control the Hopf bifurcation in the main drive delay system of rolling mill. The nonlinear state feedback controller is as follows:

$$ u=-b_{1}x_{1}-b_{2}x_{1}^{2}-b_{3}x_{1}^{3}, $$

(7)

where \(b_{1}\), \(b_{2}\), and \(b_{3}\) are positive feedback parameters. The rolling mill main drive system Eq. (2) with state feedback controller can be rewritten as

$$ \textstyle\begin{cases} \dot{x_{1}}=x_{2}-b_{1}x_{1}-b_{2}x_{1}^{2}-b_{3}x_{1}^{3}, \\ \dot{x_{2}}=-a_{1}x_{2}-a_{2}x^{2}_{2}-a_{3}x^{3}_{2}-a_{4}x_{1}+a_{5}x_{1}(t- \tau ). \end{cases} $$

(8)

By calculating, system (8) has the characteristic equation

$$ \lambda ^{2}+(a_{1}+b_{1})\lambda +a_{4}+a_{1}b_{1}-a_{5}e^{-\lambda \tau }=0. $$

(9)

Let \(\lambda = i\omega _{2} (\omega _{2}>0)\) be a root of Eq. (9), then

$$ \begin{aligned} &{-}\omega _{2}^{2}+a_{4}+a_{1}b_{1}-a_{5} \cos \omega _{2}\tau =0, \\ &(a_{1}+b_{1})\omega _{2}+a_{5}\sin \omega _{2}\tau =0 \end{aligned} $$

(10)

and

$$ \omega _{2}^{4}+\bigl(a_{1}^{2}+b_{1}^{2}-2a_{4} \bigr)\omega _{2}^{2}+(a_{4}+a_{1}b_{1})^{2}-a_{5}^{2}=0. $$

(11)

Let \(\omega _{2}^{2}=\nu,Q_{1}=a_{1}^{2}+b_{1}^{2}-2a_{4},Q_{2}=(a_{4}+a_{1}b_{1})^{2}-a_{5}^{2}\), then Eq. (11) can be rewritten as

$$ \nu ^{2}+Q_{1}\nu +Q_{2}=0, $$

(12)

and we define \(h(\nu )=\nu ^{2}+Q_{1}\nu +Q_{2},\nu >0\).

By analyzing, if \(Q_{2}<0\), then Eq. (12) has only a positive root \(\nu _{1,2}=\frac{-Q_{1}+\sqrt{\Delta }}{2}\), where \(\Delta =Q_{1}^{2}-4Q_{2}\). Thus, if \(Q_{2}<0\) holds, \(\pm i\omega _{2}^{c}\) is a pair of purely imaginary roots of Eq. (9) with \(\tau _{k2}(k=0,1,\ldots )\), where \(\tau _{k2}=\frac{1}{\omega _{2}^{c}}(\theta +2k\pi ),\theta =\arcsin ( \frac{(a_{1}+b_{1})\omega _{2}^{c}}{-a_{5}})\), \(\omega _{2}^{c}=\sqrt{\nu _{1,2}}=\sqrt{ \frac{-Q_{1}+\sqrt{\Delta }}{2}}\). Let \(\lambda (\tau )=\alpha (\tau )+i\omega (\tau )\) be the root of (3) satisfying \(\alpha (\tau _{k2})=0, \omega (\tau _{k2})=\omega _{2}^{c} (k=0,1,2, \ldots )\).

Similar to the analysis in the second part, the following conclusions can be drawn.

### Theorem 3

*If*\(Q_{1}<0, \nu \in (-\frac{Q_{1}}{2},+\infty )\)*holds*, *then*\(\operatorname{sign}(\frac{dRe\lambda (\tau )}{d\tau }\mid _{\tau =\tau _{k2}}) =\operatorname{sign}(h^{\prime }( \nu ))>0\); *if*\(Q_{1}<0, \nu \in (0,-\frac{Q_{1}}{2})\)*holds*, *then*\(\operatorname{sign}(\frac{dRe\lambda (\tau )}{d\tau }\mid _{\tau =\tau _{k2}}) =\operatorname{sign}(h^{\prime }( \nu ))<0\).

### Theorem 4

*For system* (8), *suppose that*\(Q_{2}<0\)*holds*.

- (i)
*If*\(Q_{1}<0, \nu \in (-\frac{Q_{1}}{2},+\infty )\)*holds*, \(E_{0}\)*is locally asymptotically stable whenever*\(\tau \in [0,\tau _{02})\)*and*\(E_{0}\)*is unstable whenever*\(\tau \in (\tau _{02},+\infty )\). *Moreover*, *it generates a Hopf bifurcation at*\(E_{0}\)*when*\(\tau =\tau _{k2}\).

- (ii)
*If*\(Q_{1}<0, \nu \in (0,-\frac{Q_{1}}{2})\)*holds*, \(E_{0}\)*is locally asymptotically stable whenever*\(\tau \in [0,\tau _{02})\cup (\bigcup_{i=1}^{\infty })(\tau _{i,2}, \tau _{i+1,2})\). *Furthermore*, *it generates a Hopf bifurcation at*\(E_{0}\)*when*\(\tau =\tau _{k2}\).

In the following, we will explore the nature of the Hopf bifurcation for controlled system (8) by implementing the normal form (NF) and the center manifold reduction (CMR) [22].

Let \(t\rightarrow \frac{t}{\tau }\), \(\tau =\tau _{02}+\mu \), \(\mu \in R\), then Eq. (8) can be written in a functional differential equation in \(C=C([0,1],R^{2})\) as follows:

$$ \dot{x(t)}=L_{\mu }(\phi )+F(\phi,\mu ), $$

(13)

where

$$\begin{aligned} &L_{\mu }(\phi )=(\tau _{02}+\mu ) \begin{pmatrix} -b_{1} & 1 \\ -a_{4} &-a_{1} \end{pmatrix} \begin{pmatrix} \phi _{1}(0) \\ \phi _{2}(0) \end{pmatrix} \\ &\phantom{L_{\mu }(\phi )=}{}+(\tau _{02}+\mu ) \begin{pmatrix} 0 & 0 \\ a_{5} &0 \end{pmatrix} \begin{pmatrix} \phi _{1}(-1) \\ \phi _{2}(-1) \end{pmatrix}, \\ &F(\mu,\phi )=(\tau _{02}+\mu ) \begin{pmatrix} -b_{2}\phi _{1}(0)^{2}-b_{3}\phi _{1}(0)^{3} \\ -a_{2}\phi _{2}(0)^{2}-a_{3}\phi _{2}(0)^{3} \end{pmatrix}. \end{aligned}$$

By Riesz representation theorem, let

$$\begin{aligned} \eta (\theta,\mu )=(\tau _{02}+\mu ) \begin{pmatrix} -b_{1} & 1 \\ -a_{4} &-a_{1} \end{pmatrix} \delta (\theta )- (\tau _{02}+\mu ) \begin{pmatrix} 0 & 0 \\ a_{5} &0 \end{pmatrix} \delta (\theta +1), \end{aligned}$$

where *δ* represents the Dirac delta function. Define

$$ A(\mu )\varphi =\textstyle\begin{cases} \frac{{\mathrm{d}}\varphi (\theta )}{{\mathrm{d}}\theta }, & \theta \in [-1,0), \\ \int _{-1}^{0}\,d\eta (\theta,\mu )\varphi (\theta ), & \theta =0, \end{cases} $$

and

$$ R(\mu )\varphi =\textstyle\begin{cases} 0 ,& \theta \in [-1,0), \\ F(\mu,\varphi ), & \theta =0, \end{cases} $$

where \(\varphi \in C^{1}([-1,0]R^{2})\). For \(\psi \in C^{1} ([0,1],R^{2})\), define the adjoint operator of \(A(\mu )\) as follows:

$$ A^{*}\psi (s)=\textstyle\begin{cases} \int _{-1}^{0}\,d\eta (t,0)\psi (-t),& s=0, \\ -\frac{d\psi (s)}{ds},& s\in (0,1], \end{cases} $$

and

$$ \langle \psi,\varphi \rangle =\bar{\psi }(0)\varphi (0)- \int _{-1}^{0} \int _{\xi =0}^{\theta } \bar{\psi }(\xi -\theta )\,d\eta ( \theta ) \varphi (\xi )\,d\xi. $$

\(A(0)\) and \(A^{*}\) are adjoint operators, and \(A(0)\) has a pair of purely imaginary eigenvalues \(\pm i\omega _{2}^{c}\tau _{02}\).

Define \(q(\theta )=(1,q_{2}(0))^{T}e^{i\omega _{2}^{c}\tau _{02}\theta }\), \(q^{*}(s)=D(1,q^{*}_{2}(0))^{T}e^{i\omega _{2}^{c}\tau _{02} s}\). By calculating, we obtain \(D=(1+\overline{q_{2}(0)}q^{*}_{2}(0)-\tau _{02} q^{*}_{2}(0)a_{5}e^{i \omega _{0}\tau _{02}})^{-1}\),\(q_{2}(0)=i\omega _{2}^{c}\tau _{02}+b_{1}\) and \(q^{*}_{2}(0)=\frac{1}{a_{1}-i\omega _{2}^{c}\tau _{02}}\).

On the center manifold \(\varSigma _{0}\),

$$ w\bigl(z(t),\bar{z}(t),\theta \bigr)=w_{20}(\theta ) \frac{z^{2}}{2}+w_{11}( \theta )z\bar{z} +w_{02}(\theta ) \frac{\bar{z}^{2}}{2}+\cdots, $$

we can get

$$\begin{aligned} \dot{z}&=i\omega \tau _{02}z+\bigl\langle q^{*}(\theta ),F \bigl(W+2\operatorname{Re} {z(t)q( \theta )}\bigr)\bigr\rangle \\ &=i\omega \tau _{02}z+\bar{q}^{*}(0)F\bigl(W(z,\bar{z},0)+2 \operatorname{Re} {z(t)q( \theta )}\bigr) \\ & \doteq i\omega \tau _{02}z+\bar{q_{0}^{*}}F(z, \bar{z}), \end{aligned}$$

and this equation is rewritten as

$$ \dot{z}(t)=i\omega \tau _{02}z(t)+g(z,\bar{z}), $$

(14)

where

$$\begin{aligned} g(z,\bar{z})&=\bar{q_{0}^{*}}F\bigl(W(z,\bar{z},0)+2 \operatorname{Re} {z(t)q(\theta )}\bigr) \\ &=g_{20}\frac{z^{2}}{2}+g_{11}z\bar{z}+ g_{02} \frac{\bar{z}^{2}}{2}+g_{21} \frac{z^{2}\bar{z}}{2}+\cdots \end{aligned}$$

The following coefficients are obtained by using a computation similar to that of [12, 13], which are used for determining the important qualities

$$\begin{aligned} &g_{20}=-2\overline{D}e^{2i\tau _{02}\omega _{2}^{c}\theta }\bigl[a_{2}q_{2}^{2}(0) \overline{q_{2}^{*}(0)}+b_{2}\bigr]; \\ &g_{11}=-\overline{D}\bigl[q_{2}^{*}(0)a_{2}q_{2}(0) \overline{q_{2}(0)}+b_{2}\bigr]; \\ &g_{20}=-2\overline{D}e^{-2i\tau _{02}\omega _{2}^{c}\theta }\bigl[a_{2} \overline{q_{2}^{2}(0)}\overline{q_{2}^{*}(0)}+b_{2} \bigr]; \\ &g_{21}=-2\overline{D}\bigl[b_{2}\bigl(e^{i\tau _{02}\omega _{2}^{c}\theta )}+e^{-i \tau _{02}\omega _{2}^{c}\theta )}w^{1}_{20} \bigr)+ a_{2} \overline{q_{2}^{*}(0)} \bigl(q_{2}(0)w^{2}_{11}(\theta )e^{i\tau _{02} \omega _{2}^{c}\theta )} \\ &\phantom{g_{21}=}{}+ \overline{q_{2}(0)}w^{2}_{20}(\theta )e^{-i \tau _{02}\omega _{2}^{c}\theta }\bigr)\bigr], \end{aligned}$$

where \(w_{20}\) and \(w_{11}\) satisfy

$$\begin{aligned} &w_{20}(\theta )=\frac{g_{20}}{i\omega _{2}^{c} \tau _{02}}q(0)e^{i \tau _{02}\omega _{2}^{c}\theta }- \frac{\bar{g}_{02}}{3i\omega _{2}^{c}\tau _{02}}\bar{q}(0)e^{-i\tau _{02} \omega _{2}^{c}\theta }+E_{1}e^{2i\omega \tau _{02}\theta }, \\ &w_{11}(\theta )=\frac{g_{11}}{i\omega _{2}^{c} \tau _{02}}q(0)e^{i \tau _{02}\omega _{2}^{c}\theta }- \frac{\bar{g}_{11}}{i\omega _{2}^{c}\tau _{02}}\bar{q}(0)e^{-i\tau _{02} \omega _{2}^{c}\theta }+E_{2}. \end{aligned}$$

Note that

$$\begin{aligned} &E_{1}=\biggl(2i\omega _{2}^{c}I- \int _{-1}^{0}e^{i\tau _{02}\omega _{2}^{c} \theta }\,d\eta (\theta,0) \biggr)^{-1}F_{z^{2}} \\ &E_{2}=-\biggl( \int _{-1}^{0}\,d\eta (\theta,0) \biggr)^{-1}F_{z\overline{z}}, \end{aligned}$$

we can get

$$\begin{aligned} &E_{1}= \begin{pmatrix} 2i\omega _{2}^{c} & -1 \\ a_{4}-a_{5}e^{-i\tau _{02}\omega _{2}^{c}} & 2i\omega _{2}^{c}+a_{1} \end{pmatrix}^{-1} \begin{pmatrix} 2b_{2}e^{2i\tau _{02}\omega _{2}^{c}\theta } \\ 2a_{2}q_{2}(0)^{2}e^{2i\tau _{02}\omega _{2}^{c}\theta } \end{pmatrix} \\ &E_{2}= \begin{pmatrix} 0 & -1 \\ a_{4}-a_{5} & a_{1} \end{pmatrix}^{-1} \begin{pmatrix} b_{2} \\ a_{2}q_{2}(0)\overline{q_{2}(0)} \end{pmatrix}. \end{aligned}$$

Therefore, by our previous analysis, we can obtain the following parameters that determine the nature of Hopf bifurcation:

$$\begin{aligned} &c_{1}(0)=\frac{i}{2\omega _{2}^{c}\tau _{02}}\biggl(g_{20}g_{11}-2 \vert g_{11} \vert ^{2}- \frac{1}{3} \vert g_{02} \vert ^{2}\biggr)+\frac{g_{21}}{2}, \\ &\mu _{2}=-\frac{\operatorname{Re} c_{1}(0)}{\operatorname{Re} \lambda '(\tau _{02})}, \\ &\beta _{2}=2\operatorname{Re} c_{1}(0), \\ &T_{2}=- \frac{\operatorname{Im} c_{1}(0)+\mu _{2}\operatorname{Im} \lambda '(\tau _{02}) }{\omega _{2}^{c}}. \end{aligned}$$

### Theorem 5

*For system* (8) *with*\(\tau _{k2}\), *the following results hold*:

- (i)
*If*\(\mu _{2}>0\)\((\mu _{2}<0)\), *then the Hopf bifurcation is supercritical* (*subcritical*).

- (ii)
*If*\(\beta _{2}<0\)\((\beta _{2}>0)\), *then the bifurcating periodic solutions are stable* (*unstable*).

- (iii)
*If*\(T_{2}>0\)\((T_{2}<0)\), *then the period increases* (*decreases*).