Stability and Hopf bifurcation in an inverted pendulum with delayed feedback control

Abstract

In this paper, we investigate the dynamics of the inverted pendulum with delayed feedback control. The existence and stability of multiple equilibria depending on the control strengths are studied. Taking the time delay of the control terms as a parameter, periodic oscillations induced by delay are found. By using the method of multiple scales, the effect of the control gains and the relative mass of the pendulum on the stability and direction of Hopf bifurcations are discussed. Numerical simulations are employed to illustrate the obtained theoretical results.

Appendices

Appendix A: Proof for $\tau_{0}^{+}<\tau_{0}^{-}$

Since r ₁=r ₂=k>0 for the zero equilibrium, it follows from (10) and (11) that

Define a function f of x as follows:

$$f(x)=\frac{1}{x}\arctan{ \biggl(\frac{b}{a}x \biggr)},\quad x>0, $$

and then we have

$$f'(x)=\frac{\frac{b}{a}x-\arctan{ (\frac{b}{a}x )} (1+\frac {b^2}{a^2}x^2 )}{x^2 (1+\frac{b^2}{a^2}x^2 )}. $$

Denote the denominator of the right hand of f′(x) by

$$g(x)=\frac{b}{a}x-\arctan{ \biggl(\frac{b}{a}x \biggr)} \biggl(1+ \frac {b^2}{a^2}x^2 \biggr). $$

Then we have

$$g(0)=0, \quad g'(x)=-\arctan{ \biggl(\frac{b}{a}x \biggr)} \frac{2b^2}{a^2}x<0, $$

which implies that g(x)<0 for x∈(0,∞). Thus, f(x) is a monotone decreasing function on x∈(0,∞). This, together with ω ₊>ω ₋, means that $\tau_{0}^{+}<\tau_{0}^{-}$.

Appendix B: Derivation of normal form (19)

Let p be the eigenvector of L _τ corresponding to the eigenvalue iω _c τ _c , and let q be the normalized eigenvector of the adjoint operator of L _τ corresponding to the eigenvalue −iω _c τ _c with the inner product

$$\langle\textbf{q},\textbf{p}\rangle=\sum_{i=1}^{2} \bar{q_i}p_i=1. $$

By simple calculations, we have

$$ \textbf{p}=(p_1,p_2)^{T},\quad \quad \textbf{q}=(q_1,q_2)^{T}. $$

(20)

Where

We are now in a position to apply the method of multiple scales to investigate the properties of Hopf bifurcation at the critical value τ _c . Because the nonlinearity has no second order terms, we seek a uniform second-order approximate solution of Eq. (16) in power of ϵ ^1/2. Since the periodic solution of Hopf bifurcation is a small amplitude periodic solution near the zero equilibrium, we assume that a series solution of Eq. (16) has the following form:

$$ \textbf{x}(t;\epsilon)=\epsilon^{1/2}\textbf{x}_1(T_0,T_1)+ \epsilon^{3/2}\textbf{x}_2(T_0,T_1)+ \cdots, $$

(21)

where T ₀=t, T ₁=ϵt, and ϵ is a nondimensional book-keeping parameter. Notice that the secular terms first appear at O(ϵ ^3/2). So, it is sufficient for the time scale of the second argument to be taken as the form of T ₁=ϵt. By the chain rule, we would like to introduce the following differential operator:

$$ \frac{d}{d{t}}=\frac{\alpha}{d{T_0}}+\epsilon\frac{\alpha}{\alpha {T_1}}=D_0+ \epsilon{D_1}. $$

(22)

In terms of the scales T ₀ and T ₁, with the expansion for small ϵ the delayed variable x(t−1) can be expressed as

(23)

Next, we introduce the detuning parameter δ to describe the nearness of τ to the critical value τ _c defined by

$$ \tau=\tau_c+\epsilon\delta. $$

(24)

Substituting Eqs. (21)–(24) into Eq. (16) leads to the following perturbation equations, written up to the ϵ ^3/2 order:

(25)

(26)

Equation (25) is the linear homogeneous equation and has a pair of simple imaginary roots ±iω _c τ _c and all other eigenvalues have negative real parts. The general solution is therefore given by

$$ \textbf{x}_1(T_0,T_1)=A(T_1) \textbf{p}e^{i\omega_c\tau_cT_0}+\bar {A}(T_1)\bar{\textbf{p}}e^{-i\omega_c\tau_cT_0}, $$

(27)

where the vector is defined by (20).

Due to the characteristic equation, we can obtain that ω _c and τ _c satisfy the equation

$$ \omega_c^2+k=kb(i\omega_c)e^{-i\omega_c\tau_c}+kae^{-i\omega_c\tau_c}. $$

(28)

Substituting (27) into Eq. (26) yields

(29)

where

$$ f_{*}=\left [ \begin{array}{c} 0\\ h_2 \end{array} \right ]. $$

(30)

In which

where $\mathrm{c.c}$ and NST stand for the complex conjugate of the preceding terms and terms that do not produce secular terms.

Equation (29) is the linear nonhomogeneous equation for x ₂. We seek its particular solution in the form

$$ \textbf{x}_2(T_0,T_1)= \phi(T_1)e^{i\omega_c\tau_cT_0}. $$

(31)

Substituting Eq. (31) into Eq. (29) and deleting $e^{i\omega _{c}\tau_{c}T_{0}}$, we have

(32)

where I is the 3×3 identity matrix.

It is clear that det(L _τ −iω _c τ _c I)=0 since iω _c τ _c is the eigenvalue of L _τ . So, we have a slight problem solving Eq. (32). This difficulty is easy to overcome following the idea of Kuznetsov [24]. In fact, let T ^su be the real eigenspace corresponding to all eigenvalues of L _τ other than ±iω _c τ _c . We have that y∈T ^su if and only if <q,y>=0. From [24], the restriction of the linear formation corresponding to (L _τ −iω _c τ _c I) to its invariant subspace T ^su is invertible. That is to say if the right-hand side of Eq. (32) belongs to invariant subspace T ^su and then Eq. (32) has a unique solution ϕ∈T ^su by solving so-called bordered system. Thus, the nonhomogeneous Eq. (32) has solutions provided that the right-hand side of Eq. (32) be orthogonal to every solution of the adjoint homogeneous problem. Letting the inner product of the right-hand side of Eq. (32) with q be zero yields the solvability condition (18).