Stabilizing the Oscillations of a Controlled Mechanical System with n Degrees of Freedom

Abstract

A mechanical system with n degrees of freedom subjected to the action of positional forces and a small smooth control is considered. It is assumed that in the absence of control, the system may have a family of single-frequency oscillations. A universal control—a nonlinear force that implements and simultaneously stabilizes a cycle in the system—is found. An illustrative example is given. In the previous paper [5], the universal control was designed for a two-dimensional manifold of the system.

Appendices

Appendix

1.1 Lyapunov Transformation for System of Variational Equations

A cycle of the controlled mechanical system satisfies variational equations of the form

$$\begin{array}{rcl}\delta {\ddot{q}}_{s}&=&\mathop{\sum }\limits_{j = 1}^{n}{p}_{sj}(t)\delta {q}_{j}+\mathop{\sum }\limits_{j=1}^{n}{r}_{sj}(t)\delta {\dot{q}}_{j}+\mu \delta {R}_{s}(\delta q,\delta \dot{q}),\\ {p}_{sj}(t)&=&{p}_{sj}(-t),\quad {r}_{sj}(t)=-{r}_{sj}(-t),\quad {p}_{sj}(t+2\pi )={p}_{sj}(t),\quad {r}_{sj}(t+2\pi )={r}_{sj}(t).\end{array}$$

(A.1)

For μ = 0 the system (A.1) is a reversible linear periodic system described by

$$\dot{u}={A}_{-}(t)u+{A}_{+}(t)v,\quad \dot{v}={B}_{+}(t)u+{B}_{-}(t)v,\quad u,v\in {{\rm{R}}}^{n},$$

(A.2)

with two stationary sets of the form

$${M}_{u}=\{u,v,t:v=0,\sin \,t=0\},\quad {M}_{v}=\{u,v,t:u=0,\sin \,t=0\}.$$

Here the plus (minus) sign indicates the matrices and vectors containing the even (odd, respectively) 2π-periodic functions.

The adjoint system for (A.2),

$$\dot{\xi }=-{A}_{-}^{{\rm{T}}}(t)\xi -{B}_{+}^{{\rm{T}}}(t)\eta ,\quad \dot{\eta }=-{A}_{+}^{{\rm{T}}}(t)\xi -{B}_{-}^{{\rm{T}}}(t)\eta ,$$

(A.3)

where T means transposition, is also reversible and admits of two stationary sets:

$${M}_{\xi }=\{\xi ,\eta ,t:\eta =0,\sin \,t=0\},\quad {M}_{\eta }=\{\xi ,\eta ,t:\xi =0,\sin \,t=0\}.$$

The solution (ξ(t), η(t)) of the system (A.3) is used to write the first integral

$$f={\xi }_{1}(t){u}_{1}+\ldots +{\xi }_{n}(t){u}_{n}+{\eta }_{1}(t){v}_{1}+\ldots +{\eta }_{n}(t){v}_{n}$$

(A.4)

of the system (A.2). By the way, this result shows that the function ψ(t) in the amplitude Eq. (2) is odd.

The system (A.2) is reduced to a system with constant coefficients using an appropriate Lyapunov transformation. Recall that the system (A.2) contains n solutions symmetrical with respect to M_ξ, as well as the same number of solutions symmetrical with respect to M_η. Choose these 2n solutions to obtain 2n integrals f for reducing the reversible system (A.2) to a system with constant coefficients. Following [8], select the transformation

$${x}_{s}={p}_{s}^{+}(t)u+{q}_{s}^{-}(t)v,\quad {y}_{s}={p}_{s}^{-}(t)u+{q}_{s}^{+}(t)v,\quad s=1,\ldots ,n,$$

(A.5)

with the 2π-periodic vector functions ${p}_{s}^{\pm }(t)$ and ${q}_{s}^{\pm }(t)$; the transformed system contains the stationary sets

$${M}_{x}=\{x,y,t:y=0,t=0\},\quad {M}_{y}=\{x,y,t:x=0,t=0\}.$$

The system (A.1) is transformed simultaneously.

The reference SPM has characteristics with zero real parts. Therefore, consider the following cases: (a) a pair of zero characteristics in a Jordan cell; (b) a pair  ± iω, ω > 0, of pure imaginary characteristics; (c) a pair of simple zero characteristics. Write the integrals (A.4) and the corresponding reduced equations in the three cases mentioned.

(a) A pair of zero characteristics in a Jordan cell. This pair corresponds to the first integrals

$$\begin{array}{cc}{f}_{1}={g}_{* }^{-}={\rm{const}},\quad {f}_{2}=t{g}_{* }^{-}-{g}_{* }^{+}={\rm{const}},\\ {g}_{* }^{\pm }={\xi }_{* \,1}^{\pm }(t)\delta {q}_{1}+\ldots +{\xi }_{* \,n}^{\pm }(t)\delta {q}_{n}+{\eta }_{* \,1}^{\mp }(t)\delta {\dot{q}}_{1}+\ldots +{\eta }_{* \,n}^{\mp }(t)\delta {\dot{q}}_{n}.\end{array}$$

Find the derivatives ${\dot{f}}_{1}$ and ${\dot{f}}_{2}$ along the trajectories of the system (A.1):

$${\dot{f}}_{1}=\mu \mathop{\sum }\limits_{s=1}^{n}{\eta }_{* \,s}^{+}(t)\delta {R}_{s}(\delta q,\delta \dot{q}),\quad {\dot{f}}_{2}=-\mu \mathop{\sum }\limits_{s=1}^{n}{\eta }_{* \,s}^{-}(t)\delta \,{R}_{s}(\delta q,\delta \dot{q}).$$

Denoting ${x}_{* }={g}_{* }^{+}$ and ${y}_{* }={g}_{* }^{-}$, for μ = 0 obtain ${\dot{y}}_{* }={\dot{f}}_{1}=0$, ${\dot{x}}_{* }={y}_{* }-{\dot{f}}_{2}$, and ${\dot{f}}_{2}=0$. For μ ≠ 0, from (A.1) it follows that

$${\dot{x}}_{* }={y}_{* }+\mu \mathop{\sum }\limits_{s=1}^{n}{\eta }_{* \,s}^{-}(t)\delta {u}_{s},\quad {\dot{y}}_{* }=\mu \mathop{\sum }\limits_{s=1}^{n}{\eta }_{* \,s}^{+}(t)\delta {u}_{s}.$$

(b) A pair  ± iω of pure imaginary characteristics. In this case, the first integrals have the complex representation

$$\begin{array}{rcl}{f}_{\pm }&=&\exp (\pm i\omega t){g}_{\omega }^{\pm }(\delta q,\delta \dot{q}),\\ {g}_{\omega }^{\pm }&=&{\xi }_{\omega \,1}^{\pm }(t)\delta \,{q}_{1}+\ldots +{\xi }_{\omega \,n}^{\pm }(t)\delta \,{q}_{n}+{\eta }_{\omega \,1}^{\mp }(t)\delta \,\dot{q}+\ldots +{\eta }_{\omega \,n}^{\mp }(t)\delta \,{\dot{q}}_{n}.\end{array}$$

Calculate the total derivatives of the functions f_± along the trajectories of the system (A.1):

$${\dot{f}}_{\pm }=\mu \exp (\pm i\omega t)\mathop{\sum }\limits_{s=1}^{n}{\eta }_{\omega \,s}^{\mp }(t)\delta {R}_{s}(\delta q,\delta \dot{q}).$$

Next, find

$${\dot{g}}_{\omega }^{\pm }={\dot{f}}_{\pm }\exp (\mp i\omega t)\pm i\omega \,{f}_{\pm }\exp (\mp i\omega t).$$

Then the variables ${x}_{\omega }={g}_{\omega }^{+}$ and ${y}_{\omega }=i{g}_{\omega }^{-}$ satisfy the equations

$${\dot{x}}_{\omega }=\omega {y}_{\omega }+\mu \mathop{\sum }\limits_{s=1}^{n}{\eta }_{\omega \,s}^{-}(t)\delta {u}_{s},\quad {\dot{y}}_{\omega }=-\omega \,{x}_{\omega }+\mu \mathop{\sum }\limits_{s = 1}^{n}{\eta }_{\omega s}^{+}(t)\delta \,{u}_{s}.$$

Finally, transition to the real variables yields two equations for the system (7).

(c) A pair of simple zero characteristics. Here the first integrals are given by

$${f}_{1,2}={g}^{\pm }={\xi }_{1}^{\pm }(t)\delta {q}_{1}+\ldots +{\xi }_{n}^{\pm }(t)\delta \,{q}_{n}+{\eta }_{1}^{\mp }(t)\delta {\dot{q}}_{1}+\ldots +{\eta }_{n}{(t)}^{\mp }\delta {\dot{q}}_{n}.$$

The corresponding equations of the system (A.1) take the form

$${\dot{x}}_{+}=\mu \mathop{\sum }\limits_{s=1}^{n}{\eta }_{s}^{-}(t)\delta {R}_{s},\quad {\dot{y}}_{-}=\mu \mathop{\sum }\limits_{s=1}^{n}{\eta }_{s}^{+}(t)\delta \,{R}_{s}.$$

Thus, the groups of variables (x_*, y_*), (x_ω, y_ω) and (x₊, y₋) can be adopted for reducing the system (A.1) to a convenient form for further calculation of characteristics. For this purpose, apply the transformation (A.5) in which u = δq, $v=\delta \dot{q}$, the vector x(y) consists of the vectors x_*, x_ω, x₊ (y_*, y_ω, y₊, respectively), and the functions $\delta \,R(\delta \,q,\delta \dot{q})$ and the vectors δq and $\delta \dot{q}$ are replaced by the vectors x and y using the inverse of (A.5).

Funding

This work was supported by the Russian Foundation for Basic Research, project no. 19-01-00146.