Constrained Motion of Mechanical Systems and Tracking Control of Nonlinear Systems

Abstract

This paper aims to expose the interrelations and connections between constrained motion of mechanical systems and tracking control of nonlinear mechanical systems. The interrelations between the imposition of constraints on a mechanical system and the trajectory requirements for tracking control are exposed through the use of a simple example. It is shown that given a set of constraints, d’Alembert’s principle corresponds to the problem of finding the optimal tracking control of a mechanical system for a specific control cost function that Nature seems to choose. Furthermore, the general equations for constrained motion of mechanical systems that do not obey d’Alembert’s principle yield, through this duality, the entire set of continuous controllers that permit exact tracking of the trajectory requirements. The way Nature seems to handle the tracking control problem of highly nonlinear systems suggests ways in which we can develop new control methods that do not make any approximations and/or linearizations related to the nonlinear system dynamics or its controllers. More general control costs are used and Nature’s approach is thereby extended to general control problems. A simple, unified methodology for modeling and control of mechanical systems emerges. Examples drawn from diverse areas of control are provided dealing with synchronization of multiple nonlinear gyroscopes, design of optimal Lyapunov stable controllers for nonautonomous nonlinear systems, and energy control of nonhomogeneous Toda chains.

Appendix

We shall call the column 3-vector q = [x, y, z]^T. The plant we want to control corresponds to the unconstrained system (see Table 2) whose equation of motion is given in Eq. (2) as

$$ M\;\ddot{q}(t):=\left[\begin{array}{ccc} m & 0& 0\\ 0 & m& 0\\ 0& 0 & m\end{array}\right]\kern0.5em \left[\begin{array}{l}\ddot{x}\\ \ddot{y}\\ \ddot{z}\end{array}\right]=\left[\begin{array}{l}0\\ m g\!\left( x, y, z, t\right)\\ 0\end{array}\right]:= Q $$

(62)

The trajectory requirement (constraint) is

$$ \varphi\!\left( x, y, z, t\right):={x}^2(t)+{y}^2(t)+{z}^2(t)-{L}^2= q{(t)}^T q(t)-{L}^2=0, $$

(63)

and since we may not start on this manifold initially, we consider instead the constraint

$$ \ddot{\varphi}+ c\dot{\varphi}+ k\varphi =0, c>0, k>0 $$

(64)

whose solution as t → ∞ is φ = 0. Thus we get asymptotic convergence to the trajectory requirement given in Eq. (63). Differentiating the constraint (63), Eq. (64) can be rewritten as

$$ A\ddot{q}=\left[ x\kern0.75em y\kern0.75em z\right]\left[\begin{array}{l}\ddot{x}\\ \ddot{y}\\ \ddot{z}\end{array}\right]=-{\dot{q}}^T\dot{q}- c{\dot{q}}^T q-\left( k/2\right)\left({q}^T q-{L}^2\right):= b $$

(65)

so that A = [x y z] and $b=-{\dot{q}}^T\dot{q}- c{\dot{q}}^T q- k\left({q}^T q-{L}^2\right)/2$. The control force Q ^C that minimizes at each instant of time the control cost

$$ J(t)={\left[{Q}^C\right]}^T N{Q}^C $$

(66)

where N is a user-specified positive definite 3 by 3 matrix is given by Eq. (21) as

$$ \begin{array}{l}{Q}^C{=}-{N}^{-1}\left[\begin{array}{l} x/ m\\ y/ m\\ z/ m\end{array}\right]{\left[ A{(MNM)}^{-1}{A}^T\right]}^{+}\left\{ gy{+}{\dot{q}}^T\dot{q}{+} c{\dot{q}}^T q{+}\frac{k}{2}\right({q}^T q-{L}^2\left)\right\}\\ [2pc] {}\kern1.25em =-\frac{m{N}^{-1}}{\left( A{N}^{-1}{A}^T\right)}\left[\begin{array}{l} x\\ y\\ z\end{array}\right]\left\{ gy+{\dot{q}}^T\dot{q}+ c{\dot{q}}^T q+\frac{k}{2}\left({q}^T q-{L}^2\right)\right\}.\end{array} $$

(67)