Given the inverse problem F(x,t)=0, Dynamic Inversion is a technique whereby a dynamical system is constructed whose solution is exponentially attracted to x(t). It provides a continuous time method of solving time-varying finite dimensional inverse problems.
N. H. Getz,
"Dynamic Inversion of
Nonlinear Maps with Applications to Nonlinear Control and Robotics", Ph.D. Dissertation Department of Electrical
Engineering and Computer Sciences University of California at
Berkeley December 1995.This dissertation introduces the notion of a dynamic inverse of a nonlinear map. The dynamic inverse is used in the construction of nonlinear dynamical system, called a dynamic inverter, that asymptotically solves inverse problems with time-varying vector-valued solutions. Dynamic inversion generalizes and extends many previous results on the inversion of maps using continuous-time dynamic systems. By posing the dynamic inverse itself as the solution to an inverse problem, we show how one may solve for a dynamic inverse dynamically while simultaneously using the dynamic inverse solution to solve for the time-varying root of interest. Dynamic inversion is a continuous-time dynamic computational paradigm that may be incorporated into controllers in order to continuously provide estimates of time-varying parameters necessary for control. This allows nonlinear control systems to be posed entirely in continuous-time, replacing discrete root-finding algorithms as well as discrete algorithms for matrix inversion with integration. Example applications include solving for the intersection of time-varying polynomials, inversion of nonlinear control systems, regular and generalized inversion of fixed and time-varying matrices, polar decomposition of fixed and time-varying matrices, output tracking of implicitly defined reference trajectories, end-effector tracking control for robotic manipulators, and causal approximate output tracking for nonlinear nonminimum-phase systems. For the problem of output tracking for nonminimum-phase systems, an internal equilibrium manifold is introduced. This manifold is intrinsic to the class of nonlinear nonminimum-phase systems studied. Approximate output tracking is achieved by constructing a controller that makes a neighborhood of the internal equilibrium manifold attractive and invariant. Dynamic inversion is incorporated into the controller to provide a continuous estimate of the manifold location. This estimate is incorporated into the tracking control law. It is demonstrated, by application to the tracking problem for the inverted pendulum on a cart, that the resulting internal equilibrium controller significantly outperforms a linear quadratic regulator, where the linearization of the internal equilibrium controller is made identical to the linear quadratic regulator. Internal equilibrium control is also applied to the problem of causing a nonlinear, nonholonomic model of a bicycle to track a time-parameterized trajectory in the ground plane while retaining balance.
N. H. Getz
and J.E.
Marsden, "Dynamic
Inversion of Nonlinear Maps", Center
for Pure and Applied Mathematics, Memorandum No. CPAM 621 19 December
1994 Revised 27 November 1995.We consider the problem of estimating the time-varying root of a time-dependent nonlinear map. We introduce a ``dynamic inverse'' of a map, another generally time-dependent map. The dynamic inverse is composed with the original map to form a vector-field. We prove that the flow of this vector field decays exponentially to an arbitrarily small neighborhood of the root. We then show how a dynamic inverse may be determined dynamically. Dynamic inversion is a continuous-time analog computational paradigm that may be incorporated into controllers in order to continuously provide estimates of time-varying parameters necessary for control. This allows nonlinear control systems to be posed entirely in continuous-time by eliminating the need to appeal to discrete root-finding algorithms as well as discrete algorithms for matrix inversion.
N.
H. Getz and J.E.
Marsden, "Dynamical Methods
for Polar Decomposition and Inversion of Matrices",
Linear Algebra and its Applications, vol. 258, pp311-343,
1997.We show how one may obtain the polar decomposition as well as perform inversion of fixed and time-varying matrices using a class of dynamical systems. First, a dynamical system is constructed that causes an initial approximation of the inverse of a time-varying matrix to flow exponentially toward the true time-varying inverse. The same dynamical method may be applied to the inversion of fixed matrices. By appealing to a time-parameterized homotopy from the identity matrix to a fixed matrix, and applying our result on the inversion of time-varying matrices, we show how any positive definite fixed matrix may be dynamically inverted in finite time without an initial guess at the inverse. We then construct a dynamical system that solves for the polar decomposition components of a time varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition. As a byproduct, this method gives another method of inverting time-varying matrices. Finally by using homotopy again, we show how the polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix in finite time.
See CPAM 629.
N. H. Getz
and J.E.
Marsden, "Tracking
Implicit Trajectories", Nonlinear
Control Systems Design Symposium, International Federation
of Automatic Control Tahoe City, California 26-28 June 1995. Elsevier
Science Ltd., Oxford.Output tracking of implcitly defined reference trajectories is examined. A continuous-time dynamical system is constructed that produces explicit estimates of time-varying implicit trajectories. We prove that incorporation of this "dynamic inverter" into a tracking controller provides exponential output tracking of the implicitly defined trajectory for nonlinear control systems having vector relative degree and well-behaved internal dynamics.