Approximated Stable Inversion for Nonlinear Systems with Nonhyperbolic Internal Dynamics

A technique to achieve output tracking for nonminimum phase nonlinear systems with non- hyperbolic internal dynamics is presented. The present paper integrates stable inversion techniques (that achieve exact-tracking) with approximation techniques (that modify the internal dynamics) to circumvent the nonhyperbolicity of the internal dynamics - this nonhyperbolicity is an obstruction to applying presently available stable inversion techniques. The theory is developed for nonlinear systems and the method is applied to a two-cart with inverted-pendulum example

Stable inversion based output tracking control of robotic systems

This thesis addresses stable inversion based output tracking control and its applications to robotic systems. It considers the non-causal invertibility (stable inversion) problem of control systems in its various aspects including properties of stable inverses and algorithms for constructing stable inverses. Then, the stable inversion approach is applied to solve a control problem of long-standing interest: output tracking control for non-minimum phase nonlinear systems;A minimum energy property of stable inverses is firstly established. The property claims that given any desired output trajectory, out of infinitely many possible inverse solutions, the one provided by the stable inversion process is the only one that has finite energy. Based on this property, a numerical procedure is developed to provide an efficient approach to construct stable inverses;Secondly, a new output tracking control design is developed. The design incorporates stable inverses and assumes a controller structure of feed-forward plus feedback. It achieves high precision tracking together with closed-loop stability. Furthermore, when system uncertainties are considered and assumed to satisfy the so-called matching conditions , a modified controller structure is presented and the corresponding robust tracking performance is discussed;Finally, the stable inversion based tracking control design is applied to three flexible robotic systems. The first study is output tracking control of a flexible-joint robot. The application demonstrates how the new design deals with the undesirable non-minimum phase property and achieves desired output tracking. The second application is tip trajectory tracking for a two-flexible-link manipulator. This thesis, for the first time, addresses the problem of stable tip trajectory tracking without any transient or steady-state errors for such non-minimum phase systems. In the third application, a new optimal motion control strategy for a flexible space robot is presented. The space robot system is assumed to consist of a two-link flexible manipulator attached to rigid space-craft. Optimality is in the sense that a performance index measured by maneuvering time, control effort, and structural vibrations is minimized while the interference from the manipulator to spacecraft is kept satisfactorily small;Studies on three applications demonstrate that the stable inversion based control design is very effective on output tracking for various robotic systems. This approach is expected to perform equivalently well for many other realistic non-minimum phase nonlinear systems

On the degenerated soft-mode instability

We consider instabilities of a single mode with finite wavenumber in inversion symmetric spatially one dimensional systems, where the character of the bifurcation changes from sub- to supercritical behaviour. Starting from a general equation of motion the full amplitude equation is derived systematically and formulas for the dependence of the coefficients on the system parameters are obtained. We emphasise the importance of nonlinear derivative terms in the amplitude equation for the behaviour in the vicinity of the bifurcation point. Especially the numerical values of the corresponding coefficients determine the region of coexistence between the stable trivial solution and stable spatially periodic patterns. Our approach clearly shows that similar considerations fail for the case of oscillatory instabilities.Comment: 16 pages, uses iop style files, manuscript also available at ftp://athene.fkp.physik.th-darmstadt.de/pub/publications/wolfram/jpa_97/ or at http://athene.fkp.physik.th-darmstadt.de/public/wolfram_publ.html. J. Phys. A in pres

Tippe Top Inversion as a Dissipation-Induced Instability

By treating tippe top inversion as a dissipation-induced instability, we explain tippe top inversion through a system we call the modified Maxwell--Bloch equations. We revisit previous work done on this problem and follow Or's mathematical model [SIAM J. Appl. Math., 54 (1994), pp. 597--609]. A linear analysis of the equations of motion reveals that the only equilibrium points correspond to the inverted and noninverted states of the tippe top and that the modified Maxwell--Bloch equations describe the linear/spectral stability of these equilibria. We supply explicit criteria for the spectral stability of these states. A nonlinear global analysis based on energetics yields explicit criteria for the existence of a heteroclinic connection between the noninverted and inverted states of the tippe top. This criteria for the existence of a heteroclinic connection turns out to agree with the criteria for spectral stability of the inverted and noninverted states. Throughout the work we support the analysis with numerical evidence and include simulations to illustrate the nonlinear dynamics of the tippe top