Phase-Based Gain-Modulation (PBGM) control is realized by modulating controller gains in response to the phase of the system state or tracking error. PBGM controllers have been applied to robotic hands, parallel manipulators and flexible mechanisms to give increased damping, reduced tracking error and friction compensation.A novel method is presented to establish Lyapunov stability for PBGM control. Prior PBGM stability results incorporated a constraint which limited the range of provably stable systems. The present result removes this constraint, establishing Lyapunov stability for a substantially broader class of systems. Additionally, the new approach decouples the selection of the Lyapunov function from the controller design, permitting the controls designer to independently specify a switch function which determines the application of gain modulation.The present results are applied to analyze PBGM control of the Sarcos dextrous manipulator, illuminating the stability properties of control experiments previously reported in the literature. Numerical methods for design calculations are also presented.
Nonlinear PID (NPID) control is implemented by allowing the controller gains to vary as a function of system state. NPID control has been previously described and implemented, and recently a constructive Lyapunov stability proof has been given. The controllers arising with the constructive Lyapunov method will in general depend on knowledge of the full state vector. In the present work, NPID controllers that operate without knowledge of some state variables are demonstrated. A general but conservative design method is presented with an experimental demonstration. For a special case, complete necessary and sufficient conditions are established; for this case, simulation of a robotic force control application demonstrates well-damped control with no requirement for a force-rate signal. The extension to cases of partial state knowledge is important for NPID control, which is most practical when some state variablesparticularly rate variables-are poorly known, confounding fullstate feedback or other high-damping linear control designs. Extension of NPID control to MIMO systems and computed torque control is also shown.
Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002) 430-455] developed a class of Exponential Time Differencing Runge-Kutta schemes (ETDRK) for nonlinear parabolic equations; Kassam and Trefethen [A.K. Kassam, Ll. N. Trefethen, Fourth-order time stepping for stiff pdes, SIAM J. Sci. Comput. 26 (2005) 1214-1233] have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth-order (ETDRK4) scheme. They use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials, but this approach creates new difficulties in choosing and evaluating the contour for larger problems. Neither treatment addresses problems with nonsmooth data, where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully. Such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications, including computational chemistry and financial engineering. We introduce a new version of the fourth-order Cox-Matthews, Kassam-Trefethen ETDRK4 scheme designed to eliminate the remaining computational difficulties. This new scheme utilizes an exponential time differencing Runge-Kutta ETDRK scheme using a diagonal Padé approximation of matrix exponential functions, while to deal with the problem of nonsmooth data we use several steps of an ETDRK scheme using a sub-diagonal Padé formula. The new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices, having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel. In this approach it is only required that several backward Euler linear problems be solved, in serial or parallel. Numerical experiments are described to support the new scheme.
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