The 17O, 13C, and 29Si NMR spectra of (CH3)3SiOC(O)R, CH3(XCH2)Si(OC(O)CH3)2, and R3GeOC(O)CH3 compounds are reported. In the 17O NMR spectra at 350 K the only signal is observed with the two latter series, but two well-resolved signals are displayed with the (CH3)3SiOC(O)R compounds. The equivalence of both oxygen atoms in carboxyl group on the NMR time scale is discussed from the viewpoint of a possible coordination of the oxygen atoms to the IVB group element of the periodic system.
Necessary and sufficient conditions are obtained for the stability of the following second order linear system: x˙=θ(t)x,θ(t)=θt+∑i=1lTi and θ(t) =A1,0<t<T1=A2,T1<t<T1+T2⋮=Al,∑i=1l−1Ti<t<∑i=1lTi in terms of the eigenvalues and elements of the matrices Ai, i = 1, 2…l. The conditions become very simple for the case that l = 2. An example of a pendulum with a vibrating support is included.
SUMMARYWe present an optimal gain scheduling control design for bipedal walking with minimum tracking error. We obtained a linear approximation by linearizing the nonlinear hybrid dynamic model about a nominal periodic trajectory. This linearization allows us to identify the linear model as a linear periodic system. An optimal feedback was designed using Bellman's dynamic programming. The linear periodic system allows us to determine a linear quadratic regulator (LQR) for a single period and to set the Hamilton-Jacobi-Bellman (HJB) function in a linear quadratic form. In this way, the dynamic programming yielded an admissible continuous gain scheduling that was designed with regard to the hybrid dynamics of the system. We tuned the optimization parameters such that the tracking error and the average energy consumption are minimized. Due to linearization, we were able to examine the stability of the approximated periodic system achieved by the periodic gain according to Floquet's theory, by calculating the monodromy matrix of the closed-loop hybrid system. In addition to determining stability, the eigenvalues of this approximated monodromy matrix allowed us to evaluate the settling time of the system. This approach presents a direct method for optimal solution of locomotion control according to a given reference trajectory.
Discussion: “The Stability of a Second Order Linear Periodic System” (Davison, E. J., 1969, ASME J. Basic Eng., 91, p. 207–210)
