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We consider the sensitivity of transient solutions of Markov models to perturbations in their generator matrices. The perturbations can either be of a certain structure or can be very general. We consider two different measures of sensitivity and derive upper bounds on them. The derived bounds are sharper than previously reported bounds in the literature. Since the sensitivity analysis of transient solutions is intimately related to the condition of the exponential of the CTMC matrix, we derive an expression for the condition number of the CTMC matrix exponential which leads to some interesting implications. We compare the derived sensitivity bounds both numerically and analytically with those reported in the literature.

If we assume that the space crane is a statically determinate truss with length-adjustable bars and take as controls the length adjustments of these bars, the computation of the incremental controls corresponding to the motion of a payload along a trajectory is given in terms of the inverse-transpose of matrix B of the joint equilibrium equations Bs = p, where s lists the bar forces and p the nodal loads. An algorithm with 0(w 2 ) computational complexity and 0(/z) storage demand is used for obtaining the inverse of the wth-order sparse matrix B. The compensation of the controls for elastic deformations and support movements is shown. The crane is assumed to be moving sufficiently slowly so that no vibratory motion is created during its maneuver. To simplify the computations, a zero-acceleration field is assumed in the workspace of the space crane. It is shown that the computations may be done automatically and in real time by an attached processor once the characteristics of the crane's maneuver are keyed in. I. TerminologyT HROUGHOUT this paper, an italic boldface letter indicates a free vector, while a lowercase boldface letter denotes a column matrix. Thus, r is a free vector and r a column matrix. A scalar is represented by an italic lightface letter and a matrix with more than one column by an upper case boldface letter. Thus, t and N are scalar variables, while B is a matrix with more than one column. Superscripts T and -1 stand for matrix transposition and inversion, respectively. A superscript -to any vector implies that it is derived from a larger vector by deleting a few rows. Thus, v '(t) stands for a vector that is obtained by deleting some of the rows of the vector v. Similarly, a superscript " to a matrix indicates that it is derived from a larger matrix by deleting a few columns. Thus, [B-1 ]" and [B' 7 ]" are matrices obtained from [B-1 ] and [B" 7 ], respectively, by deleting some columns.