Let wi denote the properly normalized eigenvector of N'. It follows from (4) and (6) that w; @ u;,is an eigenvector of ( N B M ) ' . Further, these vectors are a set of ns independent vectors if uk (and ak) are n-independent vectors and wi (and pi) are s-independent vectors. Also, from (4), (5), and (lo),
( w i @ u k ) ' ( p i @ a k ) = ( w : @ u i ) ( p i @ a k )It follows from (6), (7), and (1 1) that exp(N@Mt)= x x ( pi@ak)(w;@ui)eh*b' n s k i n s = x x Biw;@akuke"*Rr. ( 14) k i In particular, let N = f,: for this choice of N p. = e. = w. , I I (154 and /+= 1 (15b) for all i. It follows from (1) and (14) that n e x p [ (~, @~) r l = x~, @ a , u i e h~.Recall that a, is an eigenvector of M associated with Ak, and uk is an eigenvector of M' (properly normalized). A parenthetical comment is that it is remarkable that I,@M should have a full set of eigenvectors, because it need not be symmetric and it has only n distinct eigenvalues, each with multiplicity s.
IV. THE DERIVATIVE OF THE EXPONENTIAL MATRIXFor convenience, denoteAs is well known [9], # ( t ) satisfies d -@=Ma dr (18) and #(O) = I . Differentiate (18) with respect to some matrix B (taken to be s x p ) and use (9) to find -(O)=O. acp Use a well-known formula [9] to show that (20) and (21) lead to ==/, a# f e x p [ ( Z , @ M ) ( r -~) l~( f p @ . O (~) ) d~. aM (22) Finally, combine (1 l), (16), and (22) to show where tehl if Ak =AiRecall that s is the number of rows of B and p is the number of columns of B.Equation (23) is the main result. If Vetter's calculus [4], [8] can be used to obtain aM/aB, then (23) can be used to find the derivative of the exponential matrix. When M and B are scalars ( m and b), then A = m and u = 1/a so that (23) and (24) become V. CONCLUDING STATEMENT Equation (23) may be of more than academic interest. Many feedback-control schemes have degrees of freedom (observer and/or feedback gains that can take on a wide range of values and still provide desirable response). If M is interpreted as the closed-loop coefficient matrix and B is interpreted as the matrix of the parameters that vary, (23) could be used as a guide for selecting the degrees of freedom to minimize closed-loop sensitivity to parameter variation.
REFERENCES[ I ] M. Athans and F. C. Schweppe, "Gradient matrices and matrix calculations," [2] R. Bellman, Introduction Lo Matrix Analysis. New York: McGraw-Hill, 1960, Ch. 131 I. W. Brewer, "The gradient with respect to a symmetric matrix," I E E E Trans. (41 W. J. Vetter, "Derivative operations on matrices," I E E E Trans. Automar. Conrr., vol. [5] -, "Correction to 'Derivative operations on matrices,'" I E E E Trans. Automar. [6] -, "On linear estimates, minimum variance and least squares weighting matrices," 171 -, "An extension to gradient matrices," I E E E Trans. Sysr., Man, Cybern., vol.Absrmcr-A new procedure for reducing trajectory sensitivity for the optimal linear regulator is described. The design is achieved without increase in the order of optimization and without the feedback of trajectory sensitivity. T...
