Abstract
This paper provides the solution to the complex-order differential equation, 0 d t q x ( t ) = kx ( t ) + bu ( t ) , where both q and k are complex. The time-response solution is shown to be a series that is complex-valued. Combining this system with its complex conjugate-order system yields the following generalized differential equation, 0 d t 2 Re ( q ) x ( t ) - k ¯ 0 d t q x ( t ) - k 0 d t q ¯ x ( t ) + k k ¯ x ( t ) = p 0 d t q u ( t ) + p ¯ 0 d t q ¯ u ( t ) - ( k + k ¯ ) u ( t ) . The transfer function of this system is p ( s q - k ) - 1 + p ¯ ( s q ¯ - k ¯ ) - 1 , having a time-response 2 ∑ n = 0 ∞ t ( n + 1 ) u - 1 Re pk n Γ ( ( n + 1 ) q ) cos ( ( n + 1 ) v ln t ) - Im pk n Γ ( ( n + 1 ) q ) sin ( ( n + 1 ) v ln t ) . The transfer function has an infinite number of complex–conjugate pole pairs. Bounds on the parameters u = Re ( q ) , v = Im ( q ) , and k are determined for system stability.
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