Internal model control (IMC), which explicitly incorporates a plant model and a plant inverse model as its components, has an intuitive control structure and simple tuning procedure. Within the IMC structure, we propose composite adaptive IMC (CAIMC) which simultaneously identifies the plant and the plant inverse to minimize modeling errors and further reduce the tracking error. In this paper, the design procedure of CAIMC is generalized to an $n$-th-order SISO plant. The main challenge in the generalization is to find an identification algorithm for an $n$-th order system that satisfies the stability constraint, while assuring closed-loop stability. In the literature, stability-constrained identification has been formulated as a convex programming problem by re-parameterizing the constraint as a linear matrix inequality, but boundedness and continuity of the estimated parameters, which are critical for closed-loop stability of an adaptive control algorithm, are not guaranteed. We propose a modified stability-constrained identification method with established boundedness and continuity properties. Closed-loop stability and asymptotic performance of CAIMC are then established under proper conditions. The effectiveness of the proposed algorithm is demonstrated with an example.