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Grids in near future are going to host synchronous generators (SGs), legacy grid-following converters (GFLCs), and relatively new grid-forming converters (GFCs), where the latter two technologies support converter-interfaced generations. This change is going to introduce new challenges - one of them is the dc-link voltage collapse issue in GFCs following the outage of a generator. As a solution, this paper proposes a nonlinear backstepping control strategy that guarantees Lyapunov stability of a reduced-order model of such systems. To this end, first a reduced model of a 6-bus system with one GFC, one GFLC, and one SG is considered to develop fundamental understanding of such systems and derive a stabilizing control strategy. Then, based on certain observations, the simplified model is extended to a multimachine system, which in turn helps developing the nonlinear control strategy in presence of bounded uncertainties. Further, a dc-link voltage tracking control is introduced to ensure controllable frequency support from GFCs. The proposed controller is decentralized in nature and requires limited communication only if neighboring buses have GFCs and/or GFLCs. Detailed phasor models of the 6-bus system and the modified 68-bus New England-New York system with 2 GFCs and 2 GFLCs are used to demonstrate the effectiveness of the proposed control strategies.