Time-optimal control laws are known to be bang-bang, and well suited to relay or power electronic controllers. Closed-form solutions for the time-optimal control switching are obtained only for a limited set of low-order systems. The control approach in the paper is to determine the control value for a linear quadratic regulator, and then project this value onto the closest available switching control vector. The paper uses Lyapunov functions to show convergence of this projection approach, provided that the requested LQR control signal is limited in magnitude. A sequence of LQR solutions is used to cover the full-state space, and is found to give a control performance close to time optimal. When the system is open-loop unstable, convergence regions are determined from these Lyapunov functions. The region of guaranteed convergence is shown, in the limit, to be close to the full region of possible controllability. The sequence of LQR solutions with control projections is found to be a well defined control design for arbitrary order linear multiinput multi-output systems, leading to good switching control performance.