The construction of control Lyapunov functions for linear multivariable systems

A universal method for constructing control Lyapunov functions (CLF) for a linear multivariable system is presented.Firstly,we show that every quadratic control Lyapunov function can be obtained by solving the Lyapunov equation.Then,we prove that this method can generate all quadratic control Lyapunov functions for a given linear system.Finally,it is shown that if there is a control Lyapunov function then there must be a quadratic control Lyapunov function.Thus,the problem of constructing control Lyapunov functions for linear systems is solved completely.