In many cases, real plants have time-delay in control inputs. For example, PID controllers are designed by approximating the plant as a 1st order+time delay. In such plants, effects of feedback control inputs do not directly appear and the control of the systems is difficult. To control systems having time-delay effectively enhances safety of systems, makes mechanical systems applicable to caring of humans safely and contributes life-innovations. If a way to calculate the sum of squared tracking errors of outputs is obtained, then the minimization of the sum is easily calculated and an optimal control for such systems is straightforwardly obtained. This paper proposes a calculating method for the sum of output errors by (1) expressing the time delay system in a state-space equation, (2) calculating the sum by solving a Lyapunov matrix equation and (3) in solving the matrix equation, obtaining mathematical expressions of converted equations to linear equations and a matrix inversion by using symbolic computation software in beforehand. Also, for an example, optimal PID gains for 1st-order + time delsy system is obtained and numerical simulations are given.