## 1 Introduction  
The more modern is the machine used, the more complicated is the control required. This slogan is currently popular in various industrial societies. The objectives of this slogan, of course, are to achieve higher control efficiency, less power consumption, higher safety and so forth. In control of complicated systems, the model of adaptive control is one of the favored solutions because of its stability and high efficiency. There are two types of adaptive control: indirect and direct controls. The indirect model is the best choice for design of adaptive control, which can be changed to direct model by applying constant parameters for the denominator in main input control. In addition, the main advantage of the adaptive control is its combination with other controls such as fuzzy model, PID (Proportional-Integral-Derivative), optimal control, and sliding mode control. In [1], the indirect adaptive model was studied for MIMO (Multiple-Input and Multiple-Output) nonlinear system. In this study, three main factors of the control, namely actuator fault, unknown system function, and sensor fault, were considered and the fuzzy model was applied in the adaptation laws related to the above three factors. The controller developed in [2] is different from the controller in [1], using a new function as a reset mechanism for the adaptive control law. The objective of the reset solution was to improve the transient performance by leading the estimated parameters to the best values at the initial steps of adaptation. In fact, the reset mechanism was designed by setting a time related boundary of stability and instability times, and this boundary was solved by adaptation laws. Recently, the adaptive control has been used for control unknown phenomena such as dead-zone [3]. Basically, the unknown problem in control was solved by using Nussbaum-type function, and the parameter of the Nussbaum function was updated by adaptation law and the fuzzy model. From the above three studies on adaptive control, the Lyapunov and its modification are main functions to evaluate the stability and design of adaptation laws. The technique using PID in adaptive control [4; 5; 6; 7; 8; 9; 10] was also continuously analyzed. The sliding surface of sliding mode control was used as a first solid function for design of adaptive control and the PID control was used to ensure control robustness through adaptation laws [4]. It is noted that the technique in [4] followed the conventional design of adaptive control with non-modification of the robustness control to improve performance of the system. Several works similar to the one in [4] have been undertaken [5; 6; 7]. The technique proposed in [8] was basically similar to the work done in [4; 5; 6; 7], where the main input control (robust control) used an additive input uncertainty and regression function. Thus, the adaptation laws were derived from the matrices of the conventional Lyapunov equation and error vector. The disadvantage of this approach is less flexibility with the change of disturbances of environment, although the uncertainty of system can be controlled well. The flexible property of the adaptation laws with the Lyapunov equation through Riccati-like equation with PID was studied in [9]. This study presented a new combination of the Riccati-like equation with the conventional robustness PID control, in which the sliding surface is the main parameter of the adaptation laws. Another solution is to use exponential function as presented in [10] where the library of design PID control in adaptive control can be formulated. The application of the modified Riccati-like equation was also studied in [11; 12; 13; 14]. These studies demonstrated that the Lyapunov equation through the Riccati-like equation could improve control performance of the system subjected to uncertainties and/or disturbances. It is noted that other effective methods for calculation based on smart grid controller associated with the energy management have been developed in [15; 16; 17].