Adaptive Control is one approach which is taken for controlling systems when we are not able to use conventional P.I.D Controllers. The problem is a kind of complexity in our plant. For example there are some unexpected disturbances in our plant or the change in plant's parameters in some systems; specially systems which are changing during the time. For example if the wearing is considerable in a system or the dynamic of the system varies during time because of change in mass or other properties of the plant. So, the assumption of Linear time invariant system is not always acceptable. We have to adapt the controller to overcome the problems and perform control in range of tolerance. There are different applications for adaptive control from aviation to robotics. There are different methods for design such controllers.
There are two important categories for adaptive controllers which are more common. The first one is MRAC (Modelreference adaptive controller) and the other one is Selftuning regulators. Each one has advantages. The most important advantage of MRAC controllers is less online computational effort comparing to Selftuning regulators. In selftuning regulators we are able to use system identification in order to update the transfer function of our plant online. this is an advantage because we are able to cope with unexpected situations and also we are able to perform more robust control. Calculation cost is considerable because the parameters are under monitor continuously. In the following picture you can see a MRAC in general form. As it is clear, we have one Model we calculate the y_{m }is the output from the model. We compare this output with the real plant output. by use of some methods such as MIT rule or other approaches we try to calculate new parameters for regulator. So, we are able to tolerate changes in the plant by adjustment mechanism. On the other hand we have selftuning controller (STR). As you can see in STR, we have the outer loop which is monitor the change in the plant to change design characteristics for regulator parameters. If we ignore the estimation block the STR controller is not adoptable to plant changes. Minimum variance controllers by using system identification method recursive least square (RLS) are common approach as an adoptable minimum variance controller, a good candidate fro STR. Pictures are from lecture notes Dr. Olena Kuzmicheva Bremen university Germany. 
This article is based on books: Crage, D.P Stoten,...
Adaptive Manipulator Control IntroductionAs we discussed briefly in the introduction. Adaptive control is commonly used in robotics. There are some important reasons for using adaptive controllers.
Hence, we should cope with these problems. There are different approaches. The conventional approach which is based on accurate modeling is computed torque strategy. I discuss this strategy in detail later. The main disadvantage of computed torque strategy is accurate modelling and nonadaptable to timevarying parameters. So, aging is considerable destructive factor in this strategy.
Another approach, which is a good alternative for manipulator control specially in high change in dynamics and disturbance is STR (selftuning regulators). This approach as introduced in the introduction note is costly in computation. The main assumption of this method is a linear estimated system resulted from system identification. Then using minimum variance controller or poleplacement is applied. This approach will be also discussed later.
The third one which is respectable in sense of low computational effort and adaptable behavior is MRAC. There are some key advantages for this method as are listed below.
Motion Equation and ModelingWe start with a simple example. An inverted pendulum.
At first it is good to know the Euler–Lagrange equation.
