MRAC Theory

Again, the idea behind Model Reference Adaptive Control is to create a closed loop controller with parameters that can be updated to change the response of the system to match a desired model. There are many different methods for designing such a controller. This tutorial will cover design using the MIT rule in continuous time. When designing an MRAC using the MIT rule, the designer chooses: the reference model, the controller structure and the tuning gains for the adjustment mechanism.

MRAC begins by defining the tracking error, e. This is simply the difference between the plant output and the reference model output:


From this error a cost function of theta (J(theta)) can be formed. J is given as a function of theta, with theta being the parameter that will be adapted inside the controller. The choice of this cost function will later determine how the parameters are updated. Below, a typical cost function is displayed.


To find out how to update the parameter theta, an equation needs to be formed for the change in theta. If the goal is to minimize this cost related to the error, it is sensible to move in the direction of the negative gradient of J. This change in J is assumed to be proportional to the change in theta. Thus, the derrivative of theta is equal to the negative change in J. The result for the cost function chosen aobve is:


This relationship between the change in theta and the cost function is knwon as the MIT rule. The MIT rule is central to adaptive nature of the controller. Note the term pointed out in the equation above labeled "sensitivity derivative". This term is the partial derrivative of the error with respect to theta. This determines how the parameter theta will be updated. A controller may contain several different parameters that require updating. Some may be acting n the input. Others may be acting on the output. The sensitivity derivative would need to be calculated for each of these parameters. The choice above results in all of the sensitivity derrivatives being multiplied by the error. Another example is shown below to contrast the effect of the choice of cost function:


To see how the MIT rule can be used to form an adaptive controller, consider a system with an adaptive feedword gain. The block diagram is given below.


EXAMPLE: ADAPTIVE FEEDFOWARD GAIN





The constant k for this plant is unknown. However, a reference model can be formed with a desired value of k, and through adaptation of a feedforward gain, the response of the plant can be made to match this model. The reference model is therefore chosen as the plant multiplied by a desired constant ko:


The same cost function as above is chosen and the derivative is shown:


The error is then restated in terms of the transfer functions multiplied by their inputs.


As can be seen, this expression for the error contains the parameter theta which is to be updated. To determine the update rule, the sensitivity derivative is calculated and restated in terms of the model ouput:


Finally, the MIT rule is applied to give an expression for updating theta. The constants k and ko are combined into gamma.


The block diagram for this system is the same as the diagram given at the beginning of this example. To tune this system, the values of ko and gamma can be varied.


NOTES ON DESIGN WITH MIT

It is important to note that the MIT rule by itself does not guarantee convergence or stability. An MRAC designed using the MIT rule is very sensitive to the amplitudes of the signals. As a general rule, the value of gamma is kept small. Tuning of gamma is crucial to the adaptation rate and stability of the controller.