MODEL REFERENCE ADAPTIVE CONTROL

1. Consider an open loop control system (plant) represented by the block diagram below: -

2. Now consider an ideal plant whose characteristics are meeting all the user’s requirements of proper settling time, peak overshoot etc. (Note: These parameters are not ideal but physically realizable). This ideal plant is chosen as the model represented by the block diagram below: -

3. The user will desire that the practical plant behave in the same manner as the above model plant. In other words, the user would require the plant to follow the model. This is called model following control. Here the adaptive loop operates on the error between the actual plant output yp and the output ym of the model i.e.,

_________________________________ (1.1)If the dynamics of the model are chosen correctly, it should be possible to drive this error towards zero, thus driving the closed-loop plant behavior towards that of the model. Thus the block diagram representation of a model reference adaptive control system will be as shown below: -

PLANTINPUT(up)

OUTPUT(yp)

MODELINPUT(um)

OUTPUT(ym)

CONTROLLER

INPUT(up)

OUTPUT(yp)

MODELINPUT(um)

OUTPUT(ym)

PLANT

4. Analysis of the Model Reference Adaptive Control (MRAC) using transfer functions.

(a) Let the plant transfer function of block diagram shown in figure 1 be given by

___________________________________ (1.2)

(b) Let the model transfer function of block diagram in figure 2 be given by

____________________________________ (1.3)

(c) Thus for a given reference input to the model, the output of the model is given as______________________________ (1.4)

(d) The output of the actual plant considering up(s) as the input is given as_______________________________ (1.5)

(e) Thus substituting equations (1.4) and 1.5) in the error signal generated by the adaptive control loop (equation(1.1)), we get

______________________ (1.6)

(f) The objective of the controller will be to make this error zero. Thus the input signal for the plant which will satisfy this condition is given by equating the right hand side of equation (1.6) to zero and finding up. Thus,

______________________ (1.7)

(g) The plant output for the control signal above is given by

_______________ (1.8)

Thus we are able to achieve the desired objective of model following.

5. Problems in designing the controller.

The objective of the controller is to generate the control signal for the plant as given by equation (1.7). From the equation it is seen that the control signal depends on the inverse of the transfer function of the plant. This relationship can cause two problems (Ref. B Friedland, “Advanced Control Systems Design”): -

(a) If the transfer function of the plant is non-minimal phase i.e., it has zeros in the right half s-plane, Gp

-1 (s) has poles in the right half plane. Thus the control signal up(t) will contain increasing exponential function of time as shown in example 1.1 given below. This increasing control signal will ultimately saturate any real actuator and prevent the adaptive control from performing as intended. Moreover unless the saturation is somehow accounted for the system is vulnerable to instability.

(b) If the degree of the denominator of the plant transfer function Gp(s) is greater than the degree of the numerator, as is almost always is, and the input contains discontinuities, the control signal will contain impulses, doublets etc. reflecting that Gp

-1 (s) has a numerator of higher degree than the denominator, which implies differentiation of the input. Again, since no physical actuator can produce this kind of input, the adaptive control will fail.(Examples 1.1.2,1.1.3)

Example 1.1

Let us consider an ideal situation where the denominators of the plant and model transfer functions are the same. The three cases under study will be one without any finite zero, the next with one finite zero in the lhp and the third with one finite zero on the rhp of s-plane.

(a) Case I:Let the plant and model transfer functions be given by

If um(s) is considered to be a unit step signal, the output of the model is given by

Thus from equation (1.7), the control signal for the plant is given as

(b) Case II.Let the plant transfer function contain one zero in the left hand plane given by s=-3.

Thus for the same unit step signal as input to the model, the control signal is calculated as:

Thus the control signal now contains an additional exponential function which is decreasing with time.

(c) Case III.Let the plant transfer function contain one zero in the right hand plane given by s=+3.

Thus for the same unit step signal as input to the model, the control signal is calculated as:

Thus the control signal now contains an additional exponential function which is increasing with time. Thus the control signal is a source of instability and also may cause saturation of the actuator in this form.

Example 1.1.2

Let the plant and model transfer functions be given by

If um(s) is considered to be a unit step signal, the output of the model is given by

Thus from equation (1.7), the control signal for the plant is given as

Thus at t=0, the control signal is not zero.

Example 1.1.3

Let the plant and model transfer functions be given by

If um(s) is considered to be a unit step signal, the output of the model is given by

Thus from equation (1.7), the control signal for the plant is given as

Once again it is seen that the control signal is non-zero when time t=0.

Note: Thus it can also be seen that if the plant transfer function denominator is of degree three while that of the model is of degree two, then there will be a differentiation.