MIMOtool Help Desk

DESIGN SPECIFICATIONS

 

If we consider the two following closed loop schemes, in which the reference command r(s) acts respectively on the plant input (Fig. 1) and on the plant output (Fig. 2),

we can define the transfer matrices:

Open Loop

Fi = K G

Fo = G K

Sensitivity

Si = (I + K G)-1 = e/r

So = (I + G K)-1 = e/r

Complementary Sensitivity

Ti = Si K G

To = So G K = y/r

Control Sensitivity

Mi = G Si = y/r

Mo = K So

 

As we can see from the previous relations, the Sensitivity and the Complementary Sensitivity transfer matrices are such that:

 

Si + Ti = I

So + To = I

Si-1 = I + Fi

So-1 = I + Fo

Ti-1 = I + Fi-1

To-1 = I + Fo-1

 

In MIMOtool, the conventional closed loop configuration shown in Fig.2 is considered and the relative design specifications are: [3] [4]

   PERFORMANCE requirement

At low frequencies, where a satisfying knowledge of the plant is assumed, the most important requirements are the tracking of the reference signal and a good disturbance rejection, which can be both meet by keeping the Sensitivity low, and a Complementary Sensitivity close to the identity matrix, which implies the output signal to be close to the reference one:

 

So ® 0  Þ  To ® I  Þ  y = To r » r

   STABILITY requirement

At high frequencies, the most important requirements are sensor noise rejection and robust stability in face of uncertainties due to unmodelled dynamics, non linearities and system truncation, which usually become marked as the working frequency increases; moreover the control energy has to be reduced as much as possible at these frequencies, since no reference signal must be followed. These requirements can be meet keeping the Control Sensitivity or the Complementary Sensitivity as low as possible (note that if M(s) is low, the control signal u(s) and hence the matrix T(s) are low too):

 

Mo ® 0  Þ  To ® 0  Þ  So ® I

 

The Control Sensitivity in fact is the transfer matrix seen from an additive uncertainty to the plant, so in order to avoid un unstable closed loop between the controlled plant and the uncertainty, it has to be reduced; in addition, keeping the Control Sensitivity low means keeping the control signal low, which turns out to be a good sensor noise rejection procedure.