IV.5.3 Gain Margin and Phase Margin

The Bode stability criterion states that the maximum value of the controller gain that can be chosen for stable closed loop response is called the ultimate gain . In other words, the value of controller gain must always be less than in order to ensure stability. The gain margin (GM) is a design parameter such that

(IV.79)

Gain margin should always be chosen as greater than one (GM>1) to ensure stability .

Gain margin acts as a safety factor for model uncertainty. Since process parameters such as gain, time constant and dead time can never be estimated exactly, a safety factor of magnitude more than one is necessary for stable operation. For relatively well modeled processes, a low safety factor will be acceptable whereas poorly modeled processes need higher safety factors. For an example, let us choose GM=2 for the process we have discussed above (eq. IV.71), the design value of the controller gain is ; suppose there exists a modeling error of 50% in estimating the dead time of the process and the true value of the dead time is 0.45 instead of 0.3, then the revised value of crossover frequency is

(IV.80)

or, , and the corresponding which is still higher than the designed value of . The system is still stable despite the error by 50% we made in estimation of dead time of the process.

Phase margin is another safety factor which is used for controller design. Here we are interested to compute a frequency that satisfies the following expression,

(IV.81)

is called phase margin (PM) and it is the extra phase lag needed to destabilize a system. For an example, let us choose . can be calculated from the following expression

(IV.82)

or, . The gain is designed from the expression