3. Stability
■ The characteristic equation of the open loop or closed loop
equation helps us in finding the stability , damping and speed
response of the system
4. Niederlinski Index
■ A fairly useful stability analysis method is the Niederlinski index. It can
be used to eliminate unworkable pairings of variables at an early stage
in the design.
■ The settings of the controllers do not have to be known, but it applies
only when integral action is used in all loops. It uses only the steady
state gains of the process transfer function matrix.
■ The method is a “necessary but not sufficient condition” for stability of
a closedloop system with integral action. If the index is negative, the
system will be unstable for any controller settings (this is called
“integral instability”). If the index is positive, the system may or may
not be stable.
5.
6.
7.
8. ■ Some of the earliest work in multivariable control involved the
use of decouplers to remove the interaction between the
loops.
■ The decoupling matrix Qt9, is chosen such that each loop does
not affect the others.The decoupling element D, can be
selected in a number of ways.
■ One of the most straightforward is to set D11 = D22 = 1 and
design the D12 and D21 elements so that they cancel the
effect of each manipulated variable in the other loop.
9.
10. Relative gain matrix
■ The RGA is a matrix of numbers.The ijth element in the array
is called fiij.
■ It is the ratio of the steadystate gain between the ith
controlled variable and the jth manipulated variable when all
other manipulated variables are constant, divided by the
steady state gain between the same two variables when all
other controlled variables are constant.
11. Robustness
■ The design of a controller depends on a model of the process.
■ If the controller is installed on a process whose parameters are not
exactly the same as the modelused to design the controller.
■ The parameters of any industrial process always change somewhat
due to nonlinearities, changes in operating conditions, etc.
■ If a control system is tolerant to changes in process parameters, it
is called “robust.”
12. Doyle-stein criterion
■ For multivariable systems, the Doyle-
Stein criterion for robustness is very similar
to the reciprocal plot discussed.
■ The minimum singular value of the matrix is plotted
as a function of frequency o.This gives a measure of
the robustness of a closed-loop multivariable
system.
13.
14. ■ The peak in this curve, the maximum closed-loop log
modulus is a measure of the damping coefficient of the
system.
■ The higher the peak, the more under-
damped the system and the less margin for
changes in parameter values.
■ Thus, in SISO systems the peak in the closed-loop log
modulus curve is a measure of robustness.
