Figure 2. Behavior of output of designed control system in the case
of integrators in series at various T2.
Figure 3. Behavior of output of designed control system in the case
of integrators in series at various T1.
B. Canonical controllable form (CCF). This form is important if we would like to affect to the last term of characteristic polynomial an which corresponds to general gain of the system.
Let us consider the second order system which is identical to CCF:
.
It is known that the system will be stable if and only if the parameters a1 and a2 are positive. If for example the small perturbation will make the a2 negative then system will become unstable.
Let us set the control law in the form (1). Hence we will obtain the following equations of designed control system.
(7)
.
The system (7) has following equilibrium points:
, ; (8)
, ; (9)
Stability conditions for equilibrium points (8) and (9) respectively are
(10)
(11)
From inequalities (10) and (11) it is easy to see that here it does not matter what value except zero parameter a2 will be. Similar to above we can resume that system (7) will be stable.
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