18 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL
Now back to our control problem. Having simplified the original dynamics,
Equation (2.1) to the nicer form of Equation (2.2), we are now ready for the
analysis leading to the derivation of a control lawu(t). The strategy is this.
By examining the system under consideration and our control objective, the
form ofu(t)can be suggested by common sense or naive physics. Then the
mathematical model given by Equation (2.2) is used to determine (partially)
the control lawu(t).
In our control example,u(t)can be suggested from the following heuristic
ìIf .. then ...î rules:
Ifφis positive, thenushould be negative
Ifφis negative, thenushould be positiveFrom these common sense ìrules,î we can conclude thatu(t)should be of the
form
u(t)=−αφ(t) (2.5)
for someα> 0. A control law of the form (2.5) is called aproportional
control law,andαis called thefeedback gain. Note that (2.5) is a feedback
control since it is a function ofφ(t).
To obtainu(t),weneedαandφ(t). In implementation, with an appropriate
gainα,u(t)is determined sinceφ(t)can be measured directly by a sensor. But
before that, how do we know that such a control law will stabilize the inverted
pendulum? To answer this, we substitute Equation (2.5) into Equation (2.2),
resulting in the equation
φ®(t)−φ(t)+αφ(t)=0 (2.6)In a sense, this is analogous to guessing the root of an equation and checking
whether it is indeed a root. Here, in control context, checking thatu(t)is
satisfactory or not amounts to checking if the solutionφ(t)of Equation (2.6)
converges to 0 ast→+∞, that is, checking whether the controller will sta-
bilize the system. This is referred to as the control system (the plant and the
controller) beingasymptotically stable.
For this purpose, we have, at our disposal, the theory of stability of linear
differential equations. Thus, we examine the characteristic equation of (2.6),
namely
z^2 +α−1=0 (2.7)
Forα> 1 , the roots of (2.7) are purely imaginary: z=±j
√
α− 1 ,where
j=
√
− 1. As such, the solutions of (2.6) are all oscillatory and hence do not
converge to zero. Forα≤ 1 , it can also be seen thatφ(t)does not converge to
zero ast→+∞.Thus,u(t)=−αφ(t)is not a good guess.
Let us take another guess. By closely examining why the proportional control
does not work, we propose to modify it as follows. Only forα> 1 do we have
hope to modifyu(t)successfully. In this case, the torque is applied in the correct
direction, but at the same time it creates more inertia, resulting in oscillations
of the pendulum. Thus, it appears we need to add tou(t)something that acts