Hi,

After engaging in a quite interesting conversation on flexible manipulators I have decided to post another question that a colleague
of mine asked me today.

"What happens to system stability when you linearize a nonlinear system and find the LTI model has dominant repeated poles and then control it with a PD or PID controller? Do the repeated poles cause the system to be more stable, or less stable (more unstable)"

Lets look at plant of the form

y(s)/u(s) = g(s) = 1/(s+3)^3

Thus the plant has three repeated poles at s = -3. The plant on its own is stable, as the I/O map in the time domain is

y(t) = 1/2

Thus the exp(-3*t) is an exponential decay and the t^2 term will cause the system to start to grow exponentially at first, but eventually the exp(-3*t) term will swamp the t^2 term and the system output will decay.

Now lets wrap with a PID controller

h(s) = (Kd*s^2 + Kp*s + Ki)/s

Let [Kd Kp Ki] = [2 10 5], thus the open loop TF is

O(s) = (2*s^2 + 10*s + 5)/(s+3)^3*s

If I create a bode diagram of this, the system has about 100 deg of phase margin and infinite gain margin (you can check the infinite gain margin with a root locus plot as well, I wish I could post plots from matlab with this post).

The closed loop TF is

C(s) = (Kd*s^2 + Kp*s + Ki)/( Kd*s^2 + Kp*s + Ki + (s+3)^3*s )

and numerically C(s) is

C(s) = ( 2 s^2 + 10 s + 5 ) / (s^4 + 9 s^3 + 29 s^2 + 37 s + 5 )

All the dominator poles are negative and thus the system is stable.

SO, now lets discuss: Evidently adding another pole at s = -3 (going from 1/(s+3)^2 to 1/(s+3)^3) will cause the system to be slower to, for example, a step response. The open loop bode diagram of 1/(s +3)^2*PID has about 130 degrees of phase margin. What my friend and I are wondering is how the system robustness is effected? We have modeled some nonlinear system as a LTI model. Clearly there are other dynamics that may be effecting the system. But because we have a system that is dominated by three poles that should slow things down in the long run compared to a 1/(s+3)^2 plant. Thus, in general I think the system should be more stable and nonlinearities should be drowned out because the three repeated poles are dominant.

Any comments?

James Forbes

After engaging in a quite interesting conversation on flexible manipulators I have decided to post another question that a colleague

"What happens to system stability when you linearize a nonlinear system and find the LTI model has dominant repeated poles and then control it with a PD or PID controller? Do the repeated poles cause the system to be more stable, or less stable (more unstable)"

Lets look at plant of the form

y(s)/u(s) = g(s) = 1/(s+3)^3

Thus the plant has three repeated poles at s = -3. The plant on its own is stable, as the I/O map in the time domain is

y(t) = 1/2

***exp(-3.0***t)*t^2*u(t)Thus the exp(-3*t) is an exponential decay and the t^2 term will cause the system to start to grow exponentially at first, but eventually the exp(-3*t) term will swamp the t^2 term and the system output will decay.

Now lets wrap with a PID controller

h(s) = (Kd*s^2 + Kp*s + Ki)/s

Let [Kd Kp Ki] = [2 10 5], thus the open loop TF is

O(s) = (2*s^2 + 10*s + 5)/(s+3)^3*s

If I create a bode diagram of this, the system has about 100 deg of phase margin and infinite gain margin (you can check the infinite gain margin with a root locus plot as well, I wish I could post plots from matlab with this post).

The closed loop TF is

C(s) = (Kd*s^2 + Kp*s + Ki)/( Kd*s^2 + Kp*s + Ki + (s+3)^3*s )

and numerically C(s) is

C(s) = ( 2 s^2 + 10 s + 5 ) / (s^4 + 9 s^3 + 29 s^2 + 37 s + 5 )

All the dominator poles are negative and thus the system is stable.

SO, now lets discuss: Evidently adding another pole at s = -3 (going from 1/(s+3)^2 to 1/(s+3)^3) will cause the system to be slower to, for example, a step response. The open loop bode diagram of 1/(s +3)^2*PID has about 130 degrees of phase margin. What my friend and I are wondering is how the system robustness is effected? We have modeled some nonlinear system as a LTI model. Clearly there are other dynamics that may be effecting the system. But because we have a system that is dominated by three poles that should slow things down in the long run compared to a 1/(s+3)^2 plant. Thus, in general I think the system should be more stable and nonlinearities should be drowned out because the three repeated poles are dominant.

Any comments?

James Forbes