Confusion about the concept of stability

[mamech]

Hello

I have some confusions in my mind about the concept of stability in control systems. I know that stability is defined as "the system produces bounded output when you apply bounded input", but these words can have several meanings
for example, the bounded input, should it be step , or impulse? in case of DC motor (or any similar electrical system having same order), if we apply impulse voltage (bounded input), we will get rotation of the motor with an angle θ (bounded output, i.e stable system). But if we apply step input to the motor (still is considered as bounded input), we will get output θ increasing indefinitely (unbounded output, i.e unstable system) .

let us discuss another case, like a case of a ball resting on a convex surface, if you apply impulse or step input it will be unstable and its output will not be bounded.


for me , the case of the ball on convex surface is purely unstable, while the case of motor is not purely unstable because it depend on input (while this contradicts the definition of instability). I think that what I think is wrong, but I do not know where is the mistake. can any one help?

 

[crutschow]

My usual definition of stability is whether the system oscillates for any bounded input. Running the output to the rails (or infinity) is not normally considered an instability.

 

[albbg]

The transfer function of a DC motor is given as velocity/voltage you have to considerate this to check the stability (by means of a voltage you can control the velocity but not the position of the shaft). You used instead the position that is a wrong output variable since it's not controllable from the input variable you gave.
A ball on a convex surface is unstable since any input produce an unbounded output: you can't control the position of the sphere as you could do on a plane (simple stability).

 

[mamech]

DC motor transfer function depends on what is the input and what is the output. Dc motor can have output as angular velocity or angular displacement (position). in case of velocity/voltage it will be usually second order (always stable), and in case of position/voltage it will be third order, with having 1/s in denominator that makes part of system works as integrator (which causes instability according to the definition of bounded input and bounded output).

Actually no, you can control the position using voltage, but motor position is controlled only in closed loop, because it is unstable in open loop

and regarding the example of convex surface, I saw it in my control lectures as example of unstable systems.

 

[D.A.(Tony)Stewart]

Stability is a time domain measure of overshoot with oscillation from either stimulus; step or impulse. However Control System design is usually done in frequency domain.

It is a result of negative feedback network with a sensor based on rotational or linear position, velocity and acceleration compared with desired profile. Current controls torque, which controls acceleration, affected by mass, source impedance or current limit of driver, moment of inertia, resulting velocity ramp and back EMF from motor changing direction of acceleration, latency, etc etc.

So the stability for position is much more difficult to control than velocity ( position is integral of v) which is harder to control than acceleration ( position is 2nd integral of acceleration or current). All of these are controlled in HDD servos using various feedback with a rotary voice coil motor using a complex self-calibrated control system.


Instability or Oscillation.

Integrators cause 90 deg phase shift. Two integrators cause 180 in addition to other latency makes negative feedback turn to positive feedback and unstable (oscillating) unless compensated.

A drunk driver is considered unstable in driving a car and often oscillates left and right instead of a stable driving in the centre of the lane. This is cause by reduced or excessive gain ( mental process error) and added latency ( slow reaction) in feedback. The other effects are severe overshoot, lack of speed control ( insufficient sampling rate of speedometer or awareness) and erratic acceleration... followed by flashing lights, loss of car, license or worse.

 

[albbg]

Actually no, you can control the position using voltage, but motor position is controlled only in closed loop, because it is unstable in open loop

You can control the position using additional block (the control system) to make the position controllable via voltage. In this case the system is not the same: is motor+controller and not the motor alone. You add that block to change the transfer funtion to position/voltage.
The velocity is the first derivative of the position; due to this if you apply a voltage step to a motor alone with velocity/voltage transfer funtion, integrating you'll have the transfer function position/(integral of voltage) then actually for that system you apply a ramp that is an unbounded input.