about steady-state error

If either your plant or your controller contains a bare integrator (i.e. if the transfer function has at least one pole at s=0 or z=1) then the open-loop gain will be infinite at DC -- which means that any non-zero steady-state error will cause the output to climb indefinitely, until it reaches infinity at infinite time. This condition holds true regardless of whether your system is stable or not.
For any system with a bare integrator this property will cause the output to continue driving until the average steady-state error is zero. When you do the math you'll see this zero steady-state error, even for an unstable system. Remember that the final-value theorem is only valid for a stable system, so you can't "really" use it -- but you can have a system with a bounded instability that's whacking the stops, throwing off bits of itself and doing all sorts of other nasty things, yet still has zero _average_ error.
This is why PI and PID controllers are so popular: if you can get them to be stable and robust, then they will always drive the error to zero eventually, even in the face of parameter variations.

Dear Tim:
Very appreciate your explaination. Exactly, I have a pole locating z=0 in my equivalent transfer function. Orginally, I hope to find suitable paramaters to make my system stable. I also know steady-state theroem only used in stable system. In your explaination, I still have some sentences not understood, such as "but you can have a system with a bounded instability that's whacking the stops, throwing off bits of itself and doing all sorts of other nasty things, yet still has zero _average_ error." Could you please give me some detail explaination? According to your experience, what can I do for my design? Very appreciate your any advice!
Bests regards, Yang Hong

Dear Tim:
I just read your explaination again. Seemingly, you think I couldn't use steady-state error to design my paramaters, isn't it? Through your first graph, for any system whatever is stable or unstable, when we compute steady-state error, it is possible to get the zero steady-state error. My understanding is right? Thank you very much!

snip Hopefully you mean _s_ = 0 -- in a discrete-time system a pole at z=0 just means you have a unit delay, not an integrator.
As to the statement about unstable systems: If you have a PI or PID controller then the controller will work to keep the average error of the system at zero, and will deliver higher and higher output any time the average error isn't zero. In some cases you can have a system that's unstable and oscillating (possibly badly), yet due to the integrator action the oscillation will be centered around the setpoint, leading to an error that equals zero when averaged over time.

Hi, Tim:
Sorry, I have wrong typing. Yes, In the discrete time, I have a pole z=1;
instability

I first encountered this phenomenon in a simple simulator. I always wondered why one shouldn't used D in a flow loop. (It's OK, no further explanations are necessary.) So I tried it. The loop slammed wildly from wide open to tight shut. But over time the average was bang on setpoint. I suppose you could say I had invented pulse width modulation.
Actually the flow loop was the inner of a level flow cascade. Level control was excellent. Basically what I had was little different from a level switch direct connected to an oversized control valve. on / off / on / off.
Walter.

One of my clients has a very high performance motion control loop, with torquer motors on the inner axes, gear motors on the outer axes, the whole ball of wax. The inner axes can drive the motor to oscillate stop to stop without it showing up in any way except for the singing sound and the singeing feeling when you touch the motors.

Well, you could always call the oscillation a feature of your new sliding mode controller!
Harvey

Would the smoke and molten insulation coming from the motor be a feature as well?

Another feature, West Nile Fever mosquito repellent.
Harvey

I had a fellow at a motor rewind shop tell me that my motor was now repaired and that he had put all the smoke back in.
Walter.