Analog and Digital System Interaction

Sampled Proportional Control Systems

Our friend, Dr. Abner Mallity is working on a simple control system. He is trying to control a first order system. He has modelled it with this block diagram.

G(s) is a generic first order system, with a transfer function G_dc/(st + 1)
The sensor that measures the output of the system produces one volt per unit of output.

The system is not behaving the way he thinks it should. He has thought about the system and come to the following conclusions.

The root locus for the system is shown below.

The root locus starts at the pole (at s = -1/t ) and the closed loop pole moves to the left (i.e. the system gets faster as the closed loop time constant gets smaller.)
As the gain is increased, system performance gets better and better (i.e. faster and faster, and no oscillations!).

As the gain is increased, the system gets more accurate since the SSE (for a unit step input) is given by SSE = 1/(1 + KG_dc).

As far as Dr. Mallity is concerned, as the gain is increased the system should be better in ever way.

The problem with all this is that it's not happening. As Dr. Mallity increases the gain the system improves to a point, but it eventually oscillates and then becomes unstable. His graduate students, Willy Nilly and Millie Farad, have pointed out to him that he doesn't really have the system shown above because he is using computer control. They argue that the computer control makes this a sampled system and that the continuous analysis does not work for that kind of system.

Millie and Willy point out that the system he has is really something like the one in the diagram below.

Millie and Willy point out the following.

Everything inside the dotted line in the figure is really inside the computer.
The measurement of the output is performed by an instrument that is really an A/D. (It might be a digital voltmeter, a data acquisition unit or a data acquisition card in the computer.) In any event, the data the enters the computer is in digital form, and the conversion takes place every T seconds, where T is the value of the sampling time of the instrument.
The measured value (out of the A/D) is compared in a computer program to the desired value of the output and an error is calculated - in a computer program. The result is an error computation.
The error is multiplied by a constant gain, K, in a computer program. That produces a voltage that will be the control effort to drive the system. However, at this point, that result is still a number in a computer program.
The calculated control effort is applied to a D/A which is the device that actually produces a voltage output to be used as the control effort to drive the system.

Millie and Willy are telling Dr. Mallity that because of all this the signal that drives the system (with the transfer function, G(s) = G_dc/(st + 1), is a "Zero Order Hold" type of signal. (Click here to see an example of a zero order hold signal. The link will take you to a point in the lessons which discusses zero order hold signals.) Millie and Willy make these further claims.

Because the essential nature of the system is that it is a sampled system, you need to analyze this system using sampled concepts and Z-transform methods!
Since the system is a sampled system, the model for the linear system, G(s), should be one which describes how the system interacts with the sampled signals (the zero order hold input) that it receives. Further, the models will tell you how the system behaves at the sample instants, and will be difference equations and Z-transform transfer functions. (Willy and Millie suggest that you click here to see how to model the first order system as a sampled system if you don't know that already.)

Taking Millie and Willy's suggestions, the model for the first order system - in the sampled time-domain - is:

y_k+1 = e^-(T/^t)y_k+ G_dc(1 - e^-(T/^t)))u_k

In the Z-domain, the transfer function can also be calculated, but it may be better to define two terms to make the expression simpler. Define the following.

Let: a = e^-(T/^t)
Then: y_k+1 = ay_k + G_dc(1 - a)u_k

Then, from this difference equation, we can get a Z-transfer function:

(Y[z]/U[z]) = G[z] = G_dc(1 - a)/(z -a)

Now, we can use this sampled model in a block diagram of the system. Here is the block diagram we find.

You should note that the numerator is just a constant, albeit a more involved constant, and that the denominator has one pole in the z-plane. The root locus gain for the system is KG_dc(1 - a).

We have not lost analysis tools like the root locus. The root locus for this system is particularly simple and is shown below.

On this root locus, the pole at z = a is marked with an "x", and the root locus starts at that open loop pole and moves along the real axis to the left. At some gain - which we can easily calculate - the closed loop pole exits the unit circle - shown in green - and enters the region outside of the unit circle, which is the region of instability for a sampled system.

Now, we can begin to explain to Dr. Mallity what the problem is. We have a unique phenomenon here. This is a linear, first order system. If it were continuous it would be a wonderful system, getting better and better as the gain increases. It is not a continuous system. It is sampled and, in the sampled system, as the gain increases the closed loop pole crosses into the LHP - but still inside the unit circle - where the system exhibits oscillatory response. Eventually - at a high enough gain - the closed loop pole moves outside the unit circle and the system becomes unstable.