Control Theory: Separation principle and the Symbiosis between Controller and Plant

Relation between a Plant and a controller using signal flow modeling

Background story

A few years ago when I was a TA of undergraduate Control System courses, there was a term project in which students were asked to use state-space design to control the speed of a DC motor. I noticed something was up after a few students came up with a malfunctioning controller despite following the “design procedure”. It turns out that there was an important concept that was frequently omitted or misunderstood in the lecture.

A stable closed-loop system does not imply a “stable controller” in itself

To some extent, a plant might be necessary to “stabilize” the controller, suggested by the figure above.

Linear systems and the Separation principle [1]

To understand what's going on, we’ll need to recap some characteristics and basic theories of linear systems. Consider a linear plant and the corresponding Luenberger observer with an ideal model shown as the following equations. Notice in this simple case the observer was assumed to be an ideal one, that is the same passive dynamic(A), input matrix(B), and the output matrix(C).

Dynamics of a linear plant (first two-row) and an ideal linear estimator (last two-row)

The dynamics of the whole system can then be compactly written by stacking them together with an augmented state variable ([x, x_hat]). Followed by a linear full-state feedback controller(u=-K*x), we could further express the closed-loop system in a nice autonomous form depicted in the second row of the picture below. At last, the last row is a similarity transformation that emphasizes the dynamics of the error between states of the plant and the observer(e=x-x_hat).

Closed-loop dynamics of the state-space controller, assuming a linear controller u = -K*x

It is now clear from the last representation above, the closed-loop system poles are the individual eigenvalues of both block (A-BK) and block (A-LC). This implies a design of controller gain (K) is separated from the observer (L), hence the name “Separation principle”.

For linear systems, the design of the control law is separated from the construction of the estimator [1]

Open-loop dynamics of the Controller

Let’s considered a first-order system with a state-space controller as shown:

Closed-loop dynamics of a first-order system with state-space controller

Although this is a stable closed-loop system, the “controller itself” still has an unstable pole:

Dynamics of the first-order state-space controller, notice the unstable pole at +1

In this case, a “Plant” is necessary to stabilize a “Controller” through signal (y=C*x) and exhibit a closed-loop dynamics which is stable. No doubt this is usually not a favorable characteristic and often results in a problematic design.

We have reached our conclusion:

A stable gain (K & L) which guaranteed a stable closed-loop system by separation principle, does not imply a stable state-space controller without a plant

References

[1] S. H. Żak en S. H. Żak, Systems and Control. Oxford University Press, 2003.