Robust Control - 1
(Using the Kharitinov Stability Theorem)
Introduction
The Kharitinov Stability Theorem
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Introduction

      Whenever you have a system that you want to control, you need to consider that fact that no system you control ever stays exactly the same.  Over time, maybe as temperature or humidity changes, a system will change characteristics.  Or maybe the system will change characteristics due to changes in operating conditions.  For example, the ailerons in an airplane (one of the control surfaces) are less effective when the air is thin at high altitudes, or when the airplane's speed is lower.  For whatever the reason happens to be, it is common to encounter systems that exhibit changes in their characteristics - because all systems change for some reason.  What's more, you need to account for those changes when you design a system.

        When you design a system there are a number of things that you don't want to happen as the system.  The most important thing is that you do not want a system to become unstable as the system changes.  This problem is so important, that a concept called robust stability has been defined.

Let's take a look at what that might mean for some simple systems.
Example 1

        In this system, the system being controlled (represented by G(s)) is a third order system.

G(s) is given by:

G(s) = GDC/(s3 + 3s2 + 3s + 1)

Now, let us assume that you design the system and choose a gain for the proportional controller, K.  In the meantime, the DC gain of G(s) can vary by 20% in either direction from a nominal value of 1.  In other words:

Gmax(s) = 1.2/(s3 + 3s2 + 3s + 1)
Gmin(s) = 0.8/(s3 + 3s2 + 3s + 1)

When you design the controller (i.e. choose a value for the gain, K) you need to take that variation into account.  You might design a system for a fixed gain at the nominal values only to find that the system becomes unstable as the parameters change within their normal variation from the nominal.

        In this example, the system is stable as long as the total gain (0.8K up to 1.2K) remains less than 8.0.  In that situation, the transfer function, Gmax(s), dictates what gain you can set, and the maximum K is given by:

Kmax = 8/1.2 = 6.667

        Now, if the system has other parameters that change, the situation can get considerably more complicated.  Let's consider another example.

Example 2

        In this system, the system being controlled (represented by G(s)) is again a third order system.

This time, G(s) is given by:

G(s) = GDCt/[(s2 + 2s + 1)(ts + 1)]

The system contains a time constant that is variable, and which can be anything from 0.5 to 2.0.  (And, when the time constant changes, the open-loop DC gain does not change because of the time constant in the numerator.)  In addition, as in Example 1, the DC gain of G(s) can vary from 0.8 to 1.2 - but not because of the time constant changing..  Can we set a gain that will ensure that the closed loop system remains stable as the open loop parameters change?  Let's examine what that means exactly.

        The closed loop transfer function for this system is:

GCL(s) = K/[(s2 + 2s + 1)(ts + 1) + K]
GCL(s) = K/[(s2 + 2s + 1)(s + 1/t) + K]
GCL(s) = K/[s3 + (2 + 1/t)s2 + 3s + 1/t + K]

Now, we can define the problem a little better.  Let's put limits on the coefficients of the powers of s.

The situation isn't even that simple because the s0 term is can only take on values from 2.8 to 3.2 when the s2 term is at 4.0.  In other words, the two coefficients do not vary independently.  Because of that we might end up putting more stringent conditions on the gain than we would without that non-independent interaction.

        From the example, you can see that what this can come down to is that you need to examine the polynomial in the closed loop denominator, and you need to determine gain values that will ensure stability as the coefficients in the closed loop denominator polynomial vary.  At first, it might seem that you would need some sort of analysis that examines all possible combinations as the coefficients vary.  That would mean that you would have to examine - or account for - an infinite number of possibilities.  For example, knowing that the system is stable when all of the coefficients are large (or when they take on their smallest values) might not guarantee stability for intermediate values of the coefficients.  The task seems overwhelming.  Fortunately, there is a result in the research literature that permits you to determine stability for all possible combinations by examining stability for (at most) four polynomials.  We will look at that result next.  We'll state the result and then apply it to the examples we have put on the table above.

The Kharitinov Stability Theorem

        To use the Kharitinov Stability Theorem we need a few definitions.

Definition:

        An uncertain polynomial is one in which the coefficients can lie within specified intervals.  For example, an uncertain cubic polynomial can be represented as:

P3(s) = (p0 + q0)s0 + (p1 + q1)s1 + (p2 + q2)s2 + (p3 + q3)s3

The nominal values of the coefficients are the p's, and the variations from nominal values are the q's in the above polynomial.  Each q is within a specified interval:

qi is in the interval [q-,qi+]

        In Example 2 above, we would have:


        Now, we can see that the example is a special, very limited case of an uncertain polynomial.  Nevertheless, the Kharitinov Stability Theorem applies.  Here is a statement of the theorem.
The Kharitinov Stability Theorem

        Given an uncertain polynomial (with known bounds on the coefficients), the polynomial will have all roots in the left-half-plane (i.e., it will be the denominator of a stable transfer function) if the four Kharitinov polynomials are all stable (i.e. have all of their roots in the left-half-plane).  The four Kharitinov polynomials are given by:

If you examine these polynomials carefully, you should realize the following. The net result is that you need only test the four Kharitinov polynomials for their root locations.  If all test out to have all of their roots in the left-half-plane, then any polynomial with coefficients within the intervals of the uncertain polynomial will also have only roots in the left-half-plane.

        Let's go back and re-examine our second example.
Example 2 - Continued

We had:

P3(s) = [1.3,3.2]s0 + 3s1 + [2.5,4.0]s2 + 1s3

Notice that the odd powers never change in this example (not true in general).  Since that is the case, we have:

P--(s) = P-+(s)
and
P+-(s) = P++(s) =

and we need only consider two polynomials.  Those two polynomials are:

P3--(s) = 1.3s0 + 3s1 + 2.5s2 + 1s3

and
P3+-(s) = 3.2s0 + 3s1 + 4.0s2 + 1s3

We can check stability for these two polynomials any way at all.  One way is to check the Routh criterion.  Here is the Routh array for P3--(s).
 

s3
1
3
s2
2.5
1.3
s1
2.48
  
s0
 1.3
  

And, here is the Routh array for P3+-(s).
 

s3
1
3
s2
4
3.2
s1
2.2
  
s0
 3.2
  

Both systems are stable, so we can conclude that the original system will be stable for any variation within the limits noted above.


        Now, the example above is still relatively simple, and we have to consider what happens in higher order systems.


Problems
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