A system is described by the following open-loop and closed-loop
transfer functions.
where p is an open-loop pole that we want to establish bounds for to ensure closed-loop stability. For example, that pole may be part of a compensator that we are trying to design, and closed-loop stability is always the minimum requirement that must be achieved by any design. Therefore, the bounds that we obtain for p will tell us the only allowable locations for that compensator pole; the rest of the design would then involve determining which of the allowable locations will satisfy other requirements.
The form of G(s) implies that p > 0 means that the pole is in the left-half of the s-plane, and p < 0 means that the pole is in the right-half plane. The closed-loop characteristic equation has the variable p appearing in the coefficients of s2 and s1.
The bounds on p can be determined by using the Routh
array and finding the conditions that make all elements in the first column of
the array have the same sign. The array is
Since the elements in the first column of the s3 row and s0 row are positive, then closed-loop stability requires that the terms in the first column of the other rows also be positive. Since each of those terms depend on the value of p, it is evident that there will be restrictions on the location of the open-loop pole in order to obtain closed-loop stability.
The s2 row will be investigated first since it
is the easiest. If the first term in that row is zero or negative, closed-loop
stability is lost. Therefore, that term must be positive, which means that
Based on this term only, p can take on certain negative values (RHP), 0, and all positive values (LHP). Since p also appears in the s1 row, this constraint is not the only one that needs to be satisfied.
Now we will look at the s1 row with the
second-order term involving p. This term also needs to be positive.
Since this term is a rational function in p, we need to consider the
entire term. However, since the denominator of that term is the same as the
term in the s2 row that we have already required to be
positive, we only need to consider the numerator of the term in the s1
row. Since this is a quadratic (second-order) term, there will be two
conditions that will need to be investigated.
This
function 
is a parabola opening
upward. For other systems, the quadratic could produce a parabola opening
downward. In either case, there are two values of p at which the function
changes sign. We want those values of p that produce positive values for f(p).
The easiest way to find those values is to set the function equal to zero and
factor the polynomial using the quadratic formula (or the roots function
in MATLAB).
Thus, there is one value in the right-half plane, p1, and one value in the left-half plane, p2, where the function changes sign. We need to determine whether those limits are upper bounds or lower bounds for stability.
Since the derivative of f(p) with respect to p is negative when evaluated at p1 and f(p1) = 0, the function is going from positive to negative at p1. Therefore, p1 is an upper bound on p for stability.
Since the derivative of f(p) with respect to p is positive when evaluated at p2 and f(p2) = 0, the function is going from negative to positive at p2. Therefore, p2 is an lower bound on p for stability.
We now have three bounds on p in order to obtain
closed-loop stability. For the two bounds from the quadratic function, only one
of them can be satisfied by any particular value of p, but one of them
does have to be satisfied; they can't both be violated and have a stable
closed-loop system. In addition to satisfying one of the bounds from the
second-order function, the constraint from the first-order function must be
satisfied by the same range of p values. These bounds are
The constraint p > -1 means that the constraint p < -3.7016 cannot be simultaneously satisfied. Therefore, that upper bound on p is not usable. The constraint p > -1 and the constraint p > 2.7016 can both be satisfied by imposing the more restrictive of those two constraints. Therefore, the final and only constraint on the location of the open-loop pole p in order to obtain closed-loop stability is
This says that the open-loop pole must be in the left-half of the s-plane, to the left of s = -2.7016.
The plots
of the constraint curves graphically illustrate why this result is correct.
The parabola representing the quadratic function is seen to pass through zero at
values of p = -3.7016 and p = +2.7016. Between those two values
the function is negative, so the numerator of the term in the s1
row would be negative. The straight line representing the first-order function
passes through zero at p = -1, with the function being positive for
values to the right of that point. If this function is negative, the first term
in the s2 row is negative, and closed-loop stability is lost.
For closed-loop stability, both of the functions must be positive for the same
values of p. The graphs clearly indicate that this will occur only when p
> 2.7016.
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Lastest revision on
Wednesday, June 7, 2006 12:01 PM
