MIGRATION OF THE CLOSED LOOP POLES THROUGH THE IMAGINARY AXIS: A FIRST SIMPLE METHOD

Nevertheless, it can be remarked that the real part A*R of A* is prob­ably a good initial guess in the search of a model perturbation Ад, which renders the matrix I — М()ш)Дд singular. There is however a technical difficulty, which is to decide when a matrix X is singular. An obvious solution is to compute the magnitude of the determinant det(X), the minimal singular value g_{X), or the condition number and to de­cide that X is singular when one of these quantities is lower than a given value. The value of this threshold is however difficult to determine, and it may depend on the problem data.

If the aim is to compute a lower bound of the maximal s. s.v. over the frequency range, rather than a lower bound of n&R(M(ju>)) at a fixed frequency w, a natural solution is to remember thatMO’w) is the value of the transfer matrix M(s) at s = ju. Let (А, В, C, 0) a state-space repres­entation of M(s), which is here again assumed to be strictly proper just for the ease of notation. It can be expected that one pole of A + BA*RC is close to the point ju of the imaginary axis, if a lower bound of the regularized s. s.v. fj,^(H(ju>)) was computed.

A very simple solution is thus to increase the size of the model per­turbation until one pole of A + aBA*RC crosses the imaginary axis at the point jCj (for a > 1). Since one pole of A + BA*RC is expected to be close to the point jio of the imaginary axis, the value of a, for which a pole of A + aBA*RC crosses the imaginary axis, is possibly close to 1.

In an obvious way, a lower bound of R(M(ju)) was computed in­stead of a lower bound of цаR(M(ju)). But more importantly, a model perturbation aA*R was obtained, which brings one closed loop pole on the imaginary axis. The inverse of the size of otA*R is thus a lower bound of the maximal s. s.v. over the frequency range.

The method is thus the following. A frequency gridding is first chosen, as usually in ц analysis. A regularized fi lower bound is computed at each point of this gridding, and the real part A*R of the augmented model per­turbation, which is provided by the power algorithm, is extracted. The value of a is then increased from the initial value of 1 until one pole of A + aBARC crosses the imaginary axis.

At each point of the frequency gridding, a lower bound of the maximal s. s. v. over the frequency range was thus obtained as the inverse of the size of the model perturbation aAR. The best estimate of the maximal s. s.v. over the frequency range is finally chosen as the highest value of this lower bound over the frequency gridding.

Remarks:

(i) The problem is slightly more complex in practice. There exist indeed two ways of optimizing a. In the above approach, a is increased from its nominal value of 1, until the target pole crosses the imaginary axis (.i. e. the pole of A + BA*rC, which is the closest to the point ju of the imaginary axis, if a lower bound of the regularized s. s.v. /хд (H(jw)) was computed). It is however possible that some other poles of A 4- BA*RC were found to be strictly unstable. In this case, an alternative is to decrease a until all unstable poles cross the imaginary axis. This ap­proach provides a smaller destabilizing perturbation, but the associated frequency w is no longer guaranteed to be close to the initial value oj.

(.ii) The method can be readily extended to the problem of robust stabil­ity inside a region fi of the complex plane (typically a truncated sector).

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