Abstract
Frequency domain technique is adapted here together with an optimization scheme to design controllers for a class of linear timeinvariant singleinput singleoutput systems (SISO) with a loop delay, such that the closed loop system can be made stable in two distinct preselected delay intervals \([0,h_{1})\) and \((h_{2},h_{3})\), where \(h_{1}<h_{2}<h_{3}\). This stabilitycertifying approach is demonstrated over case studies, including an experiment.
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Notes
 1.
See discussion at the end of the section on how to extend the approach to higherorder systems.
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Appendix: selecting initial conditions for the optimization
Appendix: selecting initial conditions for the optimization
To find out the best way to select the initial condition, we first reduce the number of constraints and attempt to search for an initial condition point P satisfying some special cases. These cases are stated as follows:

I.
At \(\tau =0\), the delayfree system F(s, 1) is stable. In this case, the set of solutions to \(\chi \) is denoted by \(Q_{0}\).

II.
Denoted by \(Q_{1}\) is the set of possible solutions \(\chi \) rendering a crossing \(\omega _{c1}\) at a critical delay \(h_{1}\).

III.
Denoted by \(Q_{2}\) is the set of possible solutions \(\chi \) rendering a crossing \(\omega _{c2}\) at a critical delay \(h_{2}\).
Note that the three conditions I–III above do not guarantee the stability decomposition in Fig. 1, but only guarantee a crossing at \(h_{1}\) and \(h_{2}\), and the stability of the delayfree system. In Fig. 8, the intersection area of sets \(Q_{0}\), \(Q_{1}\) and \(Q_{2}\) is schematically pictured by
. Since the constraints corresponding to \(Q_{0}\), \(Q_{1}\) and \(Q_{2}\) are necessary conditions of \({\mathscr {C}}_{18}\), the feasible solution set satisfying \({\mathscr {C}}_{18}\), denoted by \(Q_{f}\), must lie inside the region
. Thus, a good initial condition would be to pick a point P inside of
.
When there are multiple design parameters, such as the case here, it may not be easy to select a point inside the parametric region
. Nevertheless, it is quite practical to select a point nearby the boundaries of
. Fortunately, there exist several ways to calculate the parametric settings, e.g., the points at \(P_{0}\), \(P_{1}\), \(P_{2}\). The point \(P_{0}\) satisfying Condition I is trivial to obtain from the stability analysis of the delayfree system. The points \(P_{1}\) and \(P_{2}\) follow the same type of mathematical calculations, and guarantee a single crossing either at \(h_{1}\) and \(h_{2}\). Conditions on system parameters guaranteeing a crossing can easily be obtained using frequency sweeping methods, see, e.g., [18]. Simply, set \(s=j\omega \), \(\tau =h_{1}\) or \(\tau =h_{2}\) in (15), and as \(\omega \) increases from zero to an upper bound, one can solve the real and imaginary part of the system characteristic equation from which an implicit formula can be obtained on system parameters satisfying the crossing \(s=j\omega \) at \(\tau \).
Application of the above general guidelines on the specific problem at hand leads to the following
Since it is not easy to solve (28), (29) simultaneously, we propose to pick \(\alpha _{0}={\omega _{c1}}^{2}\) and \(\alpha _{1}=0\) to simplify (28), (29) as
For the set \(Q_{1}\) and \(Q_{2}\), at \(\tau =h_{1}\), \(\omega =\omega _{c1}\) and at \(\tau =h_{2}\), \(\omega =\omega _{c2}\) respectively. Then, the following four equations obtained from (30) and (31) hold.
Since \(\omega _{c1}\), \(h_{1}\) and \(h_{2}\) are known in Step 1, a particular solution of \(\omega _{c2}\), \(\beta _{0}\), \(\beta _{1}\), \(\beta _{2}\) can be extracted from the above four equations, that is, \(\omega _{c2}\rightarrow \omega _{c1}, \beta _{0}=\beta _{2}{\omega _{c1}}^{2},\beta _{1}\rightarrow 0\). In this case, we can pick \(\beta _{2}\rightarrow 0^{+}\), then \(\beta _{0}\rightarrow 0^{+}\). With the choice of small \(\beta _{k}=\epsilon \), the coefficient of the exponential term \(e^{s\tau }\) will be vanishing, hence this point is at the boundary of the set \(Q_{1}\), e.g., point \(P_{1}\), and also at the boundary of the set \(Q_{2}\), e.g., point \(P_{2}\).
Considering the set \(Q_{0}\), with the same rules as above in picking \(\alpha _{k}\) and \(\beta _{k}\) values, when \(\alpha _{0}=\omega _{c1}\), \(\alpha _{1}=0\), \(\beta _{0},\beta _{1},\beta _{2}\rightarrow 0^{+}\) and \(\tau =0\), the characteristic equation in (15) approaches to
which comprises of an oscillating term due to \(s^{2}+\omega _{c1}^{2}\), and a stable term due to \(s^{2}+2\zeta \omega _{n}s+\omega _{n}^{2}\). Hence the system is marginally stable. This solution point is at the boundary of the set \(Q_{0}\), e.g., point at \(P_{0}\).
Since our parametric space is actually 6dimensional, the three points \(P_{0}\), \(P_{1}\) and \(P_{2}\) in Fig. 8 are indeed the same point, located at the boundary of the intersection area bounded by
. This point is therefore a good initial condition point to be used in the optimization scheme.
Furthermore, since \({\mathscr {C}}_{8}\) gives a constraint for \(\omega _{c2}\), it therefore makes sense \(\omega _{c2}\) is slightly less than \(\omega _{c1}\) for initial condition. In some sense, this lets the optimization go only in one direction (reduce \(\omega _{c2}\) further) as iterations take place along the \(\omega _{c2}\) direction, consistent with Remark 4.
For parameters \(\beta _{0}\), \(\beta _{1}\) and \(\beta _{2}\) in the numerator, they are assigned with a small positive value \(\epsilon \) as explained above. This corresponds to a weak controller, which initially guarantees that \(\alpha _{0}\), \(\alpha _{1}\) assignment above initially achieves its goal of satisfying \(Q_{0}\) as well as \(Q_{1}\) and \(Q_{2}\).
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Tian, D., Sipahi, R. Creating two disjoint stability intervals along the delay axis via controller design: a class of LTI SISO systems. Int. J. Dynam. Control 5, 1156–1171 (2017). https://doi.org/10.1007/s4043501602666
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Keywords
 Timedelay systems
 Stability
 Control design
 Optimization
Mathematics Subject Classification
 34C25
 34K20
 65K10
 93C23