Delaymargin design for the general class of singledelay retardedtype LTI systems
Abstract
For a given delay value \(\bar{\tau }\), an approach is developed here to compute the isoparameter settings of the general class of singledelay retardedtype LTI system such that this system can have a delay margin of precisely \(\bar{\tau }\), and is thus guaranteed to be asymptotically stable for all positive delays less than \(\bar{\tau }\).
Keywords
Delay margin design Stability Linear delay system1 Introduction
Time delays exist in many dynamical systems encountered in engineering, physics, biology, and economics [1, 2, 3, 4, 5, 6, 7]. As was extensively reported in the literature, when delays enter in a feedback loop in a system, e.g., via a closedloop control architecture, the system may perform poorly and even become unstable. Many studies were hence devoted to the understanding of how delay affects the stability and closedloop control of systems with delays [8, 9, 10, 11, 12, 13].
Analyzing the stability of systems with delays can be challenging even for linear time invariant (LTI) systems. This is mainly due to the difficulties in managing, solving, handling some infinite dimensional eigenvalue problems arising from the analysis. One may choose to follow either one of the two research mainstreams to investigate stability, namely, those based on timedomain analysis [8, 10, 11, 14], and those based on frequency domain [13, 14, 15]. These approaches have their respective superiorities, yet frequency domain techniques have been most preferred in cases when one wishes to analyze stability consistent with the Nyquist stability criterion, see, e.g., [12]. Along these lines, existing work is mainly focused on revealing the stabilityinstability decomposition of a given system with respect to a single or multiple system parameters. For instance, some studies [4, 12, 13, 15, 16, 17, 18, 19, 20] focus on revealing for which delay values \(\tau \) an LTI system with single delay maintains its stability, whereas those in [4, 12, 13, 21] explore the stability of the system with respect to system parameters. Moreover, the authors in [22] study the maximum delay margin achievable in a class of LTI systems with single delay, while in [4, 13, 16, 23, 24, 25, 26, 27, 28, 29] the main objective is to reveal the “regions” in multipledelay parameter space, where the LTI system is stable. The cited studies utilize various mathematical techniques to handle the difficulties inherently present in the stability analysis, all of which ultimately rely on two important fundamental theoretical results, namely, the continuity property of the arising eigenvalue problems in the parameter space [9], and the associated Ddecomposition/\(\tau \)decomposition theorem [13, 17].
Continuity property of system eigenvalues with respect to system parameters allows one to handle the stability analysis by focusing on whether or not these eigenvalues ever cross the imaginary axis of the complex plane. If they do, then the system may switch from stability to instability, or vice versa, depending on in which direction such eigenvalues move across the imaginary axis, with respect to the parameter of interest. This property then supports the Ddecomposition/\(\tau \)decomposition theorem by which one should expect to find countably many regions in the parameter space, where in each region the system can possess only a finite number of unstable eigenvalues, \(N\). Whenever \(N=0\) in a region, the system is stable for all the points in that region, and the boundaries that separate these regions are formed by some critical parameter values that impart imaginaryaxis eigenvalues.
Several stability analysis techniques stemming from frequency domain, to a certain degree, can also be adapted for parametric analysis. For instance, one can relax a system parameter \(p\in \mathbb {R}\), and investigate how the system stability along the delay axis \(\tau (p)\ge 0\) is affected as the parameter varies in a range. This can be easily implemented within a stability analysis framework, and, as expected, would require a parameter sweep for \(p\). While this approach can be satisfactory in some cases, it may not serve well for a general parametric design/synthesis problem in which one actually knows the precise amount of delay \(\tilde{\tau }>0\) in the control loop, and wishes to utilize this information to design the parameters \(p(\tau )\) of the system. This is achieved as early as in 1960s on special classes of LTI systems with single delay by the work of Popov [12] pg. 437, as well as in [30] on a secondorder system using the generalized HermiteBiehler theorem. In such cases, one would desire to know the parameter settings of the system for which the system would maintain its stability for \(\tau \in [0, \bar{\tau })\), where \(\bar{\tau }>\tilde{\tau }\). That is, the control designer could select a delay value \(\bar{\tau }\) such that \(\bar{\tau }>\tilde{\tau }\), and reveal the isoparameter curves \(\bar{p}(\bar{\tau })\) along which the system at hand is in transition from stability to instability. The settings on these curves would hence guarantee the stability of the system for \(0\le \tau <\bar{\tau }\).
To the best of our knowledge, for a given fixed \(\bar{\tau }\) as described above, and also known as “delay margin” [15], a parameter design approach for the general LTI systems with a single delay has not been thoroughly investigated, and this article is devoted to address this design problem. Here, we focus on the retardedtype systems, however the presented approach could be expanded to handling neutral type and fractional type systems as well, paying attention to the peculiarities in eigenvalue behaviors of these types of systems [11, 31].
In this article, we demonstrate that Rekasius transformation [32], which was utilized for various purposes [19, 28, 29, 33, 34, 35, 36, 37], can also be used for the objective at hand, yet the implementation requires several nontrivial steps. Once these steps are tailored together, it becomes possible to reveal the isoparameter curves of the LTI system at hand, where on the curves, the system is oscillatory at a frequency \(\omega \) for the given delay value \(\bar{\tau }\), and is stable \(\forall \in [0,\bar{\tau })\). This inverse problem is indeed difficult to solve since one should find an approach to make sure that all the system eigenvalues except those on the imaginary axis \(\mp j\omega \) are on the lefthalf plane, and at the same time, the delay value that places the eigenvalues on the imaginary axis is precisely at \(\tau = \bar{\tau }\), while making sure that stability is maintained \(\forall \in [0,\bar{\tau })\).
Notations used in the text are standard. We have \(\mathbb {R}\) for the set of real numbers, and open lefthalf of complex plane \(\mathbb {C}\) is represented by \(\mathbb {C}_\). We use \(s\in \mathbb {C}\) for the Laplace variable; the delay is denoted by \(\tau \), while \(T\) stands for the parameter of the Rekasius transformation [32]. Arguments are omitted for easier reading when no confusion occurs.
2 Preliminaries
In this section, we first formulate the problem, and present some of the wellknown fundamental concepts regarding the stability of the LTI system investigated.
2.1 Stability of a singledelay retardedtype LTI system
2.2 Rekasius substitution and some properties of stability transitions
Equation (6) indicates that for some \(T\) and \(\omega \) satisfying (5) at \(s=j\omega \), there exists infinitely many \(\tau \) solutions found from (6) that also satisfy (2) at \(s=j\omega \), see [19, 29]. In other words, each \((T, \omega )\) pair that satisfies (5) for \(s=j\omega \) maps to points \((\tau _0, \tau _1, \ldots ,\tau _q,\ldots )\) found from (6). These points mark the delay parameter space, bisecting it into intervals, where in each interval the system is either stable or unstable. These properties comply with the very fundamentals of the stability properties of retarded class systems based on \(\tau \)decomposition principle [12, 13, 17].
Remark 1
Notice that the approaches described in the literature including those stemming from Rekasius transformation are implemented to calculate the stableunstable regions of the LTI system along the delay axis, see cited studies above. That is, calculation of the delay values is an “end result” in these approaches. Nevertheless, solving the reverse problem, that is, synthesizing the stability/instability decomposition of the delay axis with \(k_z\) being the endresult of the design is not straightforward, as we detail in the following section.
2.3 Delay margin and number of crossings
The delay margin \(\bar{\tau }\) of the system (1) is the smallest positive delay value at which the delayfree stable system transitions to instability [15]. That is, it is required that the system is stable when free of delays (\(\tau = 0\)), and in that case, the delay margin is defined as \(\bar{\tau } = \min _{k}(\tau _{k,0})\).
One critical point here is the number of crossings \(m\). If \(m=1\), it is easy to detect the delay margin \(\bar{\tau } = \tau _{k,0}\); however when \(m>1\), then the delay margin is the smallest of \(\tau _{k,0}\), \(k=1,\ldots ,m\). In the next section, this will play an important role in designing the system parameters for a given delay margin.
The number \(m\) can be determined in various ways [38], one being borrowed here from [18]:
Lemma 1
[18] The number of imaginary axis crossings \(m\) can be at most \(2n^2\), where \(n\) is the dimension of (1).
Lemma 2
[18] Let \(S = \{\lambda _1(\Pi ),\ldots , \lambda _{2n^2}(\Pi )\}\). Then it holds that \(\Omega \subseteq S\).
In other words, to find the number of crossings^{1} \(m\), one can first compute the eigenvalues of matrix \(\Pi \), and next follow [18] to verify which eigenvalues are in the set \(\Omega \).
Further details regarding the detection of the delay values \(\tau _{k,q}\) in (9). can be found in [5, 13, 38] and the cited studies in the previous section regarding analyzing the stabilityinstability composition of the delay axis for fixed \(k_z\).
3 Main results
In this section, the objective is to reveal an approach by which one can design the parameters \(k_z\in \mathbb {R}\) in (2) such that the system represented by (2) is stable for \(\tau \in [0, \bar{\tau })\), given \(\bar{\tau }\). Note that we restrict this design to synthesizing the delay axis only for the delay margin \(\bar{\tau }\), and future work will be focused on designing multiple stability intervals along the delay axis.
 (a)
Equation (12) is transcendental with infinitely many solutions \((\omega ^*, k_1, \ldots , k_z, \ldots , k_{\bar{z}})\), whether or not they all lie in real domain,
 (b)
One should find a specific solution of \(\omega ^*\in \mathbb {R}\) and \(k_z\in \mathbb {R}\) such the infinitely many roots of (2) for \(\tau = \bar{\tau }\) have negative real parts, except those at \(s=j\omega ^*\).
 (c)
A feasible solution of parameters \(k_z\) guaranteeing the desired delay margin \(\bar{\tau }\) may not exist.
 (d)
The system may have multiple crossings, \(m>1\), making the design process complicated.
3.1 Derivation of design rules
The proposed design procedure is constructed based on the following logic line:
Given the known delay value \(\tau = \tilde{\tau }\) in the control loop in (1), the control designer selects a finite delay \(\bar{\tau }\), where \(\bar{\tau } > \tilde{\tau }\), and calculates the set of system parameters lying on some isoparameter curves corresponding to system’s stability margin \(\bar{\tau }\) such that the same system is for sure guaranteed to be asymptotically stable \(\forall \tau \in [0, \tilde{\tau }]\).
Let us first identify the necessary rules that need to be respected for a feasible solution to exist:
Rule 1: Since it is required that the system is stable for \(\tau = 0\), one needs to guarantee the stability of the delayfree system. This requires that \(A+B\) is a Hurwitz matrix, or equivalently, the roots of \(f(s;1)=0\) all lie in \(\mathbb {C}_\). This part of the analysis can be trivially established using the RouthHurwitz stability criterion [40].
Since we guarantee the stability of the delayfree system, we know that the system cannot have an eigenvalue at \(s=0\) for any finite delay in \(\mathbb {R}_+\) [41]. Moreover, without loss of generality, we can focus only on the positive real axis of the \(\omega \) parameter, thus we have \(\omega > 0\) in the rest of the text.
Lemma 3
Given \(\bar{\tau }\), the solution \(\omega ^*\) of (12) satisfies the condition \(0\le \omega ^*<2\pi /\bar{\tau }\).
Proof
Notice that the positive delay solutions satisfying (2) for \(\omega ^*\) when arranged in increasing order on the delay axis should have the following properties. First of all, the minimum positive delay solution, i.e., the positive delay value closest to the origin of the delay axis, must be the predetermined value \(\bar{\tau }\). Moreover, for delay \(\bar{\tau }\) to really be the minimum positive delay, it is necessary that the delay value preceding \(\bar{\tau }\) is negative, which requires, from the inspection of (6), that the inequality \(\bar{\tau }  \frac{2\pi }{\omega ^*}<0\) holds, which proves the lemma. \(\square \)
Rule 2: In light of Lemma 3, it is necessary that \(0<\omega ^*<2\pi /\bar{\tau }\) holds.
Lemma 4
Proof
Use (6) to calculate the minimum positive delay value as follows. If \(\phi >0\), then \(\pi /2>\arctan (\phi )>0\), and \(q=0\) in order to calculate the minimum positive delay value. But if \(\phi <0\), then \(\pi /2<\arctan (\phi )<0\). This requires to set \(q=1\) so that the delay value can still be made minimum and positive. Other values of \(q\) render positive delays but violate the minimum positivity condition, can hence be ignored. \(\square \)
Remark 2
Notice that \(\phi =0\) is ignored here as it requires that the characteristic equation has an imaginary root \(s=j\omega ^*\) for either \(\omega ^*=0\), or \(T=0\). Notice from [19, 32] that \(T=0\) and \(\tau =0\) give rise to identical characteristic equations as a special case. As per Rule 1, since the delayfree system is asymptotically stable, having imaginary roots \(s=j\omega ^*\) for \(T=0\) is not possible. Furthermore, it is known from [41] that \(\omega \rightarrow 0\) as \(\tau \rightarrow +\infty \), which does not comply with the assumption that \(\bar{\tau }\) is finite.
 (a)
Find the value \(m\).
 (a1)
If \(m = 1\), then proceed to step (b),
 (a2)
If \(m>1\), then compute \(\tau _{k,0}\) from (9),
 (a2.1)
If \(\bar{\tau } \le \min _{k=1\ldots m}{\tau _{k,0}}\), then \(\bar{\tau }\) is indeed the smallest possible delay causing a loss of instability, as desired in the design process, and hence one proceeds to step (b);
 (a2.2)
If \(\bar{\tau } > \min _{k=1\ldots m}{\tau _{k,0}}\), then this means there exists a different and smaller positive delay value that initiates a loss of instability, and hence one should proceed to step (c);
 (b)
The value \(k_z\) is feasible.
 (c)
The value \(k_z\) is not feasible.
Corollary 1
 (a)
\(m = 1\); or
 (b)
\(m >1\) and \(\bar{\tau } \le \min _{k=1\ldots m} {\tau _{k,0}}\) holds.
Proof
First step of the proof follows from the fact that a common root \(k_z\) of (18)–(19) is also a root of the original characteristic Eq. (12), as per the properties of Rekasius substitution [29]. Moreover, since Rule 1 holds, the system is stable for \(\tau =0\), and since Rule 2 holds, the crossing \(\omega ^*\) occurs at \(\tau =\bar{\tau }\), which is the smallest positive delay value. Since \(\omega ^*\) is enforced by design, we have \(m\ge 1\). If \(m=1\), then this is the enforced crossing, and there are no other imaginary axis crossings, and thus \(k_z\) is on the isoparameter curve (item (a)). However, if \(m>1\) for a \(k_z\) value satisfying (18)–(19), then it is necessary to check whether or not any additional crossings violate the intended delay margin design. As explained above, as long as additional crossings occur at delay values larger than \(\bar{\tau }\), the intended design will be successful (item (b)). Conversely, if there are more than one crossing, \(m>1\), with at least one delay value less than \(\bar{\tau }\), then the intended delay margin design fails, similar to when Rule 2 does not hold. If Rule 1 does not hold, then the system does not have a delay margin by definition, and since the Rekasius substitution is a twoway condition [29]; the design fails if (18)–(19) do not have common \(k_z\) roots, which would imply that a solution \(k_z\) does not exist satisfying the original characteristic Eq. (12). \(\square \)
Notice that equations in (18)–(19) are multivariate polynomials, and in general, for more than three independent parameters, the complete solution set may not be easily obtained. Nevertheless, the analytical form of these polynomials can be used to perform various parametric studies, for instance, for designing the \(PI\) gains controlling a plant with transport delay [30]. Moreover, if one wishes to explore the parametric variations in \(k_z\) starting from a known initial point \(k_z = k_{z,0}\), then it would be possible to explore the common solutions of the hypersurfaces (18)–(19) by implementing a continuation algorithm. These types of increaseddimension issues are inevitable, and arise in various multiparameter problems, where parametric sweeping is the only choice; see a discussion on this in [26, 43].
In the case when there are less than three independent variables, \(z<3\), more tractable results are possible as explained next:
3.1.1 One independent variable \(k\)
A single parameter design is the simplest case, however it carries some inherent challenges since there is only one free parameter that should satisfy (18)–(19) simultaneously. Due to this reason, an explicit solution of \(k\) is not possible in general, and one should use graphics to determine the common solutions, as detailed next.
Notice that the above approach requires to plot the error function \(e(\phi )\) with respect to \(\phi \), and to use a nonlinear solver to compute the zero crossings of the error function along the \(\phi \)axis.
3.1.2 Two independent variables \(k_1\) and \(k_2\)
3.2 Case study 1
3.2.1 \(a\) is fixed, and \(b\) is the only free parameter
Lemma 5
For a given \(a\) and \(\bar{\tau }\), the system represented by the characteristic equation in (25) can be made stable for \(\tau \in [0,\bar{\tau })\) at most by two \(b\) values.
Proof
3.2.2 \(a\) and \(b\) are free parameters
Lemma 6
 (i)
\(a+b>0\);
 (ii)
The parameter pair \((a,b)\) lies on the contours parameterized by \(\phi \),
Proof
Corollary 2
The parameters \(a\) and \(b\) that make the system represented by (25) stable for \(\tau \in [0,\bar{\tau })\) are unbounded in positive real domain.
Proof
Proof follows from the fact that, since \(\omega ^*\) is always upper bounded as per Rule 2, the term \(\phi ^2\) dominates the numerator of (32)–(33), and that a feasible \((a,b)\) pair always exists \(\forall \phi > 0\). \(\square \)
Lemma 7
There exists at least one \((a,b)\) pair for which the finite delay margin \(\bar{\tau }\) of the system represented by (25) can be made arbitrarily large.
3.3 Case study 2
Lemma 8
The \(PI\) gains, if any, that achieve a delay margin \(\bar{\tau }\) in the closedloop system with (37), are upper bounded.
Proof
The proof follows from the fact that \(\omega ^*\) is upper bounded, and that \(\phi ^2\) terms appear in both the numerators and denominators in (38)–(39). \(\square \)
The above lemma shows a contrast to the scalar case discussed in the previous subsection where we show that the \((a,b)\) parameter settings could be unbounded for a desired delay margin in the system. This is mainly because there exists an interesting interplay between the parameter \(\phi \) and the way the controller as well as the characteristic equation is structured, which eventually determines the power of \(\phi \) in the closedform expressions of the system parameters. This ultimately determines the boundedness of these parameters for ever achieving a certain delay margin in the system.
4 Computations
4.1 Example 1

\(\beta >0\),

\(\alpha +\beta >0\),

\((\beta +\alpha )^2\beta > 0\).
One key observation here is that, since the feasible \(\omega ^*\) solution is always upper and lower bounded with respect to \(\phi \), and since \(\phi ^2\) term appears both in the numerator and denominator of (41)–(42), admissible \(\alpha \) and \(\beta \) control gains can be large, if they exist, but will have to be upper bounded for a given \(\bar{\tau }\).
4.2 Example 2
We move forward with additional parametric studies. In particular, we are interested in how the isoparameter curves change with respect to \(a\), which is inversely proportional to the openloopplant bandwidth \(1/a\). For this, we investigate combinatorically \(\alpha = 4\), \(\alpha = 1\), and \(\alpha = 0.1\) values with respect to the delay values \(\bar{\tau }=1\), \(\bar{\tau }=2\), and \(\bar{\tau }=4\). The isoparameter curves are shown in Fig. 5. In this figure, firstly, we observe that, as the bandwidth of the system increases, that is, when \(\alpha \) becomes smaller, then the \(PI\) controller gains must be reduced in order to maintain the same delay margin in the closedloop system. Moreover, when the stabilizing \(PI\) gains are set in the vicinity of \(k_p = 1\) and \(k_i = 0\), i.e., the delayfree closedloop system poles are stable but they are near the imaginary axis of the complex plane, then we observe that all the contours merge into each other, showing that controller parameter space becomes extremely sensitive to small variations, which, in the case of modeling uncertainties, could destabilize the system.
Sensitivity problems arise also in the case of \(\alpha =0.1\), when \(k_p\) is selected in the vicinity of \(k_p=1\). Based on the geometric shapes observed in these figures, and/or using the closedform expressions of \(k_p\) and \(k_i\) gains, we can also study and judge the parametric space where such sensitivity issues are less dramatic, and less likely to be an issue. For instance, for \(\alpha = 4\), in order to achieve a delay margin of \(\bar{\tau }=1\), it makes more sense to select \(k_p= 5\) and \(k_i\approx 2.9\), instead of choosing \(k_p= 0.9\) and \(k_i\approx 0.1\), showing interestingly that larger gains can be more feasible for the robust operation of the closedloop system.
4.3 Example 3
Example 3. The settings that render the closedloop system in (44) marginally stable for the delay value \(\bar{\tau } = 0.5\), and stable for \(0\le \tau <\bar{\tau } = 0.5\)
\(\omega _n\)  \(\zeta \)  \(\alpha \)  \(\omega ^*\) 

1  \(0.4\)  \(0.3556\)  \(2.5206\) 
1  \(0.7\)  \(0.4872\)  \(2.9350\) 
1  \(0.9\)  \(0.5652\)  \(3.1428\) 
10  \(0.4\)  \(2.0263\)  \(7.1514\) 
10  \(0.7\)  \(3.0977\)  \(6.7518\) 
10  \(0.9\)  \(3.8177\)  \(6.5850\) 
100  \(0.4\)  N/A  N/A 
100  \(0.7\)  N/A  N/A 
100  \(0.9\)  N/A  N/A 
5 Conclusion
An approach for the reverse problem of revealing the isodelay margin contours in the system parameter and/or controller parameter space of the general singledelay retardedtype LTI system is presented. With this approach, not only one obtains the isoparameter curves that create the same delay margin in the system, but one can also stabilize such systems, simply by knowing the delay \(\tilde{\tau }\) in the closedloop system, and then selecting \(\bar{\tau }\) such that \(\bar{\tau } > \tilde{\tau }\) in order to calculate the parameters corresponding to a delay margin of \(\bar{\tau }\), which will ultimately guarantee for the same set of parameters that the system is stable at the delay value \(\tilde{\tau }\). Extensions of this work to neutral class and fractional type LTI systems can be established by considering additional features of such systems published in the literature.
Footnotes
 1.
Note that Lemma 1 considers both positive and negative values of \(\omega \). Hence, the number of positive \(\omega \) solutions is at most \(n^2\), since the roots of the characteristic equation must be symmetric, see also [38] for discussions.
 2.
The fact that \(\tau \omega \in [0, 2\pi ]\) was previously used for “analysis purposes”, within the ‘Building Block’ concept in [41].
 3.
This is analogous to “frequency sweeping” ideas introduced in [42], except that the frequency \(\omega \) here is scaled by the pseudodelay parameter \(T\). This may also resemble the Building Block concept [41], yet in the previous work, \(\tau \omega \) is the sweep parameter, and the delay margin \(\bar{\tau }\) is to be calculated as an end result but is not fixed in the problem formulation.
 4.
This is possible since the delay \(\bar{\tau }\), \(\omega ^*\), and the “scaled” frequency \(\phi \) are numerically known. We introduced this idea for a different purpose in [28] mainly for taking 2D crosssectional views of stability maps of LTI systems with more than three independent delays.
 5.
Note that by design we are enforcing one crossing in the system, hence \(m\ne 0\).
 6.
Notes
Acknowledgments
The author would like to thank Wei Qiao and Adrián Ramírez for their careful proof reading of the numerical examples in this manuscript.
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