# Spectrum design using distributed delay

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- Accepted:

## Abstract

This paper focuses on the problem of designing the rightmost eigenvalues of linear scalar distributed delay systems. We consider two different but complementary methods: generalized stability charts and matrix Lambert W functions. The generalized stability charts are based on the intersection of Hopf and fold surfaces that provide important insight into the problem, but the geometry of the surfaces may become complicated for certain delay distributions. The Lambert W function approach can be applied to general delay distributions, but requires numerical solutions which can suffer from convergence problems. We present some examples using both approaches.

### Keywords

Spectrum design Distributed delay Generalized stability chart Lambert W function## 1 Introduction

In many engineering applications one tries to achieve a desired dynamic behavior, e.g., settling time in the vicinity of an equilibrium, by designing the eigenvalues of the corresponding linear systems. Applications include vibration absorbers [1, 2], machining processes [3, 4], vehicle steering [5, 6], and connected vehicles [7, 8]. Of special interest are the rightmost eigenvalues that are also called dominant or leading eigenvalues. These determine the linear stability of the equilibrium and, for a stable system, they correspond to solutions with the slowest decay [9]. Thus, at the linear level, the desired dynamic behavior may be achieved by selecting the system parameters (including control gains) so that the rightmost eigenvalues are placed appropriately.

However, systems with time delay exhibit an infinite spectrum, which makes eigenvalue placement very challenging, even though the dynamic response is still dominated by the rightmost eigenvalues [10, 11, 12]. Here, apart from designing the system parameters, one may also tune the delays in order to place the eigenvalues appropriately [13, 14], though limitations may arise due to the achievable minimal value of the delay. Such limitations may be compensated for in systems with distributed delays where apart from system parameters one may design the delay distribution [15, 16]. Practical examples include the design of the cutting profile of helical milling tools in machining [3, 4], where the delay distributions originate from spatial force distributions. Similar ideas may also be used when designing transmission protocols for vehicle-to-vehicle communication in connected vehicle systems [7, 8]. Here the delay distributions originate from stochastic delay variations.

In this paper, for the first time, we investigate the problem of spectrum design for the rightmost eigenvalues of a linear scalar system with distributed delay. We apply two complementary approaches: the bifurcation theory approach and the Lambert W function approach. Most of this paper concentrates on the special case when we have a constant delay distribution function. For this case, the methods and results of both approaches are illustrated. In the bifurcation theory approach, we extend the idea of a stability chart [17, 18] for a general case when the leading eigenvalues have non-zero real part. That is we translate the design problem to finding the locations of generalized Hopf and fold bifurcations in parameter space. Design limitations are also investigated using numerical continuation and multi-dimensional bisection method. As for the Lambert W function approach, we reformulate the problem as a special case of a delayed system with a single constant delay and then apply the matrix Lambert W function method [12]. Here, by exploiting the structure of the arising system matrices, we accelerate the numerical computation. We also address some numerical issues arising due to multiple eigenvalues associated with this problem.

Finally, we mention that we present scenarios where the real part of the leading eigenvalues is negative (stable systems) and also when it is positive (unstable systems). The former case is related to robust stability [17], that is important in many engineering applications. The latter case may not look relevant but, as a matter of fact, unstable systems can exhibit high performance when operated by skilled operators or regulated by well-designed controllers. Examples include, nuclear reactors, steering systems, and aircrafts with forward swept wings [19].

## 2 Problem formulation

## 3 Bifurcation theory approach

We recall that when varying parameters the trivial solution \(x(t) \equiv 0\) of a linear system may lose stability in two possible ways. Either a real eigenvalue moves through zero to the right half complex plane (leading to non-oscillatory stability loss), or a pair of complex conjugate eigenvalues crosses the imaginary axis from left to right (leading to oscillatory stability loss where the frequency is given by the imaginary part of the crossing eigenvalues). In the corresponding nonlinear system, fold and Hopf bifurcations may occur. In parameter space the corresponding stability boundaries are hyper-surfaces of co-dimension one. That is, when restricting ourselves to a plane of two parameters, the boundaries are given by curves, often referred as stability curves. Referring to the corresponding bifurcations, these are often categorized as fold and Hopf curves. We adopt this terminology here even though we do not investigate nonlinear effects. Moreover, we define generalized stability curves (i.e., generalized fold and Hopf curves) that correspond to eigenvalues that cross a given vertical line in the complex plane. Note that these are not related to bifurcations in the corresponding nonlinear system and only make sense at the linear level. These generalized curves allow us to locate parameter values that correspond to given eigenvalue configurations for the leading eigenvalues.

### 3.1 Generalized stability charts for constant weight function

The concept of generalized stability charts can also be extended to higher-order weight functions when considering more terms in (4) that yield more design parameters. In this case, the generalized Hopf and fold stability boundaries become co-dimension one hyper-surfaces in parameter space.

### 3.2 Spectrum design with generalized stability charts using a constant weight function

- 1.
Two real eigenvalues (e.g., \(\lambda _1=-1\) and \(\lambda _2=-3\));

- 2.
One real eigenvalue and real part of a pair of two complex conjugate eigenvalues (e.g., \(\lambda _1=-1\) and \(\lambda _{2,3}=-3 \pm \mathrm{i} \omega \));

- 3.
Real parts of two pairs of complex conjugate eigenvalues (e.g., \(\lambda _{1,2}=-1 \pm \mathrm{i} \omega \) and \(\lambda _{2,3}=-3 \pm \mathrm{i} \tilde{\omega }\));

- 4.
One pair of complex conjugate eigenvalues (e.g., \(\lambda _{1,2}=-0.5 \pm \mathrm{i} 3\));

#### 3.2.1 Designing two real eigenvalues

A question that naturally arises is whether there is a limitation on the design for two real eigenvalues. Due to (9), the two fold curves with different \(\gamma \) are not parallel, so they always intersect at (11). Moreover, the intersection always produces the desired eigenvalue configuration \(\gamma _1>\gamma _2\), i.e., the solid segment of the \(\gamma _1\) line intersects dashed segment of the \(\gamma _2\), because other cases contradict the fact that \(\gamma _1\) corresponds to the leading eigenvalue followed by the eigenvalue at \(\gamma _2\).

#### 3.2.2 Designing one real eigenvalue and the real part of a pair of complex conjugate eigenvalues

Alternatively, we can design one real eigenvalue and a pair of complex conjugate eigenvalues with a fixed real part, that is, consider the rightmost three eigenvalues to be \(\lambda _1=\gamma _1\) and \(\lambda _{2,3}=\gamma _2 \pm \mathrm{i} \omega \) where \(\omega \) is arbitrary. As shown in Fig. 4, the fold curve for \(\gamma _1=-1\) and a Hopf curve for \(\gamma _2=-3\) intersect at \((a,b)\approx (-4.97,2.31)\) which is denoted by \(\mathrm{D}_2\) and corresponds to the eigenvalue configuration displayed at the bottom. Since \(\mathrm{D}_2\) is at the boundary of regions \((1,\cdot ),(0,\cdot )\), the rightmost eigenvalue is at \(\lambda _1=\gamma _1\). The second and third eigenvalues are at \(\lambda _{2,3}=\gamma _2 \pm \mathrm{i}\omega \), since \(\mathrm{D}_2\) is also at the boundary of regions \((\cdot ,1),(\cdot ,3)\).

#### 3.2.3 Designing the real parts of two pairs of complex conjugate eigenvalues

Based on the argument above, when the rightmost eigenvalues are given by a complex conjugate pair, a spectral gap can only be provided if these are followed by another pair of complex conjugate eigenvalues. Suppose our desired rightmost four eigenvalues are \(\lambda _{1,2}= \gamma _1 \pm \mathrm{i}\omega \), \(\lambda _{3,4} = \gamma _2 \pm \mathrm{i} \tilde{\omega }\), where \(\omega ,\tilde{\omega }\) are arbitrary. In Fig. 4, a blue Hopf curve for \(\gamma _1=-1\) and a red Hopf curve for \(\gamma _2=-3\) intersect at \((a,b)\approx (-3.20,-4.16)\) which is denoted by \(\mathrm{D}_3\) and the corresponding eigenvalue configuration is shown at the bottom. Thus, the right most eigenvalues are \(\lambda _{1,2}=\gamma _1 \pm \mathrm{i}\omega \), since \(\mathrm{D}_3\) is at the boundary of regions \((2,\cdot ),(0,\cdot )\). Similarly, the second pair of eigenvalues is \(\lambda _{3,4}=\gamma _2 \pm \mathrm{i} \tilde{\omega }\), since \(\mathrm{D}_3\) is at the boundary of regions \((\cdot ,2),(\cdot ,4)\). One may also investigate the limitation of spectrum design for this case, which remains a problem for future research.

#### 3.2.4 Designing a pair of complex conjugate eigenvalues

Finally, we can design the rightmost complex conjugate pair \(\lambda _{1,2}=\gamma \pm \mathrm{i}\omega \) without specifying a spectral gap to the others, which is equivalent to saying that there is a generalized Hopf type of stability loss at \(\gamma \) with given frequency \(\omega \). This corresponds to a point on a “primary” generalized Hopf curve, i.e., on a solid curve in Fig. 2 constituting a boundary between regions with \(0\) and \(2\) “unstable” eigenvalues. (Dashed curves separate regions with higher numbers of eigenvalues.)

### 3.3 Spectrum design with generalized stability charts using a linear weight function

In general, we are interested in the union of available regimes obtained for different values of \(d_1\). This may be found by overlapping the obtained regions. However, in each case the boundary consists of multiple branches that meet at cusp-type singularities. Each of these branches has to be computed by using a feasible initial guess at a chosen value of \(\gamma \) (from which the branch can be continued until reaching a cusp point). Moreover, for negative values of \(\gamma \) the chart may become intricate with many nearby boundaries that require separate initial guesses; cf. Fig. 5b, c. To overcome such difficulties a more efficient numerical method is needed that is presented below.

### 3.4 Finding design limitations by multi-dimensional bisection method (MDBM)

Here we present an effective way of finding design limitations. The fundamental idea is to calculate intersections of hyper-surfaces in the space spun by the real and imaginary parts of the designed eigenvalues and the design parameters. We utilize the multi-dimensional bisection method (MDBM) developed in [22] that allows us to find all solutions. Here we apply this to find the design limitations for linear weight function that was discussed in the previous section. However, as explained below, the range of applicability of MDBM goes beyond this example.

Figure 5d shows the region available for design in the complex plane for different values of parameter \(d_1 \in [-100,0]\) that is obtained by computing the region for every \(d_1 = - \ell 5\), \(\ell = 0,\ldots ,20\). Each point is colored according to the minimal value of \(d_1\) that makes the point available for design an the color goes from blue to yellow as \(d_1\) decreases. Figure 5 shows that including a third design parameter significantly increases the domain available for design. The regions gained for negative real part are important in many engineering applications.

## 4 Lambert W function method

The scalar distributed delay system (2) with an \(n\)-th order polynomial weight function (4), can be converted into a system of \(n+2\) scalar variables with discrete delay. Then the assignment of its eigenvalues can readily be handled by the matrix Lambert W function method [11], using the Lambert W toolbox [23]. The advantage of the Lambert W function approach is that the real part of the eigenvalues follows the order of the branch number [12]. This allows one to focus on a few rightmost eigenvalues which is our goal here.

Here we combine the eigenvalue calculations [23] with a root finding algorithm. This way the iterations converge to the parameter values that ensure the required eigenvalue configuration. However, these numerical iterations may be time consuming and the speed of convergence can depend on the selected initial value of \(\mathbf{Q}_\mathbf{k}\).

**Theorem 1**

This result implies that we only need to consider the first row of \(\mathbf{Q}_\mathbf{k}\) (i.e., last row of \( \mathbf{A}_\mathrm{d} \mathbf{Q}_\mathbf{k} \)). The analysis above shows that for any polynomial weight function (4), system (2) can be transformed into a sparse form and the Lambert W computations can be simplified. However, the convergence of the computation may still be sensitive to the initial value of \(\mathbf{Q}_\mathbf{k}\) as will be discussed below.

### 4.1 Lambert W approach with constant weight function

Numerical values corresponding to the panels of Fig. 6. Note that in cases (b) and (c) only the real parts of the eigenvalues are designed

Eigenvalues | Parameters | |
---|---|---|

(a) | \(\lambda _1=-1\), \(\lambda _2=-3\) | \((a,b)=(-0.26,-0.43)\) |

(b) | \(\lambda _1=-1\), \(\lambda _{2,3}=-3\pm \, \mathrm{i}\,6.15\) | \((a,b)=(-4.97, 2.31)\) |

(c) | \(\lambda _{1,2} = -1 \pm \mathrm{i}\,3.53\), \(\lambda _{3,4} = -3 \pm \mathrm{i}\,9.11\) | \((a,b)=(-3.20,-4.16)\) |

(d) | \(\lambda _{1,2} = -0.5 \pm \mathrm{i}\,3\) | \((a,b)=(-0.73,-3.46)\) |

(e) | \(\lambda _{1,2} = -0.5 \pm \mathrm{i}\,8\) | \((a,d_0,d_1)=(-1.56,-87.00,-100.00)\) |

### 4.2 Numerical issues

Note that the hybrid branch approach can only be applied when the Jordan form of \(\mathbf{A}_\mathrm{d} \mathbf{Q}_\mathbf{k}\) is diagonal (cf. Appendix) which may lead to convergence problems when the leading eigenvalues have multiplicity greater than one. To illustrate this we added the stability curves (8,9) for \(\gamma =0\) to Fig. 7 (cf. the left panel of Fig. 2). Along the straight fold line the eigenvalue 0 has multiplicity two due to the intrinsic zero eigenvalue generated when rewriting the system into vectorial form. Moreover, the green curve, given by (15), corresponds to the case where the lead eigenvalue is real and has multiplicity two (while the intrinsic zero eigenvalue still exists). At the intersection of the curves (indicated by green dot) we have a zero eigenvalue of multiplicity three. Figure 7 demonstrates that sections of these curves coincide with the strip where the Lambert W approach does not converge. However, it is not a one-to-one correspondence. Understanding further details of convergence problems is left for future research.

## 5 Concluding remarks

In this paper, we studied the problem of designing the rightmost eigenvalues of a linear scalar system with distributed delay. Our main methodological contributions to spectrum design were the introduction of generalized stability charts and the extension of the Lambert W method to distributed delay systems. Using generalized stability charts we reduced the problem to finding intersections of curves and surfaces in parameter space. In order to characterize design limitations we used two different numerical approaches. We found that numerical continuation can be used in cases involving two design parameters, while MDBM can be applied when using three or more parameters. In order to make the matrix Lambert W approach applicable we reformulated the scalar distributed delay system using multiple variables and discrete time delays. By exploiting the sparse structure of the appearing matrixes we reduced the computation time significantly. We also identified regions in the parameter space where numerical convergence is slow.

There are still some interesting problems that remain to be studied. Our future work will focus on several extensions. One is to determine design limitations in higher dimensions using MDBM. Also, finding more efficient numerical algorithms for the Lambert W method is an open question. Finally, we want to better understand the connection between these two approaches that may be achieved by expanding the infinitesimal generators of distributed delay systems using matrices.

## Notes

### Acknowledgments

The research of Wei Feng was supported by the University of Michigan through a Rackham Centennial Fellowship. The research of Gábor Orosz was supported by the National Science Foundation (Award # 1300319). The research of Dániel Bachrathy was supported by the European Union and the State of Hungary, through a European Social Fund in the framework of TÁMOP 4.2.4. A/1-11-1-2012-0001 ‘National Excellence Program’

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