#### Abstract

This paper presents a brief review on the current applications and perspectives on the stability of complex dynamical systems, with an emphasis on three main classes of systems such as delay-free systems, time-delay systems, and systems with uncertainties in its parameters, which lead to some criteria with necessary and/or sufficient conditions to determine stability and/or stabilization in the domains of frequency and time. Besides, criteria on robust stability and stability of nonlinear time-delay systems are presented, including some numerical approaches.

#### 1. Introduction

The importance of complex dynamical systems has been increasing dramatically since many real world applications have adopted this behavior. Some of them have become special areas of study such as smart grids [1–5], autonomous vehicles [6–10], biological systems [11–17], distribution networks [18–23], social interaction [24–27], communication systems [28–30], and animal monitoring [31–34], among others.

Among the structural properties in complex systems, controllability and observability are still discussion topics for a wide variety of dynamical systems, from classic linear time-invariant systems to other more complex families such as infinite-dimensional systems, stochastic systems, and hybrid systems, among others. This problem increases when they are subject to a large amount of connections, in which a classic handling is very difficult such as in complex networks [35], where the connection properties, such as symmetry [36, 37], and some computational tools have been developed to determine these properties [38].

In the case of stability of complex systems, two important aspects to consider are the time-delay between nodes (or the possible presence of time delay in the internal dynamics) and the robustness of the system (connection and internal) due to parametric variations. On the one hand, the time-delay induces infinite-dimensional dynamics whose equilibrium stability is still an active topic of research for, both, stability analysis itself and the use of the delay as a stabilizing element [39]. On the other hand, the parameter variations in a system demands a special stability treatment which has motivated some significant results such as Kharitonov’s theorem [40] or loop shaping in control design [41].

In this article, a stability review for a class of systems subject to parameter variations and time delays is addressed. The review includes a definition of the dynamics of a family of nonlinear time-delay systems, a brief historical outlook of the stability study development, and then it gives some approaches in time and frequency domain, as well as a numerical analysis to provide the stability operating regions. Then, the robustness aspect of stabilizing controls is discussed on the realm of Hurwitz polynomials families. Finally, some perspectives of applications demanding these approaches are briefly provided.

#### 2. Stability and Stabilization of Linear Systems

Characteristic root locations of linear systems or linear delay-free systems are completely related to system stability. The same criterion holds for linear time-delay system. Furthermore, many of the concepts and criteria for determining stability on time-delay system (TDS) are extensions or adaptations of the results initially proposed for linear delay-free systems. Accordingly, it is considered relevant to briefly address some classic stability criteria for this type of system.

In this section, some concepts and criteria regarding the stability of linear systems are analyzed. The section is divided as follows. In Section 2.1, the concept of stability in linear systems is addressed, Section 2.2 presents what can be the objectives in the stabilization of linear systems, and, finally, in Section 2.3, Kharitonov’s theorem is analyzed and some families of polynomials to illustrate the importance of studying the stability in systems with uncertainties in the parameters are given.

##### 2.1. Stability of Linear Systems

Lyapunov’s doctoral thesis was perhaps the first systematic work on the subject of stability [42], although the study of stability theories was started by Maxwell, around 1868, in his work “on governors” [43]. Lyapunov presented a general definition of stability which referred to the stability of a solution of a differential equation, not necessarily at the equilibrium point.

Before Lyapunov, there were works that tried to explain the stability phenomenon in applications; it is worth mentioning the works of Lagrange and Dirichlet [44–48]. After Lyapunov’s work, new concepts appeared such as uniform, global, exponential, and quasi asymptotic stability, among other types of them. In the following, some of the best known criteria for determining the stability of linear systems will be briefly mentioned.

A way to determine the asymptotic stability of continuous time-invariant systems is characterized by its eigenvalues. If all eigenvalues lie into open left half plane , then the system is stable. A polynomial that satisfies that all its roots are in is called a Hurwitz polynomial or stable polynomial. Routh and Hurwitz showed, in independent works, that the stability of a matrix system could be determined by means of the coefficients of its characteristic polynomial. Their results are currently presented under the so-called Routh–Hurwitz criterion (see [49, 50]) and can be considered as a numerical criterion. In view of that, it is reduced to the calculation of determinants. Another useful approach used to test the stability is the Hermite–Biehler criterion [51, 52], which can be considered as an algebraic criterion, which expresses the stability in terms of even and odds parts of the characteristic polynomial. A pair of polynomials and are said to be a couple positive if the principal coefficients of and have the same sign and the roots of and of alternate orderly on the real axis, i.e., are real, distinct, and negative and satisfy the interlacing property. So, the Hermite–Biehler theorem states that a polynomial that can be written in the form is stable if and only if and are a couple positive (see [49, 50], for details). There are others noncommon stability criteria; it is worth mentioning some of them. The Lienard–Chipart conditions [53] reduce the positivity evaluation of the main minors of the Routh–Hurwitz criterion to half.

The Leohnard–Mihailov criterion [54], which expresses the stability in geometric terms, analyzes the argument of the complex polynomial associated with the characteristic polynomial of the system. Routh’s scheme gives rise to a recursive algorithm for testing the stability called Hurwitz stability test. Other approaches consider the Bezoutiant or the Cauchy indexes to verify stability of polynomials; for details, see [49, 50, 55].

Not less important are those systems, where uncertainty is considered and incorporated through parameters. The robust stability of parameter uncertainty systems and the families of polynomials are associated with them, and it is addressed in the subsequent sections.

##### 2.2. Stabilizing Controls

In this section, the relation between stabilizing feedback and some families of Hurwitz polynomials is explained. Consider the controllable system: and the controllable pair is given in the canonical form:

Note that the open-loop polynomial is given by .

Now, let us define the feedback control as follows:where and . Then, the closed-loop polynomial is given by

Let us denote . Then, . Now, let us suppose that the is a Hurwitz polynomial. When a closed-loop eigenvalue, say satisfies , the other eigenvalues converge to the zeroes of , that is, when , feedback (3) is a stabilizing control (see [56, 57]).

Also note that when , then high control gains are induced in the feedback (3); hence, feedback (3) is a high-gain feedback. There are several studies to analyze the properties of high-gain controls (see [56–64]).

Returning to the analysis of the closed-loop polynomial , it can be seen that is not necessarily Hurwitz for all , even when is Hurwitz and is chosen such that is a Hurwitz polynomial (see [65–67]).

The last observation is illustrated with the following example, which was presented in [67].

Consider the system:Here, and , and are Hurwitz polynomials. However, is not a Hurwitz polynomial for ; hence, the importance of have methods for choosing vectors such that is a stabilizing control. Consequently, it is necessary to study the Hurwitz stability of the ray of polynomials with .

In the next section, a technique in terms of rays and segments of polynomials is presented.

###### 2.2.1. Relation between Stabilizing Controls and Rays and Segments of Hurwitz Polynomials

As can be observed above, the control is a stabilizing feedback if and only if the ray of polynomials is a ray of Hurwitz polynomials. On the contrary, there is an obvious relation between Hurwitz rays and Hurwitz segments of polynomials: if is a Hurwitz polynomial, then is a Hurwitz polynomial, which implies that the Hurwitz stability of the ray is equivalent to the Hurwitz stability of the segment of polynomials .

The problem to establish conditions on the Hurwitz polynomials and such that the segment of polynomials determined by is Hurwitz stable for all has been studied with different approaches (see [65–73]). The first reported work about this subject is Bialas’s paper [68]. Bialas’s theorem says that if is a Hurwitz polynomial and , then is Hurwitz for all if and only if the matrix has no eigenvalues in , where is the Hurwitz matrix of the polynomial (see [68, 74, 75]).

Other method which is known as the segment Lemma was obtained by Chapellat and Bhattacharyya (see [76, 77]). The segment lemma is an approach that presents conditions in the frequency domain. Based on the segment lemma, a computational algorithm was developed in [71] for testing the Hurwitz stability of segments of polynomials. On the contrary, Bose developed a technique to check the stability of segments of complex polynomials [78].

Based on Bose’s test, in [69], a test for checking the stability of segment of complex polynomials was obtained. Another computational method is presented in [79]. Sufficient conditions to guarantee the Hurwitz stability of segments were obtained by Rantzer (see [72, 80]). On the contrary, an approach, where sufficient conditions in terms of matrix inequalities for checking the Hurwitz stability of segments of polynomials, has been presented in [65–67, 73]. The explanation of the aforementioned approach is the following: let be a Hurwitz polynomial.

Consider the matrix defined byand consider the matrix defined by

Now, let be an arbitrary polynomial of degree with positive coefficients. If the vector satisfies the system of linear inequalities or , then is Hurwitz for every (here the symbol () means that the components of a vector are nonnegative (nonpositive) and the symbol means that all of the components of a vector are nonnegative, but there is at least one positive component).

Other interesting references about segments of Hurwitz polynomials are the works [74, 76, 81–86]. Besides, in relation with Hurwitz polynomials, it is worth to consulting paper [87].

##### 2.3. Robust Stability

The presence of several uncertain parameters in description of a LTI system manifests itself as variations in the coefficients of the characteristic polynomial. The determination of stability and stability margins under parametric uncertainty, structure uncertainty itself included, is the main purpose of the robust stability.

Perhaps, the most famous result about the Hurwitz polynomial families is the Kharitonov theorem, which is related to the interval-type polynomials. This section presents this theorem and addresses some related results.

###### 2.3.1. The Kharitonov Theorem

The problem of stability under large parameter uncertainty was strongly promoted with the advent of a remarkable theorem due to the Russian control theorist V. L. Kharitonov.

Consider the interval family of polynomial defined bywhere

Consider the following four elements of the family, named the Kharitonov polynomials:

Kharitonov’s theorem establishes that every polynomial in families (8) and (9) is Hurwitz if and only if the four Kharitonov polynomials 10(10) are Hurwitz. This result has been the motivation of different extensions, alternate proofs, and applications to some classes of families of polynomials, for instance,(i)The problem to find conditions for family (8) to be Hurwitz was planted by Faedo [88](ii)The original proof was given by Kharitonov in [89], but different authors have presented other proofs (see [70, 77, 90–93])(iii)Kharitonov extended his result to the complex case in 1979 [94](iv)Generalizations of Kharitonov’s theorem are presented in [56, 95, 96](v)Applications of Kharitonov’s theorem can be consulted in [97](vi)Recent information about Interval Families and Kharitonov’s theorem was published in papers [65, 98, 99]

The appearance of Kharitonov’s theorem led to a resurgence of interest in the study of robust stability under real parametric uncertainty. In Section 2.2, some results related to stability of certain families of polynomials such as the segments and the rays of polynomials were mentioned. Some other families worth mentioning are the ball of stable polynomials and polytope of polynomials. The ball of stable polynomials is a way of characterizing the largest region where the stability of a family of polynomials is preserved. Soh et al. [100] in 1985 adopted a point of view opposite to Kharitonov. Starting with an already stable polynomial , they gave a way to compute the radius of the largest stability ball in the space of polynomial coefficients around . The estimation for the -norm stability ball in the space of coefficients was calculated by Soh et al. For -norm, the calculation was realized by Tsypkin and Polyak [101].

Otherwise, the main robust stability result related with polytope of polynomials is the celebrated Edge theorem of Bartlett et al. [102], which considers more general stability regions, and it is not restricted to Hurwitz stability. They considered a family of polynomials whose coefficients vary in an arbitrary polytope: on , with its edges not necessarily parallel to the coordinate axes as in Kharitonov’s problem. They proved that the root space of the entire family is bounded by the root loci of the exposed edges. In particular, the entire family is stable in and only if all the edges are proved to be stable. The key idea behind this result is that we can reduce a multidimensional uncertainty problem into a finite number of one-parameter problems whose solution requires less effort.

One of the most used tools in the analysis of robust stability in families of polynomials, where the coefficients depend continuously on a set of parameters, is the *Boundary Crossing theorem* and its computational version the *Zero Exclusion principle*. Consider a family of polynomials of degree , where the real parameter ranges over a connected set . If it is known that one member of the family is stable, a useful technique of verifying robust stability of the family is to ascertain that for all . This can also be written as the *zero exclusion condition*, for all . This zero exclusion condition has been exploited to derive various types of robust stability and performance margins.

#### 3. Stability and Stabilization of Time-Delay Systems

This section begins with a brief classification of nonlinear TDS, followed by a recurring classification in which the nonlinear TDS has a nominal part (linear part). The above allows to show basic and pillar results existing in the literature for the stability and stabilization analysis of linear TDS in two domains, frequency and temporal. Although it is typically believed that an analysis in the frequency domain is only limited to linear systems, recent results have shown that a study on polytope of quasi-polynomials (generalized characteristic quasi-polynomial) can determine stability conditions for a class of nonlinear TDS. Furthermore, in the time domain, the use of complete type Lyapunov–Krasovskii functionals can provide necessary and sufficient stability conditions for a class of linear TDS, while reduced type functionals only give sufficient conditions.

The understanding, analysis, and prediction of the dynamics of a system are topics that generate considerable interest in the scientific community. When this system presents nonlinearities and delays in its structure, the useful information is even richer, since it is more consistent to the dynamics observed in the systems/processes/prototypes of the physical world. These types of systems are known as nonlinear time-delay systems or nonlinear systems with delays. Sometimes, nonlinear dynamics are phenomena that often introduce unpredictable chaotic behaviors into a system, whereas the delays are due to the fact that the rate of variation in the system dynamics depends on past states, which implies an analysis in an infinite-dimensional space, and this is in mathematical terms. Thus, the dynamics observed in communication networks [103–105], teleoperation [106, 107], chemical processes [108], population dynamics [109], biological phenomena [110], game theory and economic applications [111, 112], unmanned aerial vehicles [113], haptic interfaces [114], and robotic systems [115, 116], among others, can be mathematically modeled using nonlinear time-delay systems.

The nonlinear TDS are usually represented by functional differential equations (FDEs) also known as delay differential equations. Among functional differential equations, one may distinguish some particular classes as retarded functional differential equations (RFDEs) (or functional differential equations of retarded type), neutral functional differential equations (NFDEs) (or functional differential equations of neutral type), distributed functional differential equations (DFDEs) (or functional differential equations of distributed type), and differential-difference equations (DDEs). For illustrative purposes and to characterize the research space of the FDEs, the form of RFDE with one delay is presented below:where is continuous and satisfies a local Lipschitz condition regarding the second element of the argument. For , denote by the system solution with initial function (condition) , and by the Banach space with norm . Here, denotes the Euclidean norm. As a natural extension of the initial function, a solution segment of in a time interval is denoted byand called state of system (11). In turn, the above RFDE can be classified as follows:(i)Time-invariant RFDE if the first term of the argument is omitted:(ii)RFDE with multiple delay if , that is,(iii)State-dependent delay RFDE if depends on state, :(iv)Time-varying delay RFDE if the delay depends on :(v)RFDE with distributive delay, also known as DFDE, if the delay is represented as a continuously distribution and it is not instantaneous:(vi)RFDE with neutral delay or NFDE if the system also depends on the time derivative of the state:(vii)RFDE with discrete delay or DDE if is the interval between the successive sample instants and , , and . Here, is called the sample period, and is an -dimensional discrete mapping, and using any integral scheme we obtain

The above, as well as the possible combinations between these, are just some types of TDS. Currently, there is a broader classification of systems that are outside this basic classification. In addition, each type of TDS needs more appropriate concepts and descriptions for the research space and existence and uniqueness of the solution, among others. For further information, the reader is referred to the following authoritative references [52, 117–127]. Although these themes are very interesting, the focus of this review is on stability and stabilization. Therefore, it will be the next topic to discuss.

##### 3.1. Stability of Time-Delay Systems

Undoubtedly, one of the most important research topics for the TDS research community is the analysis of stability. In this context, obtaining sufficient and/or necessary conditions to determine when the studied system remains stable or when it gains and/or loses stability is an important topic for the community. Furthermore, the knowledge of these conditions allows to solve other problems associated with this topic such as analysis of robustness/adaptability/uncertainty/perturbation, design of observers, synthesis and tuning of controllers, determination of attraction regions estimates, and study of chaotic/hyperchaotic behaviors, to name just a few, [128–137].

Typically, the stability of the TDS is studied on two main frameworks: frequency domain and time domain. The fundamental results were proposed by Pontryagin [138, 139], Wright [140, 141], Bellman [142, 143], and Cooke [144] in the 1940s and 1950s. These results are in the frequency domain, which is based on a study of its corresponding characteristic equation (exponential polynomial, quasi-polynomial, and analytic function with transcendental terms) to determine the location of its roots in the complex plane or from the nontrivial solutions of a delayed Lyapunov matrix function. Later, Razumikhin [145] and Krasovskii [146] proposed to extend Lyapunov’s results to analyze stability of TDS in the time domain.

The stability of nonlinear TDS has been studied for almost 80 years and most of the results proposed by the research community are about nonlinear systems with a specific structure. Among which, it is possible to apply techniques that benefit/facilitate the analysis of the stability of complex systems around an operating (equilibrium) point or even in a sector of it. Since, these allow to rewrite exactly or approximately a complex system in a more accessible system to study. Techniques such as sector nonlinearity [147], tangent linearization [148], feedback linearization approach (Lie derivative) [149, 150], passification [151–154], backstepping [155–158], immersion and invariance [159–162], and differential flatness [163–165], among others. On the contrary, in some cases, a linearization is proposed around an operating point. While in other cases, the nonlinear systems is represented with a dominant part (dominant linear part) plus nonlinear part is proposed. In this way, in the nonlinear part, disturbances, nonmodel dynamics, and parametric variations, among others, are usually introduced. Below, it is a classification of the aforementioned systems.

Let a nonlinear TDS be given in (11); in some cases, this system can be represented aswhere is a linear operator and is a nonlinear operator, both properly defined. Typically, if satisfies certain conditions (bounded, Lipschitz, and quasi-Lipschitz, among others), then is considered as an uncertainty or/and perturbation of the nonlinear TDS, so the stability analysis of (20) focuses on the nominal system . A classification of the nominal part proposed here is as follows:(i) is a linear time-variant (LTV) delay system of retarded type(ii) is a linear parameter varying (LPV) system with time delay, where are uncertain time-variant real parameters which satisfy (iii) is a linear system with time-varying delay, where and , for all (iv) is a linear time-invariant (LTI) TDS of retarded type(v) is a linear NFDE (linear neutral time-delay systems or linear time-delay system of type neutral), where , for any matrix norm (vi) is a linear DFDE or linear distributed time-delay system; here, is a continuous matrix on of appropriate dimensions(vii) is a LTI-TDS with multiple delays of retarded type,

To learn more about the systems described above, the reader can consult the following references [52, 166–169]. Although the nonlinear part is also important and its structure depends on the type of studied nonlinear system, it is more important to know the studies regarding the stability of the nominal part. Since the stability of the nominal part can contribute the obtaining of robust stability conditions in the presence of the nonlinear part. Therefore, some research studies on stability analysis of linear TDS in the frequency domain and time domain are presented below.

##### 3.2. Stability in the Frequency Domain

It is well known that the analysis of stability and stabilization of a TDS in the frequency domain is based on a study of its corresponding characteristic equation to determine the location of its roots in the complex plane. This concept is inherited from the stability analysis of delay-free systems, mainly from Hurwitz’s concept [170, 171]. However, when the delays are considered in a system, it involves the inclusion of transcendental terms in the characteristic equation, changing the analysis of a polynomial (free-delay systems) to a quasi-polynomial (TDS). This complicates the analysis of a finite number to an infinite number of roots [172], also this analysis is usually limited to LTI-TDS. However, in contrast with the time domain, this analysis allows obtaining necessary and sufficient stability conditions; recent results have shown that an analysis in the frequency domain can be applied to a class of nonlinear TDS, see [173, 174].

Given a LTI-TDS with multiple delays of retarded type,then its quasi-polynomial is of the formwhere is the identity matrix of *n*-dimension, , , , are polynomials with real coefficients and , i.e., , . Although given in 3.2 has an infinite number of roots, it is enough to know the location of the dominant roots as shown below.

*Definition 1 (see [166]). *The LTI-TDS with multiple delays of retarded type (21) is said to be -stable (exponentially stable) if the system response satisfies the following inequality:where , , and is the initial condition.

Under consideration, for LTI-TDS of retarded type, the exponential stability and asymptotic stability are equivalent.

*Definition 2 (see [166]). *Consider the quasi-polynomial 3.2, a positive constant andwhere denotes the real part of . Then, the LTI-TDS with multiple delays of retarded type (22) is -stable if (relative stability, [175]).

As can be seen in the above definitions, the stability of an LTI-TDS with multiple delays depends on the dominant roots of 3.2, which determine the abscissa (spectral abscissa) or vertical line on in the complex plane. Furthermore, the roots have continuous variations with respect to parametric variations of the system. This is known as continuity property.

Theorem 1 (see [176]). *If the matrices or the delays , , are varied, then a loss or acquisition of exponential stability of the solution of LTI-TDS with multiple delays (21) is associated with the dominant roots of the quasi-polynomial (22).*

This allows to obtain conditions of robustness when there are parametric variations and also the design and tuning of control laws for *s*-stabilize TDS are as shown in Section 3.4.

However, in the framework of TDS stability in the frequency domain, there are many results/criteria; among the first and most important are the following:(i)The Pontryagin criterion [177] is considered as one of the most general analytical criteria, and it gives necessary and sufficient conditions for the stability of (22). However, it has strong limitations and may become very complicated for systems with more than one delay.(ii)The Yesipovich–Svirskii criterion [178] is for systems with one discrete delay. The necessary and sufficient condition of the stability of (22) is given by means of the expression: where is a further transformation , , of , is a function that does not contain transcendental terms, and are the real roots of .(iii)The -decomposition method [179] requires the transformation of the quasi-polynomial into the form where is a ration of two polynomials. This method is for systems with one discrete delay, and it is based on the analysis of the contour , around the unit circle in the complex plane.(iv)The principle of argument [121] is used to determine the number of roots of inside of an closed curve , where Here, denotes the changes of the argument of along .(v)The Chebotarev criterion is the direct generalization of the Routh–Hurwitz. The analytical criterion needs to calculate an infinite number of Hurwitz determinants and the stability of the system with long delay is determined by determinants of high dimension, whereby it is not effective practically.(vi)The D-partition (D-subdivision) method [180] is a geometric method to construct stability charts (regions) in the parameter space of the quasi-polynomial. It is very effective to determine stability based on system parameters. However, the number of parameters used is reduced.(vii)The Nyquist [181] method is also a geometric method and the stability of the systems is determined by the relative position of the point and the contour , , where is a transformation of the quasi-polynomial.(viii)The Bode and Nichols criteria and some others are transformations of the Nyquist criterion as Satche mentions in [182].(ix)The Mikhailov criterion [183] is a consequence of the Cauchy Residue theorem in complex analysis. It can be applied to RFDE if there exists a scalar such that the quasi-polynomial is bounded and analytical in any closed domain in ; this is also known as Satche’s diagram [184].(x)The Hermite–Biehler criterion [51, 52] mentioned in the previous section can be applied for TDS using the imaginary and real parts of the quasi-polynomial, ; the stability of the system is determined by a continuous alternation between transformations of the real functions and , when increasing phase condition , for any . See its extensions in [185, 186].(xi)The Edge theorem [187], zero exclusion principle, and concept of convex direction [188] are graphical methods to determine stability of a set of quasi-polynomial family or convex polytope family.

The above criteria are the most recurrent on stability analysis of quasi-polynomials, see [176, 189–194], but there are quasi-polynomial classes of larger complexity that require special attention. Among this class, the quasi-polynomial of NFDE [195], polynomial family which is described by convex polytope in the coefficient space [196], and stability analysis of LPV-TDS through a generalized characteristic quasi-polynomial [173] can be found.

Consider a LTI-NFDE with multiple delays of the form:where , and . The corresponding quasi-polynomial is given by

A TDS has an infinite number of characteristic roots, but is an entire function, which implies that there can only be a finite number of characteristic roots within any bounded domain. These characteristic roots form root chains that are rather easy to describe. The quasi-polynomial has two types of root chains. The first type is retarded chains; here, the roots fall in the region , for some and . In other words, there may only be a finite number of roots on the right of the abscissa in the complex plane for any given [195]. The second type is neutral chains; here, the roots are bounded by two abscissas . The positions of such abscissas are determined by

Additionally, if is a solution of (31); then, there is a series of roots of quasi-polynomial (30) such that and . Due to all the above, the stability of (30) is associated with the stability of fd38(31). Furthermore, (31) can be sensitive to infinitesimal delay perturbations, which strongly affect the continuity property of the roots of (30). Moreover, in contrast with the retarded case, this property cannot be ensured with respect to parametric variations of the system. For more details on this type of systems, see [176, 195].

Another interesting study in the TDS research community is the stability analysis of convex sum of quasi-polynomials, known as polynomial family or convex polytope or polytope of quasi-polynomials. These are quasi-polynomials that are entire functions which include both degree of the independent variable and exponential functions and they appear when several subsystems with delays are interconnected. Consider a convex hull of quasi-polynomials of the form:where the vertex quasi-polynomial is of form (22). The stability of this class of the quasi-polynomials of family **Q** is studied using the zero exclusion principle, the concepts of convex direction, and the Edge theorem [196–199]. Here, a coefficient vector is associated with every element ; then, family (32) can be described by the convex polytope:where the vector corresponds to the vertex quasi-polynomial . Thus, is the set of edges and is the set of vertexes of the polytope . Every edge corresponds to the one-parameter family of quasi-polynomial of the form , , i.e., the stability analysis for the families is reduced to a finite number of simpler problem stability to convex couples.

Theorem 2 (see [197]). *The family is stable if and only if all members of one-parameter family corresponding to the edges are stable.*

On stability analysis of LPV-TDS, it seems that this topic is one of the most relevant topics and the best opportunity field to direct the current research. An aircraft is one classic physical system where the mathematical model can be represented by a LPV system [200], while a system of distributed type can be seen in [201] and a LPV-TDS in [173].

Consider a LPV-TDS of the form:where . Now, consider the LTV-TDS of the form:and the nonlinear TDS as follows:

In terms of stability, the previous three systems (34)–(36) can have the same properties. In other words, the convex representation of the uncertainty of and the nonlinear functions and are equivalent in a stability analysis if only the manipulation range of the previous variables is considered. This undoubtedly provides the opportunity to obtain stable conditions for a wide variety of types of systems. Furthermore, these conditions can be obtained using LMIs and by studying a generalized characteristic quasi-polynomial of the formwhere and are the coefficients of a polynomial with a finite set of bounded uncertainties, which depend on the uncertainties . The quasi-polynomial (37) is exactly rewritten as a polytope whose interpolating functions exhibit mutual dependency. Therefore, the stability analysis of this type of polytopes implies the stability analysis of a class of nonlinear TDS, see [173, 174].

On the contrary and to finish this section, it is worth mentioning that some members of the scientific community have preferred to employ transformations, approximation methods or pseudo-delays to avoid the transcendental terms in the stability analysis of a TDS instead of using the direct methods presented above. Although, in many occasions, a direct approach to analyze the stability of a TDS is more efficient [202]. Among these, they can be found the Smith predictors [203], Rekasius transformation [204], and Padé approximation [205]. The Smith predictors allow to use a controller structure which takes the delay out of the control loop, which reduce the stability analysis to the one of a free-delay system. The employment of the Rekasius transformation implies an infinity-to-one holographic mapping (the mapping is asymmetric), and it is also impossible to track all of the infinitely many roots, especially, since the dominant root cannot be declared, as mentioned in [206]. The Padé approximation has been used to approximate the exponential function , , through rational approximation of the form , wheresee [207].

##### 3.3. Stability in the Time Domain

In this section, a brief description of the most well-known criteria for the stability analysis of TDS in the time domain will be given. Emphasize the results using two types of Lyapunov–Krasovskii functionals: reduced type and complete type. While the first type of functional is usually the favorite of the scientific community, perhaps due to the relative flexibility to propose the functional candidate and to accomplish the requirements of the system studied, it seems that this type of functional only can provide sufficient conditions of stability and stabilization. The second type of functional is used by a narrowed community, perhaps due to the relative complexity compared to the first, but this has been shown to be closer to obtaining the necessary and sufficient conditions of stability and stabilization, see [208].

This approach is based primarily on two methods: the use of Lyapunov–Krasovskii (L-K) functional [146] or Lyapunov–Razumikhin (L-R) functions [145]. Both methods are an extension of the Lyapunov direct method [42] for free-delay systems. These stability criteria usually provide sufficient stability conditions in terms of linear matrix inequalities (LMIs), [209], which can be effectively solved by means of convex optimization techniques [210]. Although these two methods have received a great deal of attention, the results only offer conservative and sufficient stability conditions. The Razumikhin results allow one to obtain stability results based on adapted Lyapunov functions to analyze the stability of the TDS, while the Krasovskii results employ Lyapunov functionals as a natural extension to TDS. Despite the two methods provide interesting results for stability studies, the last method is the predominant one in research.

The main idea of Krasovskii consists in proposing an appropriate functional which can satisfy extensions of concepts and criteria of Lyapunov for TDS. One of these stability concepts is the definition, while the most used criterion is given below.

Theorem 3 (see [122]). *Consider the TDS given in (11) and that there are continuous nondecreasing functions, where and are positive for and .*

(i)If there exists a continuous differentiable functional such that then the trivial solution of (11) is uniformly stable.(ii)If the trivial solution of (11) is uniformly stable, and for , then the trivial solution of (11) is uniformly asymptotically stable.(iii)If the trivial solution of (11) is uniformly asymptotically stable and if , then the trivial solution of (11) is globally uniformly asymptotically stable.

Consider the LTI-TDS of the form:where are matrices of appropriate dimensions. The functional ones proposed to satisfy the above conditions are known as L-K functional candidates and their basic form is as follows:where and are positive definite matrices (symmetric matrices where every eigenvalue is positive). This functional satisfies the conditions of Theorem 3 if there are and which satisfy or

If this is true, then the functional is called L-K functional. However, this type of L-K functional only provides delay-independent sufficient stability conditions for LTI-TDS with one delay, namely, sufficient conditions that can only be applied to LTI-TDS that are stable for all . When the primary objective is to propose L-K functional candidates that provide delay-dependent stability conditions and that these conditions may be necessary and sufficient conditions to determine stability in a TDS, as well as for linear free-delay systems. Unfortunately, the above is still an open problem of the TDS. Therefore, most of the research studies carried out focus not only on the type of L-K functional, but also on mathematical properties that reduce the conservatism of the stability conditions (LMI-based stability conditions) and/or these can relax the conditions. Some of these properties follow immediately.

Lemma 1 (Schur complement). *Consider a given symmetric matrix , where ; then, the following conditions are equivalent:*

Lemma 2. *Let , and given matrices with appropriate dimensions. Then, holds for all such that if and only if there exists such that .*

Lemma 3 (Jensen inequality, see [166]). *For any constant matrix , , scalar , vector function such that the integrations concerned are well defined, then*

Lemma 4 (see [122]). *Let , and be real matrices with appropriate dimensions, and let and . Then, the following propositions are true:*(i)*For any , *(ii)*For any and any , *(iii)*For any satisfying , *

For decades the TDS research community has proposed different types of L-K functional candidates and/or also different mathematical properties in order to satisfy the postulated in (iii) of Theorem 3.3. One of the most observed trends is to add quadratic or cross terms to the functional candidate as follows.

Consider the following L-K functional candidate:where each term , can be of the form: cross terms , quadratic terms , quadratic terms for exponential terms , cross terms , and quadratic terms

Typically, these terms (quadratic or crossed) are introduce depending on the type of system analyzed (RFDE, DFDE, and NFDE, among others) or the type of conditions required to obtain (robustness and exponential estimates, among others) as an effort to obtain the functional one that grants less restrictive conditions and more types of systems can be analyzed. Also, comparisons between various criteria of delay-dependent stability can be observed in the literature, to demonstrate the efficiency and loss of conservatism with the proposed conditions, see [211]. In this context, there are miles of contributions giving necessary conditions of stability, among which the following can be mentioned [39, 212–238]. In most of these contributions a type of functional known as reduced type functional is used. However, until now it is unknown what type of reduced-type functional is suitable for the type of TDS analyzed, [214, 239–241]. Therefore, some criteria for the construction of full size (complete) type L-K functionals have been developed with the intention of solving these problems.

The construction of the complete-type functional requires a prior proposal of the quadratic derivative of the functional and the construction of the so-called delay Lyapunov matrix. The first results were proposed in [242], followed by some interesting results such as those given in [243, 244] for RFDE with one delay, while in [245] some of the results have been extended to a general case of LTI-TDS. In the latter, it is also shown that the constructed functional requires additional information to admit a lower quadratic bound. In [213], an interesting numerical scheme for the construction of complete type L-K functionals has been proposed using the LMI approach. In [246, 247], properties are clarified and completed for the construction of a functional with upper and lower quadratic bounds. This technique can be summarized in the following result.

Theorem 4 (see [123]). *Consider a prescribed quadratic functional of the form:where ,and are positive definite matrices of appropriate dimensions. If the LTI-TDS (40) is stable, then there is only one functional such that*

This functional is known as complete type L-K functional and it is given bywhereis a counterpart of the classical Lyapunov matrix equation in the context of Lyapunov quadratic forms for the linear delay-free systems; therefore, it is called Lyapunov matrix for TDS (delay Lyapunov matrix). Here, and is the fundamental matrix of LTI-TDS (40) and the solution of the matrix equation:with initial condition for *y*. In addition, the delay Lyapunov matrix satisfies the following conditions:

Equations (52)–(54) are known as dynamic property, symmetry property, and algebraic property, respectively.

One of the key points to use this type of functional are the existence, uniqueness, and numerical calculation of the delay Lyapunov matrix (50), for which several manuscripts have been dedicated to this sense, see [248–254]. Some of the functionals prescribed to construct complete type L-K functionals are as follows.(i)For LTI-RFDE [255],(ii)For LTI-DFDE [256],(iii)For RFDE with time-varying delay [257],(iv)For RFDE with uncertain coefficients and an uncertain time-varying delay , with and [258],

This type of functional and the delay Lyapunov matrix are giving good results for the design of control laws such as linear quadratic suboptimal controllers [259]; recently, these have also been used to designing dynamic stabilizing controllers (predictor-based controls) for preserving the exponential stability of the closed-loop system after the replacement of the integrals by finite sums [260–263], robotic systems with constant input time delay through the active disturbance rejection paradigm and generalized proportional integral observers [264], design of delayed output-feedback controllers that optimize a quadratic cost function [265], necessary and sufficient exponential stability condition for systems with multiple delays [266], and partial differential equations [267]. Furthermore, these types of functionals can provide necessary and sufficient stability conditions for linear systems with pointwise and distributed delays [208]. Here, a stability criterion is presented for the exponential stability of systems with multiple point and distributed delays. These conditions are in terms of the delay Lyapunov matrix (50) using the evaluation of a complete type L-K functionals at a pertinent initial function that depends on the system fundamental matrix. Undoubtedly, this topic is solving important open problems in the area of TDS, whereby it is highly recommended to direct current and future research.

##### 3.4. Numerical Methods (Applications)

One of the most important aspects in the stability analysis of time-delay linear systems is the calculation of characteristic roots of the linear/linearized dynamics, being the root with the maximum real part a very important one [268]. The characteristic equation is a quasi-polynomial equation whose roots are computed through a numerical method for nonlinear equations.

In the contribution of Engelborghs [269], it is proposed a Matlab package for numerical bifurcation analysis of TDS, and the computation of the rightmost characteristic roots is carried out by using a linear multistep method (LMS method) [270].

The monograph of Breda et al. [271] presents a comprehensive set of pseudospectral techniques (Pseudospectral Differentiation Method and Piecewise Pseudospectral Differentiation Method) to analyze the stability of the solution of linear TDS with numerical implementations in Matlab.

In [272], Louisell establishes a method for determining the stability exponent and eigenvalue abscissas of a linear delay system based on examining the endpoint values of the solution to a functional equation occurring in the Lyapunov theory of delay equations. Other interesting related works are [273, 274]. In [275, 276], it is presented a methodology for calculating the Lyapunov matrix with a distributed delay, whose algorithm consist in solving a two points boundary value problem for a delay-free system.

Olgac and Sipahi provide an alternative procedure based on the cluster treatment of characteristic roots to analyze the stability of multiple time-delayed LTI dynamics. This methodology detects all the stable regions precisely, in the space of the time delays, by means of a set of curves (kernel and offspring) which count the possible imaginary root crossings for the system (see [206, 277–282] and references therein for a comprehensive treatment).

Concerning the eigenvalue problem, Michiels has come up with different procedures. In [283], it is provided a characterization of the solutions to an arbitrary nonlinear eigenvalue problem as the reciprocal eigenvalues of an infinite-dimensional operator, and the resulting algorithm is completely equivalent to the standard Arnoldi method, including many of its properties. An extension of last contribution [284] is used to compute the partial Schur factorization of a nonlinear eigenvalue problem. In [285], it is presented a procedure to compute solutions to a type of nonlinear eigenvalue problem with low-rank structure. The algorithm turns out to be equivalent to the Arnoldi method (even in the numerical behavior). In [286], first, the formula for the sensitivity of a simple eigenvalue with respect to a variation of a parameter is extended to the case of multiple non-semisimple eigenvalues. Also, it is provided a dual treatment of the delay eigenvalue problem, in one hand, at the level of the finite-dimensional nonlinear eigenvalue problem and, on the other hand, at the level of a standard operator eigenvalue problem. A numerical procedure to compute the pseudospectral abscissa is given in [287], whose main feature is that the approach is a pioneering applicable procedure for nonlinear problems.

The most complex systems have nonlinearities and delays, which can be easy to notice or sometimes not so much. A delay in a system is a phenomenon that can be seen as the dead time between transmitting and executing an action. In this context, delays are due to the fact that system dynamics are associated with past events. In general, delays are undesirable phenomena in a system because they can instabilize or produce a poor performance in their response. However, in recent years, it has been shown that delays can also favor stabilizing and improving the performance of the system [288, 289].

The deliberate application of delays to stabilize systems is a latent topic in the literature, one of the most important contributions in this aspect is found in [290] that drove a whole stream of research in what is known as time-delayed feedback control (TDFC). The technique is to deliberately introduce delays into the control scheme to -stabilize a system with or without inherent delays [291–308]. This is due to the advantages that they have in applications on experimental platforms, such as noise attenuation, nonimplementation of estimators, observers and speed sensors, avoiding filters, as well as its easy implementation. In addition, the -stability analysis or -stability regions allows a fragility analysis of the controllers gains, which can give a measure of the robustness of the closed-loop system under variations of the controller gains, see [303, 309–311].

#### 4. Advances in Stability and Stabilization of Nonlinear Time-Delay Systems

In recent years, different concepts have emerged that are less restrictive to classical concepts to determine stability in nonlinear time-delay systems. Next, some of them are mentioned.

##### 4.1. Complex Delay Complex Networks

One topic which has attracted attention from the scientific community is the analysis and control of complex networks [312, 313], which describe a wide variety of physical and social systems [314], from population interactions, brain activity, and language patterns to Internet traffic behavior among other interesting phenomena. One control problem derived from the complex systems control is the synchronization of delay coupled networks [315, 316]. In this sense, a formal stability analysis for the synchronization of complex networks, in particular for a set of oscillators, is usually given for linearized systems in a vicinity of the equilibrium point [317] by computing the stability regions in the delay parameters or using the circle criterion [318]. Some reported contributions can be found in [169, 238, 319–328].

##### 4.2. Robotic Teleoperation and Predictor-Based Control

Robotic teleoperation and haptic teleoperation has been a highly active topic in last years, specially now with the active development of virtual reality and haptic interfaces [329–334]. The time delay in these applications arises mainly to the latency aspects of the virtual reality system [335–338], and the network traffic in a teleoperation system [339–342].

The stability analysis for teleoperation has the following general approaches:(i)The use of predictors working on the stability of the system in the feedforward dynamics with nominal or robust criteria, see [203, 343–360](ii)Passivity-based approaches whose stability tests can be given in the Lyapunov–Krasovskii sense (see [106, 342, 361–374])(iii)Robust and predictor-based schemes whose stability is given for time-delay dynamics tested by means of time- or frequency-based approaches [264, 375–378](iv)Other schemes which assume the delay as a disturbance to be compensated by robust or adaptive techniques [379–384]

##### 4.3. Nonlinear and Fractional Order Systems

Lyapunov methods have shown very useful for the study of stability and the design of nonlinear control laws [130, 385, 386]. Here, discontinuous [387] and fractional order systems [388] represent an important challenge. Particularly, sliding mode control techniques is an active field of research where Lyapunov methods have been a key factor [389–393]. The main advantages of sliding mode control, including robustness and finite time convergence, are supported by no conventional Lyapunov methods.

On the contrary, fundamental research has shown that a large variety of physical signals can be described by means of functions with more varied topological properties [394–397], which can be continuous but not necessarily differentiable in any integer-order sense [398]. This has motivated the design of fractional order sliding mode-based controllers which have been proved to be robust against Hölder disturbances [399].

Fractional calculus has become an emerging approach for modeling complex systems, which has attracted the attention of several areas of study, including control systems. The use of fractional versions of PID controllers [400–406] has increased the interest and development of fractional control designs. As a natural consequence, this approach has been extended to a wider class of systems, especially time delay ones which has come up with stability studies and analyses.

Since classic methods for stability testing integer systems such as Routh–Hurwitz are not universally valid, the stability analysis for even linear fractional systems is more challenging. In particular, among the approaches of stability study for fractional time-delay systems, some pioneering contributions are given by Hwang and Cheng. In one hand, the Lambert W function is proposed for the stability analysis [407] and, on the other hand, in [408], the characteristic equation of the system is numerically analyzed for a BIBO stability test by means of Cauchy’s integral theorem. In [409], a numerical procedure to obtain the delay values where there is a root crossing (from the left half plane to the right half and vice versa), for a further procedure of finding the stability zones in terms of the delay value. Other fundamental contributions are given in [410–415]. In [416], a Matlab toolbox for the -stability analysis of fractional systems with commensurate delays is provided. L-K stability approaches for fractional systems are given in [417–419]. In [325, 420–424], some L-K stability approaches are given in the realm of neural network framework.

Concerning using Smith predictor-based stabilizing controllers, several applications are reported, see [425–432].

#### 5. Conclusions

This paper has presented a set of criteria concerning robust stability of dynamical systems with or without delay, which is important for the analysis of complex systems which may not provide complete information, involving parameter uncertainties. On the one hand, the ideas of robust stability through families of polynomials were addressed and the main criteria discussed. On the other hand, an emerging topic in the area of complex systems such as time-delay systems stability was analyzed, including the motivation, basic results for its understanding, the difficulties involved with respect to systems without delay, the time and frequency approaches, applications involving complex oscillators, current trends in this field of research, and some interesting open problems. This review explored the stability as a fundamental structural property which is crucial in the analysis and development of studies and applications of complex dynamical systems, in which the couplings and dynamical behavior may come up with new developments in a wide variety of research areas.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

In memory of Oscar Villafuerte-Segura, my brother, my partner, and unconditional friend. I will never forget you. This work was partially supported by Secretaría de Investigación y Posgrado-IPN under grant SIP20201675 (A. Luviano-Juárez).