Topology property analysis and application of stable time‐delay regions for linear multiple time‐delay systems

This study examines the relationship between the topology of stable time‐delay regions and the stability analysis of an linear multiple time delay system. To analyze the topology of stable time‐delay regions, we construct a function with a value equal to zero for the time‐delay points on the boundaries of stable time‐delay regions. The function is continuous and differentiable in the whole defining field with a global minimum of zero so that we can locate the boundaries by minimizing the value of the function. Based on the topology analysis, we proposed a performance validation approach for controllers that are designed to stabilize the system using feedback signals with time delays. The method based on the topology analysis is simple and reliable so that can deal with a linear time‐invariant system with high order and multiple time delays (more than 3). The example case study shows that the above method is reliable and enables ultra‐low latency coordination and control for the future power grid with ubiquitous power electronics.

In the space of time delays, the time delays of a system vary in a particular region, which is determined by the performance of the system's communication channels.If in the space of time delays, there is a certain region in which the system can remain asymptotically stable when the time delays vary, we call that region a stable time-delay region.Because the stability of nonlinear systems with time delays is related to the initial values of state variables, it is difficult to describe the stable time-delay regions of nonlinear systems with time delays.Here, we concentrate only on the stable time-delay regions of linear time-invariant (LTI) systems with time delays.
It is well known that if an LTI system with a single time delay, τ, is stable for τ = 0, then there must exist a positive number τ for which the system is stable for 0 ≤ τ ≤ τ.Many researchers have simply extended this idea to a system with multiple time delays.However, this simple extension may lead to conservatism.For example, Fridman and Shaked 5,6 dealt with a linear system containing two delays: The upper bounds τ 1 and τ 2 on τ 1 and τ 2 , respectively, are selected such that (1) is stable for 0 ≤ τ 1 ≤ τ 1 and 0 ≤ τ 2 ≤ τ 2 .However, the ranges of τ 1 and τ 2 , which ensure the stability of the system (1), are quite conservative because they are guaranteed from zero to the upper bound, even though it may not be necessary for them to start from zero.The results of, 16,[25][26][27] show that in the space of time delays there may be infinite isolated regions far away from the origin of zero, and the system can remain asymptotically stable when the system's time delays in those regions.
Based on the Rekasius substitution, 13 the method in [14][15][16] can determine stable time-delay regions in the time-delay space without conservativeness by computing the boundaries on which the characteristic equation of the system has purely imaginary roots.At present, the method deals mainly with systems with orders and time delays of no more than 3, because the method involves complicated symbol calculation, and the boundaries will consist of hypersurfaces and their intersections for the systems with more than 3 delays.
This study analyzes the topology of stable time-delay regions and demonstrates the important application value of the topology analysis.Specifically, based on the topology analysis, we proposed a performance validation approach for controllers that are designed to stabilize the system using feedback signals with time delays.The method based on the topology analysis is simple and reliable so that can deal with an LTI system with high order and multiple time delays (more than 3).
The remainder of this paper is organized as follows: Section 2 presents concepts about stable time-delay regions and their boundaries.Section 3 provides the problem statements.In Section 4, we analyze the topology of stable time-delay regions and its influence on the stability analysis of an LTI system with time delays.In Section 5, based on the above topology analysis, we propose a performance validation approach for controllers designed to stabilize an LTI system with time delays in feedback signals.Then, we present a dedicated case study to demonstrate the important application value of the topology analysis and verify the validity of the proposed method in Section 6.We conclude the paper in Section 7.

| Stable delay regions
Consider the following linear system (2), with multiple time delays: where x(t) is an n-th order state vector; Notice that the equation CP = 0 is transcendental when the time delays in (3) are distinct.There are infinite roots for CP = 0.The system (2) is asymptotically stable if all roots of CP = 0 are in the left-half complex plane.The point (τ 1 , τ 2 , … τ n ) is called a stable time-delay point in the space of time delays if for any sC such that Re(s) ≥ 0, CP(s, τ 1 , τ 2 , … τ n ) ≠ 0. An infinite number of stable time-delay points connecting together can constitute a stable time-delay region in the space of time delays.The results of, 16,[25][26][27] show that in the space of time delays, there may be infinite isolated stable time-delay regions for an LTI system with multiple time delays.

| Boundaries of stable time-delay regions
For a fixed point (τ 1 , τ 2 , … τ n ) in the space of time delays, if sC exists such that Re(s) = 0, CP(s, τ 1 , τ 2 , … τ n ) = 0, and CP(s, τ 1 , τ 2 , … τ n ) ≠ 0 for any sC such that Re(s) > 0, then the point (τ 1 , τ 2 , … τ n ) is on the boundary of a stable timedelay region.In other words, if CP(s, τ 1 , τ 2 , … τ n ) = 0 has roots on the imaginary axis and no roots in the right-half complex plane, then in the space of time delays, (τ 1 , τ 2 , … τ n ) is a point on the boundary of a stable time-delay region.Conversely, if (τ 1 , τ 2 , … τ n ) is a point on the boundary of a stable time-delay region, there must be a ω such that (4) holds: The results of, 16,[25][26][27] show that in the space of time delays there may be infinite isolated stable time-delay regions for an LTI system with multiple time delays.However, in engineering, it is unnecessary to describe the whole stable time-delay region for a system with time delays because the ranges of time delays of the system's communication channels correspond only to a particular region in the space of time delays.The region Ω(τ 1 , τ 2 , … τ n ), determined by the performance of channels, can be defined as , where τ min i and τ max i are the minimum and maximum delays of the i-th channel of the system, respectively, and we assume they are known.Here, Ω is the actual time-delay region of the system.Thus, we need only care about the system's stability in Ω.
Next, we provide a simple demonstration of the importance of the topology of stable time-delay regions in analyzing the stability of an LTI system with time delays.For example, in Figure 1, Ω is the actual time-delay region of a system with two delays.Suppose stable time-delay regions have the topology characteristic of denseness (there are no holes in them).Then, we can get Ω & e Ω, as shown in Figure 1, and the system can remain asymptotically stable in Ω, where Ω is a stable time-delay region of the system, if all the time-delay points on the boundary of Ω are verified to be stable The actual time-delay region and stable time-delay region.
time-delay points.This means that we can judge whether an LTI system with time delays can remain asymptotically stable for all time-delay points in Ω by computing only the system's stability at the time-delay points on the boundary of Ω.Then, we can judge whether the system can remain asymptotically stable when its time delays in Ω by computing only the system's stability at the time-delay points on the boundary of Ω because the stability of LTI systems with time delays has no relation to the initial values of state variables.As a result, we can judge whether the performance of the system's communication channels is allowable by computing only the system's stability at the time-delay points on the boundary of Ω because Ω is determined only by the performance of the channels.There are many methods 9,17 that can easily compute the stability of an LTI-MTD (multiple time delay) system for a fixed time-delay point.This study intends to provide a practical method to stabilize an LTI system using feedback signals transmitted in communication channels with delays.

| TOPOLOGY ANALYSIS OF STABLE DELAY REGIONS
In Section 3, we provide a method to judge whether the performance of a system's channels is allowable using the topology characteristics of stable time-delay regions.However, this method will be infeasible if either of the following situations occur: 1. Stable time-delay regions are not dense, as shown in Figure 2, where Ω is an unstable time-delay region, Ω is a stable time-delay region, and Ω is the actual time-delay region of the system.
2. There are time-delay regions connecting together, as shown in Figure 3, where Ω1 and Ω2 are two stable time-delay regions connecting by the boundary C, and Ω is the actual time-delay region of the system.Strictly speaking, if all

stable time-delay region with holes (unstable time-delay regions).
F I G U R E 3 Two stable time-delay regions connecting together.
points on the boundary of Ω are stable time-delay points, this situation will not occur.However, there are infinite points on the boundary, making it impracticable to investigate all the points on the boundary.Thus, the work can only be done by sampling points at intervals.
In Figure 3, on the boundary Ω, there are only 2 points (points of intersection of C and the boundary of Ω) at which the system cannot remain asymptotically stable.It is very possible to skip these two points in the process of investigating points on the boundary at intervals.However, this is not allowable because the system cannot remain asymptotically stable when its time-delays are on the boundary C. The worst case scenario for this situation is shown in Figure 4, in which Ω1 , Ω2 , Ω3 , and Ω4 are four stable time-delay regions connecting together; Ω is an unstable time-delay region; and Ω is the actual time-delay region of the system.
To overcome this limitation of the method proposed in Section 3, we provide remediation for the method as follows.
Notice that the common characteristic of Figure 2 through Figure 4 is that there are boundaries in the actual timedelay region Ω.Therefore, the asymptotical stability of the system in Ω can still be judged by investigating the points on the boundary of Ω at intervals, if we can ensure there are no boundaries in Ω.
Next, we construct a function: where CP Ã is the conjugate of CP in (3).Notice that F is continuous and differentiable with respect to ω, τ 1 , τ 2 , … τ n in the entire defining field, and F = 0 is the global minimum of F. So, when F = 0, and ∂F/∂ω = ∂F/∂τ 1 = ∂F/∂τ 2 = … = ∂F/ ∂τ i = … = ∂F/∂τ n = 0 hold.From the analysis in Section 2.2, it can be deduced easily that if F gets its minimum then in the space of time delays, there must exist a boundary passing through the point (τ 1 , τ 2 , … τ n ) so that in Ω the system cannot remain asymptotically stable, else if min (F) > 0 in Ω and all the points on the boundary of Ω are stable time-delay points, then the system can remain asymptotically stable in Ω.The process of obtaining the minimum of F is a process of nonlinear programming.Because F is continuous and differentiable in the whole defining field, there are many optimal algorithms that can compute the global minimum of F in Ω. Incidentally, the defining field of the independent variable ω can be obtained by analyzing Re[CP(ω i , τ 1 , τ 2 , … τ n )] = 0 and Im[CP(ω i , τ 1 , τ 2 , … τ n )] = 0, as shown in the case study.

| APPLICATION OF THE TOPOLOGY ANALYSIS
In practice, after a controller is designed, we need to validate its performance before implementation.However, for a system with multiple delays, because there are infinite delay points in Ω, it is not feasible to verify the controller performance for each delay point in Ω.Here we proposed a topology analysis-based approach, which can be illustrated as follows.Consider an LTI system: Four stable time-delay regions enclosing an unstable time-delay region.
where x is a vector of state variables of the system, y is a vector of output variables of the system, and u is a vector of control variables.Suppose the system is unstable and needs to be stabilized with feedback signals, and the feedback signals are transmitted in communication channels with delays.Now, we illustrate the application of the topology analysis by classifying feedback into two categories: state feedback with time delays and output feedback with time delays.

| State feedback with time delays
In the system (6), when the state variables are used as feedback signals to stabilize the system, we can let where K is a feedback gain matrix.Substituting (7) for u in ( 6), ( 6) becomes When the time delays in feedback signals are taken into account, (7) becomes Substituting (9) for u in (6) gives the classic linear system with multiple time delays (2), and in (2) where 0 is a column vector composed of zeros and K •i is the i-th column of K in (7).To stabilize the system, one must find an appropriate feedback gain matrix K such that the closed-loop system (8) can remain asymptotically stable for the time delays in Ω, where Ω is the actual time-delay region of the system and is determined by the performance of the system's communication channels.For a given controller K, we can verify if it can stabilize the system for the delays within Ω by checking the following two conditions: 1) if the closed-loop system is stable for the delays on the boundary of Ω; 2) if the minimum value of F in ( 5) is greater than zero for the delays within Ω.A detailed flowchart is given in Section 5.3.

| Output feedback with time delays
In the system (6), if the output variables are used as feedback signals, we can let where K is a feedback gain matrix.Substituting (10) for u in ( 6), (6) becomes When the time delays in feedback signals are taken into account, (10) becomes Substituting (10) for u in (6) gives the classic linear system with multiple time delays (2), and in (2) , where 0 is a column vector composed of zeros and K •i is the ith column of K in (10).To stabilize the system, one must find an appropriate feedback gain matrix K such that the closed loop system (11) can remain asymptotically stable when the time delays vary in Ω, where Ω is the actual time-delay region of the system and is determined by the performance of the system's communication channels.For given controller K, we can verify if it can stabilize the system for the delays within Ω by checking the following two conditions: 1) if the closed-loop system is stable for the delays on the boundary of Ω and 2) if the minimum value of F in ( 5) is greater than zero for the delays within Ω.A detailed flowchart is given in Section 5.3.

| Flowchart for the application
In practice, after a controller is designed, we need to validate its performance before implementation, and usually numerical simulation is used for the validation.However, for a system with multiple delays, because there are infinite delay points in Ω, it is not feasible to verify the controller performance for each delay point in Ω.Here we proposed a topology analysis-based approach shown in Figure 5, which includes the following key steps.
Step 1: Compute the stability of ( 2) on the boundary of Ω by investigating points on the boundary at intervals.
Step 2: If the controller cannot stabilize the system for the delays on the boundary of Ω, then the controller cannot pass the validation; otherwise, go to next step.
F I G U R E 5 Flowchart for controller performance validation.
Step 3: Compute the minimum of F in Ω, if there exist F = 0 in Ω, then validation fails; otherwise, the controller passes the validation.

| CASE STUDIES
Take the system of ( 6) with The eigenvalues of the matrix A are 1 and À 1; the system cannot remain asymptotically stable and therefore needs to be stabilized with feedback signals.Suppose the feedback signals are transmitted in communication channels with delays.The time-delay characteristics of the channels are shown in Table 1.In the space of time delays, the actual timedelay region Ω corresponding to Table 1 is shown in Figure 6.Next, we use the state and output variables with time delays to stabilize the system.

| State feedback with time delays
Suppose the state variables used as feedback signals are transmitted in communication channels.The performance of the communication channels is shown in Table 1.The state variables x 1 and x 2 are transmitted in Channel 1 and Channel 2, respectively.
It is clear that the feedback gain matrix of the system is a 1 Â 2 matrix, in the form: . According to the information in Section 5.1, we know that in (2), The actual time-delay region determined by the performance of channels in Table 1.
The corresponding CP is For fixed k 1 and k 2 , the defining field of the independent variable ω in (4) can be obtained as follows.
In (13), let s = jω and substitute cos(ωτ i ) + jsin(ωτ i ) for e jωτ i .Then we have, For convenience, suppose k 1 and k 2 are positive numbers.Notice that sin τ 2 ω ð Þ j j≤ 1 and cos τ 2 ω ð Þ j j≤ 1; therefore, we can construct two polynomials in ω: , and for any given ω ≤ 0, . With the information from the above method can be extended to any nth order system with multiple time delays.
For given a controller K = [À1.2568，-0.7231],according to the flowchart in Section 5.3, first we need to test if K can stabilize the system for the delay on the boundary.Then we compute the minimum of F in Ω.The computation results show that F can reach its minimum F = 0.10329 at the point ω 0 ,τ 10 , τ 20 ð Þ , where ω 0 ¼ 2:3115, τ 10 ¼ 30:6077ms, τ 20 ¼ 699:9921ms.Fixing ω at ω ¼ ω 0 , the value of F with respect to (τ 1 , τ 2 ) in Ω is shown in Figure 7.According to the analysis in previous sections, the system can remain asymptotically stable in the actual time-delay region Ω if all the points on the boundary of Ω are stable time-delay points because the minimum of F in Ω is not equal to zero.In Step 2 of the flowchart in Figure 5, it has been verified that the points on the boundary of Ω are all stable time-delay points by sampling points on the boundary at intervals of Δτ ¼ 1ms.Therefore, the system can remain asymptotically stable in Ω.The stability of the system at any time-delay point can be verified by the integral in the time domain for any given initial values.The stability of the system at τ 10 ,τ 20 ð Þis demonstrated by Figure 8, in which the curve is the variation of as t varies from t = 0 s to t = 50s for given initial values x ¼ 1 1 ½ T .To obtain an overview of F, we fix ω ¼ ω 0 0 and plot F with respect to (τ 1 , τ 2 ).An overview of F is shown in Figure 10, and a contour map of F is shown in Figure 11.
It can be seen from Figures 10 and 11 that the curved surface is tangent to the τ 1 Àτ 2 plane periodically.This arises from the transcendental characteristic of (3).From (3), it can be determined that 10, the curved surface is tangent to the τ 1 Àτ 2 plane when τ To further demonstrate the validity of the methods proposed in this paper, we now demonstrate that the system can remain asymptotically stable for all the time-delay points in Ω.There are infinite points in Ω, so it is not practicable to verify the system's stability at all the points in Ω.Therefore, we sample the points as shown in Figure 6 and verify the system's stability at every sampled time-delay point by the time domain numerical integration (Matlab DDE23).The results of the time-domain numerical integration are shown in Figure 12, in which the curves are the variations of x k k 2 as t varies from t = 0 s to t = 50s for given initial values x ¼ 1 1 ½ T .It can be seen from Figure 12 that the (time-delayed)system can remain asymptotically stable for any one of the above time-delay points.Thus, we know that the system can remain asymptotically stable when its time delays vary in Ω because the stability of LTI systems with time delays has no relation with the initial values of state variables.

| Output feedback with time delays
Suppose the output variables used as feedback signals are transmitted in communication channels.The performance of communication channels is shown in Table 1.The output variables are transmitted in Channel 1 and Channel 2, respectively.In the space of time delays, the actual time-delay region corresponding to Table 1 is shown in Figure 6.
It is clear that the feedback gain matrix of the system is a 1 Â 2 matrix, in the form K = [k 1 k 2 ].According to the information in Section 5.1, we know that in (2), The corresponding CP is For given a controller K = [À2.2478,-0.5698], according to the flowchart Section 5.3, first we tested if K can stabilize the system for the delay on the boundary.The test result shows that K can stabilize the system for the delay on the boundary.Then we compute the minimum of F in Ω.The computation results show that minimum of F is greater than zero.For the time-delay points shown in Figure 6, the results of time domain numerical integration ((Matlab DDE23)) are shown in Figure 13, in which the curves are the variations of x k k 2 as t varies from t = 0 to t = 10s for given initial values x ¼ 1 1 ½ T .It can be seen from Figure 13 that the system can remain asymptotically stable for any one of the above time-delay points.Thus, we know that the system can remain asymptotically stable when its time delays vary in Ω because the stability of LTI systems with time delays has no relation to the initial values of state variables.
In engineering, many processes involve time-delay phenomena, such as the collecting and processing of signals as well as the long-distance transmitting of signals.Therefore, using feedback signals with time delays to stabilize a system is a significant problem.This study demonstrates the important application value of the topology analysis of stable time-delay regions.The method based on the topology analysis is simple and reliable and may enable ultra-low latency coordination and control 20,21 for the future power grid with ubiquitous power electronics, 22,23 which is a complex engineering system with high orders and multiple time delays.

T A B L E 1
Time-delay characteristics of the channels.

2 p 20 ¼
as t varies from t = 0 to t = 50s for given initial valuesx ¼ 1 1 ½ T .F I G U R E 7The value of F in Ω.Notice that the point τ 10 ,τ 20 ð Þ is a stable time-delay point approaching the boundary of Ω.This means that F reaches its minimum nearly on the boundary of Ω.As a result, F may continue to decrease if we extend the boundary.Next, we extend the upper bound of τ 2 from τ u 2 ¼ 700ms to τ u 2 ¼ 800ms; then F can reach its minimum F = 0 at the point ω 794:6984ms.According to the information in Sections 3 and 4point in the space of time delays.The critical state can be verified by Figure9, in which the curve is the variation of

F I G U R E 8 1 0
Integral results in the time domain when the system at τ 10 , τ 20 ð Þ .F I G U R E 9 Critical state of the system at τ 0 Overview of F.F I G U R E 1 1 Contour map of F. F I G U R E 1 2Variations of kxk 2 corresponding to the points in Figure6when K = [À1.2568-0.7231]in(7).