Sampling zero stability in sampled data control systems with delays using backward triangle sample and hold

Abstract

The impact of time delays on the stability of sampling zeros in sampled-data (SD) control systems is investigated. While delays, arising from communication and computational latencies, are known to critically influence zero-dynamics stability, their specific effect on sampling zeros remains less explored. This work establishes novel conditions for sampling zero stability under time delays, employing the Backward Triangle Sample-and-Hold (BTSH) method for signal reconstruction. In particular, we analyze the asymptotic behavior of sampling zeros with respect to the system’s relative degree and delay magnitude using BTSH. Moreover, through this analysis, we derive explicit stability conditions for these zeros, crucial for overall system performance. Finally, we provide a comparative analysis contrasting the stability properties under BTSH with those under the conventional Zero-Order Hold (ZOH) method in delayed settings. The theoretical findings are validated through a detailed numerical example, demonstrating the distinct advantages of BTSH in managing delay-induced zero-dynamics challenges.

Introduction

The presence of unstable zeros and poles plays a pivotal role in controller design, exerting a profound influence on the dynamic behavior and performance of linear time-invariant (LTI) control systems^1–3. In particular, unstable zeros present formidable obstacles, often undermining the implementation of robust control laws. Methods such as pole-zero cancellation^4,5 and adaptive control schemes^6,7 are especially susceptible to degradation when unstable zeros are present, thereby complicating the design landscape and limiting the achievable system performance. It is worth noting that when the system has unstable zero dynamics, it is highly vulnerable to zero-dynamics attack (ZDA)^8,9. In contrast to poles, whose stability can typically be retained through well established transformation techniques during discretization, the behavior of zeros introduces a more intricate challenge. Notably, it is possible for a continuous-time system with stable zeros to exhibit unstable zeros once discretized^1,3,8,10,11. This discrepancy between the continuous and discrete domains calls for meticulous consideration during sampling and reconstruction, as it can directly impact the fidelity and stability of digital control implementations. To address this issue, a considerable body of research has emerged, aimed at mitigating the introduction of unstable zeros during the sampling process^1,3,10–16. These investigations highlight the nuanced complexity of discretization process and affirm the critical role that zero dynamics play in system stability and even system safety^8,17. By proactively managing the emergence of undesirable zeros, control engineers can achieve more effective, reliable, and high-performance control solutions.

Time delays are a common challenge in digital control loops, primarily stemming from the latency introduced during information transmission between the system state and the controller or sensor processing units, see, e.g.,^18–23 and references therein. Such delays can markedly degrade control performance by introducing a temporal lag in the system’s response to inputs or external disturbances. In continuous-time linear systems, the presence of time delays can induce non-minimum phase characteristics, further complicating control design³. This non-minimum phase behavior poses a significant hurdle, as it limits the achievable performance and increases the difficulty of meeting design specifications. In contrast, discrete-time systems exhibit minimum-phase properties that are closely tied to the sampling process. However, this process can inadvertently introduce unstable zeros, thereby intensifying the complexity of the system’s dynamic behavior. The situation becomes particularly intricate when non-negligible time delays are involved, as they can cause a substantial increase in the system’s effective dimensionality—potentially to infinite order, depending on the nature and extent of the delay^24,25. Given these challenges, it is imperative to rigorously examine the stable characteristics of sampling zeros affected by time delays. Such an investigation is essential for developing effective digital control strategies. A deeper understanding of these zero dynamics contributes directly to enhancing the robustness, reliability, and overall performance of control systems operating under delayed conditions.

The behavior of a discrete-time system is fundamentally shaped by both the poles of its continuous-time counterpart and the sample period utilized during signal reconstruction. Notably, the stability of system poles is preserved through the discretization process, reflecting a consistent dynamic structure between the continuous and discrete domains. In contrast, the behavior of system zeros in discrete-time settings is far more intricate. These zeros are influenced by several factors, including the system’s relative degree, the selected sampling interval, and the characteristics of the employed sample-and-hold (S/H) mechanism. Ultimately, these zeros tend to the roots of a specific polynomial determined by the parameters of the system and the sampling method, reflecting a complex interdependence among these variables. Pioneering work by Åström et al.¹ significantly advanced the understanding of how different S/H schemes, particularly the widely used zero-order hold (ZOH), affect zero dynamics in SD models. Their findings revealed that with a relative degree of two or higher, the resulting sampling zeros are prone to instability. Subsequent investigations into the first-order hold (FOH) method found that, despite offering a more refined approximation of the input signal, FOH did not yield notable improvements over ZOH in terms of sampling zero stability²⁶. This underscores the inherent challenges of maintaining zero stability in discrete-time systems with higher relative degrees, irrespective of the discretization approach. Additional research has extended the analysis of sampling zeros to systems incorporating time delays^27,28. Among various approaches, the fractional-order hold (FROH) has garnered considerable attention due to its more relaxed and general stability conditions compared to ZOH^{15,16,29–32}. The stability behavior of SD systems with time delays under FROH was further investigated in³³, offering deeper insights into its applicability. More presently, the BTSH method has emerged as a novel signal reconstruction technique. Prior studies have demonstrated that BTSH can outperform ZOH in preserving the stability of sampling zeros^{10,12,34–36}. However, the presence of time delays tends to shrink the stability region of sampling zeros in discrete-time models, relative to delay-free counterparts. Despite promising preliminary results, a comprehensive theoretical analysis of sampling zeros under BTSH in the presence of time delays remains underexplored. Furthermore, it remains an open question whether the advantages observed with BTSH in discrete-time settings extend to their continuous-time analogs when time delays are present.

Inspired by the aforementioned issues, this paper focuses on the impact of time delays on the stability of zeros for SD system with BTSH. The main contributions of this paper are:

1. A novel explicit model linking system relative degree and delay magnitude via BTSH reconstruction is established, overcoming the limitations of conventional ZOH-based analysis in delayed systems.

2. We provide the stability conditions for the intrinsic and sampling zeros of the SD systems. Particularly, for the most common systems with relative degree of 2, we derive the relationship between the BTSH parameters and the time delay, ensuring the stability of sampling zeros, which is .

The following sections outline the structure of the remainder of the paper. Section 2 indicates key preliminaries, offering the necessary background to understand how BTSH operates in the presence of time delays. Sections 3 and 4 present the main theoretical results, including the formulation of an SD model for time-delay systems and a detailed analysis of the corresponding zero dynamics under BTSH. These results are central to understanding the effects of time delays on system stability and performance. To validate the theoretical developments, Section 5 provides a numerical example demonstrating the practical effectiveness. At last, Section 6 summarizes this work and discusses their implications of SD with time delays.

Notation: Before presenting the main results, we first introduce the notations used throughout this paper. Let N, , , and mean natural-number sets, complex numbers, n-dimensional real vectors, real matrices of size , and non-negative real numbers, respectively.

Preliminaries

We begin by examining a continuous-time linear system that is observable, controllable, time-invariant, and incorporates the time delay . This is represented as:

1

where the relative degree can be defined as , with . The coefficients ( ) and () mean real constants. The exponential term represents a time delay , which may arise from communication or computational sources. Typically, the delay is considered constant and has the following relation:

2

where means the natural number, and denotes the sampling period. Thus, means the integer multiple, combined with the fractional component of T. In addition, the time delay system described in Eq. (1) can be formulated in the state-space representation.

3

and

4

The system state vector, denoted by x(t), resides in an n-dimensional space and evolves within an open subset . This vector has a pivotal function in characterizing the internal dynamics. It is further determined by its input u(t) and output y(t), both of which are critical in shaping the system’s response over time. Notably, the presence of a time delay directly affects the input signal received by the system, thereby influencing its dynamic response and output behavior. These delay-included effects highlight the importance of accounting for time delays in analysis of system performance. In this regard, for the discretization process of the system, it is crucial to ensure the stability of all system zeros, that is, to ensure that the minimum phase characteristics of the system can be maintained. (It should be noted that, for continuous-time systems, if the system is of minimum phase, all zeros are located in the left half-plane. Similarly, for discrete-time systems, all zeros are situated within the unit circle.)

This study centers on analyzing the zero dynamics for an original continuous-time system that includes time delays. A primary objective is to elucidate the relationship establishing the relationship between the discrete-time zeros and the zeros of the continuous-time system with delay characteristics, particularly during the generation process of the discrete-time input via the BTSH method. Understanding this relationship is vital for improving the analysis and synthesis of control systems in which both time delays and input reconstruction strategies significantly affect performance.

The SD system under consideration comprises a sampler vis sampling period T, the BTSH for signal reconstruction, and a time delay . At discrete sampling instances iT, the digital controller samples the output of the continuous-time system and updates the BTSH at the delayed time instant . The waveform generated by the BTSH, which reconstructs the input signals, can be illustrated in Fig. 1. The corresponding mathematical formulation for this reconstruction mechanism is given as follows:

5

Here, denotes the sampling index, and represents a tunable parameter of the BTSH method that governs the duration of signal activation. The function corresponds to the signal generated by a ZOH during the interval . In accordance with the standard ZOH definition, the signal maintains a constant value throughout this interval, satisfying the equivalence:

6

By employing the signal reconstruction scheme defined in Eq. (5) to generate the system input, the corresponding SD:

7

where , and represent x(kT), y(kT) and , respectively. And

8

Fig. 1.

The signal reconstruction output of the BTSH with time delay.

Remark 1

It is noteworthy that the conventional BTSH model represents the SD system where the time delay is set to . This scenario serves as a foundational reference for analyzing system dynamics in the absence of delay effects. For readers seeking a more comprehensive exploration of the sampling zero characteristics associated with SD system generation, the study presented in our precious research¹⁰ offers substantial insights. This work contributes valuable theoretical and practical perspectives that deepen the understanding of zero dynamics and their implications in sampled-data control systems.

The SD model of time delay system

In this part of the section, we introduce a novel polynomial that emerges during the discretization process of time-delay systems utilizing the BTSH method. This polynomial, denoted as , plays a central role in characterizing the zero dynamics. The definition of this polynomial bears resemblance to the Euler-Frobenius polynomials and the Modified Euler-Frobenius polynomials, as discussed in³⁷. Subsequently, we establish and prove several key properties associated with this new polynomial, which are instrumental in understanding the influence of time delay and signal reconstruction on the system’s stability.

Definition 1

By employing BTSH as the signal reconstruction method to discretize time delay system (1), we define a new polynomial, denoted as

9

Here, , , ,

10

where

Remark 2

The authors did not locate any previous references to the new polynomial articulated in (9). Definition 1 is crucial for elucidating the SD model of a time delay system featuring BTSH, as well as for illustrating the asymptotic characteristics of the discrete-time system. Further information regarding the properties and the methodology for deriving the new polynomial will be discussed in later sections of this document.

To gain deeper insight into the interplay between discrete-time system zeros and time delay, which should be essential to analyze the properties associated with this newly introduced polynomial. These properties are pivotal in clarifying how time delays influence the zero dynamics. Through systematically establishing these properties, we aim to construct a theoretical framework that not only enhances our understanding of this relationship but also facilitates more robust analysis and design strategies for systems affected by delays.

Theorem 1

The newly introduced polynomial possesses several important properties that are instrumental in characterizing the zero dynamics. These properties are summarized as follows:

(i)

(ii) , where

11

and

Proof

(i) Based on the formulation given in Definition 1, the representation of is

Therefore, the Schur Determinant Lemma can compute the matrix determinant described above.

where represents the first item of vector .

To determine the overall determinant, it is necessary to evaluate the second term in advance. Accordingly, we first compute the component associated with the second term as follows:

The co-factor matrix of , denoted by , is defined based on the minors of the matrix. Each element of this matrix, represented as , as , where denotes the minor corresponding to the element in (i, j) position of . In particular, the first row of the co-factor matrix, is represented as . This structured relationship between the co-factor matrix and the original matrix serves as a foundational tool in deriving analytical expressions and facilitating determinant evaluations.

where

Clearly, by performing a co-factor expansion along the first column and iteratively traversing it times, one can systematically derive the determinant of the matrix, along with the corresponding sub-determinants required for the subsequent analysis.

12

By combining the previous derivations, we arrive at the following result:

13

Thus, the determinant is

14

Along with the definition of , the proof is completed.

(ii) The authors observe that establishing the properties in question proves the identity . Here, can be defined via Eq. (10). To demonstrate this identity, an inductive proof strategy is adopted. The base case, , is straightforward: one can verify that , noting that . For the inductive step, supposing the identity holds for , is valid. Under this assumption, the goal is to show that the identity also holds for , namely, .

By employing a similar proof strategy to that used in part (i), the determinant of can be equivalently reformulated as follows:

In a similar calculation process, the following can be obtained:

15

Thus, , where = , based on the induction hypothesis, assume that for is true, we have

So far, the properties of the second part were completely proved.

Next, we delve into the data model utilizing BTSH-based signal reconstruction. This forms the analytical foundation for investigating the properties of sampling zeros addressed throughout this paper. The forthcoming theorems present SD models tailored to linear time delay systems, highlighting their structural features, especially when BTSH can generate the input. We begin by examining the behavior of SD pertaining to a continuous-time integrator with delay, as the sampling period becomes infinitesimally small. The analysis can extent to more linear time delay systems.

Theorem 2

Considering an r-th order integrator systems with time delay , and it has transfer function . When BTSH is adopted as the method for signal reconstruction to obtain the input for an integrator system, the corresponding exact sampled-data system can be represented in the following state-space form:

16

where and

Meanwhile, the output of the system can be given by . Consequently, the relationship between the discrete input and the corresponding sampled output can be characterized as:

17

where represents the new polynomials as defined in (9).

Proof

Under the definition of time-delay integrator system, the corresponding state-space representation is:

18

and

By combining Eq. (8) with the continuous-time A and B, the corresponding discrete-time and is derived.

Conversely, the discrete-time state-space equations is reformulated via the forward shift operator q, yielding the following transformation:

19

where and

Cramer’s rule is applied to solve for , resulting in the following expression:

where

By computing the determinant of M, the following result is obtained:

20

From the results in Theorem 1, we know , then

One can write the value of as

21

Using the q-operator and noting that , the result can be obtained.

Next, we proceed to examine the zero characteristics of general linear time delay via the asymptotic case where the sampling period tends toward zero.

Theorem 3

For a linear system, represented by transfer function in (1), where denotes the time delay and means the delay-free component of the system , with the relative degree of r. Then, the exact SD admits the following limiting forms as the sampling period approaches zero.

22

as the sampling period .

Proof

SD model is derived the definitions of continuous- and discrete-time systems.

23

where is the Laplace transform of the output wave-forms with BTSH and time delay as shown in (5). And then

By applying the inverse Laplace transform and -transform, (23) is:

24

where , represents the system’s poles. Using the change of variables in (24), we have

25

where

Hence, as the sampling period , then

26

along a contour that circumvents the origin by an infinitesimally small margin. Additionally, according to Theorem 2, it is established that the r-th order time delay integrator system under BTSH has a limiting expression identical to (26), and the corresponding discrete-time function is:

27

By applying the same calculation process outlined in Eqs. (23)–(26) to the r-th order time delay integrator system, we ultimately obtain:

28

which completes the proof of this Theorem.

Remark 3

Let the input signal generation method be designated as BTSH. Consequently, SD of this system has n poles and m zeros, with one zero converging to , whereas the other zeros asymptotically converge to the roots of the polynomial .

Properties of zeros for time delay system with BTSH

In control theory, it is well-known that the zeros can be typically divided into 2 types: intrinsic and sampling zeros³⁸. Our earlier analysis showed that the zeros of the SD system approach the point . However, a complete understanding of the behavior of zeros using BTSH can be still lacking. We will also examine the properties and implications, which will deepen our understanding of their significance in these systems.

Theorem 4

Suppose as the point within the complex plane that represents a zero of the continuous time delay system via . Assume that as a connected bounded domain that contains only , meaning there are no other distinct zeros within or on the boundary of . Then, for the corresponding discrete-time model , then a positive constant holds and for every T in the interval , the transfer function via zeros in .

29

Proof

For the time delay discrete-time system can be written as

30

where

31

Then, by differentiating the determinant, we arrive at the following expression:

32

33

In this context, let be an arbitrary complex number. The notation represents the polynomial forming the numerator of . Crucially, if is not a zero of , then for any given positive constant , there exists a corresponding positive value such that specific desired properties of the system are preserved when the sampling period T satisfies . This result provides a foundation for analyzing the behavior of the discretized system away from its continuous-time zeros.

34

where the term arg denotes the argument of a complex number. Furthermore, we denote the boundary of a domain by , which can be parameterized by a real variable, denoted as . This parametrization allows for a more precise characterization and analysis of the boundary behavior of functions defined over .

35

where indicates a one-to-one correspondence mapping from [0, 1] to the boundary , allowing to trace around the domain in an anticlockwise direction as increases from 0 to 1. In cases where forms a closed set, one can determine such that

36

From (36), we obtain

37

namely

38

In the context where the complex variable s traverses the boundary in an anticlockwise direction, the term denotes the quantity of anticlockwise encirclements that the locus of makes around the origin in the complex plane. Likewise, the expression exhibits identical properties. Consequently, Eq. (38) suggests that

39

Because and mean integers and means arbitrary, we obtain

40

Based on the hypothesis, within the domain , the polynomial is analytic and free from poles or additional zeros aside from those under consideration. Therefore,

41

Based on the argument principle, from (40) and (41), we know

42

Furthermore, it is evident that has no zeros of the region . Therefore, according to Eq. (42), it can be concluded that contains exactly zeros within the domain .

Remark 4

Building on the results from Theorem 4, we can conclude that when mean a stable zero from , thus the intrinsic zero of the system is also stable. This establishes a direct correlation between the stability of the zeros in these two functions. Furthermore, the converse holds true: if the intrinsic zero of is stable, then the zero of must also be stable. This bi-directional relationship underscores the connection in the stability properties.

Next, we examine the stability conditions associated with sampling zeros. Since many practical systems have a relative degree of at most two^15,39, the following analysis will focus on the stability criteria for sampling zeros in systems with relative degrees no greater than two.

Theorem 5

Considering BTSH in the time delay system, the resulting SD system is denoted by . The following presents the stability conditions for the sampling zeros via T to 0.

(i)

All zeros from should be stable only when the zeros of are stable.

(ii)

When the zeros of are stable and the parameters of BTSH and the time delay meet the condition , then the zeros of will also be stable.

Proof

(i) In this scenario, only intrinsic zeros are present in the discrete-time model. Therefore, by applying the results established in Theorem 4, the stability of the zeros can be directly inferred.

(ii) As discussed earlier, when the relative degree is two, the discrete-time model contains both intrinsic zeros and a single sampling zero. The overall stability condition thus requires both the intrinsic zeros and the sampling zero to lie within the unit circle. The stable conditions from the intrinsic zeros follows from Theorem 4. As for the sampling zero, it asymptotically approaches as . By analyzing the root locations of this polynomial and ensuring they remain within the unit circle, the corresponding stable condition can be derived.

43

Solving the above equation can complete the proof of this part.

Remark 5

For the case where , the stability region corresponding to Eq. (43) is illustrated in Fig. 2. From both the analytical results and the graphical representation, it is evident that the admissible range of the parameter f is influenced by time delay. Specifically, in the absence of any time delay, the feasible interval for f lies within , which aligns with the findings reported in¹⁰.

Fig. 2.

The effective stabilization area of (43).

Numerical examples

To illustrate and validate the theoretical results presented in this paper, we introduce several representative examples. These serve as supplementary demonstrations of the analytical findings.

Consider a time-delay system characterized by a delay and described with

44

It is evident that the delay-free component has a degree of two. Under the results in^1,27, the associated SD system inherently exhibits an unstable zero as T is close to zero when ZOH can be used for input reconstruction. This conclusion can be derived from the simulation results shown in Fig. 5.

Fig. 5.

The locus of the sampling zero from (44) via ZOH and BTSH.

In order to better verify the influence of different time delays on the stability of the sampling zeros of the SD system, we select three different time delay values, namely , and , to analysis the distribution of the sampling zeros under different sampling periods, as shown in Fig. 3. Based on the above theoretical analysis and mentioned in Remark 3, we know that the sampling zero of the system (44) converge to the roots of polynomial as the sampling period approaches 0. Through calculation, the sampling zeros corresponding to the above three different time delays respectively equal , , and as the sampling period approaches 0. The simulation results are consistent with our theoretical analysis. This further demonstrates the validity and effectiveness of the conclusions obtained in the paper.

Fig. 3.

The locus of the sampling zeros about (44) with BTSH under different delays.

Furthermore, in order to better compare the distribution of samplign zeros and intrinsic zeros about system (44) under ZOH and BTSH conditions, we conducted a comparative analysis using graphical methods. We consider a time delay and set the BTSH parameter to . The asymptotic behavior of the system’s zeros-including both intrinsic and sampling zeros-is illustrated in Figs. 4 and 5. Simulation results indicate that the intrinsic zeros of the SD system remain stable under both ZOH and BTSH. However, when ZOH is employed, the sampling zero becomes unstable as , revealing a critical limitation in using ZOH for time-delay systems. In contrast, with an appropriately chosen BTSH parameter, the sampling zero consistently resides within the stable region. This highlights the pivotal role of BTSH parameter selection in preserving system stability and offers a viable strategy to address the instability issues often encountered in SD models of time-delay systems.

Fig. 4.

The locus of the intrinsic zero from (44) via ZOH and BTSH.

On the other hand, we chose the disk drives serves as an example for verification. Disk drives serves is an important data storage medium in data processing systems. It will be used to conduct a comparative analysis of the zero dynamic stability of the systems under ZOH and BTSH conditions when there is a time delay. The schematic diagram of the disk drives serves is shown in Fig. 6, which can be represented by the following differential equation¹⁰.

45

Where I , C, K, , i represent the inertia of the head assembly, the viscous damping coefficient of the bearing, the return spring constant, the motor torque constant and the input current, respectively. , , are the angular acceleration, angular velocity and the position of the head, respectively.

Fig. 6.

Computer hard disk drive¹⁰.

We also select the same numerical values Kg m, m s/rad, Nm/rad and Nm / Consistent with reference¹⁰. Here, we assume that the system has input time delay . Therefore, the disk drives serves system has transfer function with relative degree two as.

46

where and are the Laplace transform of and , respectively.

By using BTSH and ZOH as the signal reconstruction method to discretize the the disk drives serves system (46), we select the parameter of BTSH with and the time delay with , and for simulation. Finally, the zeros location of the sampling zeros with ZOH and BTSH are presented in Fig. 7. As can be seen from the Fig. 7, based on the results proposed in this paper, the sampling zeros under the BTSH condition can be stabilized, while the ZOH cannot stabilize the sampling zeros.

Fig. 7.

Comparison chart of the sampling zeros under ZOH and BTSH with different time delays.

Conclusions

The analysis of control systems with time delays poses a significant research challenge, particularly due to the loss of minimum-phase characteristics that may arise during discretization, potentially leading to zero instability. This study explores the zero dynamics of SD systems incorporating time delays under the BTSH framework. Expanding on prior studies such as^10,12,36, which primarily addressed delay-free systems, this work extends the investigation to systems with inherent delays. A key contribution is the introduction of a novel polynomial that captures the interplay between relative degree, time delay, and the adjustable BTSH parameter. The paper further discusses several essential properties of this polynomial. The corresponding mathematical formulation of the SD systems is then derived. Moreover, a sufficient condition is established to ensure the stability of all zeros in SD system with delays when the relative degree not greater than two. For sufficiently small sampling periods, BTSH enables the placement of all zeros within the stable region-a feat unattainable using the ZOH. In the future, we will extend to the study of nonlinear time delay systems, and we will expand to the control applications of actual systems based on the theoretical research results of BTSH.

Author contributions

Minghui Ou: supervision, conceptualization, formal analysis, methodology, funding acquisition, writing-original draft and editing; Yuancheng Luo and Cheng Zeng: conceptualization and writing-review; Zhenjie Yan and Yuanfei Deng: writing-review and editing; Yu Zhou : writing-review. All authors reviewed the manuscript.

Data availability

The data and graphs analysed in this study can be encoded using MATLAB based on the theoretical analysis presented in the paper. At the same time, readers can obtain the corresponding graphs through the encoded simulation. However, if there are reasonable request, these can also be provided by the corresponding author.

Declarations

Competing interests

The authors declare no competing interests.