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# Impact of Delay-Difference Approximation on PID Controllers: Stability and Performance Insights

Diego Torres-García, César Fernando Méndez Barrios, Silviu-Iulian Niculescu

► To cite this version:

Diego Torres-García, César Fernando Méndez Barrios, Silviu-Iulian Niculescu. Impact of Delay-Difference Approximation on PID Controllers: Stability and Performance Insights. MED 2024 - The 32nd Mediterranean Conference on Control and Automation, IEEE, Jun 2024, Chania, Crete, Greece. hal-04569057

HAL Id: hal-04569057

https://centralesupelec.hal.science/hal-04569057v1

Submitted on 6 May 2024

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# Impact of Delay-Difference Approximation on PID Controllers:
## Stability and Performance Insights

Diego Torres-García<fnref n="1" />,<fnref n="2" />, César-Fernando Méndez-Barrios<fnref n="1" />, and Silviu-Iulian Niculescu<fnref n="2" />

*Abstract*—This paper aims to explore the advantages and disadvantages associated with implementing derivative action in a PID controller using a delay-difference approximation scheme in continuous time. Despite the simplicity of the approximation, the characteristic function of the closed-loop system is a quasip polynomial including delay-dependent coefficients, which results in a challenging task. The proposed study primarily focuses on the way the controller gains and the delay affect the key characteristics of the dynamics such as asymptotic stability, noise rejection, and closed-loop performance for linear-time-invariant (LTI) minimum-phase second-order systems. Additionally, the paper provides some results concerning the delay margin of the delay-difference approximation of the closed-loop system. Finally, the proposed results are illustrated via numerical examples.

## I. INTRODUCTION

Proportional–integral–derivative (PID) controllers are widely prevalent in the industry due to their simplicity, straightforward implementation, and the significant advantages they offer in real-world applications by tuning a reduced number of parameters [1]–[8]. As extensively discussed in the literature, one of their primary strengths lies in the integral action, which effectively eliminates steady-state errors—a feature often required in industrial processes. Additionally, the derivative action contributes to enhancing system speed and reactivity.

Despite their extensive use and practical benefits, PID controllers have faced two significant challenges, particularly concerning the *derivative action*, in terms of implementation and noise sensitivity. Second-order systems are of particular industrial interest, where PID controllers play a crucial role in achieving optimal performance. Similarly, second-order systems are prevalent in various industrial processes.

Firstly, in numerous systems, measuring the derivative of the state proves highly challenging, if not impossible, rendering the use of derivative action in the controller infeasible [9]–[11]. Secondly, even when it is possible to measure the derivative of the state, incorporating derivative action in the control scheme often leads to the amplification of high-frequency noise [12]. Consequently, many strategies

opt to implement the derivative using a filter. However, recent results have shown that the use of filters in some situations may induce instability [11]. Such an analysis is outside of the scope of the present work.

With the above discussion in mind, a solution that, although known since the end of the 70s (see, for instance, [13]), has gained recent traction involves implementing the derivative action through a delay-difference operator [14]. Suh and Bien [13], [15] introduced a new class of controllers entitled "PMD" corresponding to "proportional- minus-delay" opening the perspective of using the delay as a control parameter. More precisely, this concept enables the implementation of an "average" derivative action by utilizing past measurements of the state [16]–[18]. For some new results in that direction, we refer to [16], [19], [20].

Notably, the aforementioned approach circumvents noise amplification and decreases the computational cost of the control scheme [12]. Although the approximation of the derivative action has been analyzed before (see, for instance [21]), it has a drawback, that is the explicit dependence of the system coefficients on the delay parameter used in the approximation scheme<fnref n="1" />. As a consequence, despite the simplicity of the implementation, such a feature is less frequently considered in stability analysis, due to the increase of the complexity that it represents (see, e.g., [22]). Moreover, recent research has shown that the inclusion of the delay-difference approximation on a closed-loop system may induce instability in some cases, due to the improperly-posed nature of the approximation [9], [10]. However, in the case of second-order systems, as long as the system does not have unstable zeros, this last problem can be avoided with the inclusion of the integral action.

The contribution of the paper is threefold: Firstly, it aims to explore additional advantages resulting from the incorporation of the delay-difference operator into the PID scheme. While existing studies have delved into the scheme, the literature often overlooks potential benefits beyond noise mitigation [12], [13]. Secondly, we provide a stability analysis of the closed-loop system, specifically considering the explicit dependence of the system's coefficients on the delay parameter.

Finally, we investigate the maximum value of the delay parameter in the approximation, ensuring the closed-loop system's stability is maintained. This parameter, referred to as the delay margin of the approximation, is a characteristic

<footnote>†The work of Torres-García is supported by the "ADI 2022" project funded by the IDEX Paris-Saclay (France), ANR-11-IDEX-0003-02 and by CONACYT-UASLP (Mexico), CVU: 929482.</footnote>

<footnote n="1">¹,²D. Torres-García and C.-F. Méndez-Barrios are with Universidad Autónoma de San Luis Potosí (UASLP), Facultad de Ingeniería, Dr. Manuel Nava No.8, San Luis Potosí, S.L.P., México. {diego.imt7, cerfranfer}@gmail.com</footnote>

<footnote n="2">²D. Torres-García and S.-I. Niculescu are with Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des Signaux et Systèmes (L2S, UMR CNRS 8506) F-91192, Gif-sur-Yvette, France. silviu.niculescu@centralesupelec.fr</footnote>

<footnote n="1">¹More precisely, the characteristic function is a quasip polynomial with delay-dependent coefficients.</footnote>
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frequently studied in delay systems (see [23] and [24]). However, whereas most studies aim to answer the question, 'How large the amount of the delay in the input/output of the system can be tolerated?'—we, on the contrary, focus on determining the maximum valid delay in the approximation scheme. Furthermore, we take into account the explicit dependence of the system coefficients on the delay parameter in our computations of the delay margin. To the best of the authors' knowledge, there does not exist a similar analysis in the open literature.

The notations employed are standard within the field.

## II. PRELIMINARIES AND PREREQUISITES

### A. Dynamical system and control scheme

We consider a class of LTI second-order systems:

$$\ddot{y}(t) + a\dot{y}(t) + by(t) = cu(t), \quad (1)$$

where, the triplet $(a, b, c)$ are system parameters $(a, b, c) \in \mathbb{R}^2 \times \mathbb{R}_+$, and $u(t)$ represents the system's input. Such second-order dynamical systems are of substantial interest due to their capacity to model a wide array of physical systems, from mass-spring-damper [25] to electrical circuits [26] (see, e.g., [27] and the references therein).

The primary objective of this study is to assess the advantages derived from an implementation based on a delay-difference operator in contrast to the classical PID control. To this end, the following control schemes are considered:

$$u(t) = k_p e(t) + k_d \dot{e}(t) + k_i \int_0^t e(s) ds, \quad (2)$$

and

$$u_{\tau}(t) = k_p e(t) + k_d (e(t) - e(t - \tau))/\tau + k_i \int_{0}^{t} e(s) ds. \quad (3)$$

Here, $e(t) = r(t) - y(t)$ is the output error, $r(t)$ denotes the reference signal and $\tau > 0$ is the delay. The incorporation of integral action is pivotal, particularly in numerous industrial systems where the elimination of steady-state error is a requirement. However, since the changes in the implementation primarily stem from the treatment of the derivative action, for the sake of brevity, we consider that $k_i$ is always chosen beforehand. It is worth noting that the derivative approximation, namely $\dot{e}(t) \approx (e(t) - e(t - \tau))/\tau$, inherently implies a modification to the proportional gain, $k_p \rightarrow k_p + k_d/\tau$, and additionally, introduces a delay block of the form $-e^{-\tau s}k_d/\tau$. While the aim is to achieve an average derivative action, this implementation naturally induces alterations in the closed-loop behavior. As we will discuss, these alterations depend not only on the chosen control gains but also on the delay parameter.

### B. Properly-posed closed-loop systems

Recent results show that the considered implementation of the derivative action may lead to *improperly-posed* systems characterized by high-sensitivity to delay variations in the difference operator. However, under the PID control it appears that, thanks to the integral action, such a situation can be avoided. For further details we refer to [9].

The analysis of the closed-loop system's stability can be conducted by examining its spectrum, represented by the roots of the characteristic function $\Delta: \mathbb{C} \times \mathbb{R}^3 \rightarrow \mathbb{C}$:

$$\Delta(s; \vec{k}) = s^3 + (k_d c + a)s^2 + (k_p c + b)s + k_i c, \quad (4)$$

where $\vec{k} \in \mathbb{R}^3$ are the control gains. Upon considering a delay-difference approximation of the derivative action, a modified characteristic function $\Delta_{\tau}: \mathbb{C} \times \mathbb{R}^4 \rightarrow \mathbb{C}$ is derived:

$$\Delta_{\tau}(s; \vec{k}_{\tau}) = s^3 + as^2 + \frac{b\tau + (\tau k_p + k_d)c}{\tau}s + ck_i - \frac{ck_d}{\tau}se^{-\tau s}, \quad (5)$$

where $\vec{k}_{\tau} = (\vec{k}, \tau) \in \mathbb{R}^3 \times \mathbb{R}_+$ represent the control gains $\vec{k}$ previously defined, and the delay parameter $\tau > 0$ introduced by the approximation scheme proposed above. We refer to $\Delta_{\tau}$ in (5) as a quasi-polynomial, owing to the presence of the exponential term. It is worth mentioning that, despite the possible existence of an infinite number of characteristic roots, its stability depends on the location of its spectral abscissa<fnref n="2" />, given by $\rho(\Delta(s; \vec{k})) = \max\{\Re(s_0) : s_0 \in \mathbb{C}, \Delta(s_0; \vec{k}) = 0\}$ (see, e.g., [28] for further details<fnref n="3" />).

## III. SPECTRAL ANALYSIS OF THE CLOSED-LOOP SYSTEM

For the sake of simplicity, we focus on stability regions defined by the control gains $(k_p, k_d)$, since, as previously mentioned, such gains are explicitly affected by the approximation scheme. In the sequel we assume $k_d \neq 0$, which is consistent with the proposed construction.

### A. Stability region in the parameter-space $(k_p, k_d)$

In order to guarantee the exponential stability of the closed-loop system, the spectrum of the characteristic function must be fully located on the left-half plane $(\mathbb{C}_-)$, that is, the spectral abscissa of the characteristic function should be negative.

On the one hand, considering the classical implementation of the PID control with a given $k_i > 0$ this is achieved as long as the following conditions hold:

$$k_d > -\frac{a}{c} \quad \wedge \quad k_p > \frac{ck_i - ab - bck_d}{ac + c^2k_d}. \quad (6)$$

The two inequalities in (6) give an explicit construction of the stable regions in the controller parameter-space $(k_p, k_d)$.

On the other hand, according to the continuity property of the closed-loop system's roots with respect to its parameters, any transition of a solution from the left-half plane (stability) to the right-half plane (instability), and vice-versa, necessitates the crossing to occur via the imaginary axis. Therefore, if we instead consider the delay-difference approach we require to find the corresponding *stability crossing* curves [29], [30], that is, the collections of all pairs $(k_p, k_d)$ such that there exist a characteristic root placed on the imaginary axis. We have the following result:

**Proposition 1:** Consider the system described by the differential equation (1), in closed-loop with the controller $u_{\tau}(t)$

<footnote n="2">²Its right-most root.</footnote>

<footnote n="3">³Although, the case of delay dependent coefficients is not covered, the definitions and related properties hold.</footnote>
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<figure><img src="image_2.png" /><figcaption>Fig. 1: Stability crossing curves comparison</figcaption></figure>

given by (3). Then, for a given $k_i > 0$, the stability crossing points are given by

$$k_p(\omega) = \frac{\tan\left(\frac{\tau\omega}{2}\right)(a\omega^2 - ck_i) - b\omega + \omega^3}{c\omega}, \quad (7)$$

and

$$k_d(\omega) = \frac{\tau \csc(\tau \omega) (ck_i - a\omega^2)}{c\omega}. \quad (8)$$

**Sketch of Proof:** The proof relies on the continuity of the roots with respect to the system's parameters. ■
As mentioned in the Introduction, our interest is to compare the stability regions produced in both situations, which in general we might expect to be similar, but not the same. Indeed, Figure 1 depicts particular cases of stability boundary for both implementations. On the one hand, we can observe that in this case there exist a gap where the system might remain stable in closed-loop with the classic implementation of the PID controller, but not with the delay-difference approach. On the other hand, there exist also a region where the opposite occurs, and the approximation stabilizes the closed-loop system, but the classical implementation does not.

## B. System's performance and optimization

One feature that allows to measure the performance of a closed-loop system is its decay rate, which can be seen as how fast does the error signal decay to zero. In general, the decay rate of a system is related to the location of its spectral abscissa. Note that, finding explicit expressions for the spectral abscissa of a quasi-polynomial is not a trivial task [28], nonetheless it can be found numerically with the help of the QPmR algorithm [31]. One way to optimize the spectral abscissa of the closed-loop system, is to consider the change of variable $s \to s - \sigma$, and study the stability of the system w.r.t. the $\sigma$-axis, for which we have the following definition:

**Definition 1 (The $\sigma$-stability):** The system described by (1) is said to be $\sigma$-stable (exponentially) if and only if for a given PID controller $u(t)$ with control gains $\vec{k}$, and

characteristic function $\Delta(s; \vec{k})$ given by (4), the following inequality holds:

$$\rho(\Delta(s; \vec{k})) \leq -\sigma,$$

where $\sigma \in \mathbb{R}_+$ represents the corresponding decay rate. With this definition in mind, it is of interest to find the maximum value of $\sigma$, such that, for a given $k_i$, there exist a pair $(k_p, k_d)$, such that the system is $\sigma$-stable (see [32] and the reference therein). We have the following result:

**Proposition 2:** Consider the controller (2) in closed-loop with the system (1), and assume that the integral gain $k_i^* \in \mathbb{R}$ is fixed. Then, the maximum value of $\sigma > 0$ such that there exists a stabilizing pair $(k_p^*, k_d^*)$ is given by the maximum positive value such that the following system of inequalities have a real solution:

$$-3\sigma + \gamma_2 > 0 \quad (9a)$$

$$\frac{8\sigma^3 - 8\sigma^2\gamma_2 + 2\sigma(\gamma_1 + \gamma_2^2) - \gamma_3}{3\sigma - \gamma_2} > 0 \quad (9b)$$

$$-\sigma^3 + \sigma^2\gamma_2 - \sigma\gamma_1 + \gamma_4 > 0 \quad (9c)$$

where $\gamma_1 = (b + ck_p)$, $\gamma_2 = (a + ck_d)$, $\gamma_3 = ab + ack_p + bck_d + c^2k_dk_p - \gamma_4$ and $\gamma_4 = ck_i$. Moreover, for $\sigma = \sigma^*$, where $\sigma^*$ is the maximum value of $\sigma$ such that the inequalities hold, the pair $(k_p^*, k_d^*)$ is given by:

$$k_p^* = \frac{-b\sigma^* + ck_i + 2(\sigma^*)^3}{c\sigma^*}, \quad k_d^* = \frac{3\sigma^* - a}{c}. \quad (10)$$

**Sketch of Proof:** The proof follows straightforwardly by applying the Hurwitz criterion. ■

**Example 1 (Explanatory example):** Consider the system described by the following differential equation:

$$\ddot{y}(t) + 2\dot{y}(t) + y(t) = u(t),$$

where $u(t)$ is a standard PID controller of the form (2). After choosing $k_i = 1$, the interest is to find the set of pairs $(k_p, k_d) \in \mathbb{R}^2$ such that the system is $\sigma$-stable for a given $\sigma > 0$. As shown in Figure 2, as we increase the value of $\sigma$, its corresponding stability region shrinks, thus the objective is to find the largest $\sigma$ such that the region is not empty. Implementing the derivative action through a delay-difference operator, while providing an approximation of the derivative, can also shift the right-most root towards the left. Consequently, after determining the optimal pair $(k_p, k_d)$, it is logical to seek the optimal value of $\tau$, such that the closed-loop system with the controller $u_\tau(t)$ given by (3) achieves its maximum exponential decay rate. This rate may surpass the decay achieved by the standard PID controller. In this vein, assume that $\vec{k}_\tau = (\vec{k}^*, \tau)$, where $\vec{k}^* = (k_p^*, k_d^*, k_i^*)$, and $(k_p^*, k_d^*)$ are the gains that achieve the maximum exponential decay rate for a given $k_i^*$. The interest then is to find the value $\tau^* > 0$ such that the system in closed-loop with the delay-difference version of the controller achieves its maximum exponential decay rate. The optimization of the spectral abscissa is not a trivial task, since it is known to be not-everywhere differentiable function [33], however in certain
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<figure><img src="image_3.png" /><figcaption>Fig. 2: Changes in the σ-stable region as the value of σ ∈ ℝ+ increases for kᵢ = 1.</figcaption></figure>

cases it is possible to find at least a local minimum. With this discussion in mind, consider the following result:

**Proposition 3:** Consider the system described by the differential equation (1) in closed-loop with the delay-difference controller $u_{\tau}(t)$ given by (3). Assume that the following conditions hold:

$$ (i) \qquad \Re \left\{ \frac{ck_d s(s\tau - e^{s\tau} + 1)}{\tau e^{s\tau}\beta + ck_d \tau(s\tau - 1)} \right\} = 0, \quad (11) $$

$$ \text{where } \beta = (\tau(s(2a + 3s) + b + ck_p) + ck_d). $$

$$ (ii) \qquad \Re \left\{ \left. \frac{d^2 s}{d\tau^2} \right|_{s=s^*} \right\} > 0, \quad (12) $$

where $s^*$ corresponds to a solution of (11).

Then, $\sigma = \Re(s^*)$ corresponds to a local minimum of the spectral abscissa.

*Sketch of Proof:* By using the Implicit Function Theorem one can obtain $ds/d\tau$, where s corresponds to a given solution $s(\tau)$. Next, conditions (i) and (ii) can be easily obtained by noting that the real part of $s(\tau)$ achieves a local minimum whenever the real part of its derivative, given by (11), nullifies and the real part of its second derivative is strictly positive (see Figure 3 for a geometric reference of such observations). ■

### C. Control effort

When we observe the control output in Figure 4b one thing we can observe is that the initial output of the PID block is *very large* in magnitude compared to the one with the delay-difference approximation. This is natural, since when considering the delay-difference operator, for $t_0 \in (0, \tau)$, $u_{\tau}(t) = (k_p + k_d/\tau)e(t_0) + k_i \int_0^{t_0} e(s)ds$, which corresponds to a PI controller, without the *aggressive* response of a derivative action. In practice, hardware limitations impose constraints on the control output, that is $|u(t)| < m$. Moreover, the

<figure><img src="image_4.png" /><figcaption>Fig. 3: Minimization of the spectral abscissa with respect to $\tau$. The arrows indicate the evolution of the root as $\tau$ increases.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4: System's response and control output with a step signal as reference. In this case the control gains correspond to $(k_p, k_d, k_i) = (\frac{8}{5}, \frac{1}{2}, 1)$ and delay $\tau = \frac{1}{2}$. (a) Time response of the system ($y(t)$) for the two different controllers. (b) Control signal of the system ($u(t)$) for different controllers.</figcaption></figure>

introduction of a saturated control can lead to deviations from the expected theoretical behavior of the system.

To illustrate the issue, consider the input constraint $|u(t)| \le 10$, a common limitation in the digital implementation of controllers. For demonstrative purposes, consider:

$$ \ddot{y}(t) + 2\dot{y}(t) + y(t) = u(t), \quad (13) $$

where, additionally, we introduce a noise component with a maximum amplitude of $\max_{t>0} |n(t)| = 0.3$ and a frequency of 500 Hz. Figure 5 illustrates the impact of control saturation on the system's response. Notably, the closed-loop system with the PID controller fails to exhibit the typical speed enhancement achieved through the use of derivative actions. In contrast, the delay-difference implementation effectively manages to address this issue.

### D. Frequency-response analysis

As previously noted, a primary concern associated with incorporating derivatives into control schemes is the amplification of high-frequency noise signals. As illustrated in Figure 6a via Bode magnitude/frequency plot, the utilization of a delay-difference operator for derivative implementation within the control scheme addresses this issue. In contrast to
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<figure><img src="image_6.png" /><figcaption>Fig. 5: Effect of control saturation on system performance: $(k_p, k_d, k_i) = (\frac{8}{5}, \frac{1}{2}, 1)$, and $\tau = \frac{1}{2}$, the control output limited to 10 and with a 200Hz noise signal presence. (a) Control output for system (13) in closed-loop (both implementations). (b) System (13) output in closed-loop (both implementations).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6: Noise rejection effect of the delay-difference implementation with $(k_p, k_d, k_i) = (\frac{8}{5}, \frac{1}{2}, 1)$, and the corresponding delay $\tau = \frac{1}{2}$. (a) Bode diagram of both control schemes. (b) Control signal of the system ($u(t)$) with a 200Hz noise signal presence and without saturation.</figcaption></figure>

the classic derivative, which exhibits an increasing magnitude with higher frequencies, the delay block remains bounded, showcasing a mitigation of this behavior.

### E. Delay margin

Despite the benefits regarding the utilization of a delay-difference operator for implementing a derivative action, a crucial consideration involves the selection of the delay value. We already discussed the possible existence of an optimal delay $\tau$. In addition to a possible optimization, there might also exist an upper limit on the delay value used on the approximation, in order to ensure the closed-loop stability of the system [23], [24]. With this in mind, we have:

**Proposition 4 (Delay Margin of the Closed-Loop System):**
Consider the dynamical system described by (1). If the system is stable for a triplet $(k_p, k_i, k_d) \in \mathbb{R}^3$ where $u(t)$ is given by (2) then, it will remain stable if we replace $u(t)$ by $u_{\tau}(t)$ given by (3) if $\tau \in (0, \hat{\tau})$, where $\hat{\tau}$ is given by

$$ \hat{\tau}(\omega) = \frac{2ck_d\omega^2(\omega^2 - b - ck_p)}{\omega^6 + \alpha_4\omega^4 + \alpha_2\omega^2 + c^2k_i^2}, \quad (14) $$

where $\alpha_4 := a^2 - 2(b+ck_p)$, $\alpha_2 := (b+ck_p)^2 - 2ack_i$, and the

frequency $\omega$ is the smallest value satisfying

$$ \tan^{-1}\left(\frac{b\tau\omega + ck_d\omega + ck_p\tau\omega - \tau\omega^3}{ck_i\tau - a\tau\omega^2}\right) + \tau\omega = \operatorname{Arg}\{-i\omega ck_d\}. $$

*Sketch of Proof:* The proof follows similar steps than those presented in [9]. ■

## IV. NUMERICAL EXAMPLES

### A. Oscillator system

Consider the system described by the differential equation:

$$ \ddot{y}(t) + \lambda y(t) = u(t), \quad (15) $$

with $\lambda = 5$. In open-loop, the system is an oscillator with characteristic roots $s_{1,2} = \pm i\sqrt{5}$. In closed-loop with $u(t)$ given by (2) the system achieve exponential stability as long as $k_i > 0$, $k_d > 0$ and $k_p > (k_i - \lambda k_d)/k_d$. From Proposition 2 taking $k_i = 1$ the gains that minimize the spectral abscissa are $(k_p^*, k_d^*) = (-2, 3)$, which bear a triple real root at $s = -1$.

**Remark 1:** Although the spectral abscissa is being minimized, the system closed-loop response is not necessarily optimal. Indeed, when observing the chosen control gains, we can observe that the transfer function of the controller is equivalent to $s^2 - \frac{2}{3}s + \frac{1}{3}$, which correspond to two unstable zeros, namely $s_{1,2} = \frac{1}{3} \pm \frac{\sqrt{2}}{3}i$; see, for instance, Figure 7a.

If we replace $u(t)$ by $u_{\tau}(t)$ given by (3) the stability crossing points can be computed using Proposition 1.

**Remark 2:** As in this example, it is not always possible to push the spectral abscissa to the left by means of the delay-difference operator. However, the control effort is still reduced with the inclusion of the delay block.

From Proposition 3, minimizing the spectral abscissa of the system through the delay difference approximation is not feasible in this case, but we can determine the delay margin using Proposition 4. Indeed, from (14) we obtain $\hat{\tau}(2.99) \approx 1.006$. The computations can be further confirmed using the QPmR algorithm, and easily verified by considering the limit case on the approximation.

In Figure 7, we illustrate the control signal and output of system (15) for both implementations. It is crucial to highlight two primary components. On the one hand, Figure 7a demonstrates the impact of the unstable zeros added to the closed-loop system with the chosen gains. This addition leads to a non-optimal response, regardless of the efforts to minimize the spectral abscissa.

On the other hand, Figure 7b showcases the disparity in the initial control effort between the two implementations, a point we have already discussed and that prove to be critical when saturation of the control output is considered.

### B. Stabilizing effect of the delay-difference operator

As previously discussed, certain scenarios exist where the classical PID controller $u(t)$, with given gains $(k_p, k_d, k_i)$, may fail to stabilize a system. However, its delay-difference counterpart $u_{\tau}(t)$, using the same set of gains, is a stabilizing controller. Keeping this insight in mind, consider the system:

$$ \ddot{y}(t) - 4\dot{y}(t) - y(t) = u(t). \quad (16) $$
• Episciences

• Sciencesconf
<figure><img src="image_8.png" /><figcaption>Fig. 7: System's response and control output with a step signal as reference. (a) Time response of system (15) in closed-loop with the two different controllers and delay $τ = 0.1$. (b) Output of the two different controllers.</figcaption></figure>

Choosing $\vec{k} = (2, 5, 1)$ results in a characteristic function with a pair of roots on the imaginary axis, i.e. $s_{1,2} = \pm i$. Now, we use $u_{\tau}(t)$ instead of $u(t)$ in (16), and maintain the same gains, with $\tau = 0.1$. The spectral abscissa of the resulting characteristic function is $s \approx -0.1$. Here, the approximation with $u_{\tau}(t)$ remains stable, while the classical PID controller fails. This unique characteristic significantly enhances the robustness of the control system, showcasing an additional advantage of the delay-difference approach in this case study.

## V. CONCLUSIONS

In this paper, we focused on various comparative criteria between conventional PID controllers and an alternative version featuring an approximation of the derivative action through a delay-difference operator and we show several advantages associated with employing the delay-difference operator to approximate the derivative within a closed-loop system. Future work may focus on the characterization of the scenarios where the inclusion of the delay allows to improve the spectral abscissa as well as the effect that unstable zeros have on the closed-loop stability when the approximation is considered.

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