# Design of Delay-based Proportional Integral Controllers
## (δ-PI) for Second-order LTI Systems

Bryan Rojas-Ricca, César Fernando Méndez Barrios, Sabine Mondié, Silviu-Iulian Niculescu

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Bryan Rojas-Ricca, César Fernando Méndez Barrios, Sabine Mondié, Silviu-Iulian Niculescu. Design of Delay-based Proportional Integral Controllers (δ-PI) for Second-order LTI Systems. 18th IFAC Workshop on Time Delay Systems, TDS 2024, IFAC, Sep 2024, Udine, Italy. hal-04701671

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Distributed under a Creative Commons Attribution 4.0 International License
Design of Delay-based Proportional
Integral Controllers ($\delta$-PI) for
Second-order LTI Systems*

Bryan Rojas-Ricca* César-Fernando Méndez-Barrios**
Sabine Mondié* Silviu-Iulian Niculescu***

* Departamento de Control Automático, CINVESTAV-IPN, Av. Instituto Politécnico Nacional 2508, Col. San Pedro Zacatenco, Gustavo A. Madero, C.P. 07360, Ciudad de México, México, (e-mail: {brojas, smondie}@ctrl.cinvestav.mx).

** Department of Electrical and Mechanical Engineering, Universidad Autonoma de San Luis Potosí (UASLP), Dr. Manuel Nava No. 8, C.P. 78290, San Luis Potosí, Mexico, (e-mail: fernando.barrios@uaslp.mx)

*** Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes (L2S), CNRS, Bât. Bréguet, 3, rue Joliot Curie, 91190, Gif-sur-Yvette, France
(e-mail: silviu.niculescu@centralesupelec.fr)

**Abstract:** This note focuses on the design of a delay-based proportional integral (PI) control scheme, proposed as an alternative to the classical PI controller. By adopting a geometric approach, we establish tuning rules to achieve closed-loop stability, ensure a specific decay rate, and assess its fragility. Specifically, through the appropriate design of the delay parameter, the proposed controller facilitates improvement in settling time and reduction of overshoot, while maintaining control efforts lower than its classical counterpart. We present several numerical examples to illustrate the effectiveness of the proposed scheme.

**Keywords:** Stability; Control of linear systems; PID control; Delay-based controllers.

# 1. INTRODUCTION

It is widely recognized that low-order controllers are the most extensively used in industry due to their simplicity in tuning and implementation. Among these controllers, the Proportional-Integral-Derivative (PID) type stands out (Silva et al., 2005). It can eliminate steady-state error and improve transient response while dealing with uncertainties or disturbances. However, the derivative action of the PID can also amplify high-frequency noise of measurements (Aström and Hägglund, 1995). Therefore, in many applications, it is convenient to omit the derivative action, resulting in implementing a Proportional-Integral (PI) controller (Mendez-Barrios et al., 2008).

There has been a growing interest in recent years in control laws that intentionally use delays as additional tuning parameters. In this regard, the delayed resonator introduced by Olgac and Holm-Hansen (1994) exemplifies how incorporating delays into systems can enhance their stability properties. Their work demonstrated that this resonator is particularly effective at absorbing harmonic disturbances, even when the disturbance frequencies vary. Furthermore, with an appropriate design of the delay parameter, the delayed resonator can effectively suppress vibrations, as reported by Vyhlídal et al. (2014) and the references therein. Mazenc et al. (2003) utilized the delayed state to stabilize a

chain of integrators with an input delay and Niculescu and Michiels (2004) demonstrated that a chain of integrators can be stabilized using a control law with multiple delays. These ideas where further applied to low-order controllers, Villafuerte et al. (2013) proposed a Proportional-Retarded (PR) controller, which improves the closed-loop response by substituting the derivative action in a classical PD controller with its Euler approximation; Hernández-Díez et al. (2018) presented a stability analysis of this scheme through a geometrical approach; Ma and Chen (2019) analyze deeply the effect of delays in a low-order systems with a classical PID controller, while (Ma et al., 2019) is centered in unstable second-order systems. Ramírez et al. (2016) introduced the Proportional-Integral-Retarded (PIR) controller, which showed an improvement in the mitigation of undesirable noise effects compared to the classical PID controller. Finally, Hernández-Díez et al. (2019) further expanded this idea by introducing the delay in the integral action instead of the derivative action.

This paper presents an alternative to the classical PI controller, named Delay-based Proportional-Integral scheme ($\delta$-PI). Here, the delay is an additional tuning parameter that affects both the derivative and integrative actions. The $\delta$-PI controller offers two main advantages over the classical PI controller: First, it improves the asymptotic behavior. Second, it can stabilize systems that remain unstable under the PI controller.

<footnote>* This work was supported by Project Conahcyt A1-S-24796.</footnote>
The contribution is structured as follows. Section 2 describes the δ-PI controller, the system under consideration, the control problem we address, and two motivating examples. Section 3 presents the stability analysis through the geometric approach, σ-stability optimization results, and the fragility analysis of the controller. Several illustrative examples are given in Section 4. Finally, Section 5 presents concluding remarks and future work.

## 2. PRELIMINARIES AND PROBLEM FORMULATION

In this section, we introduce the δ-PI controller for a class of second-order Linear Time-Invariant (LTI) systems and state the control problem we address in this paper.

### 2.1 Problem formulation

Consider the following LTI Single-Input Single-Output (SISO) system whose dynamics are described by the following ordinary differential equation:

$$ \ddot{y}(t) + a_1 \dot{y}(t) + a_0 y(t) = b_1 \dot{u}(t) + b_0 u(t) \quad (1) $$

where $u$ and $y$ are real signals defining the input and the output respectively. The coefficients $a_0, a_1, b_0, b_1$ are real constants. Thus, the system under consideration is a well-defined second-order system and it can be unstable or/and non-minimum phase in open-loop. Its transfer function simply writes as:

$$ G(s) = \frac{P(s)}{Q(s)}, \quad (2) $$

where $P(s)$ and $Q(s)$ are non-zero polynomials given by

$$ P(s) = b_1 s + b_0 \quad \text{and} \quad Q(s) = s^2 + a_1 s + a_0. \quad (3) $$

In the sequel, for the sake of brevity, we assume that the polynomials $P$ and $Q$ are coprime.

The δ-PI controller is a variant of the classical PI scheme, in which an intentional delay $\tau$ is introduced in the error signal, namely:

$$ u(t) = k_p e(t - \tau) + k_i \int_0^t e(\theta - \tau) d\theta. \quad (4) $$

where $e(t) = r(t) - y(t)$ is the error and $r$ is the reference. In the frequency domain, the control law (4) is expressed as

$$ C_{\tau}(s) = \left( k_p + \frac{k_i}{s} \right) e^{-s\tau}. \quad (5) $$

Clearly, when $\tau = 0$, the controller $C_0$ reduces to the classical PI controller. In this paper, we consider the following control problem:

**Problem 1.** Find explicit conditions on the parameters $(k_p, k_i, \tau)$ such that the δ-PI controller $C_τ(s)$ stabilizes the LTI SISO system (1).

Solving Problem 1 consists of finding the set of parameters $(k_p, k_i, \tau)$ such that the characteristic function

$$ \Delta(s) \triangleq s^3 + a_1 s^2 + a_0 s + (b_1 s + b_0)(k_p s + k_i)e^{-s\tau} \quad (6) $$

has all its roots in the left-half plane of the complex plane. An implicit requirement in solving Problem 1 is to avoid introducing a non-minimum phase zero at $s = -k_i/k_p$ in the closed-loop system, for this, both gains must either be positive or negative.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Spectral abscissa of the system (7) in closed-loop with PI controller (black dashed line) in contrast with δ-PI controller (red dashed line).</figcaption></figure>

### 2.2 Motivating examples

Next, we present two scenarios where the δ-PI controller outperforms the classical PI. The first one shows that a positive value of $\tau$ can improve the exponential decay of the closed-loop response.

Consider first a stable minimum phase SISO LTI system described by the transfer function:

$$ G(s) = \frac{1}{s^2 + 2s + 2}, \quad (7) $$

which has a pair of complex conjugate roots at $s_{1,2} = -1 \pm i$. The classical PI controller $C_0(s)$ leads to the following closed-loop characteristic function:

$$ \Delta(s) = s^3 + 2s^2 + (k_p + 2)s + k_i. \quad (8) $$

The Ruth-Hurwitz stability analysis criterion shows that the maximum exponential decay is achieved at $\sigma = -2/3$. By properly tuning the δ-PI control parameters $(k_p, k_i, \tau)$ one can see on the root locus depicted in Fig. 1 that $\tau = 0.7$ allows achieving a spectral abscissa smaller than $\sigma = -0.7$.

In the second scenario, the δ-PI controller stabilizes the systems where the classical PI-control scheme fails. Consider a system whose transfer function has a pair of complex conjugate roots on the imaginary axis:

$$ G(s) = \frac{1}{s^2 + 4}. \quad (9) $$

The characteristic function of the system in closed-loop with the PI control law is

$$ \Delta(s) = s^3 + (4 + k_p)s + k_i \quad (10) $$

It is easy to observe that there does not exist any stabilizing pair $(k_p, k_i)$. On the contrary, for $k_p = 0.2, k_i = 2$ and $\tau = 1.5$ the characteristic roots of the closed-loop system

$$ \Delta(s) = s^3 + ((4 + k_p)s + k_i)e^{-s\tau} \quad (11) $$

are located in the open left-half complex plane (see, e.g., Fig. 2).

## 3. MAIN RESULTS

In this section, we analyze the stability of the closed-loop system (2), (5) using the geometric approach. First, we derive the stability crossing boundaries and the crossing directions in the parameter-space defined by the control gains $(k_p, k_i)$. Next, we present a tuning methodology of the control parameters $(k_p, k_i, \tau)$ based on the optimization of the spectral abscissa. Finally, we study the fragility of the δ-PI control.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Spectral abscissa of the system (9) in closed-loop with δ-PI controller (5).</figcaption></figure>

## 3.1 Stability Crossing Boundaries

As in the finite-dimensional case, the characteristic function (6) is said to be (exponentially) stable if all its roots lie in the left-half plane of the complex plane. To analyze the stability of (6), we use the so-called *D-decomposition method*, detailed in Michiels and Niculescu (2007). This method consists of an appropriate decomposition of the parameter space into stability regions.

Stability crossing curves bound the stability regions. More precisely, a stability crossing curve $\mathcal{T}$ is the set of all parameters $(k_p, k_i, \tau)$ for which there exists at least one root at the imaginary axis. Considering $\tau$ constant, any point $\mathbf{k} = [k_p, k_i] \in \mathcal{T}$, is called a crossing point. For a given stability/instability region, the number of unstable characteristic roots is constant for all the parameters located inside the corresponding region. Such a number, denoted by $\eta$, is called instability degree. In particular, a degree of instability equal to zero corresponds to a stable region, while a degree greater than zero corresponds to an unstable region. Due to the continuity of the characteristic roots with respect to the parameters, the degree of instability can only change by crossing the curves $\mathcal{T}$.

To characterize the stability crossing curves, we analyze the root crossing at two key points: when $s = 0$ (denoted $\mathcal{T}_0$) and when $\omega > 0$, $s = i\omega$ (denoted $\mathcal{T}_\omega$), forming the union $\mathcal{T} = \mathcal{T}_0 \cup \mathcal{T}_\omega$.

**Proposition 2.** Consider a system whose characteristic function is given by (6). The stability crossing curves $\mathcal{T}_0$ and $\mathcal{T}_\omega$ are defined as

$$ \mathcal{T}_0 = \{\mathbf{k} \in \mathbb{R}^2 \mid k_i = 0, \forall k_p \in \mathbb{R}\}, \quad (12) $$

$$ \mathcal{T}_\omega = \{\mathbf{k}(\omega) \in \mathbb{R}^2 \mid \omega \in \mathbb{R}_+\}, \quad (13) $$

where

$$ \mathbf{k}(\omega) = \frac{1}{b_0^2 + b_1^2 \omega^2} \begin{bmatrix} \sin(\tau\omega) & \cos(\tau\omega) \\ \omega \cos(\tau\omega) & -\omega \sin(\tau\omega) \end{bmatrix} \cdot \begin{bmatrix} -\omega(b_1\omega^2 + a_1b_0 - a_0b_1) \\ (a_1b_1 - b_0)\omega^2 + a_0b_0 \end{bmatrix}. \quad (14) $$

**Proof.** On the one hand, the real root crossing at $s = 0$ is derived from (6) as follows:

$$ \Delta(0) = b_0 k_i = 0, \quad (15) $$

hence (12) immediately follows. On the other hand, when $s = i\omega$ we have

$$ \Delta(i\omega) = i\omega Q(i\omega) + P(i\omega)(ik_p\omega + k_i)e^{-i\tau\omega} = 0. \quad (16) $$

Without any loss of generality, we consider only non-negative frequencies $\omega$. Since $e^{-i\tau\omega} = \cos(\tau\omega) - i\sin(\tau\omega)$, we get

$$ (ik_p\omega + k_i)(\cos(\tau\omega) - i\sin(\tau\omega)) = -\frac{i\omega Q(i\omega)}{P(i\omega)}. \quad (17) $$

The real and imaginary parts of above are

$$ \begin{align} k_i \cos(\tau\omega) + k_p\omega \sin(\tau\omega) &= -\Re \left[ \frac{i\omega Q(i\omega)}{P(i\omega)} \right], \tag{18} \\ k_p\omega \cos(\tau\omega) - k_i \sin(\tau\omega) &= -\Im \left[ \frac{i\omega Q(i\omega)}{P(i\omega)} \right]. \nonumber \end{align} $$

Finally, expressing both equations in matrix form and solving for $k_p$ and $k_i$, we obtain (14). □

## 3.2 Crossing Directions

Now, we characterize the crossing directions of curves $\mathcal{T}_0$ and $\mathcal{T}_\omega$, defined in Proposition 2. This amount is used to determine the root tendency at the crossing of the stability crossing boundaries.

The following proposition states the crossing directions of the stability crossing curve $\mathcal{T}_0$.

**Proposition 3.** Assume that $b_0 \neq 0$. Let $\mathbf{k}_0 \in \mathbb{R}^2$ be a point on the stability crossing curve $\mathcal{T}_0$ defined as

$$ \mathbf{k}_0 := [k_{p0} \ k_{i0}]^\top = [-\frac{a_0}{b_0} \ 0], \quad (19) $$

and let $\mathbf{k} = [k_p, k_i]^\top$ be a crossing point from bottom to top of $\mathcal{T}_0$. If $k_p > k_{p0}$ ($k_p < k_{p0}$), then a real root of (6) crosses from right-to-left towards stability (left-to-right, towards instability) through the origin of the complex plane.

**Proof.** First of all, observe that any root $s \in \mathbb{C}$ of (6) must satisfy

$$ \Delta(s) = 0 \Rightarrow \Delta_r(s) \triangleq (k_p s + k_i)e^{-\tau s} + \frac{sQ(s)}{P(s)} = 0. \quad (20) $$

Hence, by fixing $s = 0$, it is clear to see from (20) that $k_i = 0$ and that $k_p$ can be any real number, implying that $k_0 \in \mathcal{T}_0$.

Bearing in mind the above discussion, let us determine the crossing direction through $\mathcal{T}_0$. In this regard, it suffices to analyze the variation of $s$ with respect to $k_i$. According to the implicit function theorem (Walton and Marshall, 1987), such a task can be performed by considering the following derivative:

$$ \frac{ds}{dk_i} = -\frac{\frac{\partial \Delta}{\partial k_i}}{\frac{\partial \Delta}{\partial s}} \equiv -\frac{\frac{\partial \Delta_r}{\partial k_i}}{\frac{\partial \Delta_r}{\partial s}}, \quad (21) $$

where

$$ \begin{align*} \frac{\partial \Delta_r}{\partial s} &= [k_p - \tau (k_p s + k_i)] e^{-\tau s} + \frac{P(s)(Q(s) + sQ'(s)) - sQ(s)P'(s)}{P^2(s)} \\ \frac{\partial \Delta_r}{\partial k_i} &= e^{-\tau s}. \end{align*} $$

Evaluating (21) at $s = 0$, we obtain that

$$ \left. \frac{ds}{dk_i} \right|_{s=0, k_i=0} = - \frac{1}{[k_p - \tau k_i] + \frac{a_0}{b_0}} \equiv - \frac{1}{k_p - k_{p0}}. \quad (22) $$

Thus, the sign of (22) determines the root tendency at the origin as $k_i$ varies. Therefore, when $\mathcal{T}_0$ is crossing from bottom to top, a real root of (6) crosses from the right half-plane to the left half-plane. A similar conclusion can be drawn if the sign is reversed. □

**Proposition 4.** Let $\mathcal{T}_\omega$ be the stability crossing curve. Define $S: \mathbb{R}^2 \to \mathbb{R}$ by:

$$ S := \operatorname{sign} \{\langle v_\sigma, v_\omega \rangle\} \quad (23) $$
where

$$v_{\sigma} = \left[ \frac{\partial k_p}{\partial \sigma}, \frac{\partial k_i}{\partial \sigma} \right]^{\top} \Big|_{\sigma=0}, \quad v_{\omega} = \left[ -\frac{\partial k_i}{\partial \omega}, \frac{\partial k_p}{\partial \omega} \right]^{\top} \Big|_{\sigma=0} \quad (24)$$

If $S = 1$, then a pair of complex roots cross from left to right of the complex plane when a set of parameters $\mathbf{k} = [k_p, k_i]^{\top}$ crosses the curve $\mathcal{T}_{\omega}$ from the right to left with respect to its positive direction.<fnref n="1" /> If $S = -1$ the crossing direction is reversed.

**Proof.** [Sketch of Proof.] Following a similar methodology that in Diekmann et al. (1995); Gu et al. (2005) we determine the crossing direction by considering the sign of the inner product between a tangent vector of $\mathcal{T}_{\omega}$ with respect to $\sigma$ and a normal vector of $\mathcal{T}_{\omega}$ with respect to $\omega$.

### 3.3 $\sigma$-Stability Optimization

A system is said to be $\sigma$-stable if all its roots are such that $\Re[s] \le \sigma$, $\sigma < 0$. The analysis is similar to the one detailed above for stability crossing boundaries. We analyze the complex roots on a vertical line $i\omega$ for all $\omega \ge 0$, but, in this case, with a fixed negative real part $\sigma \in \mathbb{R}_-$. Therefore, defining $\mathcal{T}_{\sigma}$ as the set of all $\mathbf{k}$ such that (6) has at least a root with real part equal to $\sigma$, we can say that the characteristic function (6) is $\sigma$-stable for all $\mathbf{k} \in \mathcal{T}_{\sigma}$ if $\mathbf{k}$ also lies in a stable region.

**Proposition 5.** Let $\sigma \in \mathbb{R}_-$ and $\tau \in \mathbb{R}_+$ be fixed constants. Hence, the set $\mathcal{T}_{\sigma}$ is given by

$$\mathcal{T}_{\sigma} = \mathcal{T}_{0,\sigma} \cup \mathcal{T}_{\omega,\sigma}, \quad (25)$$

where

$$\mathcal{T}_{0,\sigma} = \{\mathbf{k} \in \mathbb{R}^2 | P(\sigma)k_i + P(\sigma)k_p\sigma + Q(\sigma)e^{-\tau\sigma} = 0\}, \quad (26)$$

$$\mathcal{T}_{\omega,\sigma} = \{\mathbf{k}(\omega) \in \mathbb{R}^2 | \omega \in \mathbb{R}_+\}, \quad (27)$$

with $\mathbf{k}(\omega)$ given by

$$\mathbf{k}(\omega) = -e^{\tau\sigma} M^{-1}(\omega) \begin{bmatrix} \sigma\varphi(\omega) - \omega\psi(\omega) \\ \omega\varphi(\omega) + \sigma\psi(\omega) \end{bmatrix} \quad (28)$$

$$M(\omega) = \begin{bmatrix} \sigma \cos(\tau\omega) + \omega \sin(\tau\omega) & \cos(\tau\omega) \\ \omega \cos(\tau\omega) - \sigma \sin(\tau\omega) & -\sin(\tau\omega) \end{bmatrix} \quad (29)$$

with

$$\varphi(\omega) = \frac{(Q(\sigma) - \omega^2)P(\sigma) + (2\sigma + a_1)b_1\omega^2}{P^2(\sigma) + b_1^2\omega^2}$$

and

$$\psi(\omega) = \frac{(Q(\sigma) - \omega^2)b_1\omega + (2\sigma + a_1)P(\sigma)\omega}{P^2(\sigma) + b_1^2\omega^2}.$$

**Proof.** The proof follows steps of the proof of Proposition 2, but setting $s = \sigma$ instead $s = 0$ for real roots and $s = \sigma + i\omega$ instead $s = i\omega$ for complex roots. $\square$

Observe that $\mathcal{T}_{\sigma} = \mathcal{T}$, when $\sigma = 0$.

### 3.4 Fragility Analysis

The fragility is a measure of the sensitivity of the closed-loop system to the variation or uncertainties in the controller gains (Mendez-Barrios et al., 2008; Guel-Cortez et al., 2019; Keyumarsi et al., 2021). The least fragile controller is the one that admits the greatest deviation in its gains without losing stability. We measure this deviation with the fragility index, that is the minimum distance

<figure><img src="image_4.png" /><figcaption>Fig. 3. Stability regions, $\sigma$-stability and fragility of the closed-loop (7), (5), $\tau = 0.7$.</figcaption></figure>

between the point $\mathbf{k}^*$ and the stability crossing curves that delimit the stability region.

**Proposition 6.** Let $\mathbf{k}^* = [k_p^*, k_i^*]$ be the set of stabilizing controller gains and let $\mathbf{k}(\omega) = [k_p(\omega), k_i(\omega)] \in \mathcal{T}_{\omega}$ be as in (13).

The fragility index $d$ of $\mathbf{k}^*$ is given by $d = \min\{d_0, d_{\omega}\}$, where $d_0 = |k_i^*|$ is the distance from $\mathbf{k}^*$ to $\mathcal{T}_0$ and

$$d_{\omega} = \min_{\omega \in \Omega} \|\mathbf{k}(\omega) - \mathbf{k}^*\|, \quad (30)$$

which is the distance from $\mathbf{k}^*$ to $\mathcal{T}_{\omega}$. Here, $\Omega$ is the set of all $\omega$ that satisfy

$$\langle \mathbf{k}(\omega) - \mathbf{k}^*, \mathbf{k}'(\omega) \rangle = 0. \quad (31)$$

with $\mathbf{k}'(\omega)$ is the derivative of $\mathbf{k}(\omega)$ w.r.t. $\omega$.

**Proof.** Since the stability crossing curve $\mathcal{T}_0$, defined in (12), corresponds to the horizontal axis of the parametric plane, the minimum distance between $\mathbf{k}^*$ and the curve $\mathcal{T}_0$ is $k_i^*$. Also, the minimum distance between $\mathbf{k}^*$ and some point $\mathbf{k}(\omega)$ of the curve $\mathcal{T}_w$ occurs when for some $\omega$ the tangent of $\mathcal{T}_w$ at such point is orthogonal to $\mathbf{k}(\omega) - \mathbf{k}^*$. However, there may be multiple solutions, so the one that satisfies (30) is chosen. $\square$

## 4. ILLUSTRATIVE EXAMPLES

In this section, we showcase several illustrative examples of the tuning methodology outlined in the preceding sections. We present the results of the geometrical approach used for choosing parameters of the $\delta$-PI controller (5). We first revisit and expand the two examples introduced in section 2, and next, we add an example to further illustrate the methodology.

### 4.1 Example 1: stable open-loop second-order system

Returning to the first motivating example, the closed-loop (7), (5), we apply the proposed method to the corresponding second-order system. By using Proposition 2 and setting $\tau = 0.7$, we characterize the stability crossing curves in the space of parameters $(k_p, k_i)$, depicted in Fig. 3.

The crossing directions of the stability curves are determined by Propositions 3 and 4. This makes it easier to compute the degree of instability $\eta$ for each stability region. Also, taking into account the $\sigma$-stability detailed

<footnote n="1">¹ The increasing direction of $\omega$ is considered the positive direction of $\mathcal{T}_{\omega}$.</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Step responses of the closed-loop considering a PI controller (black dashed line), a non-minimum phase δ-PI controller (red solid line), and the least fragile δ-PI controller (blue solid line).</figcaption></figure>

in Proposition 5, we can compute the maximal exponential decay in the stable region, which is $\sigma_{min} = -0.75$. This is achieved with the control parameters $k_p = -0.06$ and $k_i = 0.4265$. It is important to note that the point of maximum decay has a real root with a multiplicity of at least three, due to the intersection of the imaginary root curve with the real root curve. However, these $k_p$ and $k_i$ values have opposite signs, indicating that the closed-loop system has a non-minimum phase. As a result, we disregard this selection. Instead, we choose $k_p = 0.04$ and $k_i = 0.5$, which correspond to an exponential decay of $\sigma = -0.7$, without introducing a non-minimal phase.

Furthermore, Proposition 6 allows us to analyze the fragility of the chosen controller, resulting in a fragility index $d = 0.5$. However, the least fragile controller is $k_p = 1.53$, $k_i = 1.21$, which has a fragility index $d_{max} = 1.21$. Fig. 4 presents the step responses of the optimal non-minimum phase PI controller, the optimal non-minimum phase δ-PI controller, and the least fragile δ-PI controller. It is evident that the δ-PI controller converges faster than the classical PI controller. Also, in this case, the least fragile controller has poor performance.

## 4.2 Example 2: open-loop oscillatory system

We follow the same steps for the geometric analysis of the motivating oscillatory system (9). Setting $\tau = 1.5$, the stable region shown in Fig. 5 is obtained. The maximum exponential decay in this region is achieved with $k_p = 0.417$ and $k_i = 1.96$, resulting in a decay rate of $\sigma = -0.63$ and a fragility index of $d = 0.67$. On the other hand, the least fragile controller is achieved with $k_p = -0.19$ and $k_i = 1.65$, which has a fragility index of $d = 1.24$, but a worst decay rate of $\sigma = -0.25$.

It is important to note that, in this case, the controller with maximum exponential decay has a complex root of multiplicity of at least two, because it is produced only by the intersection of the stability crossing curve of imaginary roots. Also, note that the least fragile controller introduces a non-minimal phase, however, as shown in Fig. 6, the step

<figure><img src="image_6.png" /><figcaption>Fig. 5. Stability regions, $\sigma$-stability and fragility of the closed-loop (9), (5).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Step responses of the closed-loop considering a non-minimum phase δ-PI controller (red solid line), and the least fragile δ-PI controller (blue solid line).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Stability regions, $\sigma$-stability and fragility of the closed-loop (32), (5).</figcaption></figure>

response performance is similar to that of the controller with maximum exponential decay.

## 4.3 Example 3: non-minimum phase second-order system

As a final example, let us consider the following unstable system:

$$G(s) = \frac{s-2}{s^2 - \frac{1}{2}s + \frac{13}{4}} \quad (32)$$

In this scenario, the optimal $\sigma$-stability is achieved with a classical PI controller by setting $k_p = 1.442$ and $k_i = -0.023$. However, this controller introduces an unstable zero. Now, by choosing $(k_p, k_i, \tau) = (-0.3775, -0.2265, 2)$
<figure><img src="image_9.png" /><figcaption>Fig. 8. Step responses of the closed-loop considering a PI controller (black dashed line), a non-minimum phase $\delta$-PI controller (red solid line), and the least fragile $\delta$-PI controller (blue solid line).</figcaption></figure>

in the $\delta$-PI controller, we achieve $\sigma = -0.1473$ without introducing unstable zeros. Such a controller has a fragility index of $d = 0.0065$. However, the best non-fragile controller is achieved with ($k_p, k_i, \tau$) = (-0.495, -0.355, 2), which results in $d = 0.162$ and a corresponding $\sigma = 0.118$. Such results are depicted in Fig.7. The system's response for these three controllers can be observed in Fig. 8.

## 5. CONCLUDING REMARKS AND FUTURE WORK

The proposed methodology enables the tuning of the $\delta$-PI controller, which enhances the performance of the classical PI controller, particularly in terms of the decay rate. Moreover, the novel control scheme has proven its ability to stabilize systems beyond the reach of a classic PI controller and to achieve superior performance when confronted with non-minimum phase behaviors.

Several interesting questions about the family of delay-based control schemes remain open. In this direction, various issues need to be addressed, such as the relationship between the location of the closed-loop roots and where the intentional delay is introduced. Additionally, a comparative analysis of the $\delta$-PI controller with other low-order delay-based and delay-free control schemes is left for future research.

## ACKNOWLEDGEMENTS

The first two authors wish to thank CONACYT and CentraleSupélec for the financial travel support for visiting Laboratory of Signals and Systems (L2S) in November 2023 and February-March 2024 respectively.

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