Automated and Concurrent Synthesis of Fractional-Order QFT Controllers for Ship Roll Stabilization Using Constrained Optimization

Abstract

Quantitative Feedback Theory (QFT) enables the control system to guarantee stability and performance in the presence of plant uncertainty, thus offering a quantitative and less conservative framework for designing robust yet practical controllers. The presented work investigates a single-stage constraint optimization-based approach for synthesizing controllers for the ship roll stabilization. The typical QFT loop shaping is a manual two-stage procedure that demands a proficient understanding of loop-shaping principles on Nichols charts. The proposed procedure simplifies the QFT synthesis process by introducing a single-stage method that allows for concurrent synthesis of both the QFT controller and pre-filter. The present work considers the synthesis of fractional order controllers (using the FOMCON toolbox). The proposed method also enables the designer to pre-specify the controller architecture at the beginning of the design procedure. A comparative analysis with the controllers obtained using the QFT toolbox, Ziegler–Nichols, ${\mathcal{H}}_{\infty }$, IMC, and MPC have also been presented in the work. The implementation has been carried out for the ship roll stabilization, which is one of the critical problems in marine engineering, as it directly impacts the vessel safety, operational efficiency, and passenger comfort, wherein excessive roll can lead to reduced propulsion efficiency. The obtained results highlight that the proposed controller performs better than the benchmark controllers, and Monte Carlo simulations have also been included to support the results.

1. Introduction

Loop shaping is a key stage used for conventional QFT design. It involves manually shaping an open-loop transfer function using Nichols charts to meet pre-defined QFT objectives and constraints. However, this process can be complex and time consuming, requiring a sequential two-stage trial-and-error-based approach. Furthermore, the attainment of an optimal controller is not always assured; for uncertain, time-delayed, and non-minimum phase systems, the conventional QFT synthesis becomes even more complicated [1]. Automating the loop-shaping process can be helpful in synthesizing optimal QFT controllers. At present, existing methods for automation of the loop shaping of QFT controllers rely on the QFT toolbox for calculating and generating bounds and follow a sequential two-stage design procedure, where the QFT controller is synthesized first to assure robustness, followed by pre-filter design to ensure good tracking performance.

To overcome these limitations, there is a need to develop QFT methodologies that offer bounds and a template-free method for synthesizing QFT controllers and pre-filters concurrently. Several researchers have tried to automate the loop shaping, and with existing methods broadly classified into two categories: (a) methods employing convex/non-convex optimization, and (b) methods choosing optimal controller from a predefined set of controller gains.

Initial attempts to automate the loop shaping followed an iterative and sequential procedure [2], wherein the poles/zeros can be added freely to the open-loop transmission (OLT) on Nichols charts. The method was not fully automated but sequential; thus, it might result in a non-optimal controller synthesis with a high order. In ref. [3], QFT design was posed as a convex optimization problem, where bounds were expressed as linear inequalities that acted as constraints in the optimization problem, and employed the use of linear programming to synthesize the controller. The method is dependent on various linear approximations, and a very high-order controller was obtained. Another drawback of the approach was fixing the poles of the controller, which imposed unrealistic constraints by approximating a non-convex optimization problem as a convex one using linearization. Despite such attempts, the reported approaches [2,3,4,5,6] still rely on the QFT toolbox for generating templates and bounds. Additionally, these methods use linear approximations to solve the complex QFT synthesis problem, and they also follow a two-stage process for QFT synthesis, with most just synthesizing the controller only by excluding the pre-filter synthesis.

The metaheuristic algorithms provide a promising means to optimally solve the complex QFT controller and pre-filter synthesis. So, to achieve this, it is required to formulate the QFT synthesis as an optimization problem. Many researchers [7,8,9,10,11] have made noteworthy efforts to do so. In refs. [7,8,9,10], a genetic algorithm (GA) has been used to synthesize the QFT controller, but the synthesis was only considered for robust closed-loop performance, and the pre-filter synthesis was not undertaken. Additionally, the reported methods relied on the QFT toolbox for computing the bounds. In ref. [9], GA has been employed for synthesizing the QFT controller, where three QFT objectives of (a) minimization of the magnitude at low frequency ranges between OLT and the desired bound, (b) inhibition of the loop transmission to enter the UHB curve and (c) controller gain at high frequency have been considered to formulate the weighted cost function. The method again relied on the QFT toolbox for calculating bounds, only addressing the controller synthesis, and pre-filter design was not incorporated. In ref. [11], the particle swarm optimization has been applied for synthesizing the optimal fractional order QFT controller; the reported method draws inspiration from [9], and modifies it by imposing the constraint that the open loop transmission must have a minimum phase at the high frequency ranges.

Recently, the QFT has found applications in many diverse areas, like in [12], and the QFT has been applied to tune the virtual synchronous generator to assure robust stability in islanded microgrids and in [13] to mitigate the multi-loop interactions for the power sharing amongst distributed generators in the control of microgrids. In ref. [14], the QFT has been applied for voltage level control for DC-DC boost converters in microgrids. The method employs the use of traditional QFT synthesis for synthesizing the controller; however, the pre-filter synthesis has not been considered in the design problem. In ref. [15], an active disturbance rejection generalized predictive control strategy for quadrotor control is presented. ADRC-GPC follows a 2DOF architecture, as in the case of QFT; hence, traditional QFT has been applied to synthesize the controller. In refs. [16,17], a decentralized MIMO QFT controller has been synthesized for ensuring good soil quality by controlling the moisture, nitrogen, and salt content [16], and also in aquaculture [17]. Both of these works model the system as an MIMO system and employ the classical MIMO QFT loop shaping to synthesize the controllers and the pre-filter. However, traditional MIMO QFT synthesis employs a sequential design process and does not guarantee the optimality of the obtained solution.

In ref. [18], a QFT controller is synthesized for the control of biomass and substrate concentrations in a CSTR system, which features a 2 × 2 MIMO plant; the QFT synthesis is performed using a conventional loop-shaping process. In ref. [19], a QFT controller with disturbance torque observer is proposed for of pump–valve compound drive system. In ref. [20], QFT controllers are proposed for the robust control of an industrial boiler. QFT control is also applied for the speed control of pipeline pigs [21], virtual inertia control in micro-grid [22], and the distillation process [23]. All of the recent existing works [18,19,20,21,22,23] still rely on the conventional QFT synthesis.

The above-mentioned approaches, however, do not guarantee optimal QFT synthesis; that too in a single step. Earlier approaches [2,3,4,5,6] try to impose unrealistic assumptions by modeling a non-convex optimization problem as a convex one and solving it using linear programming. Some approaches [7,8,9,10,11] employ the use of metaheuristic algorithms to solve the problem, but are only limited to controller synthesis and do not synthesize the pre-filter, and are also dependent on the use of the QFT toolbox for calculating templates and bounds. Some of the recent works [12,13,14,15,16,17,18,19,20,21,22,23] employ the classical QFT for some advanced applications and hence, do not guarantee the optimality of the obtained solutions. Therefore, there is still a need to develop template and bounds-free QFT methodologies that offer a more streamlined and efficient approach to synthesize optimal QFT controllers and pre-filters in a single stage.

In ref. [24], an optimization-based methodology is proposed for the concurrent synthesis of QFT controllers and pre-filters, wherein the QFT objectives of robust stability, tracking performance, and sensitivity were considered for formulating a weighted objective function. The formulation enabled the approximate template satisfaction without explicit enforcement of the classical QFT bounds, and the optimization behaves more like a heuristic search across the parameter space, rather than a constraint-driven design.

To address these issues, the aim of the proposed work is to present a single-stage approach for the synthesis of the QFT controllers and pre-filters; to achieve it, the controller synthesis is posed as a constrained optimization problem. The minimization of feedback cost at each frequency gain is considered as the objective, subject to the QFT performance indices of robust stability, plant output disturbance rejection, and tracking, which are presented as quadratic inequalities and act as the constraints. Unlike traditional methods that rely on bounds and templates, the proposed methodology uses constrained optimization to synthesize fractional-order controllers and pre-filters simultaneously [25,26]. The approach minimizes feedback cost while satisfying the predetermined QFT bounds, articulated as quadratic inequalities, at the respective design frequency. The proposed approach has been applied to design a fractional order QFT controller and pre-filter for fin stabilization of ship rolling. The proposed method also gives freedom to the control designer to prespecify the controller and pre-filter structure at the beginning of the process; as in the proposed work, it enabled us to incorporate fractional order controllers, which otherwise is not possible in the conventional QFT design. Also, a comparative analysis is presented with classical PID and manual QFT controllers reported in the literature, and also with ${\mathcal{H}}_{\infty }$, IMC, and MPC controllers to validate the efficacy of our approach. At the end, comprehensive sensitivity analysis and Monte Carlo simulations have been conducted to validate the robustness and effective transient response under uncertainties.

The key highlights of the work are as follows:

(a)

A bounds- and template-free constrained optimization-based procedure is proposed for the simultaneous synthesis of a fractional order QFT controller and pre-filter, wherein the design bounds are expressed as quadratic inequalities, which should be complied with at each design frequency, while minimizing the feedback cost constrained by QFT bounds.

(b)

The proposed method also gives freedom to the control designer to pre-specify the controller and pre-filter structure at the beginning of the process, which otherwise is not possible in the conventional QFT design.

(c)

The proposed technique examines the synthesis of a fractional QFT controller and pre-filter for fin stabilization of the ship rolling [27]. The ship roll stabilization is one of the critical problems in marine engineering, as it directly impacts the vessel safety, operational efficiency, and passenger comfort, wherein excessive roll can lead to cargo shift, reduced propulsion efficiency, and increased risk of capsizing in extreme conditions. The attained results exhibit better performance specifications when compared to classical PID, QFT, ${\mathcal{H}}_{\infty }$, IMC, and MPC controllers.

(d)

A comprehensive robustness analysis using sensitivity analysis and Monte Carlo simulations is also included to validate the robustness of the controllers by subjecting the plant to normally distributed random perturbations.

2. Background

In this section, a background on the basics of QFT, FOPID, and the mathematical model of the plant used in the present study is presented:

2.1. Quantitative Feedback Theory

QFT emerged in the 1960s as a frequency-domain robust controller method, drawing inspiration from Bode’s gain-phase integrals [28]. QFT employs a two-degrees-of-freedom (2DOF) structure with feedback controller $K\left(s\right)$ and feedforward pre-filter $F\left(s\right).$ The feedback controller $K\left(s\right)$ is synthesized to assure closed-loop system robust stability, and the pre-filter must assure tracking performance. Figure 1 demonstrates the 2DOF QFT control configuration.

2.2. Fractional Order PID Controller

Fractional-order control models plants using fractional-order differential equations. This extends the conventional control systems by allowing them to have non-integer integral and derivative terms [29]. This offers an extra degree of freedom to model and characterize the systems. This aids the fractional-order systems to precisely capture the underlying dynamics of the system, as well as aid in precise control, especially in the case of complex systems. The fractional order calculus generalizes the integral and derivative operators to a non-integer order operator ${}_{a}D_t^r$ with limits $\left\{a,t\right\}$; the operator is represented in Equation (1):

$${}_{a}D_t^r=\left\{\begin{array}{cc}{\displaystyle \raisebox{1ex}{$d^r$}\!\left/ \!\raisebox{-1ex}{$d{t}^r$}\right.}& \mathfrak{R}\left(r\right)>0\\ 1& \mathfrak{R}\left(r\right)=0\\ {\int }_a^t{\left(d\tau \right)}^{-r}& \mathfrak{R}\left(r\right)<0\end{array}\right.$$

(1)

The implementation of the fractional order controllers has been realized in MATLAB 2025b using the FOMCOM toolbox [30], wherein the Oustaloup filters are considered for the approximation of the fractional filters in a range $\left({\omega }_b,{\omega }_h\right)$ with order N. An Oustaloup recursive filter $s^{\gamma }$ for $0<\gamma <1$ is given in Equations (2) and (3):

$$G_f\left(s\right)=K\prod _{k=-N}^N{\displaystyle \frac{s+{\omega }_k^{\prime }}{s+{\omega }_k}}$$

(2)

where

$$\begin{array}{ccc}K={\omega }_h^{\gamma },& {\omega }_k^{\prime }={\omega }_b{\left({\displaystyle \frac{{\omega }_h}{{\omega }_b}}\right)}^{{\displaystyle \frac{k+N+\frac{1}{2}\left(1-\gamma \right)}{2N+1}}},& {\omega }_k={\omega }_b{\left({\displaystyle \frac{{\omega }_h}{{\omega }_b}}\right)}^{{\displaystyle \frac{k+N+\frac{1}{2}\left(1+\gamma \right)}{2N+1}}}\end{array}$$

(3)

The fractional-order PID controller is an extension of the standard PID control, where the integral and derivative terms have fractional orders instead of one. The equation of the FOPID controller is given in Equation (4):

$$K_{FOPID}\left(s\right)=K_P+{\displaystyle \frac{K_I}{s^{\lambda }}}+K_D{s}^{\mu }$$

(4)

where K_P, K_I, and K_D are gains, while λ and μ are fractional order terms for the integrator and differentiator, respectively.

2.3. Mathematical Model of the Ship Roll

The fin stabilizer is a very efficient damping device for high-speed ships, achieving a damping of 80–90% and considerably minimizing the ship’s roll motion. It adjusts the angle dependent on the ship’s velocity, generating a counteracting force against roll motion, and is very efficient, requiring minute power to operate, and is simple to integrate into the ship’s construction.

The dynamical model of the ship’s rolling motion on sea waves considers the restoring moment $M\left(\varphi \right)$, roll damping moment $M\left(\dot{\varphi }\right)$, moment of inertia $M\left(\ddot{\varphi }\right)$, wave disturbance moment $M\left({\alpha }_m,{\dot{\alpha }}_m,{\ddot{\alpha }}_m\right)$, and righting moment $M\left(\phi \right)$. The mathematical expressions of each are given in Equations (5)–(9), respectively, where $\varphi$ is the swing angle, $\dot{\varphi }$ is the angular velocity, $\ddot{\varphi }$ is the angular acceleration, $D$ is displacement, $h$ is the stability height. $I_x$ is the rolling moment of inertia, $\mathsf{\Delta }{I}_x$ is the additional moment inertia, and $\alpha$ is the effective wave angle [27].

$$M\left(\varphi \right)=-Dh\varphi$$

(5)

$$M\left(\dot{\varphi }\right)=-2N\dot{\varphi }$$

(6)

$$M\left(\ddot{\varphi }\right)=-\left(I_x+J_x\right)\ddot{\varphi }$$

(7)

$$M\left({\alpha }_m,{\dot{\alpha }}_m,{\ddot{\alpha }}_m\right)=Dh{\alpha }_m+2N{\alpha }_m+J_{xx}\ddot{\alpha }$$

(8)

$$M\left(\phi \right)=-Dh{\alpha }_f$$

(9)

The linear rolling motion of the ship with a damper is given by Equation (10):

$$\left(I_x+\mathsf{\Delta }{I}_x\right)\ddot{\varphi }+2N_u\dot{\varphi }+Dh\varphi =Dh\alpha$$

(10)

At initial conditions, when $\ddot{\varphi }\left(0\right)=\dot{\varphi }\left(0\right)=\varphi \left(0\right)=0$, and the transfer function for ship roll motion is given in Equation (11):

$$G\left(s\right)={\displaystyle \frac{\varphi \left(s\right)}{{\alpha }_e\left(s\right)}}={\displaystyle \frac{1}{T_{\varphi }^2{s}^2+2T\xi s+1}}$$

(11)

where $T_{\phi }$ and ${\xi }_{\phi }$ are given in Equation (12):

$$\begin{array}{cc}{T}_{\phi }=\sqrt{{\displaystyle \frac{I_x+\mathsf{\Delta }{I}_x}{D{h}_{\phi }}}};& {\xi }_{\phi }={\displaystyle \frac{N_u}{\sqrt{Dh\left(I_x+\mathsf{\Delta }{I}_x\right)}}}\end{array}$$

(12)

The various constants in the above equations have been considered from [27] to obtain the nominal plant transfer function given in Equation (13). In ref. [27], the NO. 32 China fisheries model has been considered having a linear roll model at a speed of 18 MPH. This model is widely recognized in marine control studies for exhibiting highly uncertain, lightly damped, and nonlinear roll dynamics, making it an appropriate and challenging benchmark for testing the proposed optimization-based QFT synthesis. The time and frequency domain response of the open-loop nominal and uncertain system is illustrated in Figure 2.

$$G_0\left(s\right)={\displaystyle \frac{1}{1.62s^2+0.47s+1}}$$

(13)

The parametric uncertain model under the influence of wind and ship speed changes is given by Equation (14):

$$G\left(s\right)={\displaystyle \frac{\left[0.528,1.496\right]}{\left[1.198,2.082\right]s^2+\left[0.295,0.603\right]s+\left[0.487,1.513\right]}}$$

(14)

3. Constrained Optimization-Based Simultaneous QFT Controllers Synthesis

The presented work proposes a single-step methodology for QFT synthesis, aimed at finding optimal gains while minimizing feedback costs, by posing the QFT synthesis as an optimization problem. The general constrained optimization problem can be expressed in Equation (15), and the flowchart is demonstrated in Figure 3.

The aim is to find the optimal solution vector $x=\left(x_1,x_2,...,x_k\right)$, such that it minimizes the objective function $\mathcal{J}\left(x\right)$ subject to inequality constraints $A_i\left(x\right)$.

$$\begin{array}{ccc}min& J\left(x\right)& \\ \mathit{subject}\mathit{to}& A_i\left(x\right)\le 0& \mathit{for}i=1,\dots ,n\\ & \underset{\_}{x_k}\le x_k\le \overline{x_k}& \end{array}$$

(15)

In the present work, QFT design objectives have been articulated as quadratic inequalities. The aim is to obtain a set of controller gains $x$ that minimizes $\mathcal{J}\left(x\right)$ given by Equation (16), subject to inequalities $g_i\left(x\right)\le 0$ and $\begin{array}{cc}\underset{\_}{x_k}\le x_k\le \overline{x_k},& i=1,\dots ,n\end{array}$:

$$\mathcal{J}\left(x\right)=\mathit{min}\sum _{i=1}^q{c}_i\left|K\left(j{\omega }_i\right)\right|$$

(16)

where $c_i$ is the set of discrete frequency ranges.

The proposed work considers the objectives of robust stability, plant output disturbance rejection, and tracking performances for articulating the constraints for the optimization. The details are shared in the subsequent sub-sections:

3.1. Robust Stability

The specification of the robust stability under the QFT framework is given by Equation (17):

$$\left|{\displaystyle \frac{\mathcal{P}\left(j\omega \right)\mathcal{C}\left(j\omega \right)}{1+\mathcal{P}\left(j\omega \right)\mathcal{C}\left(j\omega \right)}}\right|\le {\delta }_1(\omega )$$

(17)

The polar equivalent of Equation (17) is given by Equation (18):

$${\displaystyle \frac{\left|\mathcal{p}\mathcal{c}{e}^{j\left(\theta +\varphi \right)}\right|}{\left|1+\mathcal{p}\mathcal{c}{e}^{j\left(\theta +\varphi \right)}\right|}}\le {\delta }_1(\omega )$$

(18)

where $\mathcal{p}{e}^{j\left(\theta \right)}$ is the polar representation for the plant $\mathcal{P}\left(j\omega \right)$, and $\mathcal{c}{e}^{j\left(\varphi \right)}$ is the polar form of the controller $\mathcal{C}\left(j\omega \right)$.

The magnitude of Equation (18), followed by squaring both sides, is given by Equation (19):

$${\displaystyle \frac{{\mathcal{p}}^2{\mathcal{c}}^2}{{\mathcal{p}}^2{\mathcal{c}}^2+2\mathcal{p}\mathcal{c}\cos \left(\theta +\varphi \right)+1}}\le {\delta }_1^2(\omega )$$

(19)

Inverting and rearranging Equation (20), we obtain the following:

$${\mathcal{c}}^2\left[{\mathcal{p}}^2\left(1-{\displaystyle \frac{1}{{\delta }_1^2\left(\omega \right)}}\right)\right]+2\mathcal{c}\left[\mathcal{p}\cos \left(\theta +\varphi \right)\right]+1\ge 0$$

(20)

Equation (20) can be rewritten: considering ${\delta }_1\to \infty$, for robust stability, the quadratic inequality for the assurance of robust stability can be obtained in Equation (21):

$${\mathcal{c}}^2\left[{\mathcal{p}}^2\right]+2\mathcal{c}\left[\mathcal{p}\cos \left(\theta +\varphi \right)\right]+1\ge 0$$

(21)

Equation (17) is used to map the desired closed-loop specifications and the plant uncertainty as QFT bounds, and for the controller, it can be computed by varying the $\varphi$ in a certain range for each design frequency. This expression allows for the computation of the required controller magnitude in closed loop for a parametrically uncertain plant in closed form, and has a resemblance to the classical QFT synthesis.

3.2. Plant Output Disturbance Rejection

The second objective considered in the design is the sensitivity minimization or the plant output disturbance rejection, and is given mathematically as Equation (22).

$$\left|{\displaystyle \frac{1}{1+\mathcal{P}\left(j\omega \right)\mathcal{C}\left(j\omega \right)}}\right|\le {\delta }_2(\omega )$$

(22)

Equation (22) can be expressed in its polar form in Equation (23), followed by squaring and computing its magnitude in Equation (24).

$$\left|{\displaystyle \frac{1}{1+\mathcal{P}\left(j\omega \right)\mathcal{C}\left(j\omega \right)}}\right|\le {\delta }_2(\omega )$$

(23)

$${\displaystyle \frac{1}{{\mathcal{p}}^2{\mathcal{c}}^2+2\mathcal{p}\mathcal{c}\cos \left(\theta +\varphi \right)+1}}\le {\delta }_2^2(\omega )$$

(24)

The quadratic inequality for the plant output disturbance rejection can be obtained by inverting Equation (24) and simplification, and can be given by Equation (21):

$${\mathcal{c}}^2\left[{\mathcal{p}}^2\right]+2\mathcal{c}\left[\mathcal{p}\cos \left(\theta +\varphi \right)\right]+\left(1-{\displaystyle \frac{1}{{\delta }_2^2\left(\omega \right)}}\right)\ge 0$$

(25)

3.3. Tracking Performance

In QFT, the robust tracking performance is expressed as Equation (26), where ${\mathcal{T}}_{LB}\left(j\omega \right)$ and ${\mathcal{T}}_{UB}\left(j\omega \right)$, are the lower and upper bounds, respectively.

$${\mathcal{T}}_{LB}\left(j\omega \right)\le \left|{\displaystyle \frac{\mathcal{F}\left(j\omega \right)\mathcal{P}\left(j\omega \right)\mathcal{C}\left(j\omega \right)}{1+\mathcal{P}\left(j\omega \right)\mathcal{C}\left(j\omega \right)}}\right|\le {\mathcal{T}}_{UB}\left(j\omega \right)$$

(26)

Equation (26) does not pose any constraint on the phase of the system, and thus can be rewritten as Equation (27):

$${\displaystyle \frac{{\mathcal{T}}_{LB}\left(j\omega \right)}{\mathcal{F}\left(j\omega \right)}}\le \left|{\displaystyle \frac{\mathcal{P}\left(j\omega \right)\mathcal{C}\left(j\omega \right)}{1+\mathcal{P}\left(j\omega \right)\mathcal{C}\left(j\omega \right)}}\right|=\mathcal{T}\left(j\omega \right)\le {\displaystyle \frac{{\mathcal{T}}_{UB}\left(j\omega \right)}{\mathcal{F}\left(j\omega \right)}}$$

(27)

Thus, from Equation (27), the magnitude difference between any two closed-loop functions ${\mathcal{T}}_i\left(j\omega \right),{\mathcal{T}}_k\left(j\omega \right)$ must lie within the max. and min. values of the transfer function ${\mathcal{P}}_i\left(j\omega \right),{\mathcal{P}}_k\left(j\omega \right)\in \mathbb{P}$ and must satisfy Equation (28), considering $\left|{\mathcal{T}}_i\left(j\omega \right)\right|>\left|{\mathcal{T}}_k\left(j\omega \right)\right|$:

$$\left|{\mathcal{T}}_i\left(j\omega \right)\right|-\left|{\mathcal{T}}_k\left(j\omega \right)\right|\le {\displaystyle \frac{{\mathcal{T}}_{UB}\left(j\omega \right)}{\mathcal{F}\left(j\omega \right)}}-{\displaystyle \frac{{\mathcal{T}}_{LB}\left(j\omega \right)}{\mathcal{F}\left(j\omega \right)}}$$

(28)

Thus, to synthesize the controller, the expression could be expressed as Equation (29), where ${\delta }_3\left(\omega \right)={\mathcal{T}}_{UB}\left(j\omega \right)/{\mathcal{T}}_{LB}\left(j\omega \right)$:

$${\displaystyle \frac{\left|{\mathcal{T}}_i\left(j\omega \right)\right|}{\left|{\mathcal{T}}_k\left(j\omega \right)\right|}}\le {\delta }_3(\omega )$$

(29)

Rewriting Equation (29),

$${\displaystyle \frac{\left|{\mathcal{P}}_i\left(j\omega \right)\right|\left|1+\mathcal{C}\left(j\omega \right){\mathcal{P}}_k\left(j\omega \right)\right|}{\left|{\mathcal{P}}_k\left(j\omega \right)\right|\left|1+\mathcal{C}\left(j\omega \right){\mathcal{P}}_i\left(j\omega \right)\right|}}\le {\delta }_3(\omega )$$

(30)

The polar equivalent of Equation (30) can be obtained in Equation (31), given as follows:

$${\displaystyle \frac{{\mathcal{p}}_i\left|1+\mathcal{c}{\mathcal{p}}_k{e}^{j\left(\varphi +{\theta }_k\right)}\right|}{{\mathcal{p}}_k\left|1+{\mathcal{c}\mathcal{p}}_i{e}^{j\left(\varphi +{\theta }_i\right)}\right|}}\le {\delta }_3(\omega )$$

(31)

Further squaring and inverting Equation (31), we obtain the quadratic inequality for the tracking performance in Equation (32), which must be satisfied at each possible pair of the uncertain plant $\mathbb{P}.$

$${\mathcal{c}}^2\left[{\mathcal{p}}_k^2{\mathcal{p}}_i^2\left(1-{\displaystyle \frac{1}{{\delta }_3^2\left(\omega \right)}}\right)\right]+2\mathcal{c}\left[{\mathcal{p}}_k{\mathcal{p}}_i\left({\mathcal{p}}_k\mathit{cos}\left(\varphi +{\theta }_i\right)-{\displaystyle \frac{{\mathcal{p}}_i}{{\delta }_3^2\left(\omega \right)}}\mathit{cos}\left(\varphi +{\theta }_k\right)\right)\right]+\left[{\mathcal{p}}_k^2-{\displaystyle \frac{{\mathcal{p}}_i^2}{{\delta }_3^2\left(\omega \right)}}\right]\ge 0$$

(32)

The proposed constrained optimization-based formulation is fundamentally distinct from [24], wherein the proposed method explicitly defines the QFT bounds as quadratic inequalities; these inequalities are defined as constraints in the optimization problem, thereby ensuring their exact satisfaction across all design frequencies. The proposed method inherits the original intent of the QFT, i.e., the incorporation of frequency domain templates directly into a rigorous constrained optimization structure. The expression of the QFT bounds as inequality constraints in the optimization ensures their satisfaction across all design frequencies, thus guaranteeing compliance with robustness margins, tracking specifications, and stability requirements. Therefore, the proposed synthesis methodology remains fully aligned with the classical QFT philosophy of explicitly meeting loop-shaping templates. In contrast, the approach in [24] employs QFT concepts only in a soft, weighted error-minimization formulation, where template compliance is approximate and dependent on penalty weights. As a result, the optimization in [24] behaves more like a heuristic search, rather than a constraint-driven design that enforces the classical QFT requirements explicitly. Importantly, despite the methodological differences in these approaches, both methodologies contribute to the advancement of QFT-based design by offering template-free and bounds-free formulations for the concurrent synthesis of QFT controllers and prefilters, thereby expanding the scope for automated and systematic QFT loop-shaping.

4. Results

The work involves the use of a genetic algorithm to determine the controller gains. A fractional-order PID controller given by Equation (4) and a first-order pre-filter given by Equation (33) have been considered. The goal here is to find the optimal gains $[K_P,K_I,K_D,\lambda ,\mu ,a,\gamma ]$, which minimizes the objective function, Equation (16), subject to inequality constraints (21), (25), and (32). A finite set of frequency values has been chosen: ${\omega }_i=\left\{\mathrm{0.4,0.8,1.2,1.7,2.1}\right\}$ rad/s, and the choice of the frequency range has been estimated from the open-loop frequency response of the system (Figure 2), as it can be observed that the system exhibits notable variations within this interval, including a resonant-like behavior around 0.8–1 rad/s. Therefore, the choice of the frequency band enables an effective capture of the dominant dynamics of the plant and ensures that the controller synthesis is performed over the most influential frequency range, which is consistent with recommendations reported in the literature.

$$F\left(s\right)={\displaystyle \frac{a}{s^{\gamma }+a}}$$

(33)

An optimal controller and pre-filter parameters have been obtained using a genetic algorithm. The bounds, given by Equations (30) and (31).

$$T_{UB}\left(s\right)={\displaystyle \frac{1}{0.1s+1}}$$

(34)

$$T_{LB}\left(s\right)={\displaystyle \frac{25}{s^2+10s+25}}$$

(35)

The algorithmic specific parameters have been chosen: the population size has been chosen as 70, tournament-based selection, and random crossover with a rate of 0.2. The set of optimal FOPID and fractional-order pre-filter obtained after the optimization process is given by Equations (36) and (37).

$$K_{QFT-FOPID}\left(s\right)=100.035+{\displaystyle \frac{0.514}{s^{1.117}}}+46.082s^{1.179}$$

(36)

$$F\left(s\right)={\displaystyle \frac{7.482}{s^{1.017}+7.482}}$$

(37)

4.1. Closed Loop Response with Nominal Plant Parameters

The QFT-FOPID controller and pre-filter given in Equations (36) and (37) are used to assess the system’s closed-loop response, and the response is illustrated in Figure 4 for the nominal plant. Both plots indicate that the response complies with design limits in the time and freq. domains.

Table 1. Closed-loop performance metrics for a nominal and uncertain system.

Performance Metric	Nominal Plant	Uncertain Plant
Rise Time (s)	0.310	0.320
Settling Time (s)	0.780	1.570
Overshoot Percentage	2.01	5.40%
Gain Margin	Inf.	Inf.
Phase Margin	179.88	102.88
Peak Complementary Sensitivity	1.00	1.00
Peak Sensitivity	1.059	1.06

provides the closed-loop performances. From Figure 4 and Table 1, it is evident that the synthesized controller offers a satisfactory response.

4.2. Closed Loop Response with Uncertain Plant Parameters

To assess the performance of QFT controllers designed for a plant with parametric uncertainties, we considered the response of the uncertain plant described in Equation (14), which is illustrated in Figure 5. Table 1 provides details regarding the closed-loop performances. The obtained results validate that the proposed QFT-FOPID controller offers a satisfactory response within the design bounds.

The stability analysis of the designed QFT-FOPID controller is performed using the FOMCON toolbox, wherein the stability can be determined using $\left|\arg \left(eig\left(A\right)\right)>\gamma {\displaystyle \frac{\pi }{2}}\right|$, where $0<\gamma <1$ is the equivalent order of the fractional-order state space system, and $eig\left(A\right)$ are the eigenvalues for matrix $A$ [30]. A system would be deemed to be stable if the condition for stability, in the equation above, is satisfied. FOMCON also allows for the visualization of the stability, as shown in Figure 6, wherein the poles of the corresponding rational order system are plotted; for a system to be stable, none of these poles must lie in the shaded (red) region, and thus, from Figure 6, it can be concluded that the closed-loop system is stable.

5. Comparison of the Proposed Controllers with Existing Works

To evaluate the performance of the controller with the existing literature, a comparison has been considered for a classical PID controller, a QFT controller obtained using manual loop shaping, $H_{\infty }$, IMC, and MPC controllers. Figure 6 and Figure 7 illustrate the time domain step response and the magnitude plot for the sensitivity and complementary sensitivity function of the nominal system. Figure 8 illustrates the time domain response of the QFT-FOPID controller when subjected to a plant with parametric uncertainties. It is observed from Figure 7, Figure 8 and Figure 9 that these responses lie within the design bounds even under parametric uncertainties. Various closed-loop performances are given in

Table 2. Compared closed-loop performance metrics for the system.

Performance Metric	Proposed	PID [27]	QFT [27]	${\mathit{H}}_{\mathit{\infty }}$	IMC	MPC
Rise Time (s)	0.310	0.651	0.591	1.613	1.679	1.110
Settling Time (s)	0.780	17.313	10.279	2.814	2.918	3.30
Overshoot Percentage %	2.01	0	71.045	0	0	5.786
Gain Margin	Inf.	Inf.	7.392	122.684	Inf.	-
Phase Margin	179.88	150.299	4.099	76.229	76.365	-
Peak Comp. Sensitivity	1.00	1.038	0.858	1	1	-
Peak Sensitivity	1.059	1	1.194	1.16	1.154	-

. From Figure 7 and Figure 8, and from Table 2, it is observed that the proposed QFT-FOPID controller proposes a much better performance as compared to the existing ones. The QFT-FOPID offers a better transient as well as frequency domain response. The classical PID controller and the QFT controller, as given in [27], are given by Equations (38) and (39), respectively. The synthesized $H_{\infty }$ and IMC controllers are given by Equations (40) and (41), respectively. The $H_{\infty }$ controller is synthesized considering using mixed-sensitivity optimization with weighting functions $W_s(s)={\displaystyle \frac{0.6667s+1}{s+0.001}}$ and $W_u\left(s\right)=0.1$. The IMC controller is synthesized considering a second-order IMC filter $F(s)={\displaystyle \frac{1}{(0.5s+1{)}^2}}$, with tuning parameter $\lambda =0.5$. Also, a linear discrete-time MPC controller has been synthesized by minimizing a quadratic cost function penalizing output deviation, MV magnitude, and MV rate with weights 1, 0.1, and 0.3, respectively, for comparison, and considers a prediction horizon of 20, a control horizon of 3, and a sampling time of 0.05 s.

$$K_{PID}\left(s\right)=3.5+{\displaystyle \frac{0.6}{s}}+4.2s$$

(38)

$$K_{QFT}\left(s\right)={\displaystyle \frac{13.339s+9.33}{s^2+4.3s+18.49}}$$

(39)

$$K_{H_{\infty }}\left(s\right)={\displaystyle \frac{867.3s^2+251.6s+535.4}{s^3+126.7s^2+518.4s+0.5183}}$$

(40)

$$K_{IMC}\left(s\right)={\displaystyle \frac{6.48s^4+3.76s^3+8.545s^2+2.321s+2.469}{s^4+4.29s^3+1.778s^2+2.469s+8.882\times {10}^{-16}}}$$

(41)

Figure 9 shows the compared step response of the closed-loop step for a parametrically uncertain plant. From Figure 9, it can be determined that the synthesized QFT-FOPID controller performs satisfactorily, even when subjected to parametric uncertainties. The response of the controller is tightly bound within acceptable limits. On the other hand, the manually tuned QFT controller exhibits poor performance with high levels of steady-state error and overshoot. In the case of the IMC and the $H_{\infty }$ controllers, the variations in the outputs are evident. Thus, it can be concluded that the proposed automation of the loop shaping procedure using constrained optimization has multiple benefits; i.e., it simplifies the QFT design process by synthesizing both controller and pre-filter concurrently; in addition, it offers the freedom to specify the controller structure before the synthesis, offers an appropriate procedure to design QFT controllers, and making the overall design process more streamlined.

In terms of the transient behavior, the proposed controller offers a superior behavior in terms of both the tracking speed and damping. The rise time of the proposed controller is 0.31 s which is an ~47–81% reduction over the other controllers; similarly, the settling time offered is 0.78 s, which is ~70–95% reduction over the others indicating that the system stabilizes much faster without oscillations or delay; the overshoot too is 2%, which is comparable to the one offered by the $H_{\infty }$ and IMC controllers, while a significant reduction of ~97% over the QFT, thus highlighting the superior transient dynamics. When comparing the robustness, the peak sensitivity of 1.059 is lower than QFT/H∞/IMC controllers, signifying an 8–11% improvement in robustness against external disturbances and measurement noise. Likewise, the peak complementary sensitivity of 1.0 confirms that the system maintains a balanced trade-off between robustness and performance.

6. Sensitivity Analysis and Monte Carlo Simulations

In order to validate the performance of the controllers, a statistical analysis using sensitivity analysis and Monte Carlo simulations has also been included to validate the robustness of the controllers when subjected to normally distributed random perturbations.

6.1. Sensitivity Analysis to Parametric Uncertainties

To validate the robustness of the proposed controller, a sensitivity analysis is conducted, wherein the gains of the plant transfer function are perturbed with random parametric variations, as given by Equation (42), over a range of 5–20% for 2000 iterations. This has been performed to mimic the real-world variations in the plant dynamics.

$${\mathcal{G}}_{var}\left(s\right)={\displaystyle \frac{1}{\left(1.62\left(1+{\mathsf{\delta }}_i·r\right)\right)s^2+\left(0.47\left(1+{\mathsf{\delta }}_i·r\right)\right)s+\left(1+{\mathsf{\delta }}_i·r\right)}}$$

(42)

The evaluation of the average time domain performances for 2000 iterations is compared in

Table 3. Compared average time domain performance metrics for the system subjected to random parametric uncertainties in the range of 5–20%.

Uncertainty	Controller	Rise Time (s)	Overshoot (%)	Settling Time (s)
5%	QFT-FOPID	0.414	0.63	0.656
	PID [27]	0.650	0.00	17.327
	QFT [27]	0.593	71.10	10.536
	$H_{\infty }$	1.626	1.10	3.472
	IMC	1.693	1.10	3.574
	MPC	1.103	8.55	3.289
10%	QFT-FOPID	0.414	0.61	0.656
	PID [27]	0.651	0.02	17.473
	QFT [27]	0.588	71.96	11.018
	$H_{\mathrm{\infty }}$	1.669	1.85	5.503
	IMC	1.743	1.87	5.603
	MPC	1.103	8.55	3.289
15%	QFT-FOPID	0.414	0.61	0.656
	PID [27]	0.648	0.08	17.419
	QFT [27]	0.588	72.26	11.137
	$H_{\infty }$	1.747	3.00	7.381
	IMC	1.850	3.02	7.575
	MPC	1.103	8.55	3.289
20%	QFT-FOPID	0.414	0.62	0.656
	PID [27]	0.651	0.18	17.513
	QFT [27]	0.590	73.11	11.279
	$H_{\infty }$	1.909	4.01	9.815
	IMC	2.021	4.04	9.948
	MPC	1.103	8.55	3.289

and in Figure 10. From Table 3, it can be observed that the proposed QFT-FOPID controller consistently outperformed the other control schemes when subjected to all levels of uncertainties. The proposed controller offered good transit behavior with the fastest rise time (~0.41 s) and shortest settling time (~0.65 s), with negligible overshoots of <1%. In contrast, the conventional PID and conventional QFT controllers exhibited slower time domain response, while for the conventional QFT controller, large overshoots (>70%) are evident. $H_{\infty }$ and IMC controllers exhibited slower but smoother dynamics, with rise settling times exceeding 1.6 s and 3 s, respectively, while in the case of the higher overshoot is evident. Thus, it can be concluded that the proposed QFT-FOPID controller demonstrated superior robustness, faster transient response, and minimal sensitivity to uncertainty compared to all benchmark controllers.

6.2. Monte Carlo Simulations for Uncertain Plant

To further validate the robustness of the proposed QFT-FOPID controller, Monte Carlo simulations have been incorporated, by taking into consideration a random variation in the plant dynamics within ±20% of their nominal values to emulate structured uncertainty in the process dynamics. A total of 2000 independent trials were executed to statistically characterize closed-loop performance. In each trial, the corresponding closed-loop response of the system with different controllers is evaluated considering the uncertain plant. The time domain indices of rise time, settling time, and percentage overshoot, as well as the robustness metric of the peak sensitivity, are extracted in each trial. After completion of the trials, the results were statistically summarized using median, mean, and standard deviation metrics, as in

Table 4. Compared summary of Monte Carlo simulations for controller performance metrics under uncertainty.

Controller	Metric	Median	Mean	Std. Dev.
QFT	Rise time (s)	0.304	0.304	0.001
	Settling time (s)	0.974	0.832	0.280
	Overshoot (%)	2.129	2.125	0.268
	Peak Sensitivity	1.00	1.00	0
PID [27]	Rise time (s)	0.654	0.651	0.049
	Settling time (s)	17.317	17.303	1.066
	Overshoot (%)	0	0.245	0.571
	Peak Sensitivity	1.00	1.00	0.00
QFT [27]	Rise time (s)	0.300	0.303	0.028
	Settling time (s)	10.879	11.203	1.678
	Overshoot (%)	71.019	71.401	7.215
	Peak Sensitivity	1.202	1.203	0.020
${\mathcal{H}}_{\infty }$	Rise time (s)	1.630	1.83	0.582
	Settling time (s)	8.526	10.634	7.494
	Overshoot (%)	3.591	4.456	3.346
	Peak Sensitivity	1.164	1.188	0.102
IMC	Rise time (s)	1.707	1.941	0.640
	Settling time (s)	8.546	10.317	7.280
	Overshoot (%)	3.467	4.293	3.253
	Peak Sensitivity	1.160	1.184	0.104
MPC	Rise time (s)	1.114	1.106	0.094
	Settling time (s)	2.866	3.480	1.036
	Overshoot (%)	5.924	5.958	0.808

, to reflect key performance tendencies and robustness spread. Additionally, the kernel density maps (Figure 11), the histogram visualization (Figure 12) for the performance metrics (e.g., rise time vs. $M_s$, overshoot vs. $M_s$) is also included to present the performance variability and correlation between sensitivity and transient dynamics.

The compared performance of the controllers through Monte Carlo simulations, reported in Table 4, is visualized in Figure 13, and it can be observed that the proposed QFT-FOPID controllers offer a superior performance when compared to other benchmark controllers under parametric uncertainties. The proposed controllers achieve a faster rise time (~0.304 s) and minimal overshoot (~2.1%), with a peak sensitivity of 1.0, thus exhibiting robust tracking and low sensitivity regarding the plant uncertainties. The conventional PID controller exhibits slow dynamics with rise and settling times of ~0.651 s and ~17.3 s, respectively. The conventional QFT controller exhibits very large overshoots of the order of ~71%. The $H_{\infty }$ and IMC controllers offer robust yet slow responses, whereas MPC offers moderate rise times ~2.87 s and an overshoot of ~5.9%. Thus, it can be concluded that the proposed QFT-FOPID controller outperforms the other benchmark controllers in terms of the transient response and robustness.

7. Conclusions

The process of automating QFT loop-shaping is still an open challenge that requires attention. Doing so will simplify the synthesis of QFT controllers by allowing them to be designed in a single stage, as an alternative to the conventional manual process on Nichols charts. In this study, a constrained optimization is explored for concurrent synthesis of QFT controllers and pre-filters. Work focuses on synthesizing the QFT-FOPID controller and fractional order pre-filter, which highlights the benefits of pre-specifying their architecture at the beginning of the design process. The formulation of QFT synthesis as a constrained optimization problem also gives choice to the designer to specify the controller architecture, as well as aid in synthesizing in a single stage. When compared to current methods, the results clearly demonstrate the work’s superiority, and the same has been validated through sensitivity analysis and Monte Carlo simulations. The potential limitation of the proposed method is that it involves higher computational complexity due to constrained optimization, and future work can also focus on incorporating phase information directly into the controller synthesis to further enhance performance and design flexibility.