1 Introduction

Nowadays, drivers are eager to have their vehicles integrated with vehicle cruise control systems, as such mechanisms provide comfort in long journeys and significantly reduce driver-related accidents [1]. Not only does the safety factor come into play, but there is also a considerable reduction in fuel consumption, making the cruise control system a vital requirement for consumers [2]. However, a convenient control mechanism is crucial for operating such a system with the desired efficiency [3]. Therefore, in the previously reported works, the use of different controller mechanisms such as proportional–integral–derivative (PID) [4], PID with reference model [5], fractional-order PID (FOPID) [6], real PID plus second-order derivative (PIDD2) [7], fuzzy logic [8], adaptive [9] and model predictive [10] controllers can be encountered for operation of vehicle cruise control systems.

Despite the existence of those mechanisms, there is still room for newer controllers for reaching more excellent performance characteristics [7]. The latter is an important challenge that needs attention. In this regard, this paper proposes proportional, fractional order integral, derivative plus double derivative with filter (\(P{I}^{\lambda }DN{D}^{2}{N}^{2}\)) controller [11] as a new and more efficient solution for a vehicle cruise control system compared to PID and FOPID controllers. The reason of the proposal of \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller is due to its efficiency in reducing the overshoot, settling and rise time of the system [12]. Moreover, the \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller has two additional low pass filters which let the derivative terms to reduce the high frequency gain and noise appropriately [11]. It is also worth noting that this paper is also the first report demonstrating the implementation of a \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller for efficient operation of a vehicle cruise control system.

The proposed \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller has seven parameters (\({K}_{P}\), \({K}_{I}\), \({K}_{D}\), \({K}_{DD}\), \(\lambda\), \({N}_{1}\) and \({N}_{2}\)) which require to be adjusted similar to the case for PID [13] and FOPID [14] controllers. Therefore, an efficient adjustment method is also required to tune those parameters to optimal values. In terms of tuning such parameters, metaheuristic algorithms have been demonstrated as the most convenient options in the reported works [15]. With regards to vehicle cruise control system, enhanced reptile search algorithm [16], ant lion optimization algorithm [5], differential evolution algorithm [17], atom search optimization algorithm [18], genetic algorithm [19], hunger games search algorithm [20], arithmetic optimization algorithm [21] and Harris hawks optimization algorithm [22] can be found as the recent metaheuristic approaches for adjustment of different controllers.

The examples mentioned above, and the reported good results were considered in this study in order to come up with an even more efficient tuning mechanism for the adjustment of the proposed \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller parameters. Therefore, a novel metaheuristic algorithm is also proposed in this paper for the first time by boosting the structure of a recent metaheuristic algorithm named weighted mean of vectors (INFO) algorithm [23] with the aid of elite opposition-based learning scheme [24] and the pattern search method [25]. The proposed boosted INFO (b-INFO) algorithm was constructed such that excellent balance between exploration and exploitation stages was achieved. In this way, a more efficient algorithm was constructed to adjust the parameters of the \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller to optimal values for the vehicle cruise control system.

To verify the effectiveness of the proposed b-INFO algorithm, classical benchmark functions having unimodal, multimodal and fixed-dimensional multimodal properties (twenty-three test functions in total) were employed and the performance evaluation of the b-INFO algorithm was comparatively performed against original form of INFO algorithm [23], marine predators algorithm [26] and gravitational search algorithm [27]. The assessment on the test functions confirmed the excellent ability of the proposed b-INFO algorithm in terms of exploration and exploitation, thus, reaches improved solution quality. On the other hand, the comparative time complexity analysis showed a negligible higher run time for the proposed b-INFO algorithm compared to the original INFO algorithm which is expected due to inclusion of elite opposition-based learning and pattern search mechanisms, however, has less run time compared to marine predators and gravitational search algorithms.

A sensitivity analysis was performed to reach the good performance, and the best suitable values for the parameters of the algorithm were determined. A suitable objective function was then employed, and its value was minimized to reach optimal parameters of the \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller. Several analyses such as statistical, time response and robustness for different operating speeds were performed and the superior performance of the algorithm was demonstrated. To further evaluate the efficiency of the proposed b-INFO algorithm based \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller for the vehicle cruise control system, the most recently reported methods of genetic algorithm based PID [4], ant lion optimizer based PID [5], atom search optimization algorithm based PID [18], arithmetic optimization algorithm based PID [20], slime mould algorithm based PID [20], Lévy flight distribution algorithm based PID [20], manta ray foraging optimization based PID [20] and hunger games search based PID [20] along with Harris hawks optimization based PID and FOPID [22] controllers were employed. The related comparisons further demonstrated the excellent ability of the proposed method for the vehicle cruise control system. Considering the above discussion, the contributions of this work can briefly be listed as follows:

  • A state-of-the-art metaheuristic algorithm is proposed by augmenting the performance of the INFO algorithm using PS and EOBL mechanisms.

  • The resulting b-INFO algorithm surpasses the original INFO, marine predators, and gravitational search algorithms in terms of performance on benchmark functions, including unimodal, multimodal, and fixed-dimensional multimodal functions.

  • A novel \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller is proposed as a more efficient control structure for vehicle cruise control system.

  • This work presents the first report of a \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller's implementation for vehicle cruise control system, underscoring the novelty and significance of this research.

  • The proposed method's performance is compared against a range of recent approaches. Results demonstrate that the b-INFO algorithm-based \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller is the most efficient and superior method for controlling a vehicle cruise control system.

  • The proposed method's exceptional ability is further confirmed by comparisons with the genetic algorithm, ant lion optimizer, atom search optimizer, arithmetic optimization algorithm, slime mold algorithm, Lévy flight distribution algorithm, manta ray foraging optimization, and hunger games search-based PID, along with Harris hawks optimization-based PID and FOPID controllers.

  • Considering the above list, this work marks a remarkable milestone toward safer and more efficient vehicle cruise control systems.

2 INFO Algorithm

The INFO algorithm in question is a cutting-edge, population-based metaheuristic approach that operates by considering vectors as potential solutions [23]. Through sophisticated calculations involving the weighted mean of a set of vectors within the search space, the INFO algorithm generates a dynamic sequence of steps that involve updating the positions of these vectors. The process is refined through a series of intricate stages that include vector combining, local search, and updating rule, which all contribute to the overall efficacy of the algorithm. The initialization phase of the INFO algorithm is a crucial stage that relies on the power of Eq. (1), a sophisticated equation that randomly generates a population of vectors. At this stage, the algorithm harnesses the full potential of its arsenal, with \(Np\) representing the population of vector, \(D\) symbolizing the dimension of the search space, and \(l\) taking on the values of \(\mathrm{1,2},\dots ,Np\). With these complex mechanisms working in tandem, the INFO algorithm stands as a true testament to the cutting edge of computational science.

$${X}_{l,j}^{g}=\left\{{x}_{l,1}^{g},{x}_{l,2}^{g},\dots ,{x}_{l,D}^{g}\right\}$$
(1)

The weighted mean factor (\(\delta\)) and the scaling factor (\(\sigma\)) are also employed in the initialization stage. The latter two variables are respectively defined in Eqs. (2) and (3) where \(\beta =2{e}^{-4(g/Maxg)}\) and \(\alpha =c{e}^{-d(g/Maxg)}\).

$$\delta =2\beta \times rand-\beta$$
(2)
$$\sigma =2\alpha \times rand-\alpha$$
(3)

In the latter definitions, \(g\) denotes the current generation, \(Maxg\) represents the maximum number of generations (iterations) and \(c\) is a constant number of \(2\) whereas \(d\) is a constant number of \(4\). The variables of \(\delta\) and \(\sigma\) are not adjusted by the user and they change dynamically through generations. The scaling factor serves to amplify the resulting vector by utilizing the updating procedure, which relies on the size of the search domain. It is employed to scale the weighted average of vectors. Its value is determined based on the feasible search space of the problems at hand and is subsequently reduced following an exponential formula. The population diversity is increased with the \(MeanRule\), defined in the following equation.

$$MeanRule=r\times {WM1}_{l}^{g}+(1-r)\times {WM2}_{l}^{g}$$
(4)

The terms of \({WM1}_{l}^{g}\) and \({WM2}_{l}^{g}\) have the following definitions involving several terms, including a random number \(r\) within \([\mathrm{0,0.5}]\), a small constant number \(\varepsilon\), a normally distributed random value rand, and randomly selected integer numbers \(\alpha 1\), \(\alpha 2\), \(\alpha 3\) (where \(\alpha 1\ne \alpha 2\ne \alpha 3\)) from [\(1, Np\)].

$${WM1}_{l}^{g}=\delta \times \frac{{w}_{1}\left({x}_{\alpha 1}-{x}_{\alpha 2}\right)+{w}_{2}\left({x}_{\alpha 1}-{x}_{\alpha 3}\right)+{w}_{3}\left({x}_{\alpha 2}-{x}_{\alpha 3}\right)}{{w}_{1}+{w}_{2}+{w}_{3}+\varepsilon }+\varepsilon \times rand$$
(5)
$${WM2}_{l}^{g}=\delta \times \frac{{w}_{1}\left({x}_{bs}-{x}_{bt}\right)+{w}_{2}\left({x}_{bs}-{x}_{ws}\right)+{w}_{3}\left({x}_{bt}-{x}_{ws}\right)}{{w}_{1}+{w}_{2}+{w}_{3}+\varepsilon }+\varepsilon \times rand$$
(6)

Additionally, the wavelet functions \({w}_{1}\), \({w}_{2}\) and \({w}_{3}\) [23] are used to manipulate these terms, where \({x}_{bs}\), \({x}_{bt}\) and \({x}_{ws}\) represent the best, better, and worst solutions amongst the vectors, respectively. Furthermore, the updating rule also integrates a convergence acceleration (\(CA\)) component, as defined below:

$$CA=randn\times \frac{\left({x}_{bs}-{x}_{\alpha 1}\right)}{\left(f\left({x}_{bs}\right)-f\left({x}_{\alpha 1}\right)+\varepsilon \right)}$$
(7)

where \(randn\) is a random number with normal distribution. The latter definition can be used to calculate new vector using the following expression.

$${z}_{l}^{g}={x}_{l}^{g}+\sigma \times MeanRule+CA$$
(8)

The updating rule based on \({x}_{bs}\), \({x}_{bt}\), \({x}_{l}^{g}\) and \({x}_{\alpha 1}^{g}\) is defined as follows in order to obtain new vectors of \({z1}_{l}^{g}\) and \({z2}_{l}^{g}\) in the \({g}^{th}\) generation. For a random number less than \(0.5\), the Eqs. (9) and (10) are used; otherwise, Eqs. (11) and (12) are used for calculation.

$${z1}_{l}^{g}={x}_{l}^{g}+\sigma \times MeanRule+randn\times \frac{\left({x}_{bs}-{x}_{\alpha 1}^{g}\right)}{\left(f\left({x}_{bs}\right)-f\left({x}_{\alpha 1}^{g}\right)+1\right)}$$
(9)
$${z2}_{l}^{g}={x}_{bs}+\sigma \times MeanRule+randn\times \frac{\left({x}_{\alpha 1}^{g}-{x}_{b}^{g}\right)}{\left(f\left({x}_{\alpha 1}^{g}\right)-f\left({x}_{\alpha 2}^{g}\right)+1\right)}$$
(10)
$${z1}_{l}^{g}={x}_{\alpha }^{g}+\sigma \times MeanRule+randn\times \frac{\left({x}_{\alpha 2}^{g}-{x}_{\alpha 3}^{g}\right)}{\left(f\left({x}_{\alpha 2}^{g}\right)-f\left({x}_{\alpha 3}^{g}\right)+1\right)}$$
(11)
$${z2}_{l}^{g}={x}_{bt}+\sigma \times MeanRule+randn\times \frac{\left({x}_{\alpha 1}^{g}-{x}_{\alpha 2}^{g}\right)}{\left(f\left({x}_{\alpha 1}^{g}\right)-f\left({x}_{\alpha 2}^{g}\right)+1\right)}$$
(12)

The vector combining stage of the algorithm amalgamates the obtained vectors of \({z1}_{l}^{g}\) and \({z2}_{l}^{g}\) with the vector of \({x}_{l}^{g}\) to produce the new vector \({u}_{l}^{g}\), which adheres to certain pre-defined conditions. The parameter \(\mu\), which signifies a normally distributed random value multiplied by \(0.05\), is also taken into account in this process.

if rand < 0.5

         if rand < 0.5

$${u}_{l}^{g}={z1}_{l}^{g}+\mu \cdot \left|{z1}_{l}^{g}-{z2}_{l}^{g}\right|$$
(13)

         else

$${u}_{l}^{g}={z2}_{l}^{g}+\mu \cdot \left|{z1}_{l}^{g}-{z2}_{l}^{g}\right|$$
(14)

         end

else

$${u}_{l}^{g}={x}_{l}^{g}$$
(15)

end


The exploitation process is performed through the vector combining stage; nevertheless, the INFO algorithm goes the extra mile to boost its effectiveness by implementing a local search stage to evade the pitfalls of being trapped in local optima. In this later stage, the algorithm harnesses the power of the global best position (\({x}_{best}^{g}\)) and the mean-based rule described in Eq. (16) to perform the local search operator.

$$WM=\frac{{{x}_{1}\times w}_{1}+{{x}_{2}\times w}_{2}}{{w}_{1}+{w}_{2}}$$
(16)

A new vector around \({x}_{best}^{g}\) can be produced as follows:

if rand < 0.5

         if rand < 0.5

$${u}_{l}^{g}={x}_{bs}+randn\times \left(MeanRule+randn\times \left({x}_{bs}^{g}-{x}_{\alpha 1}^{g}\right)\right)$$
(17)

         else

$${u}_{l}^{g}={x}_{rnd}+randn\times \left(MeanRule+randn\times \left({v}_{1}\times {x}_{bs}-{v}_{2}\times {x}_{rnd}\right)\right)$$
(18)

         end

end

The INFO algorithm aims to increase the diversity of the search space by introducing a new solution, denoted as \({x}_{rnd}\), which is randomly generated using a combination of the best and better solutions. Specifically, \({x}_{rnd}\) is defined as \({x}_{rnd}=\varphi \times {x}_{avg}+\left(1-\varphi \right)\times \left(\varphi \times {x}_{bt}+\left(1-\varphi \right)\times {x}_{bs}\right)\). Here, the term \(\varphi\) is a randomly generated number within the range of (\(\mathrm{0,1}\)) and \({x}_{avg}=\left({x}_{a}+{x}_{b}+{x}_{3}\right)/3\), while \({v}_{1}\) and \({v}_{2}\) are two additional random numbers that are used to further emphasize the impact of the best position on the vector. The values of \({v}_{1}\) and \({v}_{2}\) are determined using a randomly generated number \(p\) within the range of (\(\mathrm{0,1}\)).

$$v_1 = \left\{ {\begin{array}{*{20}l} {2 \times rand, } \hfill & {p > 0.5} \hfill \\ {1,} \hfill & { otherwise} \hfill \\ \end{array} } \right.$$
(19)
$$v_2 = \left\{ {\begin{array}{*{20}l} {rand,} \hfill & { p < 0.5} \hfill \\ {1,} \hfill & { otherwise} \hfill \\ \end{array} } \right.{ }$$
(20)

The flowchart of the INFO algorithm is provided in Fig. 1.

Fig. 1
figure 1

Flowchart of INFO algorithm

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3 A Novel Approach for Improving Performance of INFO Algorithm

3.1 Elite Opposition-Based Learning Mechanism

The opposition-based learning (OBL) scheme, proposed by Tizhoosh in 2005 [28], has been an efficacious machine learning technique to enhance the performance of metaheuristic algorithms, as suggested by recent studies [29]. A more advanced version of the OBL mechanism, known as the elite OBL (EOBL) strategy, has also been introduceed [24]. The OBL approach leverages the opposite solutions of the current individuals in conjunction with their current states, resulting in more effective exploration [30]. In contrast, the EOBL strategy combines the best individuals with the current ones, generates the opposite solutions of the elite individuals [31], and evaluates their fitness values. From a mathematical standpoint, if we consider \(X=\langle {x}_{1}, {x}_{2}\dots ,{x}_{v}\rangle\) as an elite candidate solution with \(v\) decision variables, then the elite opposition-based solution (\({X}^{o}\)) is defined as follows:

$${X}^{o}=\langle {x}_{1}^{o}, {x}_{2}^{o}\dots ,{x}_{v}^{o}\rangle$$
(21)

where \({x}_{i}^{o}=\delta \left(d{a}_{i}+d{b}_{i}\right)-{x}_{i}\) and \(\delta\) is a parameter within (0, 1) which is used to control the opposition magnitude. The dynamic boundaries are denoted by \(d{a}_{i}\) and \(d{b}_{i}\) which are defined as

$$d{a}_{i}=min\left({x}_{i}\right), d{b}_{i}=max\left({x}_{i}\right)$$
(22)

The EOBL strategy employs the following rule to prevent exceeding the boundaries [\(L{b}_{i}, U{b}_{i}\)] for any opposite decision variable where \(rand\left(L{b}_{i}, U{b}_{i}\right)\) denotes a random number within (\(L{b}_{i}, U{b}_{i}\)).

$$x_i^o = rand\left( {Lb_i , Ub_i } \right),\quad {\text{if}}\quad x_i^o < Lb_i \quad {\text{or}}\quad x_i^o > Ub_i$$
(23)

3.2 Pattern Search Method

The pattern search (PS) is a derivative-free method having good exploitation capability [25]. In the PS method, an initial point (\({S}_{0}\)) is defined by the user which becomes the starting point for the search [32]. The size of the mesh, for the first iteration, is considered as 1 and the pattern (direction) vectors are constructed as: \(S_0 + [0 \,1]\), \(S_0 + [1 \,0]\), \(S_0 + [ - \,1 \,0]\) and \(S_0 + [0\, - \,1]\). This procedure helps generating new mesh points. The objective functions are then calculated for those generated points which continues until a smaller value than \({S}_{0}\) is found. Finding a smaller value such as \(f\left({S}_{1}\right)<f\left({S}_{0}\right)\) means the poll is successful. In such a case, the related point is set as a source point. After the successful poll, the PS algorithm performs the expansion operation in the second iteration by multiplying the current mesh size with 2 such that the new points are produced as \({S}_{1}+2\times [0 \,1]\), \({S}_{1}+2\times [1\, 0]\), \({S}_{1}+2\times [-1\, 0]\) and \({S}_{1}+2\times [0\,-\,1]\). The expanding stage keeps operating provided a newer point with lower objective function is achieved. Otherwise, the contraction stage is performed which reduces the mesh size by multiplying it with 0.5. The overall process continues until the termination condition is met. In this study, the following parameter values were used: initial mesh size = 1, mesh expansion factor = 2, mesh contraction factor = 0.5, all tolerances = 10–6. The flowchart of the PS algorithm is provided in Fig. 2.

Fig. 2
figure 2

Flowchart of PS method

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3.3 Proposed Boosted INFO Algorithm

In this study, with the aim of reaching a better exploration and exploitation capability, the structure of the original form of the INFO algorithm was improved by the integration of the EOBL strategy and the PS method. The proposed b-INFO algorithm uses the PS method to aid enhanced exploitation and employs the EOBL strategy to reach further exploration power. A detailed flowchart of the proposed b-INFO algorithm is provided in Fig. 3. As shown in this flowchart, the proposed b-INFO algorithm starts with the operation of the original INFO algorithm to generate a best solution which is then further improved with the integration of the EOBL strategy. After obtaining \({N}_{P}\) best solutions, the PS method operates with the aim of reaching better exploitation power. At this stage, instead of performing the PS method in each iteration, it is operated only twice through complete operation of the algorithm and for each operation it runs for \(100\times D\) iterations where \(D\) denotes the dimension size of the problem. Such an overall procedure was constructed after extensive simulations and consequently boosted the ability of the original INFO algorithm significantly.

Fig. 3
figure 3

Flowchart of b-INFO

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4 Initial Performance Evaluation

4.1 Employed Benchmark Functions

In this study, the performance of the constructed b-INFO algorithm was tested initially against benchmark functions. In this regard, the unimodal, multimodal, and fixed dimensional multimodal benchmark functions were employed in order to demonstrate the more excellent capacity of the proposed b-INFO algorithm in terms of exploration, exploitation and solution quality as those benchmark functions provide a good platform to test the performance of the algorithms. The name, equation, range, dimensions (\(D\)) and optimum point (\({f}_{min}\)) details of the unimodal, multimodal and fixed-dimensional multimodal benchmark functions are respectively provided in Tables 1, 2 and 3.

Table 1 Unimodal benchmark functions
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Table 2 Multimodal benchmark functions
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Table 3 Fixed-dimensional multimodal benchmark functions
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4.2 Compared Metaheuristic Algorithms

In this study, marine predators algorithm (MPA) [26] and gravitational search algorithm (GSA) [27] were employed in addition to the original INFO algorithm in order to provide a comparative assessment. Table 4 lists those algorithms and their respective parameters. In addition, each algorithm was run for 30 times.

Table 4 Parameter settings of algorithms for benchmark functions
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4.3 Assessment of the Exploitative Behavior

To assess the exploitation power of the proposed b-INFO algorithm, a suite of unimodal benchmark functions listed in Table 1 were employed. The attained results are comprehensively presented in Table 5, where the superior performance of the b-INFO algorithm is evidently manifested by the remarkably low values obtained.

Table 5 Statistical results of the unimodal benchmark functions from b-INFO and the other three algorithms
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4.4 Assessment of the Explorative Behavior

In the present study, the exploration power and solution quality of the proposed b-INFO algorithm were assessed using both multimodal and fixed-dimensional multimodal benchmark functions provided in Tables 2 and 3. The results of the experiments are presented in a comparative manner in Tables 6 and 7. The numerical results indicate a remarkable performance of the proposed b-INFO algorithm, highlighting its superiority in terms of exploration and solution quality. Overall, the findings provide strong evidence for the excellent capability of the b-INFO algorithm.

Table 6 Statistical results of the multimodal benchmark functions from b-INFO and the other three algorithms
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Table 7 Statistical results of the fixed-dimensional multimodal benchmark functions from b-INFO and the other three algorithms
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4.5 Wall-Clock Time Analysis

The proposed b-INFO algorithm was also investigated comparatively in terms of the run-time. In this regard, all the 23 benchmark functions were employed, and a comparative analysis was performed by running all algorithms. The respective results are numerically presented in Table 8. As can be observed, for each benchmark function the INFO algorithm takes the shortest time per run. The proposed b-INFO algorithm has a slightly higher run time, due to inclusion of EOBL and PS mechanisms (between 3 and 5% depending on the benchmark function), compared to INFO algorithm. however, it reaches a significantly improved solution quality. In addition, the proposed b-INFO algorithm has less run time compared to MPA and GSA algorithms.

Table 8 Comparison of the run-time of b-INFO and other algorithms
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4.6 Sensitivity Analysis

A sensitivity analysis was also carried out in this study in order to determine the best suitable values for the adjustment of \(c\) and \(d\) parameters of the proposed b-INFO algorithm. In this regard, \(c\) and \(d\) parameters were assigned five different values (\(2, 4, 6, 8, 10\)) separately which created a combination of \(5\times 5=25\) different designs. The analysis was performed against \({f}_{5}(x)\) (unimodal), \({f}_{12}(x)\) (multimodal) and \({f}_{20}(x)\) (fixed-dimensional multimodal) benchmark functions. The respective parameter combinations and the obtained statistical metrics are shown in Table 9. Considering the presented numerical results, it can easily be spotted that there are different specific combinations (shown in bold for case no 2, 5, 6, 11, 16, 18 and 22) to reach good values for \({f}_{20}(x)\), however, the best combination for the parameter settings is achieved with \(c=2\) and \(d=4\) (case no 2) which provided the best results for all types of the benchmark functions. Therefore, in this study, the \(c\) and \(d\) parameters were adjusted to be \(2\) and \(4\), respectively.

Table 9 Sensitivity analysis of b-INFO’s control parameters with different benchmark functions
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5 Vehicle Cruise Control System’s Mathematical Model

The precise control of a vehicle's velocity is achieved through the utilization of its cruise control system, which continuously adjusts the engine throttle angle (\(u\)) to regulate the vehicle's speed (\(v\)) to a pre-set reference speed (\({v}_{ref}\)). The underlying dynamics of a vehicle's longitudinal motion can be mathematically formulated as per the established work of Lewis and Houghton [33]:

$${F}_{d}=M\frac{dv}{dt}+{F}_{a}+{F}_{g}$$
(24)

where \({F}_{d}\), \({F}_{a}\), \({F}_{g}\) and \(M\left(dv/dt\right)\) stand for the engine’s drive force, aerodynamic drag, the climbing resistance and the inertia force, respectively. The dynamic model of the vehicle cruise control system is shown in Fig. 4 where the terms of \(\theta\), \({v}_{w}\), \({C}_{\mathrm{a}}\) and \(M\) stands for road angle, wind gust speed, the aerodynamic drag coefficient and the total mass of the vehicle and the passengers.

Fig. 4
figure 4

Dynamic model of the system

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For the initial assessment, the vehicle is assumed to operate at a speed of \(30\,{\text{km/h}}\) without any wind gust speed and climbing resistance. The system’s state model can then be described as [33]:

$$\dot{v}=\frac{1}{M}({F}_{d}-{C}_{\mathrm{a}}{v}^{2})$$
(25)
$${\dot{F}}_{d}=\frac{1}{T}({C}_{1}u\left(t-\tau \right)-{F}_{d})$$
(26)

For an equilibrium state with a nominal operating speed (\({v}_{ref}={v}_{0}=30\,{\text{km/h}}\)), a nominal drive force (\({F}_{d0}\)) with a nominal throttle position (\({u}_{0}\)), which are defined as follows, are required.

$${F}_{d0}={C}_{\mathrm{a}}{{v}_{0}}^{2}$$
(27)
$${u}_{0}={F}_{d0}/{C}_{1}$$
(28)

Linearizing the model around the stated set points would help obtaining the following definitions [33]:

$$\delta \dot{v}=-\frac{2{C}_{\mathrm{a}}{v}_{0}}{M}\delta v+\frac{1}{M}\delta {F}_{d}$$
(29)
$$\delta {\dot{F}}_{d}=-\frac{1}{T}\delta {F}_{d}+\frac{{C}_{1}}{T}\delta u\left(t-\tau \right)$$
(30)

The following transfer function can be obtained for the linearized model of the vehicle cruise control system where \(C=\frac{{C}_{1}}{MT\tau }\), \({p}_{1}=-2\frac{{C}_{a}{v}_{0}}{M}\), \({p}_{2}=-\frac{1}{T}\) and \({p}_{3}=-\frac{1}{\tau }\).

$$G\left(s\right)=\frac{\Delta V(s)}{\Delta U(s)}=\frac{C}{(s-{p}_{1})(s-{p}_{2})(s-{p}_{3})}$$
(31)

In this paper, the parameter values presented in Table 10 were employed in order to present a fair comparison with the works presented in the literature.

Table 10 The employed parameters of the vehicle cruise control system [20]
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6 A Novel Controller for Vehicle Cruise Control System

6.1 Proposed \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) Controller

A PID controller has a wider implementation due to several advantages such as being easy to implement, having a simple structure and providing good performance [34]. A PID controller has three parameters (\({K}_{P}\), \({K}_{I}\) and \({K}_{D}\)) which yields the following transfer function.

$${C}_{PID}\left(s\right)={K}_{P}+\frac{{K}_{I}}{s}+{K}_{D}s$$
(32)

Despite its wider implementation, a PID controller cannot provide best performance for the dynamic and more complex problems as effectively as its fractional counterpart called FOPID controller [35]. The latter one has additional integration (\(\lambda\)) and derivative (\(\mu\)) orders which helps increasing the stability and robustness [36]. The transfer function of a FOPID controller can be defined as follows by considering the latter statement.

$${C}_{FOPID}\left(s\right)={K}_{P}+\frac{{K}_{I}}{{s}^{\lambda }}+{K}_{D}{s}^{\mu }$$
(33)

Apart from a FOPID controller, a PIDD2 controller [37] having four parameters (\({K}_{P}\), \({K}_{I}\), \({K}_{D}\) and \({K}_{DD}\)) can also alternatively be used. The transfer function of the latter controller is provided as follows.

$${C}_{{PIDD}^{2}}\left(s\right)={K}_{P}+\frac{{K}_{I}}{s}+{K}_{D}s+{K}_{DD}{s}^{2}$$
(34)

The additional second-order derivative (\({K}_{DD}\)) term helps reaching improved phase margin, steady state accuracy, and stability [38]. Considering the abilities of those controllers, this study employs a \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller [11] which has an additional fractional order for the integrator and low pass filters for derivative terms. The transfer function of the employed controller is provided as follows.

$${C}_{P{I}^{\lambda }DN{D}^{2}{N}^{2}}\left(s\right)={K}_{P}+\frac{{K}_{I}}{{s}^{\lambda }}+{K}_{D}\frac{s{N}_{1}}{s+{N}_{1}}+{K}_{DD}{\left(\frac{{sN}_{2}}{s+{N}_{2}}\right)}^{2}$$
(35)

The advantage of the employed \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller is that it can effectively reduce the overshoot, settling and rise time of the system [12]. Besides, the additional two low pass filters allow the derivative terms to appropriately reduce the high frequency gain and noise [11].

The calculations for the fractional integrator were performed with the FOMCON toolbox [39] and the limits for the controller parameters were arranged as \(1\le {K}_{P}\le 6\), \(0.1\le {K}_{I}\le 0.5\), \(1\le {K}_{D}\le 6\), \(0.1\le {K}_{DD}\le 0.5\), \(0.5\le \lambda \le 1.5\), \(10\le {N}_{1}\le 1000\) and \(10\le {N}_{2}\le 1000\). The block diagram of the \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller is shown in Fig. 5. In the respective figure, the input represents the error between the reference speed and actual speed.

Fig. 5
figure 5

Block diagram of \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller

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6.2 \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) Controlled Vehicle Cruise Control System

The block diagram shown in Fig. 6 presents a \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controlled vehicle cruise control system. As can be seen, the system consists of a closed loop feedback loop where \(\Delta {V}_{ref}\left(s\right)\) represents the commanded speed and \(\Delta V\left(s\right)\) is the actual operating speed. In this study, the respective block diagram was considered to perform analysis.

Fig. 6
figure 6

Block diagram of vehicle cruise control system with \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller

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7 A Novel Design Approach and Comparative Simulation Results

7.1 Objective Function and Implementation of b-INFO Algorithm

To make use of the advantage of the proposed b-INFO algorithm and the \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller for the vehicle cruise control system, the \(F\) objective function defined in Eq. (36) was employed.

$$F=\left(1-{e}^{-\sigma }\right)\left(\frac{\%OS}{100}+{e}_{ss}\right)+{e}^{-\sigma }({t}_{s}-{t}_{r})$$
(36)

where \(\sigma\) denotes a weighting parameter and set to 1 [40], \(\%OS\) represents maximum percent overshoot, \({e}_{ss}\) is the steady state error whereas \({t}_{s}\) and \({t}_{r}\) are respectively the settling and rise times. Figure 7 depicts the application of the proposed b-INFO algorithm to design a \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller for the vehicle cruise control system. The figure shows that the b-INFO algorithm updates the controller's parameters by considering the \(F\) function. The algorithm continues its operation until the maximum number of generations (iterations) is reached, resulting in the optimal parameters of the controller.

Fig. 7
figure 7

Proposed \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller design procedure via b-INFO algorithm for vehicle cruise control system

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7.2 Statistical Analysis and the Obtained Best Parameters of the Controller

Table 11 presents the parameter settings of different algorithms employed in this study in order to perform the comparative analysis for the vehicle cruise control system. In the beginning, the time complexity analysis was performed for the employed algorithms. The respective numerical results for the time complexity are also provided in Table 11. As can be seen, the original INFO algorithm has less time complexity than all other algorithms. On the other hand, the proposed b-INFO algorithm has a slightly higher time complexity than the original form of INFO algorithm which is expected due to integration of EOBL and PS mechanisms. As mentioned earlier, the proposed b-INFO algorithm reaches significant solution quality at the expense of this slight increase in time complexity. However, it still occupies the second place in terms of less time complexity which the most convenient algorithm for the vehicle cruise control system.

Table 11 Parameter settings and time complexity of algorithms for vehicle cruise control system
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The comparative statistical results obtained from the \(F\) objective function are provided numerically in Table 12. As can be seen from the latter table, the proposed b-INFO algorithm reaches the best statistical metrics for the employed objective function making it the most suitable approach for vehicle cruise control system.

Table 12 Statistical results of the \(F\) objective function
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Figure 8 presents the results of a boxplot analysis that highlights the superior performance of the proposed b-INFO algorithm. The plot shows that the proposed algorithm outperforms the other algorithms by achieving the minimum value in the objective function. Moreover, the worst result obtained by the proposed b-INFO algorithm is still better than the best results obtained by the other algorithms, indicating the robustness and stability of the proposed algorithm.

Fig. 8
figure 8

Comparative boxplot analysis

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The obtained \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller parameters (\({K}_{P}\), \({K}_{I}\), \({K}_{D}\), \({K}_{DD}\), \(\lambda\), \({N}_{1}\) and \({N}_{2}\)) for the best runs of the algorithms are presented in Table 13. Those values were used to perform the analysis presented in the following subsections for the respective algorithms-based controllers employed in a vehicle cruise control system.

Table 13 Optimized parameters of \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller
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7.3 Time Response Analysis

In this study, the time domain analysis was performed by considering the speed change step responses as demonstrated in Fig. 9. As can be observed, the proposed b-INFO algorithm is able to reach smoother response with no overshoot, less rise and settling times making it the most convenient approach amongst the other algorithms.

Fig. 9
figure 9

Step responses of the system

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Table 14 presents a numerical comparison of the overshoot, rise time, settling time, and peak time of the proposed b-INFO algorithm-based \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller for the vehicle cruise control system with other algorithms. The results confirm the superior time response profile of the proposed controller, as it achieves lower values for rise time, settling time, and peak time while completely eliminating overshoot. This is consistent with the illustrative performance demonstrated in Fig. 9.

Table 14 Comparison of transient response specifications
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7.4 Robustness Analysis Under Different Operating Speeds

In this study, an operating speed of \(30\,{\text{km/h}}\) was initially considered for the analyses. In this section, the performance of the proposed approach is further evaluated by considering the robustness of the proposed b-INFO algorithm based \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller for vehicle cruise control system by separately considering different operating speeds. In this regard, operating speeds of \(20\,{\text{km/h}}\) (Case A) and \(40\,{\text{km/h}}\) (case B) were separately considered. Figure 10 illustrates the step response of the system for Case A while Fig. 11 shows for Case B. As can be observed from those figures, the proposed b-INFO algorithm demonstrates an overall better characteristic.

Fig. 10
figure 10

Step responses of the system for \({v}_{0}=20\, \mathrm{km}/\mathrm{h}\)

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Fig. 11
figure 11

Step responses of the system for \({v}_{0}=40\, \mathrm{km}/\mathrm{h}\)

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In addition, comparative numerical values for Case A and B are also provided in Table 15. As presented in this table, the proposed b-INFO algorithm achieves the lowest values for the overshoot, rise time, settling time and peak time for Case A demonstrating greater robustness. In case of Case B, all algorithms were able to reach no overshoot, however, the proposed b-INFO algorithm was also able to reach the shortest rise and settling time values confirming the excellent robustness of the proposed approach for different operating speeds.

Table 15 Comparison of transient response specifications under different operating speeds
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7.5 Comparison with the Most Recent Algorithms

Comparisons using the most recent reported methods of genetic algorithm (GA) based PID controller [4], ant lion optimizer (ALO) based PID controller [5], atom search optimization (ASO) algorithm based PID controller [18], arithmetic optimization algorithm (AOA) based PID controller [20], slime mould algorithm (SMA) based PID controller [20], Lévy flight distribution (LFD) algorithm based PID controller [20], manta ray foraging optimization (MRFO) based PID controller [20] and hunger games search (HGS) based PID controller [20] along with Harris hawks optimization (HHO) based PID and FOPID controllers [22] were employed, as well in order to provide an indication from a wider perspective.

Table 16 showcases the adopted parameters for the algorithms evaluated in this study, alongside their respective time complexity analyses. The numerical results presented in this table indicate that the proposed b-INFO algorithm-based \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller exhibits the lowest time complexity, highlighting its superior efficiency over the most recently reported approaches in terms of run time. Additionally, a comparison of the time domain performances of the different algorithms is illustrated in Fig. 12. The depicted figure demonstrates the exceptional effectiveness of the proposed b-INFO algorithm-based \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller among the most recently reported ones, achieving no overshoot along with minimum rise, settling, and peak times.

Table 16 Different reported methods along with their respective parameter values and the time complexity
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Fig. 12
figure 12

Percent overshoot, rise time, settling time and peak time for different approaches

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8 Conclusion

This paper presents a breakthrough in the field of vehicle cruise control systems by introducing a novel control approach that promises to revolutionize the industry. The proposed \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller, a first-of-its-kind in the literature, has demonstrated outstanding effectiveness in controlling vehicle speed. To optimize the controller's parameters, the paper also introduces a cutting-edge metaheuristic algorithm called b-INFO, a boosted version of the INFO algorithm, leveraging the EOBL and PS mechanisms. The proposed b-INFO algorithm has been rigorously tested and validated through unimodal, multimodal, and fixed-dimensional multimodal benchmark functions, exhibiting exceptional performance. Furthermore, to attain the optimal parameters of the \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller, a well-designed objective function has been employed, leading to a highly efficient and robust method for controlling vehicle cruise control systems. Comparative analysis with other state-of-the-art algorithms, including INFO, MPA, and GSA algorithms, has demonstrated the superiority of the proposed b-INFO algorithm-based \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) the controller in controlling vehicle speed. The paper also compares the proposed method with the most recently reported approaches, demonstrating the remarkable capability of the proposed b-INFO based \(P{I}^{\lambda }DN{D}^{2}{N}^{2}\) controller to achieve unparalleled performance. In summary, this paper's innovative contribution has opened up new avenues for designing highly effective and efficient vehicle cruise control systems control approaches.