Enhanced Output Tracking Control for Direct Current Electric Motor Systems Using Bio-Inspired Optimization

Abstract

In this paper, an efficient output reference trajectory tracking control scheme for direct current electric motor systems based on bio-inspired optimization is proposed. The differential flatness structural property of the electric motor along with dynamic tracking error compensation is suitably exploited for the backstepping control design. Off-line optimal selection of control parameters, implementing bio-inspired ant colony and particle swarm optimization algorithms, is addressed by minimizing an objective function where the decision variables are the tracking error and control input effort. A novel adaptive version of the control approach based on B-spline artificial neural networks is provided as well. The introduced flat output feedback tracking control design approach can be further extended for other differentially flat dynamic systems. Considerably perturbed, diverse velocity and position reference trajectory tracking scenarios are developed for demonstrating the acceptable closed-loop system performance. The results prove the efficient and robust tracking of the position and velocity reference profiles planned for the operation of the controlled electric motor system under variable torque disturbances using bio-inspired optimization.

1. Introduction

Nowadays, there is a wide range of applications and processes that demand increasingly advanced automatic systems capable of providing movement with high precision while being subjected to various unwanted and unknown external disturbances in multiple operating scenarios. A large part of these automatic systems depends on the proper control of actuation subsystems that provides to the main system the ability to perform specific regulation and trajectory tracking tasks. Direct current (DC) electric motors are ideal for a multitude of industrial and service applications where high torque and variable speed are required [1,2]. These range of applications stand from the construction of educational prototypes to the development of advanced systems where precise movements are required, such as in robots [3,4].

The development of DC motor control systems remains an active and multifaceted research field driven by several motivations. Researchers persistently endeavour to enhance the efficiency, performance, and reliability of these systems by refining control algorithms, sensor technologies, power electronics, and overall system integration to achieve heightened energy efficiency and responsiveness. Constant advancements in control theory lead to the discovery and exploration of novel techniques, such as predictive, adaptive, model-based, and optimal control, aimed at achieving superior precision and robustness in DC motor control. Simultaneously, the mitigation of noise and vibration, often undesirable in motor systems, prompting research into control algorithms and mechanical enhancements to diminish these effects, particularly crucial in noise-sensitive applications like medical equipment and precision machinery. Additionally, researchers delve into sensorless control methods, reducing reliance on external sensors and bolstering cost-efficiency and system dependability.

The synergy between DC motor control systems and renewable energy sources like solar panels and wind turbines becomes a focal point, necessitating the optimization of control strategies to harness variable and intermittent energy inputs efficiently. Innovation in sustainable energy sources is crucial for ensuring the availability of clean and dependable energy, with electric motors playing a pivotal role in this endeavor [5]. The ever-evolving landscape of applications, encompassing robotics, drones, electric vehicles, and automation, compels researchers to tailor and innovate control systems to meet specific requirements. In tandem, the development of robust fault detection and tolerance mechanisms assumes paramount importance, particularly in critical applications, as researchers work on enhancing the systems’ ability to detect and respond to faults or anomalies.

Hardware innovations, driven by advances in power electronics and semiconductor technology, lead to the exploration of novel materials, designs, and manufacturing techniques to engender more efficient and reliable motor control hardware. Lastly, in scenarios necessitating coordinated efforts of multiple motors, real-time and distributed control systems take precedence, with researchers concentrating on developing algorithms and architectures facilitating seamless coordination and communication among multiple DC motor control units.

Important velocity and position control approaches for the DC electric machine have been proposed in the literature for improving the closed-loop performance of this electromechanical system. The authors of [6,7,8] propose different robust controllers and parameter estimation schemes for suitably integration of output tracking controllers, where considerably perturbed case studies are evaluated by simulation and experimental set-ups. In the same breath, in [9] an adaptive backstepping approach is proposed for the speed control of a separately excited DC motor. Comparison results regarding a conventional Proportional Integral (PI) controller demonstrate an improved response of the overall system by using the adaptive backstepping controller. In the meantime, an observer-based Active Disturbance Rejection Controller for robust tracking performance is proposed for a separately excited DC motor in [10]. The introduced control scheme concentrates disturbance estimation to feed it back in the control loop, achieving a robust and stable performance. Despite several important contributions having been made, the DC motor control design is an open research area due to its complexity, interdisciplinary nature, diverse applications, adaptability to uncertainty, and the constant emergence of new technologies and ethical considerations. Researchers are continually pushing the boundaries of our understanding of motor control and its practical applications.

On the other hand, optimization is a keystone of engineering, playing a pivotal role in ensuring that the systems, processes, and designs we create are efficient, cost-effective, and sustainable. By fine-tuning and enhancing various parameters and variables, engineers strive to find the best possible solutions, maximizing performance, minimizing waste, and ultimately shaping a more innovative and resource-conscious daily world. The reader can find more detailed information about these kinds of optimization applications in [11,12,13,14] and references therein. Simultaneously, bio-inspired computation is a broader field that encompasses various computational techniques inspired by biology, while bio-inspired optimization algorithms are a specific subset of these techniques tailored for solving optimization problems. Both fields use biological metaphors and principles to develop algorithms that can solve complex real-world problems in diverse domains, such as engineering, finance, biology, among others. In this context, interesting approaches have been proposed in specialized literature for improving the performance of several types of electric machines by an optimal selection of control design parameters. The parameter tuning problem addresses the issue for finding an appropriate parameter settings of algorithms such a way performance can be optimized [15]. The authors of [16] use a metaheuristic technique based on genetic algorithms to adjust the parameters of a Proportional Integral Derivative (PID) controller to attain the optimal behavior of a DC motor. On the other hand, in [17] an optimal gain selection scheme is presented for the tuning of a PI speed controller of a Permanent Magnet Synchronous Motor. Here, the performances of the Bat Algorithm, Biogeography-Based Optimization, the Cuckoo Search Algorithm and the Flower Pollination Algorithm, which are nature inspired algorithms, are suitably evaluated during the optimization process.

At the same time, the authors of [18] proposed a PID controller for the speed control of a DC motor by using different metaheuristic techniques such as Genetic Algorithms, Particle Swarm Optimization (PSO), and a simulated annealing algorithm, which was inspired by the crystal formation process when solids are cooled down from a high temperature. Similarly, in [19], an Ant Colony Optimization (ACO) outline is employed for tuning the parameters of PID controllers. An interesting comparative of the yielded results in contrast with a classic approach based on the Ziegler–Nichols methodology and a metaheuristic approach based on the genetic algorithms is also presented. Moreover, from a different perspective, an adaptive control strategy is introduced in [20], which is rooted in a bioinspired optimization methodology for the speed regulation of a DC motor, wherein an online optimization problem is formulated and subsequently solved through the implementation of a modified differential evolution optimizer. All the previously research aims to include intelligent algorithms to further improve the system’s performance.

The present research explores the capabilities of backstepping and differential flatness theories to handle the tracking velocity and position control problem of electric direct current machines. Moreover, it discusses the implementation of ACO and PSO algorithms for the purposes of enhancing the system’s closed-loop performance by optimizing both the tracking error and the control signal effort. A demonstration of the effectiveness and efficiency of DC electric motors’ control for the tracking of reference trajectories is assessed by numerical simulation scenarios. Addtionally, smoothly transitioning from initial to desired operating points is properly addressed by introducing Bézier polynomial reference profiles. Finally, the performance of an adaptive integral backstepping control scheme based on the B-spline neural networks is evaluated where the gains tuning process is constantly performed online. It is worth mentioning that, to the best knowledge of the authors, there are no previous reports on this control design approach for electric DC motors in the specialized literature.

The organization of this paper is as follows: in the first section, a brief introduction to the state-of-the-art of the optimization of DC motor output tracking control is presented. Section 2 introduces the mathematical modelling of the DC machine, where position and velocity representations are derived. Subsequently, a control design scheme is formulated based on the backstepping as well as differential flatness theories. Moreover, the analysis of the system and subsystems stability is properly addressed by implementing Lyapunov functions. Finally, intelligent algorithms are utilized for enhancing the system output tracking performance based on the error and the control input effort. Several simulation scenarios are introduced to outcomes to delineate the operational efficiency of the control approach outlined in this paper, presenting it as a viable alternative for effectively managing the angular position and velocity control in direct current electric motors.

2. Direct Current Electric Machine Modelling

A frequently employed actuator, grounded in the principles of motors, generators, and control systems, is the direct current motor. A DC generator is a device that transforms mechanical energy into DC electrical power, while when it is employed to convert DC electrical power into mechanical energy, it is referred to as a DC motor [21]. Its purpose is to generate rotary motion. The magnets employed can either be electromagnets or, in the context of compact motors, permanent magnets [22]. Permanently-magnetic DC machines are commonly encountered in a diverse range of low-power applications, featuring a simplified construction where the field winding is substituted with a permanent magnet [23]. The following set of differential equations constitutes the elemental mathematical model of the system which has been vastly examined by the scientific and engineering community [22]:

$$\begin{array}{cc}\hfill J_m\ddot{\theta }& =-b_m\dot{\theta }+k_t{i}_a\hfill \\ \hfill L_a\frac{d{i}_a}{dt}& =-k_e\dot{\theta }-R_a{i}_a+u,\hfill \end{array}$$

(1)

where $\theta$ is the rotor angular position and $J_m$ stands for the rotor inertia moment. The electrical parameters of the armature circuit, $L_a$ and $R_a$, quantify the inductance and resistance. Additionally, we employ the symbol $k_t$ to represent the motor torque constant, while $k_e$ is utilized to denote the back electromotive force constant. The previous model can be further extended for a representation of the dynamic model of a DC motor actuator with a gearhead,

$$\begin{array}{cc}\hfill \left(J_g+n^2{J}_m\right)\ddot{\theta }& =-\left(b_g+n^2{b}_m\right)\dot{\theta }+n{k}_t{i}_a\hfill \\ \hfill L_a{\displaystyle \frac{d{i}_a}{dt}}& =-k_{e}n\dot{\theta }-R_a{i}_a+u,\hfill \end{array}$$

(2)

with the inertia moment of the gearhead denoted by $J_g$, the viscous damping as $b_g$, and n signifies the speed reduction ratio intrinsic to the gearhead. Common applications of electric motors with gearhead mechanisms include robotics, conveyor systems, automotive systems, industrial machinery, and household appliances. In these cases, the combination of an electric motor and a gearhead enhances the motor’s performance and makes it better suited to the specific demands of the application. Moreover, the load torque can be considered, for purposes of control design and system performance evaluation, in expression (2) as follows:

$$\begin{array}{cc}\hfill \left(J_g+n^2{J}_m\right)\ddot{\theta }& =-\left(b_g+n^2{b}_m\right)\dot{\theta }+n{k}_t{i}_a-{\tau }_L\hfill \\ \hfill L_a{\displaystyle \frac{d{i}_a}{dt}}& =-k_{e}n\dot{\theta }-R_a{i}_a+u.\hfill \end{array}$$

(3)

Additionally, the system mathematical model can be expressed by considering a generalized inertia moment $J_e$ and an equivalent viscous damping as $b_e$; therefore, the system (2) can be rewritten as follows:

$$\begin{array}{cc}\hfill J_e\ddot{\theta }& =-b_e\dot{\theta }+n{k}_t{i}_a\hfill \\ \hfill L_a{\displaystyle \frac{d{i}_a}{dt}}& =-k_{e}n\dot{\theta }-R_a{i}_a+u,\hfill \end{array}$$

(4)

with $J_e=\left(J_g+n^2{J}_m\right)$ and $b_e=\left(b_g+n^2{b}_m\right)$. Observe that, by performing a variable change as $\dot{\theta }=\omega$, the second order position model given by equations in (4) can be implemented for velocity control design,

$$\begin{array}{cc}\hfill J_e\dot{\omega }& =-b_e\omega +n{k}_t{i}_a\hfill \\ \hfill L_a{\displaystyle \frac{d{i}_a}{dt}}& =-n{k}_e\omega -R_a{i}_a+u.\hfill \end{array}$$

(5)

Notice that if the inductance value $L_a\approx 0$ is depreciated in Equation (4), the following simplified model can be used for purposes of position control design:

$$\begin{array}{cc}\hfill \ddot{\theta }& =-\left({\displaystyle \frac{b_e}{J_e}}+{\displaystyle \frac{n^2{k}_t{k}_e}{J_e{R}_a}}\right)\dot{\theta }+{\displaystyle \frac{n{k}_t}{J_e{R}_a}}u.\hfill \end{array}$$

(6)

Then, by using the state variables $z_1=\theta$ and $z_2=\dot{\theta }$, the mathematical model in Equation (6) can be expressed as follows:

$$\begin{array}{cc}\hfill {\dot{z}}_1& =z_2\hfill \\ \hfill {\dot{z}}_2& =-a{z}_2+bu,\hfill \end{array}$$

(7)

with

$$\begin{array}{cccccc}\hfill a& ={\displaystyle \frac{b_e}{J_e}}+{\displaystyle \frac{n{k}_t{k}_e}{J_e{R}_a}}\hfill & \hfill \mathrm{and}& \hfill & \hfill b& ={\displaystyle \frac{n{k}_t}{J_e{R}_a}}.\hfill \end{array}$$

Despite the exhibited dynamic behaviour being linear, mismatching load torque disturbance ${\tau }_L$ in (3) might affect the system’s performance. Dealing with rejecting mismatched disturbances poses a greater challenge. These discrepancies in disturbances are prevalent in various real-world systems, such as missile roll autopilots, electric motors, and flight control systems, which are susceptible to such disparities. Unlike matched disturbances, these mismatched disturbances cannot be effectively converted into input channels that act upon the system, making their direct elimination through inputs infeasible.

As a result, regardless of the chosen control strategy, completely eradicating the impact of mismatched disturbances on the system’s state remains unattainable. Consequently, a pragmatic approach involves mitigating the effects of mismatched disturbances in specific variables of interest that describe the controlled state. Backstepping control can efficiently deal with mismatching disturbances by using virtual controllers.

Differential Flatness Modelling

Differential flatness is a structural property of a wide class of dynamical systems in which the system variables can be expressed in terms of a set of flat outputs and a finite number of their time derivatives [24,25]. In a differentially flat system, the state variables and control input can be described as formulae in terms of flat or linearizing outputs and their time derivatives. The integration of any differential equation is not required in the differential parameterizations of the system variables. Thus, systems that exhibit this property have outputs that are easily controllable and allow for the generation of feasible control inputs. Essentially, it is a property of certain systems that can simplify control design by allow for the transformation of the system into a linear or simpler form. This property is particularly useful when dealing with complex systems because it easily enables the design of controllers for them. The structural property of differential flatness has been exploited to derive solutions to robust and efficient tracking control problems of desired motion profiles and simultaneous active vibration suppression in linear and nonlinear, disturbed vibrating systems [26]. A flat output of the system model (5) is given by the angular velocity $y_f=\omega$. Time derivatives of the flat output are functions of system variables as well. Then for purposes of control design the system in (5) can be expressed as

$$\begin{array}{cc}\hfill y_f& =\omega \hfill \\ \hfill {\dot{y}}_f& =-\frac{b_e}{J_e}\omega +\frac{n{k}_t}{J_e}{i}_a\hfill \\ \hfill {\ddot{y}}_f& =-\frac{b_e}{J_e}\dot{\omega }-\frac{n{k}_t{R}_a}{J_e{L}_a}{i}_a-\frac{n^2{k}_e{k}_t}{J_e{L}_a}\omega +\frac{n{k}_t}{J_e{L}_a}u.\hfill \end{array}$$

(8)

Therefore, from expression (8), the differential parametrization yields

$$\begin{array}{cc}\hfill \omega & =y_f\hfill \\ \hfill i_a& =\frac{J_e}{n{k}_t}{\dot{y}}_f+\frac{b_e}{n{k}_t}{y}_f\hfill \\ \hfill u& =\frac{L_a{J}_e}{n{k}_t}{\ddot{y}}_f+\left(\frac{b_e{L}_a+R_a{J}_e}{n{k}_t}\right){\dot{y}}_f+\left(\frac{n^2{k}_e{k}_t+R_a{b}_e}{n{k}_t}\right)y_f.\hfill \end{array}$$

(9)

The flat output $y_f$ then satisfies the following perturbed input–output differential equation:

$$\begin{array}{cc}\hfill {\ddot{y}}_f+a_1{\dot{y}}_f+a_0{y}_f& =b_{c}u,\hfill \end{array}$$

(10)

with

$$\begin{array}{cc}\hfill a_0& =\frac{n^2{k}_t{k}_e+R_a{b}_e}{J_e{L}_a}\hfill \\ \hfill a_1& =\frac{b_e}{J_e}+\frac{R_a}{L_a}\hfill \\ \hfill b_c& =\frac{n{k}_t}{J_e{L}_a}.\hfill \end{array}$$

It is worth noting that the previous model can be employed for control design purposes. In doing so, we can leverage its capabilities to achieve the control goals. In the next sections, the derived dynamic models will be used for both position and velocity in the output backstepping tracking control design.

Table 1. Parameters of the RE 40 150 watt DC motor and a planetary gearhead GP 52 from Maxon®.

Parameter	Units	Value
$R_a$	$\Omega$	2.5
$J_e$	kg m2	0.0774
$b_e$	Nm s/rad	6.562
$L_a$	mH	0.612
n	-	81
$k_t$	mNm/A	82.2
$k_e$	mV/rad/s	82.3215

summarizes the system parameters used for the numeric simulation experiments.

3. Direct Current Electric Machine Control

3.1. Bio-Inspired Algorithms for Control Design

Bio-inspired optimization, also known as nature-inspired optimization, draws its inspiration from the principles and behaviors observed in the natural world (please refer to [27,28,29,30] and references therein for further detailed information on optimization algorithms). This approach has gained significant attention and popularity in various fields, including automatic control design. Some of the more significant benefits of incorporating bio-inspired optimization techniques in the realm of automatic control are improved performance, adaptability, enhanced robustness, reduced development time, handling complex nonlinearities, and reduced dependency on prior knowledge and energy efficiency. Utilizing bio-inspired optimization in automatic control design offers a multitude of advantages.

These techniques enhance control system performance, adaptability, and robustness by mimicking nature’s processes, making them particularly effective in handling dynamic and uncertain environments. They succeed in multi-objective optimization scenarios, reducing development time and mitigating the need for extensive prior knowledge. Bio-inspired methods also excel in managing complex nonlinearities, and can be scaled for systems of various sizes, promote sustainability, and enhance energy efficiency. Overall, these nature-inspired approaches represent a versatile and promising avenue for creating more efficient, resilient, and adaptive control systems in diverse applications within automation and control engineering.

Particle Swarm Optimization [31] and Ant Colony Optimization [32] are valuable tools for control design due to their abilities to handle nonlinear and complex systems, design robust controllers, optimize for complex objectives, adapt to changing conditions, and explore diverse control strategies. They excel in multimodal optimization, online adaptation, and parameter tuning, making them suitable for a wide range of control challenges. Their versatility, speed of convergence, and nature-inspired foundations also offer advantages in control applications, particularly in cases where accurate system modelling is difficult or where innovative control strategies are required.

ACO is a probabilistic optimization method designed for scenarios where the objective is to find the optimal path within a graph. This technique draws its inspiration from the foraging behavior of ants as they search for the most efficient route between their colony and a food source. In contrast, PSO draws inspiration from the collective behavior seen in bird flocks and fish schools, emphasizing information sharing and social interactions. It serves as a global optimization algorithm, which is especially effective when tackling problems involving multidimensional parameter spaces (real-valued optimization). In the realm of computational science, PSO stands as a computational approach aimed at enhancing a candidate solution iteratively in pursuit of improving its quality as measured against a predefined criterion [33]. The inspiration from these natural analogies, such as schooling and flocking, is reflected in the fact that agents, represented as particles, possess not only a position but also a velocity, enabling them to explore and navigate within the search space [34,35].

The following steps summarize how ACO works:

• Initialization: a set of artificial ants is created, each representing a potential solution to the optimization problem. A pheromone matrix is initialized, typically with small values, to represent the desirability of different paths or solutions. Pheromones are used to communicate between ants.
• Ant movement: each ant explores the problem space by iteratively making decisions based on a combination of pheromone levels and a heuristic function. The heuristic function guides ants towards potentially better solutions. Ants construct solutions incrementally by selecting one element (e.g., a path or a solution component) at a time.
• Pheromone update: after all ants have constructed their solutions, the pheromone levels are updated. Ants deposit pheromone on the components of their solutions based on the quality of their solutions. Better solutions receive more pheromone. Pheromone levels also decay over time to simulate the natural evaporation of pheromones.
• Solution evaluation: the quality of the solutions constructed by the ants is evaluated based on the objective function of the optimization problem.
• Global information exchange: ants communicate indirectly through the pheromone matrix. Good solutions result in higher pheromone levels on the components used in those solutions. This indirect communication guides other ants towards exploring similar paths or solutions.
• Iteration: steps 2 to 5 are repeated for a certain number of iterations or until a termination condition is met.
• Termination: the algorithm finishes when a stopping criterion is reached, such as a maximum number of iterations or when no significant improvement is observed.
• Output: the best solution found by the algorithm is returned as the output.

ACO is particularly useful for solving optimization problems with a large search space and complex constraints, such as the travelling salesman problem (TSP) or the vehicle routing problem (VRP). By simulating the way ants explore and communicate in nature, ACO can efficiently find high-quality solutions to these problems. In this work, we propose a control design scheme where the ACO and PSO algorithms are used for tuning of the control parameters. Figure 1 portrays a graph representing how the ACO is implemented. Here, ${\lambda }_1$ to ${\lambda }_i$ stands for a finite number of control parameters to be selected.

The algorithm explores paths concerning the numbers of columns i, or control parameters, and rows, associated with the possibly j number of values than each parameter can adopt, and which is selected based on the experience of the designer (e.g., $\lambda \in \left(0,\infty \right)$ for ensuring stability). Line colours represents different combinations or paths chosen for each ant. Notice that for each iteration k a set of individuals or ants $\mathcal{H}$ are tasked to probe a specific and different path or combination that minimizes the objective function $f_o(e_y,u)$.

From Figure 2 a hypothetical case can be observed in which the l-th individual from the colony has selected the best path (highlighted in green) based on information from previous iterations. No other element repeats the same path which helps with the algorithm convergence and its objective function record is properly stored for future comparisons.

On the other hand, the step summary of how PSO works is presented below:

• Initialization: PSO starts by initializing a population of particles. Each particle represents a potential solution to the optimization problem. Each particle has a position and a velocity, which are randomly assigned at the beginning.
• Objective function evaluation: the objective function of the optimization problem is evaluated for each particle, and the fitness or quality of each particle’s solution is determined.
• Particle movement: each particle adjusts its velocity and position based on its own best-known solution (individual best or “pBest”) and the best-known solution of the entire population (global best or “gBest”). The velocity of each particle is updated by considering its current velocity, its distance from its pBest, and its distance from gBest. This update encourages particles to move towards better solutions. The particle’s position is updated based on its updated velocity.
• Updating pBest and gBest: after the position update, each particle compares its new solution to its pBest. If the new solution is better, it updates its pBest. The algorithm also updates the gBest by comparing the pBests of all particles in the population.
• Termination: the algorithm continues to iterate through steps 3 and 4 for a specified number of iterations or until a termination condition is met (e.g., a target fitness level is achieved).
• Output: the best solution found by any particle in the population, typically the gBest, is returned as the final output.

PSO is known for its ability to efficiently explore solution spaces and find optimal or near-optimal solutions in a wide range of optimization problems. It is particularly useful for continuous and high-dimensional search spaces. PSO’s strength lies in its ability to balance exploration (searching broadly) and exploitation (focusing on promising areas), making it a popular choice for optimization in various fields, including engineering, machine learning, and economics.

Figure 3 represents the main search space used in this work (as a result of a combination of parameter values $\lambda$) for the PSO. Here, each sphere represents a possible solution and could be evaluated by a particle in some iteration. As observed in Figure 4, each particle, represented by a dark sphere, will seek the better solution based on the previous iterations knowledge. Initially, they are located randomly and will move towards the best solution. The algorithm will carry the particles, after each iteration, towards the fitness solution, represented by the gold particle in the diagrams, see Figure 5.

In this research, the ACO is utilized for tuning of the position control parameters and the PSO for velocity control design. Simultaneously, in this study, data from the output tracking error and the control input magnitude are used as design parameters of the following objective function:

$$\begin{array}{c}\hfill f_o=\left[\left(1+e^{\alpha }\right)\mathrm{ITAE}+\kappa \left(\mathrm{ISCI}\right)\right],\end{array}$$

(11)

where the coefficients $\alpha$ and $\kappa$ are weighting factors that penalize the error and the magnitude of the voltage control input. Additionally, the ITAE index (Integral Time Absolute Error) is given by

$$\begin{array}{c}\hfill \mathrm{ITAE}={\int }_0^{t}t\left|e_y\left(t\right)\right|dt\end{array}$$

(12)

Here $e_y$ is the output tracking error and t is the time variable. Additionally, the ISCI term stands for the Integral Squared Control Input index introduced in Equation (13) and which is associated with the voltage control input.

$$\begin{array}{c}\hfill \mathrm{ISCI}={\int }_0^t{u}^{2}dt.\end{array}$$

(13)

3.2. Backstepping-Based Position Control

In the field of control theory, backstepping [36] is a control technique introduced around 1990 by Petar V. Kokotovic and his colleagues. It is employed to design stabilizing controllers for a particular class of dynamical systems. These systems consist of interconnected subsystems, with one core subsystem that can be stabilized using an alternative method. Due to this recursive arrangement, the controller designer initiates the design process from the already stable core subsystem and systematically goes backward to develop new controllers that progressively stabilize each successive outer subsystem. The procedure concludes when the ultimate external control layer is established. As a result, this method is properly named backstepping [37].

In this section, a backstepping based control is introduced for the DC electric motor machine with a planetary gearhead. The objective is to find a control law such that the system output tracks the reference trajectory. This method guarantees asymptotic stability for the system and offers the advantage of easy applicability to dynamic systems, providing flexibility in the design of feedback control laws [38].

Considering the system in (7), it is possible to design an output tracking controller by using backstepping control theory. The intention is to track a reference trajectory given with acceptable level of tracking error. Then, for purposes of control design, lets consider the following representation of the model as follows:

$$\begin{array}{cc}\hfill {\dot{x}}_0& =x_1\hfill \end{array}$$

(14a)

$$\begin{array}{cc}\hfill {\dot{x}}_1& =x_2\hfill \end{array}$$

(14b)

$$\begin{array}{cc}\hfill {\dot{x}}_2& =-{\ddot{y}}^{★}-a\left(x_2+{\dot{y}}^{★}\right)+bu,\hfill \end{array}$$

(14c)

with

$$\begin{array}{cc}\hfill x_0& ={\int }_0^t{e}_{\theta }dt\hfill \\ \hfill x_1& =e_{\theta }=z_1-z_1^{★}\hfill \\ \hfill x_2& =z_2-{\dot{z}}_1^{★},\hfill \end{array}$$

(15)

where $y=z_1=\theta$ is the system output variable to be controlled. $e_{\theta }$ represents the tracking error of the desired angular position reference trajectory $z_1^{★}=y^{★}={\theta }^{★}$. Then, for the previous extended 3D linear system, a backstepping control design is introduced as follows:

• System 1: from Equation (14a) $x_1$ can be seen as a virtual controller as follows:

$$\begin{array}{cc}\hfill {\dot{x}}_0& ={\gamma }_1.\hfill \end{array}$$

(16)

For the stability analysis of expression (16), let us consider the next candidate Lyapunov function,

$$\begin{array}{cc}\hfill V_1& =\frac{x_0^2}{2}.\hfill \end{array}$$

(17)

It is evident that $V_1$ is quadratic and positive definite and its derivative

$$\begin{array}{cc}\hfill {\dot{V}}_1& =x_0{\dot{x}}_0=x_0{\gamma }_1;\hfill \end{array}$$

(18)

therefore, the virtual control input ${\gamma }_1$ is proposed in such a manner to ensure the negative definitiveness of ${\dot{V}}_1$,

$$\begin{array}{c}\hfill {\gamma }_1=-{\beta }_0{x}_0,\end{array}$$

(19)

with ${\beta }_0>0$. Thus, by Lyapunov stability theory [39], the following closed-loop system matches the globally asymptotically stable condition

$$\begin{array}{c}\hfill {\dot{x}}_0=-{\beta }_0{x}_0.\end{array}$$

(20)

In order for the state $x_0$ to be globally asymptotically stable the state $x_1\to {\gamma }_1$. In this regard, let us introduce the following error variable,

$$\begin{array}{cc}\hfill e_1& =x_1-{\gamma }_1=x_1+{\beta }_0{x}_0,\hfill \end{array}$$

(21)

where

$$\begin{array}{cc}\hfill x_1& =-{\beta }_0{x}_0+e_1;\hfill \end{array}$$

(22)

then, Equation (14a) can be rewritten as

$$\begin{array}{cc}\hfill {\dot{x}}_0& =-{\beta }_0{x}_0+e_1,\hfill \end{array}$$

(23)

while

$$\begin{array}{cc}\hfill {\dot{e}}_1& ={\dot{x}}_1+{\beta }_0{\dot{x}}_0=x_2+{\beta }_0{\dot{x}}_0,\hfill \end{array}$$

(24)

by substituting expression in (23) it yields

$$\begin{array}{cc}\hfill {\dot{e}}_1& =x_2+{\beta }_0\left(-{\beta }_0{x}_0+e_1\right)=x_2-{\beta }_0^2{x}_0+{\beta }_0{e}_1.\hfill \end{array}$$

(25)

Then, the original system can be expressed as follows:

$$\begin{array}{cc}\hfill {\dot{x}}_0& =-{\beta }_0{x}_0+e_1\hfill \end{array}$$

(26a)

$$\begin{array}{cc}\hfill {\dot{e}}_1& =-{\beta }_0^2{x}_0+{\beta }_0{e}_1+x_2\hfill \end{array}$$

(26b)

$$\begin{array}{cc}\hfill {\dot{x}}_2& =-{\ddot{y}}^{★}-a\left(x_2+{\dot{y}}^{★}\right)+bu.\hfill \end{array}$$

(26c)

• System 2: Notice from expression (26b) that the state variable $x_2$ can be adopted as a virtual control input for the subsystem. Then, let us work with the following set of expressions:

$$\begin{array}{cc}\hfill {\dot{x}}_0& =-{\beta }_0{x}_0+e_1\hfill \end{array}$$

(27a)

$$\begin{array}{cc}\hfill {\dot{e}}_1& =-{\beta }_0^2{x}_0+{\beta }_0{e}_1+{\gamma }_2\hfill \end{array}$$

(27b)

and introduce the following Lyapunov function and its time derivative:

$$\begin{array}{cc}\hfill V_2& =V_1+\frac{e_1^2}{2}\hfill \\ \hfill {\dot{V}}_2& ={\dot{V}}_1+e_1{\dot{e}}_1,\hfill \end{array}$$

(28)

after elemental mathematical manipulation and by substituting Equation (35a,b) it yields

$$\begin{array}{cc}\hfill {\dot{V}}_2& =-{\beta }_0{x}_0^2+e_1\left(x_0+{\gamma }_2-{\beta }_0^2{x}_0+{\beta }_0{e}_1\right),\hfill \end{array}$$

(29)

where, in order to ${\dot{V}}_2$ be negative definite, the virtual controller ${\gamma }_2$ is proposed as follows:

$$\begin{array}{cc}\hfill {\gamma }_2& =-x_0+{\beta }_0^2{x}_0-{\beta }_0{e}_1-{\beta }_1{e}_1,\hfill \end{array}$$

(30)

with ${\beta }_1>0$, and utilizing (30) in (29) it results in

$$\begin{array}{cc}\hfill {\dot{V}}_2& =-{\beta }_0{x}_0^2-{\beta }_1{e}_1^2.\hfill \end{array}$$

(31)

Therefore, the linear system (35a,b) is globally asymptotically stable by the virtual controller effect. Similarly, as in the previous system, the state $x_2\to {\gamma }_2$. Let us introduce the following tracking error:

$$\begin{array}{cc}\hfill e_2& =x_2-{\gamma }_2,\hfill \end{array}$$

(32)

then

$$\begin{array}{cc}\hfill {\dot{e}}_2& ={\dot{x}}_2-{\dot{\gamma }}_2={\dot{x}}_2+{\dot{x}}_0-{\beta }_0^2{\dot{x}}_0+{\beta }_0{\dot{e}}_1+{\beta }_1{\dot{e}}_1,\hfill \end{array}$$

(33)

and

$$\begin{array}{cc}\hfill x_2& =e_2+{\gamma }_2=e_2-x_0+{\beta }_0^2{x}_0-{\beta }_0{e}_1-{\beta }_1{e}_1,\hfill \end{array}$$

(34)

and by substituting (33) in (34) in the (26a–c), the system can be expressed by the following set of first order differential equations:

$$\begin{array}{cc}\hfill {\dot{x}}_0& =-{\beta }_0{x}_0+e_1\hfill \end{array}$$

(35a)

$$\begin{array}{cc}\hfill {\dot{e}}_1& =-{\beta }_1{e}_1-x_0+e_2\hfill \end{array}$$

(35b)

$$\begin{array}{cc}\hfill {\dot{e}}_2& =-{\ddot{y}}^{★}-a\left(x_2+{\dot{y}}^{★}\right)+bu+(1-{\beta }_0^2){\dot{x}}_0+\left({\beta }_0+{\beta }_1\right){\dot{e}}_1.\hfill \end{array}$$

(35c)

• System 3: Finally, let us introduce the total Lyapunov function given by

$$\begin{array}{cc}\hfill V& =\frac{x_0^2}{2}+\frac{e_1^2}{2}+\frac{e_2^2}{2}\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill \dot{V}& =x_0{\dot{x}}_0+e_1{\dot{e}}_1+e_2{\dot{e}}_2\hfill \end{array}$$

(37)

and after mathematical manipulations

$$\begin{array}{cc}\hfill \dot{V}& =x_0\left(-{\beta }_0{x}_0+e_1\right)+e_1\left(-x_0-{\beta }_1{e}_1+e_2\right)+e_2{\dot{e}}_2.\hfill \end{array}$$

(38)

Therefore, by considering the above expression, it is possible to propose an integrator backstepping control law u given by

$$\begin{array}{cc}\hfill u& =\frac{1}{b}\left[-e_1+{\ddot{y}}^{★}-a\left(x_2+{\dot{y}}^{★}\right)-\left({\beta }_0^3-2{\beta }_0-{\beta }_1\right)x_0-\left(1-{\beta }_0^2-{\beta }_0{\beta }_1\right)e_1\right.\hfill \\ \hfill & \left.-\left({\beta }_0+{\beta }_1+{\beta }_2\right)e_2\right]\hfill \end{array}$$

(39)

and after substituting all the design variables, the real control input voltage is as follows:

$$\begin{array}{cc}\hfill u& =\frac{1}{b}\left[{\ddot{y}}^{★}+a{z}_2-\left({\beta }_2+{\beta }_0{\beta }_1{\beta }_2+2{\beta }_0{\beta }_1^2-2{\beta }_0\right)x_0\right.\hfill \\ \hfill & \left.-\left(1+{\beta }_0{\beta }_1+2{\beta }_1^2+{\beta }_0{\beta }_2+{\beta }_1{\beta }_2\right)z_1-{\beta }_2{z}_2\right].\hfill \end{array}$$

(40)

Hence, by Lyapunov stability theory, it is corroborated that the linear system in (31) is globally asymptotically stable.

Notice the control design intention for including an integral action in the system closed-loop error dynamics as an extended system. Integral action helps the controller to deal with the disturbances existing in the motion control and to improve the system’s transient and steady state performance [40,41].

3.3. Backstepping-Based Velocity Control

The differentially flat electric motor system model (10) can be used for velocity reference trajectory tracking control design. A state space representation of the flat output dynamics is then given by

$$\begin{array}{cc}\hfill {\dot{z}}_{1_f}& =z_{2_f}\hfill \\ \hfill {\dot{z}}_{2_f}& =-a_1{z}_{2_f}-a_0{z}_{1_f}+b_{c}u.\hfill \end{array}$$

(41)

The state variables are denoted as $z_{1_f}=\omega =y_f$ and $z_{2_f}={\dot{y}}_f$. For derivation of some backstepping control law based on differential flatness, the following state space description for the extended velocity trajectory tracking error dynamics, $e_{\omega }=y_f-y_f^{★}$, is considered:

$$\begin{array}{cc}\hfill {\dot{x}}_{0_f}& =x_{1_f}\hfill \end{array}$$

(42a)

$$\begin{array}{cc}\hfill {\dot{x}}_{1_f}& =x_{2_f}\hfill \end{array}$$

(42b)

$$\begin{array}{cc}\hfill {\dot{x}}_{2_f}& =-{\ddot{y}}_f^{★}-a_1\left(x_{2_f}+{\dot{y}}^{★}\right)-a_0\left(x_{1_f}+y_f^{★}\right)+b_{c}u.\hfill \end{array}$$

(42c)

The velocity reference trajectory is here represented by $y_f^{★}$. The error state variables are thus defined as follows:

$$\begin{array}{cc}\hfill x_{0_f}& ={\int }_0^t{e}_{\omega }dt\hfill \\ \hfill x_{1_f}& =e_{\omega }\hfill \\ \hfill x_{2_f}& ={\dot{e}}_{\omega }.\hfill \end{array}$$

(43)

Similarly to the above described procedure, an angular velocity trajectory tracking controller from a backstepping approach results in

$$\begin{array}{cc}u\hfill & =\frac{1}{b}\left[-e_{1_f}+{\ddot{y}}_f^{★}+a_1\left(x_{2_f}+{\dot{y}}_f^{★}\right)+a_0\left(x_{1_f}+y_f^{★}\right)-\left({\beta }_{0_f}^3-2{\beta }_{0_f}-{\beta }_{1_f}\right)x_{0_f}\right.\hfill \\ \hfill & -\left.\left(1-{\beta }_{0_f}^2-{\beta }_{0_f}{\beta }_{1_f}\right)e_{1_f}-\left({\beta }_{0_f}+{\beta }_{1_f}+{\beta }_{2_f}\right)e_{2_f}\right]\hfill \end{array}$$

(44)

with

$$\begin{array}{cc}\hfill e_{1_f}=& x_{1_f}+{\beta }_{0_f}{x}_{0_f}\hfill \\ \hfill e_{2_f}=& x_{2_f}+x_{0_f}-{\beta }_{0_f}^2{x}_{0_f}+({\beta }_{0_f}+{\beta }_{1_f})e_{1_f}.\hfill \end{array}$$

(45)

Tuning a backstepping controller efficiently can be a challenging task, especially for complex systems with nonlinear dynamics and uncertainties. There are various methods and algorithms that can be used to tune the controller parameters effectively. Two popular optimization algorithms for controller tuning are the PSO and ACO optimization algorithms.

On the other hand, differential flatness and backstepping are two distinct theory concepts, but they can be related in some control system scenarios and utilized satisfactory in control design, especially when dealing with complex systems. Recently, the authors of [42] introduce an interesting approach for controlling a mechanical system where trajectory planning and backstepping is proposed for the antisway problem of an underactuated overhead cranes. Here, the planned trajectory is obtained by solving for the optimal parameters of the selected flat output. Different from that proposal, in this paper, the DC electric motor differential flatness property is properly used as a starting point for the design of a backstepping controller. The flat output of the system serves as intermediate virtual control inputs in the backstepping design process. This simplifies the design of control laws for several complex systems. The flatness property helps for selecting suitable virtual control inputs that, combined with the backstepping approach, ease the closed-loop control design.

4. Numeric Simulation Results

During the experiments, the system is assigned to perform velocity and position output tracking tasks. Here, in order to obtain smooth transitions between initial and final operation states, the following motion scheme is adopted [7]:

$${\Gamma }^{★}=\left\{\begin{array}{cc}{\Gamma }_0& 0\le t<T_1\\ {\Gamma }_0+\left({\Gamma }_f-{\Gamma }_0\right){\mathcal{B}}_z(t,T_1,T_2)& T_1\le t\le T_2\\ {\Gamma }_f& t>T_2,\end{array}\right.$$

(46)

where ${\Gamma }_0$ and ${\Gamma }_f$ stand for the initial and final desired values, respectively. On the other hand, $T_1$ is the time when the transition begins and $T_2$ when it finishes. Therefore,

$$\begin{array}{c}\hfill {\mathcal{B}}_z(t,T_1,T_2)=\sum _{k=0}^n{b}_k{\left(\frac{t-T_1}{T_2-T_1}\right)}^k;\end{array}$$

(47)

here, $n=6$, y $b_1=252,b_2=1050,b_3=1800,b_4=1575,b_5=700,b_6=126$.

4.1. Case: 1 Backstepping Position Control

In the first case study, the controller proposed in (39) is implemented, where it is intentionally provided a vibrating load torque given by

$$\begin{array}{c}\hfill {\tau }_L=\mathcal{D}cos\left(2t\right)\mathcal{F}sin\left(0.5t\right),\end{array}$$

(48)

which is smoothly injected (for $15s<t\le 35s$) to the system with amplitude parameter values $\mathcal{D}=3.5$ and $\mathcal{F}=0.8$. Notice that, despite the load torque not being explicitly considered in the control design, the controller is capable of attenuating a bounded load torque due to the integral term.

From Figure 6 an acceptable output position tracking performance can be corroborated. The position set-point was established at 45 degrees ($\frac{\pi }{4}$ rad), and by using expression (47), it is possible to achieve soft transitions between operational points, as can be seen in the figure. Moreover, the control input voltage is portrayed as well as the tracking error. In the presented figures, dashed lines denote the trajectory references.

Table 2. Optimization parameters and fitness value.

Algorithm	ISCI	ITAE	${\mathit{f}}_0$	Size	Iterations
ACO	35.72	0.14	11.42	50	100

summarizes the optimization data utilized in this case study when the ACO is suitably featured for control design.

4.1.1. Noisy Measurements in the Backstepping Position Control

Additionally, for the purposes of closed-loop performance assessment, the system is tested when noisy position measurements $y_n={\eta }_w\times \left|y\right|$ are utilized in the control input computation, with

$$\begin{array}{cc}\hfill {\eta }_w& =\mathcal{A}\left[\mathcal{U}\left(0,1\right)+\mathrm{sgn}(\mathcal{G}\left(\mu ,\sigma \right)-0.25)\right],\hfill \end{array}$$

(49)

where $\mathcal{U}(0,1)$ stands for noisy components with uniform distributions in the interval $[0,1]$. Moreover, white Gaussian noise $\mathcal{G}\left(\mu ,\sigma \right)$ with mean value $\mu =0$ and standard deviation $\sigma =1$ are considered. Lastly, $\mathcal{A}$ represents the noise amplitude in the output measurement and sgn stands for the signum function. Figure 7 presents two cases when the system output measurements are corrupted with noise. In both of them, it is clear that the system performance is deteriorated. Notwithstanding, the controller allows the system to perform the tracking in an acceptable fashion.

4.1.2. Convergence of Virtual Errors

Now let us consider the initial conditions of the system that are different from zero, in order to visualize how, as expected, the virtual errors $e_1$ and $e_2$ converge quickly to zero. Consider the initial conditions $y_0={\theta }_0=0.3$ and ${\dot{y}}_0={\dot{\theta }}_0=0.01$. Here, a small time simulation was selected for purposes of presentation as can be seen in Figure 8. Notice, the system is capable of recovering efficiently from an initial condition that is different from the desired position through the designed virtual controller for each subsystem.

4.2. Case 2: Backstepping Velocity Control

In this simulation case, the DC machine is tasked to track a desired velocity profile. The motor stars from the rest and has to reach a velocity of 2.1 rad/s (approximately 20 rpm) in a short time. Observe from Figure 9 how the control input and the electric current try to compensate the load torque undesired effects. Additionally, notice an acceptable level of velocity tracking error. From deriving that the designed controller takes care of the asymptotic stability of the system, an acceptable performance is achieved despite the presence of completely unknown load torques.

Table 3. Optimization parameters and fitness value.

Algorithm	ISCI	ITAE	${\mathit{f}}_0$	Size	Iterations
PSO	1.61 × ${10}^3$	0.79	4.84 × ${10}^3$	45	100

summarizes the optimization data utilized in this case study when the PSO is suitably featured for velocity output tracking control design.

4.3. Case 3: Performance Comparative Regarding a LQR-like Controller

The LQR (Linear Quadratic Regulator) control is of paramount significance in the field of automatic control, as it enables the design of optimal control systems that efficiently minimizes a quadratic cost function. Its ability to consider both system dynamics and control constraints makes it an essential choice in applications ranging from aerospace engineering to industrial automation [43,44]. For the purposes of system responses comparison, an integral LQR controller is implemented based on model (41) for minimizing the following quadratic cost function for the system [45,46,47]

$$\begin{array}{c}\hfill J={\int }_0^{\infty }\left({\tilde{\mathbf{x}}}^{T}Q\tilde{\mathbf{x}}+{\tilde{u}}^{T}R\tilde{u}\right)dt,\end{array}$$

(50)

where Q is an output error weighting matrix and R is the matrix that penalizes the rate of change of control input. Additionally,

$$\begin{array}{c}\hfill \tilde{\mathbf{x}}=\left[\begin{array}{c}{e}_{y_f}\\ {\dot{e}}_{y_f}\\ {\eta }_{e_{y_f}}\end{array}\right]\end{array}$$

(51)

and

$$\begin{array}{c}\hfill \tilde{u}=u_c-u^{★}.\end{array}$$

(52)

Thus, the feedback control law that minimizes the value of J is given as follows:

$$\begin{array}{c}\hfill u_c=u^{★}-k_1{e}_{y_f}-k_2{\dot{e}}_{y_f}-k_3{\eta }_{e_{y_f}},\end{array}$$

(53)

with the control parameters $k_1,s k_2$ and $k_3>0$, $e_{y_f}=z_{1_f}^{★}-z_{1_f}^{★}$ and ${\eta }_{e_{y_f}}$ as the tracking error integral used as an extended state. Finally,

$$\begin{array}{c}\hfill u^{★}=\frac{{\dot{z}}_{2_f}^{★}+a_1{\dot{z}}_{1_f}^{★}+a_0{z}_{1_f}^{★}}{b_c}.\end{array}$$

(54)

It is worth mentioning that the PSO algorithm has been utilized for tuning the controller parameters in both cases with the same optimization criteria and features, which are summarized in

Table 4. Optimization parameters and fitness value for case study 3 using PSO, $\alpha =0.95$, and $\kappa =0.05$.

Controller	ISCI	ITAE	${\mathit{f}}_0$	Size	Iterations
LQR	1.860 × ${10}^3$	21.62	987.34	35	80
Backstepping	1.863 × ${10}^3$	2.52	939.12	35	80

. In order to further assess the system’s closed-loop performance, a completely unknown vibrating load torque is intentionally injected in both scenarios as portrayed in Figure 10:

In Figure 11 it can be seen the system output tracking responses using the LQR and backstepping controllers. Notice it is used the subscript $bks$ to differentiate the backstepping response. Additionally, it is observed that despite the load torque information not being included, nor the output tracking controller’s designs, the system achieves acceptable levels of load torque attenuation but, remarkably, its best performance when using the introduced backstepping control approach. The proposed backstepping controller demonstrates a superior performance in comparison with the integral LQR control approach for the velocity control of a DC electric machine, where the backstepping controller exhibits a higher degree of adaptability and robustness. As a result, it can achieve enhanced velocity control, maintaining greater stability and accuracy even in the presence of varying external factors, thus making it a more promising choice for DC motor velocity control applications. Take into account that the concept of differential flatness can be effectively harnessed and suitably extended in the design of a variety of control schemes, as exemplified herein through the application of the flat integral LQR control approach.

4.4. Case 4: Adaptive Integral Backstepping Control

B-spline artificial neural networks (Bs-ANN) represent intelligent agents that exhibit the ability to adapt and optimize control system performance in response to significant variations in the plant dynamics and operational conditions, all while adhering to stringent controller specifications. These specifications typically encompass objectives such as minimizing system output errors and controlling input effort. As system complexity increases, accompanied by rising model uncertainties and the presence of unwanted external disturbances during system operation, BsNNs effectively facilitate continual learning through their adaptive learning rules and synaptic weight updates, thereby ensuring the reliable operation of the control system.

The existence of unpredictable perturbations, whether originating from internal dynamics or external factors, can present formidable obstacles to the attainment of satisfactory process or system control. Consequently, there is a burgeoning interest in the exploration and development of advanced and intricate control strategies. In this research, the authors propose the innovative integration of B-spline artificial neural networks for real-time computation of backstepping control parameters in the context of output tracking control for a DC electric machine. This forward-looking approach seeks to augment the control system’s capacity to adapt dynamically to evolving conditions and unforeseen disturbances, thereby making a substantial contribution to the enhancement of overall control system performance.

The selection of the inputs in the neuron architecture is based on a close relationship that exists between the controller output error and the control input signals [35]. A weighted linear combination of the transformed inputs constitutes the neuron output. Figure 12 introduces a generalized architecture for a Bs-ANN when a couple of basis functions are considered per input signal.

The proposed adaptive controller using B-spline artificial neural networks is designed to adapt to varying system dynamics and disturbances adjusting its control parameters based on the changing system operational conditions. This type of artificial neural network employs an instantaneous learning rule structure as a perspective of online and continuous learning. The constant updating depends on the presented changes of the selected input signals; in this paper, this input is defined by the error between the desired trajectory and the actual system output. In this fashion, the output is then given by:

$$\begin{array}{c}\hfill \beta \left(t\right)={\mathbf{a}}^T\mathbf{w},\end{array}$$

(55)

where the weight and transformed input (or basis function outputs) column vectors, defined by the number of synaptic weights n, are given as follows:

$$\begin{array}{c}\hfill {\mathbf{w}}^T=\left[w_1{w}_2\dots w_n\right],{\mathbf{a}}^T=\left[a_1{a}_2\dots a_n\right],\end{array}$$

(56)

with

$$\mathbf{w}\left(t\right)=\mathbf{w}\left(t-1\right)+\Delta \mathbf{w}\left(t-1\right).$$

(57)

In this study, we utilize a B-spline for each of the control parameter’s calculus, where the architecture observed in Figure 12 is adopted. Moreover, the output tracking errors are used as main inputs throughout the simulation experiments and, by using the following instantaneous learning rule, the neuron is continuously trained:

$$\Delta \mathbf{w}\left(t-1\right)=\frac{\lambda e\left(t\right)}{\Vert \mathbf{a}\left(t\right){\Vert }_2^2}\mathbf{a}\left(t\right).$$

(58)

In the context of our study, the notation $e\left(t\right)$ represents the instantaneous system output tracking error, while $\lambda$ denotes the learning rate index. The utilization of instantaneous error correction rules enables the iterative adjustment of the neural network’s weight vector, aimed at reducing output errors following the presentation of each training sample. This adaptation process is facilitated through the mechanism of backward output error propagation, a technique well-documented in the literature [48,49].

Furthermore, to comprehensively assess the functionality of the Bs-ANN, we employ a Functional Flow Block Diagram methodology [50]. This visualization technique is exemplified in Figure 13, illustrating a scenario where the output tracking error serves as the correction input signal for real-time computation of control parameters within the Bs-ANN.

In this process, the measured tracking error assumes the role of the central input to the neuron and undergoes transformation via the B-spline basis functions, yielding the transformed input vector $\mathbf{a}$ within function block 1.0. Notably, to enhance the convergence of the learning rule during the update procedure, a dead band may be incorporated. Consequently, weight factor values are preserved without modification if the error magnitude falls below a predetermined threshold, as described in functions 2.0 and 3.0. This continuous storage of current weight values ensures their availability for the subsequent update stage in the subsequent iteration, as outlined in function block 4.0. Lastly, the adaptive output is determined as the weighted sum, representing the scalar product of weights and weighted input vectors. The magnitudes of these weights dictate the strength of the connections between each input and the output. In this research, the artificial network is mainly used for the on-line computing of the control parameters related to the output tracking control based on the tracking error, adopting the structure presented in Figure 12.

4.5. B-Spline Off-Line Training by PSO

The utilization of intelligent agents, referred to as “particles”, enables this algorithm to iteratively discover the optimal solution within a defined search space. As a result, the capabilities of the PSO have been effectively harnessed in various engineering and research applications, including fine-tuning automatic controllers and training artificial neural networks. The training process is performed while the system is commanded to reach a desired velocity. This methodology has been adopted from [35] under the following outline sumarized in Algorithm 1, where the pseudocode for the training process is presented, and a simulation time of 10 s is adopted.

The adaptive scheme has been adopted for the designed backstepping flat controller. Firstly, the system close-loop response is presented when there is no load disturbing external torque, as shown in Figure 14. Notice from this figure a properly closed-loop system performance where soft behavior of the variables of interest is achieved. As can be observed, an adaptive evolution of the control parameters is exhibited as a result of the online learning process, which is a invaluable premise when a detailed system model is not available or when significant environmental changes deteriorate the system’s performance.

Table 5. Optimization parameters and fitness value.

Algorithm	ISCI	ITAE	${\mathit{f}}_0$	Size	Iterations
PSO	25.65 × ${10}^3$	1.84	7.95 × ${10}^3$	35	80

summarizes the optimization data utilized in this case study when the PSO is suitably featured for velocity output tracking control design.

Algorithm 1: Evaluation of the objective function $f_o$.

Simultaneously, in the same experiment, a slow time varying bounded load torque with high frequency components is intentionally included. The yielded response can be appreciated in Figure 15. By using the adaptive approach, the system can effectively perform trajectory tracking tasks in a satisfactory fashion even when unpredictable changes occur, providing a comprehensive understanding of its capabilities and advantages.

5. Conclusions

A new output feedback control approach for the efficient and robust tracking of planned reference trajectories for both position and velocity in DC motor systems has been introduced in this research. Differential flatness, backstepping, and bio-inspired optimization were integrated to derive a novel alternative solution to the tracking control problem. In the presented flat output feedback control scheme, intelligent bio-inspired algorithms were performed for enhancing the closed-loop electric motor system’s performance. Multiple case studies demonstrated the feasibility and efficiency of the new tracking control perspective. Numerical results proved the accurate and robust tracking of position and velocity reference profiles planned for a realistic DC electric machine with a coupled gearhead. Moreover, undesired and completely unknown disturbance load torques and noisy measurements were intentionally integrated for performance evaluation. Acceptable levels of vibrating load torque attenuation were achieved by using the introduced control technique despite the presence of high frequency components in the feedback output signal measurements. Furthermore, an extra case study was conducted, where an adaptive integral backstepping design was introduced for updating the control parameters by using B-spline artificial neural networks that are trained off-line using the PSO algorithm. The future direction of this research is to further extend the integral backstepping control to improve the system’s robustness using the central ideas of Generalized Proportional-Integral (GPI) control theory as well as experimental implementation. Integral reconstructors of unavailable state variables from the GPI control perspective will be used to eliminate dependence on asymptotic or numerical differentiation with respect to the time of flat output signals. In future research, the adaptive robust tracking control problem will also be considered based only on the output feedback of highly uncertain vibrating mechatronic systems using electric motors as motion actuators, in which the structural differential flatness property is exhibited. In this sense, the backstepping control approach, bio-inspired optimization algorithms, and artificial neural networks will be integrated for the construction of innovating strategies to efficiently regulate differentially flat systems toward the desired reference trajectories in the presence of significant dynamic disturbances.