Application of Particle Swarm Optimization to a Hybrid H_∞/Sliding Mode Controller Design for the Triple Inverted Pendulum System

Abstract

The robotics field of engineering has been witnessing rapid advancements and becoming widely engaged in our lives recently. Its application has pervaded various areas that range from household services to agriculture, industry, military, and health care. The humanoid robots are electro–mechanical devices that are constructed in the semblance of humans and have the ability to sense their environment and take actions accordingly. The control of humanoids is broken down to the following: sensing and perception, path planning, decision making, joint driving, stability and balance. In order to establish and develop control strategies for joint driving, stability and balance, the triple inverted pendulum is used as a benchmark. As the presence of uncertainty is inevitable in this system, the need to develop a robust controller arises. The robustness is often achieved at the expense of performance. Hence, the controller design has to be optimized based on the resultant control system’s performance and the required torque. Particle Swarm Optimization (PSO) is an excellent algorithm in finding global optima, and it can be of great help in automatic tuning of the controller design. This paper presents a hybrid H_∞/sliding mode controller optimized by the PSO algorithm to control the triple inverted pendulum system. The developed control system is tested by applying it to the nominal, perturbed by parameter variation, perturbed by external disturbance, and perturbed by measurement noise system. The average error in all cases is 0.053 deg and the steady controller effort range is from 0.13 to 0.621 N.m with respect to amplitude. The system’s robustness is provided by the hybrid H_∞/sliding mode controller and the system’s performance and efficiency enhancement are provided by optimization.

1. Introduction

Humanoids are robots designed in the shape of humans to carry out a variety of tasks. They are built to receive data from their environment through suitable sensors and to take actions by moving their joints in a certain way to accomplish the required task (Figure 1a). Their sophisticated control system is partitioned into smaller dedicated-purpose controllers. The most crucial part of the overall control system is the one that is responsible for generating the motors’ commands to maintain balance and stability. For this control goal, a triple inverted pendulum system is used to develop, test, and enhance the control design. Mechanically, the torso, thighs, and shanks of the humanoid are represented by three links connected by joints that are driven by electric direct current (DC) motors, as shown in Figure 1b. Hence, the triple inverted pendulum system describes the basic dynamics of a humanoid. The angles that are made by the three links with the vertical axis are to be controlled by the torques produced by two DC motors. The torques are provided by the drivers to the upper two joints through two belt pulleys. The angles are measured by three potentiometers. The horizontal bars are added to ease the balance by increasing the moment of inertia [1].

The triple inverted pendulum is an unstable system which is difficult to control. Besides being an unstable system, it is subject to perturbations, due to uncertainties in the system’s friction parameters and moments of inertia. In addition, external disturbances like wind gusts affect the system response. Noise in measurement readings is another source of uncertainty that affects the system response. In such systems, where perturbations occur, the controller used must take the uncertainties into account during its design to provide robustness against them. The other challenge of controlling multiple degrees of freedom (DOFs) by fewer control signals is encountered practically. This may occur by the failure of one actuator during operation or may even be a goal in itself for the sake of minimizing the consumed energy, cost, or size of the system. The underactuation makes disturbance rejection more difficult, because the applied disturbance to the unactuated joint would not fulfill the matching condition.

In the last few years, both optimal and non-optimal control methods have been suggested and developed to control the inverted pendulums and robotic systems. Among the researchers who utilized non-optimal control methods are Sharma et al. [3]; they presented a decoupling sliding mode algorithm to control a single inverted pendulum on a cart. The poles of the reduced order system have been placed once near the imaginary axis and in another case away from the imaginary axis, and in the dominant region. It has been shown that locating the poles in the dominant region reduces steady state error and produces better disturbance rejection. Bonifacio et al. [4] used the Attractive Ellipsoid Method (AEM) to stabilize the triple inverted pendulum and compared it to the sliding mode control (SMC) system. It was found that AEM-based control rejects external perturbations more smoothly, while the SMC consumes less energy. Nguyen [5] developed an SMC system for the two-link fully actuated robot arm. In system response, the angles of the arms reach the desired value within 1.5 s. Kharabian et al. [6] proposed a hybrid sliding mode/H-infinity control approach using a fuzzy neural network weighting method to reduce nonlinearity, provide precise trajectory tracking, and to the enhance noise rejection capability of a single-link flexible manipulator system. The proposed hybrid controller reduces the total system nonlinearity, leading to higher performance with respect to noise cancellation, compared to sliding mode controller alone. Yet, the proposed control approach was not applied on manipulators with two or three links, whereas the system’s complexity rises and its control becomes harder.

Saif et al. [7] utilized synergetic control theory and fractional calculus to develop a fractional synergetic control (FSC) strategy for a four-DOF robot manipulator; the synergetic control approach was implemented to obtain fast convergence to the equilibrium point. The results of combining synergetic control and fractional calculus in control system design showcased good tracking performance in both joint space and workspace trajectories. However, the method’s robustness had not been investigated. Ahmed et al. [8] utilized Time Delay Estimation (TDE)-based model-free control to estimate the external disturbance and the unknown friction parameters of a Puma 560 rigid manipulator, and terminal SMC to obtain system robustness. The system responses show that the control method can suppress the effect of uncertainty and produce effective tracking. However, the effect of measurement noise that can affect the overall control performance has not been considered. Likewise, the effect of measurement noise was not determined by Anjum et al. [9] when they incorporated a fixed-time adaptive sliding mode observer into the fixed-time non-singular terminal SMC design for the Puma 560 manipulator.

For the single-link manipulator, Liu et al. [10] proposed an adaptive tracking controller based on the mismatched disturbance observer. The adaptive approach and disturbance-observer design contribute to effective disturbance rejection and smoother control of the robotic manipulator link. The closed-loop signals are found to be globally uniformly bounded with asymptotically stable tracking error, though the proposed control method is not generalized to more complex multi-link systems. In the same context, Qiu et al. [11] developed a disturbance observer-based adaptive fuzzy control method for the single-link manipulator; the proposed control system acquires finite-time prescribed performance, by which the tracking error enters a prescribed bounded set within finite time, even though the prescribed performance control strategy which is debated may not fully account for all practical prospects.

Flatness-based control has been employed by Rigatos et al. [12] in successive loops for both the three-DOF rigid-link robotic manipulator and three-DOF autonomous underwater vessel. The proposed method separates the controlled system into two differentially flat subsystems, connected in cascade. The response of the two case studies exhibits fast and precise tracking of the desired set points; still, the robustness to uncertainty has not been inspected.

Jabbar et al. [13] proposed a modified backstepping control method to stabilize the rotary double-inverted pendulum, where the control law is a combination of the backstepping control action and uncertainty compensation control action. The simulation results show the control system’s ability for exponential stabilization in the presence of uncertainties and disturbances. Yet, the proposed method involves complex design steps and lacks tunability to become consistent within practical applications. Siradjuddin et al. [14] used pole placement to obtain the feedback gain that stabilizes the single-link inverted pendulum at the desired position of the cart. The results show that placing the poles at more negative locations yields faster response at the expense of higher overshoot and greater control action, yet does not afford robustness. Pristovani et al. [15] employed a Multi Input/Multi Output (MIMO) decoupled control system method to implement the push-recovery strategy for the triple inverted pendulum by simplifying the control design into three serial Single Input Single Output (SISO) systems with known and uncertain disturbance models in each inverted pendulum, where the PID controller was used in each link to damp the external force applied on it. The proposed control system achieved 85.71% success in withstanding external forces, but did not consider its suitability for under-actuated systems. Although Masrom et al. [16] implemented Interval Type-2 Fuzzy Logic Control (IT2FLC) to a triple inverted pendulum on two wheels and the developed controller managed to withstand 16% greater disturbance than the type-1 fuzzy logic controller, the researchers considered disturbance applied on the third link only. The control system’s robustness against possible disturbances on other links is missing in the study.

Since optimization provides enhancements in control system design, it has been employed in different ways to provide optimal solutions for engineering problems. Soltanpour et al. [17], for instance, presented optimal fuzzy SMC for a two-DOF robotic manipulator. In order to compensate for the information shortages with respect to the system’s uncertainties, PSO was used to adjust the parameters of fuzzy membership functions. The proposed control method outperformed the classic SMC in terms of control input smoothness, though it had slightly higher tracking errors. The two-DOF robotic manipulator has also been controlled by an optimal integral SMC based on the pseudo spectral method. Liu et al. [18] applied integral SMC to restrain disturbance and adopted the pseudo spectral method to deal with the constraints. The proposed controller demonstrated the ability to track the desired reference signals accurately within 2 s. The results illustrated that the robotic manipulator system exhibited good robustness and anti-disturbance capabilities when subjected to external disturbances. However, the measurement noise for the system has not been considered.

Oliveira et al. [19] used Grey Wolf Optimization (GWO) with chaotic basis to tune the parameters of a higher-order SMC for the position control of a two-DOF rigid robot manipulator. Tent and Singer maps were applied to the optimization method to balance the exploration/exploitation phases of the algorithm, based on the algorithm itself and the chosen cost function. It was shown that the general repeatability of the algorithm was improved using chaotic maps in the higher-order SMC optimization. However, the suitability of the proposed method has not been tested for three-DOF manipulators. For the electro–hydraulic actuator system, Soon et al. [20] applied PSO to tune the Proportional–Integral–Derivative (PID) sliding surface of the SMC. The use of optimization in controller design has improved the system’s performance by 0.6407%. However, uncertainty was not taken into consideration in the study. On the other hand, Jibril et al. [21] used a Linear– Quadratic Regulator (LQR), where the cost is defined through a quadratic function, and pole placement, where the closed-loop poles of the plant are positioned in desired locations in the s-plane. Both control methods were applied for the stabilization of a triple inverted pendulum system and compared; the comparison shows that the pole-placement controller improves the stability of the system more, but with no indications of its robustness against perturbations. As for the triple-link rotary inverted pendulum, Hazem et al. [22] made a comparative study of the Neuro-Fuzzy Friction Estimation Model (NFFEM) and Adaptive Friction Estimation Model (AFEM) methods used to estimate the friction coefficients of the plant. The NFFEMs are trained by a radial-basis function artificial neural network. It has been deduced, based on the root mean square error of the joints’ position, that NFFEMs produce much better estimation results than AFEMs, although the researchers had not address the robustness issue.

Singh et al. [23] applied H_∞ and μ-synthesis control for the double inverted pendulum on a cart to achieve disturbance rejection and robust stability. The H_∞ controller seeks to minimize the mixed-sensitivity cost function, while the μ-synthesis controller implements the D-K iteration method to find the stabilizing gain that minimizes the upper structured singular value of the system. The simulation results show that the μ-synthesis control system has a more robust performance. In [24,25], Shafeek et al. proposed a method for enhancing the H_∞ and μ-synthesis control systems’ robustness and performance for the triple inverted pendulum, by incorporating PSO and the Gazelle Optimization Algorithm (GOA), respectively, into the controllers’ design. It has been shown that utilizing the optimization in control system design allows the possibility of balancing robustness and performance aspects within the system.

Meta-heuristic optimization algorithms are known to be well-suited for a wide range of complex optimization problems. Their key power lies in their ability to handle large search spaces, constraints, and multi-objective cost functions. The following studies are recommended to researchers interested in the field of meta-heuristic optimization applications. In fact, various meta-heuristic optimization algorithms (such as PSO [26,27,28], Ant Colony Optimization [29], the Bees Algorithm [30], Grey Wolf Optimization [31], the Whale Optimization Algorithm [32], Sunflower Optimization [33], the Gorilla Troops Algorithm [34], and the Chimp Optimization algorithm [35]) have been applied successfully in many engineering problems. In addition, it is also noteworthy that Rubio et al. [36] tuned the high-gain observer and controller gains to enhance the position and velocity perturbation attenuation for inverted pendulums using a genetic optimizer. The suggested method resulted in better perturbation attenuation compared to simplex and Bat optimizers. As for robots, Rubio [37] applied the Bat Algorithm and the Modified Bat Algorithm to minimize both tracking error and control energy consumption by optimizing the control gain. Sorcia-Vázquez et al. [38] also managed the minimization of the tracking error and the control effort for the experimental two-tank system through applying the genetic algorithm for the tuning of the PID and Fractional Order PID (FOPID) controller gains. The tuned FOPID control system performed better than the tuned PID control system in terms of overshoot, settling time, and control signal smoothness.

The literature review shows a gap in a fully robust analysis of the triple inverted pendulum which considers all the possible sources of uncertainties in the system. To fill the gap, this paper contributes from three important perspectives:

• Formulating a hybrid H_∞/SMC design for a triple inverted pendulum system that is robust to parameter variations, external disturbances, and measurement noise;
• Utilizing PSO to tune parameters in the control action based on the Integral Time Absolute Error (ITAE) performance index of the three controlled variable errors and the Integral Square Control Signal (ISCS) performance index of the two actuators’ torques;
• Evaluating the robustness, performance, and efficiency of the proposed optimized control system in different cases.

The rest of the paper introduces the mathematical model of the triple inverted pendulum system in Section 2, presents the PSO algorithm in Section 3, develops the hybrid H_∞/SMC design and optimization in Section 4, and tests the control system response to different cases in Section 5. Finally, it discusses the outcomes of the developed control system and the most noticeable findings, and suggests directions for future work in Section 6.

2. Mathematical Model of the Plant

The Mathematical model of the triple link inverted pendulum shown in Figure 1b is derived based on Lagrangian mechanics established by the scientist Joseph-Louis Lagrange. The Lagrangian (L) of many mechanical systems represents the difference between their kinetic (T) and potential (V) energies

$$L=T-V,$$

(1)

For the triple inverted pendulum model of Figure 1b, the kinetic and potential energies are defined as:

$$T={\displaystyle \frac{1}{2}} m_1\left[{\left({\displaystyle \frac{d }{dt}}\left(h_1 \mathit{sin}{\theta }_1\right)\right)}^2+{\left({\displaystyle \frac{d }{dt}}\left(h_1 \mathit{cos}{\theta }_1\right)\right)}^2\right]+{\displaystyle \frac{1}{2}} I_1{\dot{\theta }}_1^2+{\displaystyle \frac{1}{2}} m_2\left[{\left({\displaystyle \frac{d}{dt}}\left( l_1 \mathit{sin}{\theta }_1+h_2 \mathit{sin}{\theta }_2\right)\right)}^2+{\left({\displaystyle \frac{d }{dt}}\left(l_1 \mathit{cos}{\theta }_1+h_2 \mathit{cos}{\theta }_2\right)\right)}^2\right]+{\displaystyle \frac{1}{2}} I_2{\dot{\theta }}_2^2+{\displaystyle \frac{1}{2}} m_3\left[{\left({\displaystyle \frac{d}{dt}}\left( l_1 \mathit{sin}{\theta }_1+l_2 \mathit{sin}{\theta }_2+h_3 \mathit{sin}{\theta }_3\right)\right)}^2+{\left({\displaystyle \frac{d }{dt}}\left(l_1 \mathit{cos}{\theta }_1+l_2 \mathit{cos}{\theta }_2+h_3 \mathit{cos}{\theta }_3\right)\right)}^2\right]+{\displaystyle \frac{1}{2}} I_3{\dot{\theta }}_3^2,$$

(2)

$$V=M_3 g \mathit{cos}{\theta }_1+M_2 g \mathit{cos}{\theta }_2+M_3 g \mathit{cos}{\theta }_3,$$

(3)

respectively, where $g$ represents the acceleration of gravity, and

$$M_1=m_1 h_1+m_2 l_1+m_3 l_1,$$

(4)

$$M_2=m_2 h_2+m_3 l_2,$$

(5)

$$M_3=m_3 h_3,$$

(6)

Then, by applying the stationary action principle,

$${\displaystyle \frac{d}{dt}} \left({\displaystyle \frac{\partial L}{\partial {\dot{\rho }}_k}}\right)={\displaystyle \frac{\partial L}{\partial {\rho }_k}},$$

(7)

where ${\rho }_j$ represents the $k^{th}$ position vector in the generalized coordinates, the equations of motion of the system are obtained [1]:

$$\begin{array}{l}\left[\begin{array}{ccc}{J}_1+I_{p1} & l_1 M_2\mathit{cos}\left({\theta }_1-{\theta }_2\right)-I_{p1}& l_1 M_3\mathit{cos}\left({\theta }_1-{\theta }_3\right)\\ l_1 M_2\mathit{cos}\left({\theta }_1-{\theta }_2\right)-I_{p1}& J_2+I_{p1}+I_{p2}& l_2 M_3\mathit{cos}\left({\theta }_2-{\theta }_3\right)-I_{p2}\\ l_1 M_3\mathit{cos}\left({\theta }_1-{\theta }_3\right)& l_2 M_3\mathit{cos}\left({\theta }_2-{\theta }_3\right)-I_{p2}& J_3+I_{p2}\end{array}\right]\left[\begin{array}{c}{\ddot{\theta }}_1\\ {\ddot{\theta }}_2\\ {\ddot{\theta }}_3\end{array}\right]\\ +\left[\begin{array}{ccc}{C}_1+C_2{+C}_{p1} & -C_2-C_{p1}& 0\\ -C_2-C_{p1}& C_{p1}+C_{p2}+C_2+C_3& -C_3-C_{p2}\\ 0& -C_3-C_{p2}& C_3+C_{p2}\end{array}\right]\left[\begin{array}{c}{\dot{\theta }}_1\\ {\dot{\theta }}_2\\ {\dot{\theta }}_3\end{array}\right]\\ +\left[\begin{array}{c}{l}_1{M}_2\sin \left({\theta }_1-{\theta }_2\right){\dot{\theta }}_2^2+l_1{M}_3\sin \left({\theta }_1-{\theta }_3\right){\dot{\theta }}_3^2-M_{1}g \mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_1\right)\\ l_1{M}_2\sin \left({\theta }_1-{\theta }_2\right){\dot{\theta }}_1^2+l_2{M}_3\sin \left({\theta }_2-{\theta }_3\right){\dot{\theta }}_3^2-M_{2}g \mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_2\right)\\ l_1{M}_3\sin \left({\theta }_1-{\theta }_3\right){(\dot{\theta }}_1^2-2 {\dot{\theta }}_1{\dot{\theta }}_3)+l_2{M}_3\sin \left({\theta }_2-{\theta }_3\right){(\dot{\theta }}_2^2-2 {\dot{\theta }}_2{\dot{\theta }}_3)-M_{3}g \mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_3\right)\end{array}\right]\\ +\left[\begin{array}{cc}{K}_1& 0\\ {-K}_1& K_2\\ 0& {-K}_2\end{array}\right]\left[\begin{array}{c}{t}_{m1}\\ t_{m2}\end{array}\right]=\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ 0& 0& 1\end{array}\right] \left[\begin{array}{c}{d}_1\\ d_2\\ d_3\end{array}\right]\end{array}$$

(8)

where $\mathit{\theta }={\left[\begin{array}{ccc}{\theta }_1& {\theta }_2& {\theta }_3\end{array}\right]}^T$ represents the vector of angles of each link from the vertical line, as depicted in Figure 1b.

$$J_1=I_1+m_1 h_1^2+m_2 l_1^2+m_3 l_1^2,$$

(9)

$$J_2=I_2+m_2 h_2^2+m_3 l_2^2,$$

(10)

$$J_3=I_3+m_3 h_3^2,$$

(11)

$I_i, m_i, h_i, \mathrm{and} l_i$ represent the $i^{th}$ link’s moment of inertia around its center of gravity, mass, distance from its bottom to the its center of gravity, and length, respectively.

$$I_{pi}=I_{p_i^{\prime }}+K_i^2 I_{mi},$$

(12)

$I_{p_i^{\prime }}, \mathrm{and} K_i$ represent the $i^{th}$ hinge’s belt–pulley system’s moment of inertia, and ratio of teeth, respectively, $I_{mi}$ represents the $i^{th}$ motor’s moment of inertia, and $C_i$ represents the viscous friction coefficients of the $i^{th}$ hinge,

$$C_{pi}=C_{p_i^{\prime }}+K_i^2 C_{mi},$$

(13)

$C_{mi}, C_{P_i^{\prime }}$ represent the viscous friction coefficients of the $i^{th}$ motor and the $i^{th}$ hinge’s belt–pulley system, respectively, $t_{mi}$ represents the control torque of the $i^{th}$ motor, and $d_i$ represents the disturbance torque to the $i^{th}$ link.

Linearizing the model described in Equation (3) [1] around the operating point $\begin{array}{ccc}{\theta }_1& ={\theta }_2& ={\theta }_3=0\end{array}$, results in the following:

$$\begin{array}{l}\left[\begin{array}{ccc}{J}_1+I_{p1} & l_1 M_2-I_{p1}& l_1 M_3\\ l_1 M_2-I_{p1}& J_2+I_{p1}+I_{p2}& l_2 M_3-I_{p2}\\ l_1 M_3& l_2 M_3-I_{p2}& J_3+I_{p2}\end{array}\right]\left[\begin{array}{c}{\ddot{\theta }}_1\\ {\ddot{\theta }}_2\\ {\ddot{\theta }}_3\end{array}\right]\\ +\left[\begin{array}{ccc}{C}_1+C_2{+C}_{p1} & -C_2-C_{p1}& 0\\ -C_2-C_{p1}& C_{p1}+C_{p2}+C_2+C_3& -C_3-C_{p2}\\ 0& -C_3-C_{p2}& C_3+C_{p2}\end{array}\right]\left[\begin{array}{c}{\dot{\theta }}_1\\ {\dot{\theta }}_2\\ {\dot{\theta }}_3\end{array}\right]\\ +\left[\begin{array}{ccc}-M_{1}g& 0& 0\\ 0& -M_{2}g& 0\\ 0& 0& -M_{3}g\end{array}\right]\left[\begin{array}{c}{\theta }_1\\ {\theta }_2\\ {\theta }_3\end{array}\right]+\left[\begin{array}{cc}{K}_1& 0\\ {-K}_1& K_2\\ 0& {-K}_2\end{array}\right]\left[\begin{array}{c}{t}_{m1}\\ t_{m2}\end{array}\right]\\ =\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ 0& 0& 1\end{array}\right] \left[\begin{array}{c}{d}_1\\ d_2\\ d_3\end{array}\right]\end{array}$$

(14)

Arranging the model in state-space representation, given that the system has two inputs ($u_1$, $u_2$) and three outputs (${\theta }_1$, ${\theta }_2$, ${\theta }_3$), and substituting the nominal values of the parameters given in

Table 1. Nominal values of model’s parameters [1].

Parameter	Value	Unit
$I_1$	0.654	kg.m2
$I_2$	0.117	kg.m2
$I_3$	0.535	kg.m2
$m_1$	3.25	kg
$m_2$	1.9	kg
$m_3$	2.23	kg
$h_1$	0.35	m
$h_2$	0.181	m
$h_3$	0.245	m
$l_1$	0.5	m
$l_2$	0.4	m
$I_{p{1}^{\prime }}$	7.95 × 10−3	kg.m2
$I_{p{2}^{\prime }}$	3.97 × 10−3	kg.m2
$K_1$	30.72	dimensionless
$K_2$	27	dimensionless
$I_{m1}$	2.4 × 10−5	kg.m2
$I_{m2}$	4.90 × 10−6	kg.m2
$C_1$	6.54 × 10−2	N.m.s
$C_2$	2.32 × 10−2	N.m.s
$C_3$	8.80 × 10−3	N.m.s
$C_{m1}$	2.19 × 10−3	N.m.s
$C_{m2}$	7.17 × 10−4	N.m.s
$C_{p{1}^{\prime }}$	0	N.m.s
$C_{p{2}^{\prime }}$	0	N.m.s
$g$	9.81	m.s−2

[1], yields

$$\begin{gathered} \dot{\mathbf{x}}\left(t\right)=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 12.54& -8.26& -0.39& -0.043& 2.75& -0.36\\ -4.38& 36.95& -3& 0.086& -9.57& 2.29\\ -6.82& -22.94& 11.93& -0.034& 6.82& -2.86\end{array}\right] \mathbf{x}\left(t\right)+\left[\begin{array}{cc}0& 0\\ 0& 0\\ -50& 6.12\\ 174.41& -38.93\\ -124.2& 48.62\end{array}\right] \mathbf{u}\left(t\right) \\ \mathbf{y}\left(t\right)=\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\end{array}\right]\mathbf{x}\left(t\right)+\left[\begin{array}{cc}\begin{array}{c}0\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{c}0\\ \begin{array}{c}0\\ 0\end{array}\end{array}\end{array}\right]\mathbf{u}\left(t\right) \end{gathered}$$

(15)

where the state vector x is ${\left[{\theta }_{1 }{\theta }_2 {\theta }_3 {\dot{\theta }}_1 {\dot{\theta }}_2 {\dot{\theta }}_3\right]}^{\mathrm{T}}$, the control input vector $u$ is ${\left[\begin{array}{cc}{u}_1& u_2\end{array}\right]}^{\mathrm{T}}$, and the output vector y is ${\left[{\theta }_{1 }{\theta }_2 {\theta }_3\right]}^{\mathrm{T}}$. Then, for this MIMO system, $n=6$ states and $m=2$ inputs.

3. PSO Algorithm

The PSO algorithm is a population-based metaheuristic search method that is valuable in engineering design optimization (and many other applications, as stated in [35]). The algorithm was developed by James Kennedy and Russell C. Eberhart, inspired by the organized trajectory adjustment used in bird swarms and fish schools, based on their own experiences and the experiences of others in the group (Figure 2).

The algorithm (Algorithm 1) starts by spreading its agents randomly in the predefined search space, to look for the best solution. Then, in each iteration, the agents update their velocities and positions based on mathematical formulas that incorporate their current positions, velocities, and historical information. These update formulas for the agents’ velocity ($V_{PSO}$) and position ($X_{PSO}$) are the following:

Algorithm 1: PSO algorithm
Step 1:	Initialize the algorithm’s parameters ($w_{PSO}, c_{PSO1}, r_{PSO1}, c_{PSO2}, r_{PSO2},$ dimension, bounds, number of search agents, iteration counter, and maximum number of iterations)
Step 2:	Set the cost function according to the required application of optimization
Step 3:	Initialize the population
Step 4:	Evaluate the cost function for all candidate solutions
Step 5:	Specify each candidate solution as its best personal solution primarily ($p_{best}$)
Step 6:	Specify the candidate solution that produces the minimum cost function among all agents as the best global solution ($g_{best}$)
Step 7-a:	While (iteration number < maximum number of iterations), do:
Step 7-b-1:	For (all agents of all dimensions), do the following:
Step 7-b-2:	Update agents’ velocity (Equation (16))
Step 7-b-3:	Update agents’ position (Equation (17))
Step 7-b-4:	Evaluate the cost function for all candidate solutions
Step 7-b-5:	Specify the candidate solution that produces the minimum cost function in its history as the best personal solution obtained so far ($p_{best}$)
Step 7-b-6:	End for
Step 7-c:	Specify the candidate solution that produces the minimum cost function in all agents’ history as the best global solution obtained so far ($g_{best}$)
Step 7-d:	Increase the iteration counter by one
Step 7-e:	End while
Step 8:	Return the best solution ($g_{best}$)

$$V_{PSOi}\left(t+1\right)=w_{PSO}\times V_{PSOi}\left(t\right)+c_{PSO1}\times r_{PSO1}\times \left(p_{besti} \left(t\right)-X_{PSOi}\left(t\right)\right)+c_{PSO2}\times r_{PSO2}\times \left(g_{best} \left(t\right)-X_{PSOi}\left(t\right)\right),$$

(16)

and

$$X_{PSOi} \left(t+1\right)=X_{PSOi} \left(t\right)+V_{PSOi} \left(t+1\right),$$

(17)

respectively, where $w_{PSO}$ represents the inertia weight, which is, instead of being a constant value as in [41], chosen to change its value through iterations from ${\overline{w}}_{PSO}$ to ${\underset{\_}{w}}_{PSO}$ according to the formula

$$w_{PSO}={\overline{w}}_{PSO}-\left({\overline{w}}_{PSO}-{\underset{\_}{w}}_{PSO}\right)\times {\displaystyle \frac{t}{maximum number of iterations}},$$

(18)

to provide a thorough exploration of the search space in the early stages of optimization. Then its value keeps decreasing, to reduce the particles’ tendency to move far away from their current positions, such that the optimization method converges to a solution more efficiently; $r_{PSO1}$ and $r_{PSO2}$ are random numbers between 0 and 1, $c_{PSO1}$ and $c_{PSO2}$ are cognitive and social coefficients, respectively, and $g_{best}$ and $p_{besti}$ represent the best global and personal solutions, respectively. Subsequently, the continued updating guides the agents towards better solutions till a specified stopping criterion is met (usually, the maximum number of iterations). The PSO agents’ updating is visualized in Figure 3.

4. Hybrid H_∞/SMC Controller Design

The control of uncertain systems is required to provide robustness against parameter variations, external disturbances, and measurement noise. Nevertheless, seeking extreme robustness is not advised, because it leads to unnecessary conservativeness that affects the system’s performance characteristics. Optimization can be a great aid in enhancing the controller design to calibrate the system’s robustness/performance outcome. H_∞ and SMC are well known robust controllers that have shown their effectiveness in many uncertain control systems, so the controller design process is composed of the steps shown in Figure 4 and detailed below:

First, the H_∞ control is used to design the sliding manifold of the SMC to provide a more powerful hybrid controller. To account for parameter variations, the system in Equation (15) is represented as

$$\dot{x}\left(t\right)=\left(A+∆A\right) x\left(t\right)+B u\left(\mathrm{t}\right)$$

(19)

where ΔA represents the uncertainty matrix bounded by the Q matrix:

$$\mathrm{Q}=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0.2& 0.2& 0.2& 0.2& 0.2& 0.2\\ 0.2& 0.2& 0.2& 0.2& 0.2& 0.2\\ 0.2& 0.2& 0.2& 0.2& 0.2& 0.2\end{array}\right]$$

(20)

In order to apply decoupling for MIMO systems, a similarity transformation is required to transform the state variable $x\left(t\right)$ to a new state variable $q\left(t\right)$, defined as:

$$q\left(t\right)=H x\left(t\right)$$

(21)

where $H$ is the $n\times n$ similarity transformation matrix, obtained as

$$H={\left[\begin{array}{cc}N& B\end{array}\right]}^T$$

(22)

where the $n\times \left(n-m\right)$ matrix $N$ is the null space of the transpose of $B$ matrix. Thus:

$$H=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{c}0\\ \begin{array}{c}0.227\\ \begin{array}{c}-0.793\\ \begin{array}{c}0.564\\ \begin{array}{c}0\\ 0\end{array}\end{array}\end{array}\end{array}\end{array}& \begin{array}{c}0\\ \begin{array}{c}0.399\\ \begin{array}{c}-0.452\\ \begin{array}{c}-0.796\\ \begin{array}{c}0\\ 0\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}1\\ \begin{array}{c}0\\ \begin{array}{c}0\\ \begin{array}{c}0\\ \begin{array}{c}0\\ 0\end{array}\end{array}\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}0\\ \begin{array}{c}0.788\\ \begin{array}{c}0.361\\ \begin{array}{c}0.19\\ \begin{array}{c}-50\\ 6.12\end{array}\end{array}\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}0\\ \begin{array}{c}0.361\\ \begin{array}{c}0.165\\ \begin{array}{c}0.087\\ \begin{array}{c}174.4\\ -38.93\end{array}\end{array}\end{array}\end{array}\end{array}& \begin{array}{c}0\\ \begin{array}{c}0.19\\ \begin{array}{c}0.087\\ \begin{array}{c}0.045\\ \begin{array}{c}-124.2\\ 48.62\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right]$$

(23)

The transformed state space representation is

$$\dot{q}\left(t\right)=\overline{A} q\left(t\right)+∆\overline{A} q\left(t\right)+\overline{B} u\left(t\right)$$

(24)

where

$$\overline{A}=H A H^{-1},$$

(25)

$$∆\overline{A}=H ∆A H^{-1},$$

(26)

$$\overline{B}=H B,$$

(27)

The new state variable representation becomes

$$\left[\begin{array}{c}{\dot{q}}_{\mathbf{1}}\left(\mathbf{t}\right)\\ {\dot{q}}_{\mathbf{2}}\left(\mathbf{t}\right)\end{array}\right]=\left[\begin{array}{cc}{\overline{A}}_{11}& {\overline{A}}_{12}\\ {\overline{A}}_{21}& {\overline{A}}_{22}\end{array}\right]\left[\begin{array}{c}{q}_{\mathbf{1}}\left(\mathbf{t}\right)\\ q_{\mathbf{2}}\left(\mathbf{t}\right)\end{array}\right]+\left[\begin{array}{cc}∆{\overline{A}}_{11}& ∆{\overline{A}}_{12}\\ ∆{\overline{A}}_{21}& ∆{\overline{A}}_{22}\end{array}\right]\left[\begin{array}{c}{q}_{\mathbf{1}}\left(\mathbf{t}\right)\\ q_{\mathbf{2}}\left(\mathbf{t}\right)\end{array}\right]+\left[\begin{array}{c}0\\ \overline{B}\end{array}\right]u\left(t\right)$$

(28)

In this representation, the $m$ control signals $u\left(t\right)$ act as an input to the lower subsystem only (controlling the $m$ new state variables $q_2\left(\mathit{t}\right)$ directly), then, during the sliding phase, $q_2\left(\mathit{t}\right)$ acts as an input to the upper subsystem (controlling the remaining $(n-m)$ new state variables $q_1\left(\mathit{t}\right)$ by feedback gain):

$$q_2\left(t\right)=-K_{\infty }{q}_1\left(t\right)$$

(29)

The $m\times \left(n-m\right)$ feedback gain matrix $K_{\infty }$, which is related to the sliding manifold, is to be designed using H_∞ control to provide robustness to parametric uncertainty. Equation (29) and the upper subsystem:

$${\dot{q}}_1\left(t\right)=\left({\overline{A}}_{11}+∆ {\overline{A}}_{11}\right)q_1\left(t\right)+\left({\overline{A}}_{12}+∆ {\overline{A}}_{12}\right)q_2\left(t\right)$$

(30)

have to be represented in the Linear Fractional Transformation (LFT) [42] shown in Figure 5, in which the uncertainty is pulled out in the Δ block, I denotes the identity matrix, $B_{\infty }, C_{\infty }, D_{\infty }$ are matrices obtained by singular value decomposition, and $z_{\infty }$ and $d_{\infty }$ represent the disturbed output and input, respectively.

Upper LFT is applied to the system, so that the H_∞ problem becomes one of finding the stabilizing feedback gain $K_{\infty }$ that minimizes $\gamma$ > 0 in the quadratic objective function:

$$J_{\infty }={\displaystyle \frac{1}{2}}{\int }_0^{\infty }\left[{\mathbf{z}}_{\mathbf{\infty }}^{\mathbf{T}}\left(t\right) {\mathbf{z}}_{\mathbf{\infty }}\left(t\right)-{\gamma }^2 {\mathbf{d}}_{\mathbf{\infty }}^{\mathbf{T}}\left(t\right) {\mathbf{d}}_{\mathbf{\infty }}\left(t\right)\right] dt.$$

(31)

From Figure 5,

$$J_{\infty }={\displaystyle \frac{1}{2}}{\int }_0^{\infty }[{\mathbf{q}}_{\mathbf{1}}^{\mathbf{T}}\left(t\right)C_{\infty }^T C_{\infty } {\mathbf{q}}_{\mathbf{1}}(t)+2{\mathbf{q}}_{\mathbf{1}}^{\mathbf{T}}(t)C_{\infty }^T D_{\infty } {\mathbf{q}}_{\mathbf{2}}(t)+{\mathbf{q}}_{\mathbf{2}}^{\mathbf{T}}(t)D_{\infty }^T D_{\infty } {\mathbf{q}}_{\mathbf{2}}(t)-{\gamma }^{\mathbf{2}}{\mathbf{d}}_{\mathbf{\infty }}^{\mathbf{T}}(t){\mathbf{d}}_{\mathbf{\infty }}(t)]dt.$$

(32)

The Ricatti-like matrix equation for this linear quadratic problem is

$$P_{\infty }\left({\overline{A}}_{11}-{\overline{A}}_{12}{O}_{22}^{-1}{O}_{12}^T\right)+\left({\overline{A}}_{11}^T-O_{12}{O}_{22}^{-1}{\overline{A}}_{12}^T\right)P_{\infty }-P_{\infty }\left({\overline{A}}_{12}{O}_{22}^{-1}{\overline{A}}_{12}^T-{\gamma }^{-2}{B}_{\infty } B_{\infty }^T\right)P_{\infty }+\left(O_{11}-O_{12}{O}_{22}^{-1}{O}_{12}^T\right)=0.$$

(33)

where $O_{11}$ = $C_{\mathrm{\infty }}^T{C}_{\mathrm{\infty }}$, $O_{12}$ = $C_{\mathrm{\infty }}^T{D}_{\mathrm{\infty }}$, $O_{22}$ = $D_{\mathrm{\infty }}^T{D}_{\mathrm{\infty }}$. The solution of the above equation ($P_{\mathrm{\infty }})$ is used to find the feedback gain $K_{\infty }$:

$$K_{\infty }=O_{22}^{-1}({\overline{A}}_{12}^T{P}_{\infty }+O_{12}^T),$$

(34)

Thus,

$$K_{\infty }=\left[\begin{array}{cccc}-2447& -12790& 14632& -6474\\ 392& 2015& -2301& 1024\end{array}\right],$$

(35)

which can be used to define the sliding manifold $\mathbf{s}\left(t\right)$:

$$\mathbf{s}\left(t\right)=K_{\infty }{\mathbf{q}}_{\mathbf{1}}\left(t\right)+{\mathbf{q}}_{\mathbf{2}}\left(t\right)$$

(36)

When the state variables reach and stay on the sliding manifold ($\mathbf{s}\left(t\right)=0$), then the system dynamics is governed by Equation (29). Equation (36) can be written as

$$\mathbf{s}\left(t\right)=\left[\begin{array}{cc}{K}_{\infty }& I\end{array}\right]\mathbf{q}\left(t\right)$$

(37)

From Equation (21), the sliding manifold is expressed in terms of the original states as

$$\mathbf{s}\left(t\right)=G \mathbf{x}\left(t\right)$$

(38)

and

$$G=\left[\begin{array}{cc}{K}_{\infty }& I\end{array}\right]H$$

(39)

Next, the control law of the SMC is suggested to comprise a nominal term (${\mathbf{u}}_{°}\left(t\right)$) and a discontinuous term (${\mathbf{u}}_{\mathbf{d}}\left(t\right)$):

$$\mathbf{u}\left(t\right)={\mathbf{u}}_{°}\left(t\right)+{\mathbf{u}}_{\mathbf{d}}\left(t\right)$$

(40)

The nominal term is responsible for driving the states towards the sliding manifold defined by Equation (38). It is obtained by equating the derivative of the sliding variable to zero, that is

$$\dot{\mathbf{s}}\left(t\right)=0=G \dot{\mathbf{x}}\left(t\right)$$

(41)

Substituting the nominal terms of Equation (19),

$$\dot{\mathbf{s}}\left(t\right)=0=GA \mathbf{x}\left(t\right)+GB {\mathbf{u}}_{°}\left(\mathrm{t}\right)$$

(42)

from which

$${\mathbf{u}}_{°}\left(\mathrm{t}\right)=-{\left(GB\right)}^{-1}GA \mathbf{x}\left(t\right).$$

(43)

provided that the $GB$ matrix is non-singular; thus, it is invertible. To obtain the reaching condition of the SMC, a candidate quadratic Lyapunov function is suggested to be

$$V_L\left(t\right)={\displaystyle \frac{1}{2}} {\mathbf{s}}^2\left(t\right)$$

(44)

To ensure the stability of the sliding manifold and the overall system rigorously, the time derivative of $V_L\left(t\right)$ has to be negative definite for all $\mathbf{s}$ ≠ 0, so that $V_L\left(t\right)$ keeps dec-reasing along any trajectory in the state space which, in turn, guarantees the stability of the sliding mode. It implies that

$$\mathbf{s}\left(t\right)\dot{\mathbf{s}}\left(t\right)<0$$

(45)

The discontinuous term of the control law (${\mathbf{u}}_{\mathbf{d}}\left(t\right)$), which ensures the systems converge to the desired equilibrium point by imposing the negative definiteness of ${\displaystyle \frac{d{V}_L\left(t\right) }{dt}}$ in the presence of parametric uncertainty, disturbance, and noise, can be obtained by substituting Equations (19), (40) and (43) in Equation (45) and replacing the uncertainty matrix by its upper bound $Q$:

$$\mathbf{s}\left(t\right)[G Q\mathbf{x}\left(t\right)+GB {\mathbf{u}}_{\mathbf{d}}]<0$$

(46)

$${\mathbf{u}}_{\mathbf{d}}\left(\mathbf{t}\right)=-\left({GB)}^{-1}\left[(\left|G\right| Q\left|\mathbf{x}\left(t\right)\right|+\mathsf{\eta }\right)sgn\right(\mathbf{s}\left(t\right)\left)\right]<0$$

(47)

where the sign function ($sgn$()) is used to bring the states back to the sliding manifold whenever they attempt to leave it due to uncertainty, and $\mathsf{\eta }$ ($2\times 2$ diagonal matrix in this system) is a design parameter added to provide protection against disturbance and noise:

$$\mathsf{\eta }=\left[\begin{array}{cc}{\eta }_{11}& 0\\ 0& {\eta }_{22}\end{array}\right]$$

(48)

Instead of using trial and error to find good values of the diagonal elements of $\eta$, optimization is suggested, to tune them by searching for their optimal values that lead to both good robustness and performance of the system.

Finally, PSO is applied to enhance the performance of the robust control system in terms of response errors and the overall energy of the needed torques. These characteristics are used to form the cost function

$$J_O={\sum }_{i=1}^3{\int }_0^{t_f}t \left|e_i\left(t\right)\right|dt+{\sum }_{i=0}^2{\int }_0^{t_f}{t}_{mi}^2\left(t\right)dt .$$

(49)

where $J_O$ represents the optimization cost function, $e_i\left(t\right)$ represents the $i^{th}$ link’s angle response error, and $t_f$ represents the final time used in simulations.

The ITAE performance index of error is used to penalize the steady-state errors more than transient errors. Since the lower the control energy required, the more efficient the control system, then the ISCS performance index of the control signal is used.

5. Results

This section presents the system and optimization settings and the results of simulating the proposed method for different cases to test its effectiveness. The developed control system is compared with the H₂ control system, whose 3 × 3 performance weighting matrix is also optimized by PSO. This weighting matrix consists of three second-order transfer functions (each with three coefficients in its numerator and the same in its denominator, and a gain) on its diagonal, and is used for weighting the system’s tracking accuracy and robustness to uncertainty. The optimization problem of the two control systems is set as given in

Table 2. Setting of the PSO algorithm.

Parameter	Values for Hybrid H∞/SMC Control System Optimization	Values for H2 Control System Optimization
dimension	2	21
lower bound	100	10
upper bound	2000	100
number of search agents	10	10
maximum number of iterations	100	100
${\overline{w}}_{PSO}$	0.7	0.7
${\underset{\_}{w}}_{PSO}$	0.2	0.2
$c_{PSO1}$	2	2
$c_{PSO2}$	2	2
$r_{PSO1}$	random number in the interval (0, 1)	random number in the interval (0, 1)
$r_{PSO2}$	random number in the interval (0, 1)	random number in the interval (0, 1)

, with the cost function given by Equation (49).

The hybrid H_∞/SMC system (built in MATLAB/Simulink R2023b) used in optimization is given in Figure 6, where, in each iteration, the candidate solutions (${\eta }_{11}$ and ${\eta }_{22}$) are passed from the optimization code (written in MATLAB script) to the controller block (written in MATLAB S-function); the control law is calculated and applied by the actuator to the system block (written in MATLAB S-function), and the states x₁(t), x₂(t), x₃(t) and the actuator signals $t_{m1}$(t) and $t_{m2}$(t) are passed back to the optimization code to calculate the cost function of that candidate solution and to update the solutions accordingly. The convergence progress of the optimization algorithm is depicted in Figure 7. The PSO algorithm converges in the 59th iteration to the cost value of 1488.76. The best solution returned by PSO is [${\mathsf{\eta }}_{11}$ = 168.4755, ${\mathsf{\eta }}_{22}$ = 1.0173 $\times {10}^3$].

The H₂ control system optimization results in the following performance weighting matrix $W\left(s\right)$:

$$W\left(s\right)=\left[\begin{array}{ccc}10{\displaystyle \frac{51.123s^2+94.763s+10}{10s^2+100s+10}}& 0& 0\\ 0& 39.14{\displaystyle \frac{17.412s^2+10s+10}{53.782s^2+55.429s+14.368}}& 0\\ 0& 0& 10{\displaystyle \frac{53.8s^2+100s+100}{100s^2+100s+73.998}}\end{array}\right]$$

The performance index of the controller is 6.102.

5.1. Nominal System Response

First, the developed control system is tested by applying it to the nonlinear model of the triple inverted pendulum given in Equation (3), with the nominal values of its parameters given in Table 1. The initial condition is 0.5 deg, 1 deg, and 2 deg for the first, second, and third angle, respectively. The three angles respond as shown in Figure 8, the phase-plane trajectories of the angles and their derivatives are shown in Figure 9, and the applied torques are shown in Figure 10. Please note that data tips in regular font correspond to hybrid H_∞/SMC system, while data tips in italic font correspond to the H₂ control system in all responses.

The ${\theta }_1$ and ${\theta }_2$ absolute overshoot difference between the two control systems is 0.1 deg and 1.59 deg, respectively, while ${\theta }_3$ overshoots 17.323 deg higher in the H₂ control system. The difference between the errors of the two systems is small: 0.062, 0.151, and 0.025 deg for ${\theta }_1$, ${\theta }_2$, and ${\theta }_3$, respectively.

The above phase-plane trajectories show the direct sliding nature of the states towards their desired equilibrium point, driven by the hybrid H_∞/SMC control law. The H₂ control system lacks this preference.

The start-up torques t_m1 and t_m2 in the H₂ control system are 9.633 and 11.872 N.m times higher, respectively, than in the hybrid H_∞/SMC system. On the other hand, the differences in steady torques are 0.153 and 0.404 N.m.

5.2. Robustness to Parameter Variation

Second, the developed control system is tested by applying it to the nonlinear model of the triple inverted pendulum given in Equation (3), with perturbed uncertain parameters, $I_1$,$I_2$, and $I_3$ = nominal value + 10%, $C_1$,$C_2$, $C_3$, $C_{m1}$, and $C_{m2}$ = nominal value + 15%. The three angles respond as shown in Figure 11, and the phase-plane trajectories of the angles and their derivatives are shown in Figure 12.

The first and third angles overshoot by 1.299 and 22.47 deg, respectively, in the H₂ control system, more than in the hybrid H_∞/SMC system. Only the second angle overshoots higher in the hybrid H_∞/SMC system, by 1.503 deg. The difference in errors between the two systems are found to be 0.004, 0.05, and 0.027 deg.

The three phase-plane trajectories clearly depict the fluctuations at the start of the response in the H₂ control system.

The applied torques are shown in Figure 13. The start-up torques t_m1 and t_m2 in the H₂ control system are again higher than in the hybrid H_∞/SMC system, but by 8.227 and 10.2 N.m, respectively, this time. On the other hand, the differences in steady torques are 0.13 and 0.38 N.m. In this case, the H₂ controller fluctuates significantly at the beginning.

5.3. Robustness to External Disturbance

Third, the developed control system is tested by applying it to the nonlinear model of the triple inverted pendulum given in Equation (3), with external disturbance in the form of a sine wave of 0.1 N.m amplitude applied to each link. The three angles respond as shown in Figure 14; the phase-plane trajectories of the angles and their derivatives are shown in Figure 15.

The first, second and third angle responses overshoot slightly higher, moderately lower, and significantly higher, respectively, in the H₂ control system, while the errors are close to each other in all responses.

The phase-plane trajectories show the direct alignment of the systems’ states with the desired point in the state space.

Figure 16a shows that t_m1 at start-up in the H₂ control system is higher by 13.567 N.m, and t_m2 by 21.839 N.m., while the differences in steady torques of the first and second actuators are only 0.132 and 0.405 N.m, respectively.

5.4. Robustness to Measurement Noise

Finally, the developed control system is tested by applying it to the nonlinear model of the triple inverted pendulum given in Equation (3), with measurement noise in the form of random white noise and with amplitude ±0.4 V applied to each reading. The three angles respond as shown in Figure 17; the phase-plane trajectories of the angles and their derivatives are shown in Figure 18, and the applied torques are shown in Figure 19.

In this case, the hybrid H_∞/SMC system produces less overshoot in the first and third angles’ responses and fewer errors in the second and third angles’ responses.

Figure 19 shows that, compared to the H₂ controller, the proposed hybrid H_∞/SMC controller requires much less effort at start-up, and almost the same effort is needed at steady state.

The characteristics of the proposed hybrid H_∞/SMC system performance in all tested cases are summarized in the three charts in Figure 20, Figure 21 and Figure 22, and will be discussed in the next section.

6. Discussion and Conclusions

This section explores the interpretation and implications of the obtained results. Regarding the optimization aspect, Figure 7 shows that PSO exhibited a very fast convergence rate (majorly within 16 iterations, and completely within 59 iteration). The application of optimization to the controller design helped in improving the control system, through minimizing the response error (as shown in Figure 8, Figure 11, Figure 14 and Figure 17) and the applied torque needed for the system (as shown in Figure 10, Figure 13, Figure 16 and Figure 19).

In relation to the robustness attribute, the developed hybrid H_∞/SMC control system was able to stabilize the nominal, perturbed by parameter variation, perturbed by external disturbance, and perturbed by measurement-noise system. In all cases, the system responded in the same manner (except for very small differences, as shown in Figure 14 and Figure 17).

Concerning performance characteristics, it has been shown in Figure 20 that, in all cases and among all the angles, the error was below 0.152 deg and the average error was 0.053 deg, which is an indication of the system’s accuracy. The overshoot in all cases is almost unaffected by the type of uncertainty, as shown in Figure 21. For the first, second, and third angle, the overshoot ranged from 6.035 to 6.472 deg, 9.263 to 10.072 deg, and 11.052 to 11.561 deg, respectively. The least overshoots were in the nominal system case, while the highest were in the perturbed-by-parameter-variation case. Figure 9, Figure 12, Figure 15 and Figure 18 show that the phase plane trajectories of the angular positions and velocities are driven successfully by the developed controller, from the initial condition to a very small neighborhood of the origin, in all cases. Figure 15 and Figure 18 show small fluctuations in the late stage of the trajectory, which come from forcing the states by the controller to reach the origin, in spite of disturbance and noise.

From the system’s efficiency point of view, the required steady controller effort in all cases ranged from 0.13 to 0.621 N.m in amplitude, owing to the inclusion of the torque in the objective function of optimization. A higher torque was required only at start-up; the average start-up torque for the first actuator is 1.663 N.m and it is 2.158 N.m for the second actuator. This contributes to a more efficient control system. The small chattering caused by the sign function in Equation (47), which is necessary for driving the states, can be attenuated to obtain a smoother control signal, through using boundary-layer or neural networks.

As compared to the H₂ control system, the proposed hybrid H_∞/SMC system produces fewer overshoots and fewer start-up torques, and smooth states driving in general. The robustness is provided in both control systems with small disparities, as illustrated in the results.

So, instead of having unnecessary conservativeness in the system, which degrades the performance with respect to providing robustness against uncertainties, the application of metaheuristic optimization to the controller design for such an under-actuated nonlinear MIMO system is beneficial in enhancing both performance and efficiency of the control system, besides its robustness.

Another point to be made is that for this specific electro-mechanical control system-design optimization problem, a swarm size of 10 was suitable, as has been presented in the results. A larger swarm size has to be considered in other optimal designs of engineering problems, especially if more parameters are required to be tuned in the application. Also, the adaptation of the cost function can be considered in future, to further enhance the optimization of the design problem, as proposed and described in [43].

Having great impact on control systems’ performance and efficiency, metaheuristic optimization algorithms are advised to be applied to enhance other control systems designs, as in the following: modelling predictive control to optimize the control sequence over a specified prediction horizon, hybrid model-based and data-driven control to adapt model parameters and controller gains in real-time, and distributed control systems to optimize the control signals’ distribution in large scale systems.