Stabilization control of rotary inverted pendulum using a novel EKF-based fuzzy adaptive sliding-mode controller: design and experimental validation

This article methodically develops an improved self-regulating fuzzy-adaptive Sliding Mode Controller (SMC) that strengthens the disturbance compensation capacity of the nonlinear rotary pendulum systems while effectively attenuating the chattering content and curbing the control energy consumption. The article contributes to augmenting the SMC with online adaptation tools to achieve the said objectives. It employs the conventional Gao’s power-rate reaching law as the baseline. The scaling gain and the power rate of the said reaching law are adaptively modulated via a pre-calibrated two-input state-error-driven fuzzy nonlinear function. Additionally, the sign function in the law is also replaced with an odd-symmetric nonlinear fuzzy function to address the hard limits imposed by the former. Finally, the membership functions of the fuzzy function are self-regulated using the Extended-Kalman-Filter to improve the compensator’s adaptability in handling the system’s rapidly changing control requirements under exogenous disturbances. The aforementioned propositions are verified by performing customized and reliable hardware-in-loop experiments on the Quanser single-link rotary pendulum platform. As compared to baseline SMC law, the proposed control procedure contributes a ∼45.2%, ∼48.5%, and ∼34.8% reduction in position-regulation errors, control energy consumption, and peak overshoots, respectively. The experimental assessment validates the proposed control system’s enhanced robustness and chattering-suppression capability.


Introduction
Single-link Rotary-Inverted-Pendulum (RIP) is an under-actuated mechatronic platform that is widely favored in the domain of experimental physics and control systems engineering due to its complex geometry, multiple degrees of freedom, nonlinear dynamics, kinematic instability, and under-actuated configuration [1,2].These properties make it a perfect experimental setup to test the agility and resilience of the nonlinear control law [3,4].The stabilization control problem of the RIP becomes quite difficult to tackle under bounded exogenous disturbances and parametric uncertainties [5].This challenging problem is evident in various practical scenarios, such as the stabilization of Segway platforms, the satellites in orbit, the attitude of underwater vessels, the gait of legged robots, the vertical takeoff of helicopters and rotorcrafts, the building under structural vibrations (tremors or earthquakes), and the launching of rockets and missiles, etc [6,7].

Literature review
Several robust stabilization control strategies have been scientifically investigated for standard RIP systems [8][9][10].Fixed-gain PID controllers are considered reliable in self-balancing robotic applications [11,12].However, the simplicity of their structure prevents them from fully realizing and efficiently addressing nonlinear disturbances [13,14].The fuzzy-logic controllers require a pre-calibrated set of fuzzy rules that are defined by the control expert [1,15].Their membership functions (MFs) can be calibrated offline to eliminate any discrepancies in the heuristically established logical rules to formulate a robust fuzzy control law [16,17].The neural controllers, despite their flexibility, require extensive training to deliver a reliable control effort, which inevitably affects their computational economy [18].The dependence of the optimal state-feedback linear quadratic regulator (LQR) on the dynamic system's mathematical model affects the optimality of its control yield when the system encounters model uncertainties and identification errors in real time [19,20].Extensive research has been done to improve the LQR's capability by augmenting it with auxiliary self-adaptation tools to effectively compensate for bounded exogenous disturbances [21][22][23].
The Sliding Mode Controller (SMC) is a widely favored nonlinear feedback compensator [21].It delivers robust control effort and efficiently compensates for exogenous disturbances and parametric uncertainties that cannot be practically handled via the aforementioned control schemes otherwise [24,25].It belongs to the class of variable structure control, wherein the control law commutes between the sliding manifolds using a pre-calibrated switching function to address the rapid state variations [26,27].The SMC scheme and its variants have been extensively proposed in the available literature to develop an agile self-balancing control strategy for RIPs [28].The sliding-mode control schemes offer robust disturbance rejection while guaranteeing finite-time convergence [29,30].However, the SMC's redundant switching phenomenon generates highly disruptive control input, which inevitably injects chattering and ripples in the state response.This behavior leads to high oscillations of the RIP's mechanical parts as well as dynamic power losses in the reactive electrical devices [31,32].Furthermore, it is required to effectively manipulate the control stiffness as per the state error variations to amplify the damping against disturbances and to increase the transient response speed while economizing the control energy and preserving stability.
The chattering effect is generally caused by any of the following three reasons: (1) the un-modeled intrinsic nonlinearities; (2) the ill-calibrated switching gain; and (3) the discontinuous switching function [33].Some of the notable techniques devised to effectively minimize chattering are the adaptive SMC, DSMC, and Fuzzy SMC [34].Filtering out the high-frequency noise components from the SMC's control signals has gained a lot of traction [35].Replacing the conventional signum function with a fuzzy-logic inference mechanism to avoid the signum function's hard nonlinearity tends to improve the chattering suppression behavior [36].However, to achieve the desired chattering suppression, these schemes tend to compromise the SMC's robustness and affect its adaptability against external perturbations [37].Furthermore, the poor calibration of the fuzzy membership functions or the inaccuracies (caused by the expert's lack of knowledge) in the fuzzy logic rules incapacitate the fuzzy inference mechanism to yield robust and agile control effort under rapidly changing error conditions.
The disturbance-observer-based adaptive sliding mode controller designs have also been proposed in the scientific literature to counteract unknown disturbances actively [38,39].The memory-based fuzzy systems and reinforcement learning-based adaptation strategies used in these works enhance the controller's robustness at the cost of higher computational complexity and, thus, higher computational burden on the embedded processing unit.

Main contributions
To address the aforementioned problems, this article presents the systematic formulation and performance validation of an innovative self-regulating fuzzyadaptive SMC design for the RIP systems to minimize chattering while maintaining the agility of the SMC against under bounded disturbances.Gao's power-rate SMC procedure is used as the baseline reaching law, as expressed below [34].
where, k represents the preset positive scaling gain, s(t) is the sliding surface that is computed as the weighted sum of all state errors, β is the preset power rate of the magnitude of s(t), and f (s, ṡ) is an odd-symmetric fuzzy nonlinear function that is bounded between −1 and 1 as the state error conditions vary.This limiter function f (s, ṡ) avoids the hard nonlinearity of the sign(s) function, and thus suppresses the chattering content in the state response(s).In this article, the aforementioned FSMC law is methodically evolved by augmenting it with a customized self-regulation block that improves its disturbance-rejection ability, economizes control expenditure, and attenuates the chattering in the response.The salient features of the proposed self-regulating SMC law are as follows: (1) The power rate, β, and scaling gain, k, of the reaching law are adaptively modulated online via the pre-calibrated nonlinear fuzzy functionf (s, ṡ) by using well-postulated meta-rules.(2) The fuzzy MFs of the function f (s, ṡ) are selfregulated online via Extended Kalman Filter (EKF) to efficiently adapt to the abrupt changes in the state errors.(3) The efficacy of the proposed scheme is investigated against the conventional SMC laws, in the physical environment, by conducting reliable and customized Hardware-In-Loop (HIL) experiments on the Quanser RIP.
The proposed FSMC law variant reinforces the compensator's dexterity to automatically regulate control stiffness as per the state error variations while (concurrently) attenuating the chattering content.The selforganizing fuzzy MFs aid in removing any shortcomings or residual inaccuracies in the heuristically defined rule base due to the imprecise knowledge of the expert.Furthermore, it reconfigures the waveform of the fuzzy function in real time to accurately alter the relevant controller gains to address the system's requirements.Consequently, the proposed SMC law increases the response speed, strengthens the system's immunity against disturbances, improves chattering suppression behavior, and preserves the control economy by avoiding peak servo requirements.
The idea of employing an EKF-based self-regulating nonlinear fuzzy function to adaptively modulate the odd limiter function, power rate, and scaling gain of the SMC reaching law to simultaneously enhance the disturbance-rejection and chattering attenuation capability has not been investigated in the open literature.Hence, this novel idea is the main focus of this article.
The article is structured as follows: The dynamic model and the baseline SMC law for the RIP system are presented in Section 2. The FSMC law augmented with online adaptive parameters is derived in Section 3. The proposed EKF-based self-regulation fuzzy adaptive SMC law is formulated in Section 4. Section 5 presents the comparative experimental analysis of the designed SMC variants.Finally, Section 6 concludes the article.

System description
The proposed SMC law is validated by benchmarking its performance using the standard single-link RIP system.The single-link RIP system is diagrammatically illustrated in Figure 1.It has a stable and unstable equilibrium state.The rod is vertically downright in the stable state, whereas the rod is balanced vertically upright in the unstable state.Thus, an active balancing control effort is required to maintain the rod in the equilibrium state.The arm of the RIP system is rotated by mounting it on the DC servo motor's shaft.The motor's shaft is equipped with a rotary encoder that records the arm's position α in real-time.The pendulum's rod is attached to the other end of the arm via a hinge that is instrumented with another rotary encoder that records the apparatus's rod position θ.The data acquired from each encoder is transmitted to the control software application over a serial link.The software application processes the data to compute the control command.The resulting motor control torque rotates the pendulum arm, which continuously provides the necessary swinging energy to the rod to invert it and eventually balance it.

Mathematical model derivation
The single-link RIP system is modeled in terms of α and θ via the Euler-Lagrange method [40].The computation of RIP's Lagrangian, L, is presented in Eq. 2. where E P = M p l p g(cos θ), and 2 where E K and E P represent the RIP's total kinetic and total potential energy, respectively.The parameters mentioned in the above expressions are quantified in Table 1.The computed Lagrangian is expressed as follows [36].
The nonlinear equations of motion are derived by evaluating the following partial derivatives [40].
where τ represents the torque applied by DC motor.
It is a function of the motor's terminal voltage, V m , as shown below.
After linearization, the following state equations are obtained.
such that, H = J e J p + M p r 2 J p + M p l 2 p J e The generalized state-space representation of a linear dynamical system is shown in Eq. 7. where A is the system matrix, B is the input matrix, C is the output matrix, D is the feed-forward matrix, and y(t) is the output vector.The state vector x(t) and the control input vector u(t) of the proposed system are presented in Eq. 8.
The RIP system's model is represented in state-space form as shown below [36]. where, As mentioned earlier, the model parameters associated with Quanser RIP are quantified in Table 1 [41].

Baseline SMC law
The SMC law functions by driving the state trajectories of the RIP system toward a sliding surface, s(t).The sliding surface utilized in this article is shown in Eq. 11. where, where G T is a 1 × 4 vector containing the state error weights and x ref represents the RIP system's reference state vector of the form T .The arm's initial position is set as its reference position, α ref , before the beginning of every experiment.Owing to its rapid convergence, Gao's reaching law is used in this article [42].The said reaching law is expressed in Eq. 12.
where k is the preset positive scaling gain, βis the preset power rate of the absolute value of s(t) that is restricted between 0 and 1, and sign(.) is a "sign" (or signum) function that is expressed in Eq. 13.
The reaching law guarantees the convergence of the sliding surface s(t) in finite time, as proven later in this section.The proposed finite-time SMC law is not required to address the variable time delays, as observed in bilateral teleoperators and similar nonlinear systems [30].Hence, the said control law is formulated via a single term of exponential and limiter function to tackle the single sliding variable associated with the RIP system.This arrangement significantly simplifies the computational realization of the control structure.The first derivative of Eq. 11 is expressed in Eq. 14. where, The expressions ẋref = 0 0 0 0 T and ẋ(t) = Ax(t) + Bu(t), when substituted in Eq. 14, yield, The comparison of Eq. 12 and Eq. 15 delivers the expression in Eq. 16.
By making u(t) as the subject, the state-space-driven SMC law is derived as shown below.
The stability of the SMC law is verified by using the positive-definite Lyapunov function in Eq. 18.
The first derivative of V(t) is derived in Eq. 19.
Substituting the ṡ(t) expression from Eq. 15, the derivative V(t) is expressed as shown in Eq. 20.
Substituting the u(t) expression from Eq. 17, the derivative V(t) is simplified as shown below.
The derivative function V(t) is simplified as shown in Eq. 22.
Hence the expression of V(t) < 0 if k > 0. This specification satisfies the stability requirements of the SMC law.The SMC law's finite-time convergence is analyzed as per Lemma 1 discussed below.
where δ and μ are predetermined constants such that δ > 0 and 0 < μ < 1.The function z(t) takes a finite time t s to converge to zero.The time t s is given in Eq. 24.
Using the expression in Eq. 22, the derivative V(t) can be rewritten as shown in Eq. 25.
Then, according to Lemma 1, the function V(t) and the sliding variable s(t) converge in a finite time that is given by Eq. 26, as long as k > 0 and 0 < β < 1.
To ensure accurate position-regulation behavior, the SMC parameters are optimized by iteratively minimizing the following objective function J c .
In this cost function, an equal (unity) weight is allocated to the state error and the control criteria.This is done to apply equal impact on minimization of the position regulation errors and the control input to prevent wind-up.The selection range of each coefficient of G T is bounded within [−100, 100].The parameter optimization is initialized with G T = 1 1 1 1 .The tuning algorithm explores the range space for the best-fit solution in the direction of the objective function's decreasing gradient.In every trial, the RIP apparatus rod is inverted and balanced for 10.0 s.The corresponding cost is computed at the trial's conclusion.The optimization process is concluded when the global minimum is acquired for a given parameter.The parameters β and k are also tuned offline by using the aforementioned technique.The power rate β is bounded within [0, 1] and the scaling gain k is chosen from the range [0, 10] as discussed in Eq. 10.The tuning algorithm is initialized with β = 1 and k = 1.The parameter values thus selected are k = 4.07, β = 0.45, and G T = −3.1673.48 −3.02 9.25 , (See [40]).The utilization of linear models for control design is preferred over nonlinear models primarily due to their simplicity and tractability.The linear differential equations are computationally simpler to solve, which leads to easier analysis, design, and realization of the control laws.Furthermore, it is also much simpler to assess the stability and robustness of a control design without having to simulate the entire nonlinear system.The computational efficiency of linear systems is especially important in real-time applications that require quick response speeds.The control designers can focus on the behavior of the system near specific operating points by linearizing the nonlinear systems to create linear models.This streamlines the design process and is adequate for a variety of applications.Hence, owing to the aforementioned benefits, the SMC law expressed in Eq. 17 uses the system's linear state space model.This arrangement also increases the scalability of the SMC.The intrinsic nonlinear properties of the underactuated systems are difficult to model and identify owing to their complex geometry.
The SMC discussed in this work is formulated using a linear state space model.Despite working with a linear system model, the proposed SMC law is augmented with auxiliary online adaptation tools that are realized using model-free nonlinear fuzzy functions.These functions autonomously self-tune the critical parameters as the error conditions vary, which aids in amplifying the controller's resilience against exogenous perturbations.This arrangement allows the designer to easily modify and extend the SMC law to other mechatronic systems, as long as the linearized state space model is known a priori.

Baseline linear control law
The conventional LQR is a state space compensator that minimizes the QCF provided in Eq. 28 to provide optimal control decisions [19,20].
]dt (28) where R ∈ R ≥ 0 is the control input weighting factor and Q ∈ R 4×4 > 0 is the state weighting matrix linked with the single-link RIP system.The R and Q matrices are expressed in Eq. 29.
where ρ and q x are the preset positive coefficients of the respective matrices.These coefficients are calibrated offline by minimizing the objective function in Eq. 27 using the procedure discussed in 2.2.The Algebraic-Riccati-Equation (ARE) uses the pre-calibrated R and Q matrices to evaluate P matrix as expressed in Eq. 30.
where P ∈ R 4 × 4 > 0. The corresponding LQR gain vector K is computed as expressed in Eq. 31. where The fixed-gain LQR law is formulated in Eq. 32.
The closed-loop asymptotic stability of LQR is verified via the following Lyapunov function [21,22].
The first derivative of M(t) is derived in Eq. 34.
By making necessary substitutions from Eq. 25, V(t) simplified as expressed in Eq. 35.(35) The function

Fuzzy-Adaptive SMC methodology
The utilization of sign(s) in Eq. 17 imposes a hard nonlinearity that induces abrupt switching in the generated control signals as the state errors vary.The repetitive switching causes highly disrupted control behavior that unavoidably contributes to chattering in the response.
To minimize the chattering, the sign(s) in Eq. 17 is replaced with a smooth fuzzy nonlinear function [43].
The function f (s, ṡ) smoothly commutes between the sliding surfaces as an odd-symmetric function of s and ṡ.It is designed such that it is bounded between −1 and 1 so that it complies with the control law requirements.
It uses s and ṡ as its inputs, which improves the agility of the controller design to better address nonlinear disturbances.To further enhance agility and resilience, the SMC law is also retrofitted with online gain modulators that self-tune the controller's scaling gain and the power rate.By retrofitting the SMC control law with the aforementioned tools, the resulting Fuzzy Adaptive SMC law (or FA-SMC) takes the following form.
where β(s, ṡ) and k(s, ṡ) represent the time-varying power rate and scaling gain of the control law that is realized by using the aforesaid fuzzy function f (s, ṡ).These modifications enhance the SMC's adaptability to efficiently compensate for the disturbance while attenuating the chattering phenomenon and maintaining smooth control activity.This scheme is computationally economical because it uses the prescribed f (s, ṡ) to online adapt k(.) and β(.) instead of formulating a new gain-adaptation function.This control procedure is referred to as FA-SMC in this article.Its methodical construction is discussed below.

Fuzzy function formulation
The fuzzy-logic inference system is a computational intelligence paradigm that uses predefined qualitative rules to make logical decisions for solving problems in engineering cybernetics [44].It utilizes heuristically fabricated logical rules to yield an agile control effort that undertakes to reject bounded external disturbances.The rules of f (s, ṡ) are empirically designed to ensure the state trajectories always converge to the (1) When both s(t) and ṡ(t) have the same polarities, the response is considered to be in the perturbed state.The fuzzy system applies a tight control effort to reject the disturbances during this phase.(2) When s(t) and ṡ(t) have opposite polarities, the response is considered to be in the equilibrium or quasi-equilibrium state.The fuzzy system applies a soft control effort during this phase to minimize the state fluctuations as the response settles to the reference position.
These rules preserve the robustness of the conventional SMC law while suppressing the chattering content.The rule base synthesized to constitute f (s, ṡ) is presented in Table 2.The Mamdani inference method is used to implement the fuzzy implication, as shown below.
where μ represents the MF degree, n represents the rule number, and h ij (.) represents the triangular-shaped input MF as formulated in Eq. 38.
where g represents the normalized value of inputs (s or ṡ), and b ij − , c ij , and b ij + are the left-half width, centroid, and right-half width of the j th input MF of the i th input respectively.The decisions are defuzzified using the center of gravity approach, and the associated crisp output f (s, ṡ) is computed as shown in Eq. 39. (39) where w is the output MF's centroid.Asymmetrical waveforms of the MFs are utilized in this work to strengthen the fuzzy-augmented SMC's robustness against exogenous perturbations.The fuzzy MFs are empirically optimized by utilizing the tuning procedure discussed in Section 2.2.The resulting triangularshaped MFs linked with input and output variables are depicted in Figure 2. The execution of the fuzzy inference scheme does not involve any recursive computations, and thus, it can be easily programmed with modern embedded microprocessors.

Power rate adaptation
The power rate β of the reaching law influences the stiffness of control activity.A fixed β cannot address the ever-changing control requirements of the system.On the one hand, it yields insufficient control resources under disturbance conditions.Conversely, it applies surplus control energy during the equilibrium state.This arrangement renders the SMC law wasteful.Hence, in this article, β is dynamically reconfigured as a nonlinear scaling function of s and ṡ that uses the following meta-rules.
(1) The power rate is decreased under large state-error conditions (disturbances).This helps in amplifying the influence of |s(t)| β(s,ṡ) in the control law, thus applying an aggressive damping control force to attenuate the overshoots resulting from exogenous disturbances.
(2) The power rate is enlarged under small state-error conditions (equilibrium).This helps in reducing the influence of |s(t)| β(s,ṡ) in the control law, thus applying a gentle control force to allow for smooth stabilization behavior around the equilibrium.
This rationale improves the disturbance attenuation of the system while preventing large servo requirements, thus maintaining a low chattering content in the response.The following nonlinear function is thus formulated in this work as per the aforementioned rationale.
where β(s, ṡ) represents the time-varying power rate.The power rate is self-adjusted between 0 and 1.

Scaling gain adaptation
Similar to the power rate, the scaling gain also directly manipulates the magnitude of the applied control force.A fixed scaling gain lacks the flexibility to address the abruptly varying control requirements as the operating conditions change.To robustify the system, the scaling gain is also adaptively modulated by using the following meta-rules.(1) The scaling gain is enlarged under large state errors to amplify the damping control force.(2) The scaling gain is decreased under small state errors to soften the applied control force.
As per the aforementioned rules, the following smooth adaptation function is formulated.
where k(s, ṡ) represents the time-varying scaling gain, and k o is the nominal value of the scaling gain that is calibrated offline and set at k o = 6.84 by using the tuning procedure prescribed in Section 2.2.

Self-Regulating fuzzy-adaptive SMC methodology
The EKF is a well-established state estimation and filtering technique that adapts to real-time changes in system dynamics [15].If the system undergoes variations or if the model parameters change, the EKF will update the state estimate and covariance accordingly, leading to "self-regulating" behavior [46].Hence, its adaptability to changing dynamics to yield real-time updates in the target parameters makes it well-suited for the proposed application.In this article, the EKF is used to adapt the output MFs of the function f (s, ṡ) [15].Hence, the weights (w) and centers of the MF are probabilistically modified.The proposed fuzzy function uses the updated estimates of the said weights (w) and centers of the MF to reconfigure the shape of MF waveforms online.The updated MFs are used to re-compute the controller parameters of the SMC law after every sampling interval [45].Consider the nonlinear system in Eq. 42.
where, q n represents the state vector of the system at n th instance, g n represents process noise, d n represents observation vector, p(.) is a nonlinear state function, and v n represents observation noise.To reform the shape of output MFs online, the system's state vector is comprised of the left-half width, centroid, and righthalf width of j th input MF of i th input, respectively [15].
The EKF is realized with the aid of the following set of equations.
where, qn represent the updated estimate of the state, H n represents the estimated state's measurement model, K n represents the Kalman gain vector, P n refers to the prediction error's covariance, R n is the covariance matrix of v n , and M n represent the covariance matrices of v n .The system's partial derivative as per the MF's weight is evaluated as shown below [15]. where, , The partial derivative of the system output for fuzzy MF centers is computed by using the following chain rule [15]. where, The partial derivative of the MF's right and left halfwidths is computed, as shown below [15]. where, The augmentation of the FA-SMC law with EKF yields the Self-Regulating FA-SMC (or SFA-SMC) law presented in Eq. 52.
The asymptotic stability proof and the finite-time convergence analysis prescribed for the SMC law in Section 2.2 are also valid for the proposed SFA-SMC law.
The proposed augmentation further increases the controller's agility to flexibly reconfigure the applied control force.The SFA-SMC is shown in Figure 3.

Results and discussions
This section discusses the specifics of the hardware configuration and the experiments used to assess the efficacy of the aforementioned controllers.

Hardware platform
In this research, the Quanser single-link RIP setup is used as the benchmark platform to carry out the HIL experiments.The Quanser RIP setup used to conduct experiments is shown in Figure 4.The state measurements related to θ and α are acquired via their respective rotary encoders.These NI-ELVIS II data acquisition board is used to acquire samples of raw state measurements at a rate of 1.0 kHz.Upon digitization, the measurement data is transmitted serially at 9600 bps to a customized control application in LABVIEW.The software's "Block Diagram" tool and functions palette are used to create the control application.A 2.0 GHz laptop with 8.0 GB of RAM is used to run the said control program.Apart from using the built-in functions, libraries, and blocks, the algebraic equations are realized by programming them using C-language code.The control program re-calibrates the function f (s, ṡ) online as per the updated state measurements and makes the necessary parameter adjustments to produce the new control signal.The onboard motor driver circuit receives the control signals serially that are amplified to activate the DC servo motor.The graphical user interface of this control application serves to display and record (download) the real-time variations in the states and control requirements of the RIP system.

Experimental evaluation
The RIP system's balance control performance under the control of the LQR, SMC, FA-SMC, and SFA-SMC schemes is individually investigated via five specific test cases that emulate practical disturbance conditions.In each test case, the time-domain profiles of θ(t), θ(t),  investigates the flexibility of each control law to effectively suppress the perturbations caused by parametric variations, identification errors, or model changes in the RIP system.The experiment is carried out by suspending a 0.10 kg mass underneath the horizontal link of the arm at t ≈ 7.0 s as shown in Figure 4.This attachment permanently modifies the constituents of the matrix A and, hence, the system's model.The contradiction in the system's updated and reference mathematical model perturbs the pendulum's state and control responses.The resulting variations in the states, control input, and f (.) are illustrated in Figure 9. (F) Gaussian noise attenuation: This test case examines each controller's agility against the measurement noise contributed by the sensors.The experiment is conducted by applying a simulated white Gaussian noise signal having a mean of zero and variance of 0.2 in the V m (t) signal.The changes in the states, control input, and f (.) are depicted in Figure 10.

Results and discussions
The following key performance metrics (KPMs) are used to analyze the experimental outcomes: RMS value of sensitivity that is evaluated as the ratio of θ to V m .
The performance metrics described above are used in this article to analyze the RIP's disturbance attenuation ability, position regulation accuracy, and control efficiency in the physical environment [47].The results of the HIL experiments are quantified in Table 3.
In Experiment A, the LQR exhibits poor position regulation accuracy and economic control activity.The SMC-regulated system demonstrates deficient reference-tracking activity with large state fluctuations, high chattering content, and highly disrupted control activity.The FA-SMC demonstrates moderately improved position-regulation behavior with relatively less chattering and fluctuations.However, it imposes large control requirements on the actuator.The SFA-SMC surpasses the other controller variants by robustly minimizing position-regulation errors while economizing the overall control input requirements, which significantly suppresses the chattering content as well.In Experiment B, the LQR demonstrates deficient disturbance-rejection behavior.The SMC demonstrates a mediocre improvement as compared to LQR.However, it still exhibits slow transits with fragile control effort against the simulated disturbance signals, significant chattering content in the responses, and the imposition of peak servo demands upon the actuator.The FA-SMC exhibits reasonably improved damping against disturbances and chattering but displays  pendulum's observed states and highly disrupted control activity.The FA-SMC shows considerable enhancement in the closed-loop system's control input activity and disturbance attenuation ability.The SFA-SMC contributes the strongest disturbance-rejection behavior by exhibiting minimal offset and α pp in the arm.It also ensures a smooth and economical control activity with minimal peaks recorded.In Experiment D, the LQR yields insufficient control effort.The SMC yields an exceedingly disrupted control behavior, which unavoidably introduces high chattering content in the states.The FA-SMC shows considerable improvement in the lumped disturbance attenuation behavior while moderately economizing the control energy demands.The SFA-SMC efficiently attenuates the impact of the simulated sinusoidal disturbance by minimizing steady-state fluctuations and curbing the input requirements.In Experiment E, the SMC and FA-SMC contribute relatively weak damping control efforts against the modeling error, which results in large In Experiment F, the LQR underperforms as compared to the rest of the designed controllers.The SMC yields a discontinuous control effort, which  inevitably introduces large ripples in state responses.The FA-SMC shows considerable improvement in white noise suppression behavior while expending a reasonable amount of control energy.The SFA-SMC robustly attenuates the white noise by minimizing the oscillations in the state responses as well as the control energy requirements.The benefits afforded by SFA-SMC are attributed to the dynamic reconfiguration of the shape and form of f (s, ṡ).In every test case, the variations in f (s, ṡ) under  FA-SMC are relatively more disrupted.The function is redundantly and abruptly switching between +1 and −1, which inevitably results in a very sporadic control action and the consequent chattering phenomenon.On the contrary, the variations in the EKF-regulated f (s, ṡ) under SFA-SMC are smooth.The functional variations are abrupt, but they only commute to +1 and −1 under large error conditions.The variations are mostly condensed around the intermediate values of the function under normal conditions.This behavior economizes the system's control input requirements, which leads to smooth control activity and reduces chattering.The SFA-SMC avoids the inherent shortcomings of the conventional fuzzy system by employing EKF to dynamically reconfigure the MFs, which in turn supplements the controller's self-tuning capability in real time.The sensitivity analysis quantified in Table 3 also validates the superior robustness of SFA-SMC to disturbances and noise.The phase diagrams for test case A, plotted between the angular velocity and angular displacement associated with the arm and the pendulum rod, are depicted visually in Figure 11 and Figure 12, respectively.The phase portraits linked with θ show that the state variations in SFA-SMC are restricted within −179.0 and 181.0 degrees while exhibiting relatively high response speed.The phase portraits linked with α show that the state variations in SFA-SMC are restricted within −14.0 and 14.0 degrees while exhibiting reasonable response speed.The phase diagrams validate the agility of the proposed SFA-SMC law.

Comparative analysis using modern controllers
The proposed SFA-SMC is also benchmarked against state-of-the-art controllers to verify its robustness in addressing the RIP's balancing control problem.The state-of-the-art control schemes considered for the sake of comparison are the Complex-Fractional-Order Linear-Quadratic-Integral-Regulator (CFO-LQIR) [21], Improved Variable-Structure Linear-Quadratic-Integral-Controller (IVS-LQIC) [22], and Immune-Adaptive Degree-of-Stability LQR (IDoS-LQR) [23].The aforementioned controllers were experimentally analyzed on the same RIP system, as modeled in Table 1, via test cases (A to E) under the same operating conditions.This arrangement ensures a fair comparison.
The graphical visualization of the experimental results associated with the aforementioned (selected) control schemes is clearly illustrated in the articles [21][22][23] respectively.The comparative performance analysis is presented in Table 4, which justifies that the disturbance compensation, position regulation behavior, time optimality, and chattering suppression of the proposed SFA-SMC are on par with modern controllers.Most of the KPMs associated with the SFA-SMC surpass the respective KPMs of other controllers, as quantified (and highlighted) in Table 4.

Conclusion
This study discusses the methodical development and experimental verification of a novel self-tuning SMC procedure that undertakes to strengthen the disturbance attenuation ability of an RIP system while simultaneously minimizing the chattering in the response as well as the control energy expenditure.For this purpose, the SMC is methodically modified by retrofitting it with auxiliary adaptation tools.The proposed augmentations increase the controller's durability against external disturbances and suppress the chattering while upholding the closed-loop stability of the RIP system.The EKF-driven dynamic reconfiguration of the fuzzy nonlinear function's waveform to self-adjust the critical controller parameters significantly improves the controller's agility to achieve the desired objectives.Consequently, the system exhibits minimum transitional times while effectively attenuating the disturbanceinduced overshoots (and oscillations) and economically managing the control input demands.The experimental findings successfully validate the enhanced selfbalancing behavior of the SFA-SMC in the physical environment.In the future, the SFA-SMC law can be appropriately modified and extended to other electromechanical and energy-conversion systems as well.Furthermore, apart from exogenous disturbances, the proposed augmentations can be retrofitted with other variants of SMC laws to address hardware limits and parametric uncertainties.Finally, a rigorous experimental comparison must be done to benchmark the efficacies of the SFA-SMC law against the ubiquitous disturbance-observer-based SMC laws.

Figure 1 .
Figure 1.Schematic diagram of a single-link RIP system.

Figure 3 .
Figure 3. Block diagram of the proposed SFA-SMC.
α(t), α(t), V m (t), and f (s, ṡ) are visualized for analysis.The testing procedures and the corresponding results are presented below: (A) Reference-tracking: This experiment examines the pendulum rod's position regulation and arm's station-keeping behavior under no-disturbance conditions.The state responses, control input response, and the variations in f (.) are illustrated in Figure 5. (B) Impulse disturbance compensation: This experiment analyzes the controller's sturdiness against the exogenous impulsive disturbances by applying a simulated signal, having a peak of −5.0 V and a span of 100.0 ms, to the V m (t) signal.The disturbances are injected whenever the arm displaces to its local maximum position.The variations in the states, control input, and f (.) are depicted in Figure 6.(C) Step perturbation compensation: This test examines the resistance of the control laws against abrupt and constant load or exogenous torque variations.The disturbance scenario is simulated by administering a step signal of −5.0 V in V m (t) signal at t ≈ 7.0 s mark.The consequent variations in the states, control input, and f (.) are demonstrated in Figure 7. (D) Sinusoidal perturbation attenuation: This test case examines each controller's agility against the disturbances contributed by mechanical perturbations or air resistance.The experiment is performed by applying a simulated sinusoidal signal, d(t) = 1.5 sin(20π t), in the V m (t) signal.The changes in the states, control input, and f (.) are depicted in Figure 8. (E) Model error compensation: This experiment

Figure 10 .
Figure 10.RIP's response under Gaussian white noise.
6hese conditions sufficiently prove the asymptotic stability of LQR.Based on the aforementioned stability specifications and the tuning technique discussed in Section 2.2, the weighting matrices of QCF are chosen by minimizing Eq. 27.The selected matrices are R = 1 and Q = diag 38.2 52.65.3 2.1 .The LQR gains thus evaluated are K = −6.21130.56 −4.22 17.83 .

Table 2 .
Rule base of the fuzzy function.

Table 3 .
Quantitative summary of experimental findings.

Table 4 .
Comparison of SFA-SMC with state-of-the-art controllers.Bold values indicate the optimum value of a given performance metric yielded by the designed controllers.