The Ball and beam system (BBS) is an attractive laboratory experimental tool because of its inherent nonlinear and open-loop unstable properties. Designing an effective ball and beam system controller is a real challenge for researchers and engineers. In this paper, the control design technique is investigated by using Intelligent Dynamic Inversion (IDI) method for this nonlinear and unstable system. The proposed control law is an enhanced version of conventional Dynamic Inversion control incorporating an intelligent control element in it. The Moore-Penrose Generalized Inverse (MPGI) is used to invert the prescribed constraint dynamics to realize the baseline control law. A sliding mode-based intelligent control element is further augmented with the baseline control to enhance the robustness against uncertainties, nonlinearities, and external disturbances. The semi-global asymptotic stability of IDI control is guaranteed in the sense of Lyapunov. Numerical simulations and laboratory experiments are carried out on this ball and beam physical system to analyze the effectiveness of the controller. In addition to that, comparative analysis of RGDI control with classical Linear Quadratic Regulator and Fractional Order Controller are also presented on the experimental test bench.

The ball and beam system (BBS) is a standard laboratory experimental setup and serves as an essential benchmark to investigate and validate the performance of various control strategies [

The control of an under-actuated system is an active research field having fewer control inputs than the state variables that need to be controlled. The experimental setup of BBS deals with the two loops control strategy. The outer loop is responsible for ball position control, whereas the inner loop is engaged for servo angle control. Due to its double loop control strategies, it is not easy to control this system using linear controllers [

A feedback linearization method is applied for BBS, which produces singularity and does not produce good results [

Recent work was proposed designing a controller comprising two embedded loops based on linear fractional-order calculus [

In addition to that, Nonlinear Dynamic Inversion (NDI) was also implemented for this ball and beam underactuated system, see [

Therefore, NDI control may possess certain shortcomings and limitations while developing the algorithms. This includes the simplified approximations to introduce the inverse model of plant dynamics, square dimensionality condition, useful nonlinearity cancellations, etc.

Recently a control law focused on the inversion principle is proposed known as Generalized Dynamic Inversion (GDI), which addresses the limitations of the classical NDI approach [

A two-loop control architecture is employed in this paper to solve the tracking and stabilization problem for two degrees of freedom under actuated BBS. A proportional derivative (PD) control is implemented in the outer loop to generate rotary servo angle commands based on the ball position on the beam. An intelligent version of GDI, Intelligent Dynamic Inversion (IDI), is used to process the inner loop's angular command. Identical (baseline) control and switching (continuous) control elements make up this IDI control system. The baseline control enforces the constraint dynamics based on the deviation function of the rotary servo angle for stable attitude tracking, while the switching control ensures robustness against nonlinearities and external perturbations. As a result, the proposed control guaranteed a semi-global attitude tracking system and bounded tracking errors. Quanser's BBS is used to simulate and perform laboratory experiments on the controller performance and compare the controller performance to existing control strategies.

The remaining sections of the paper are described as follows. The modeling of BBS is presented in Section 2. The two-loop control structure is discussed in Section 3. The classical PD control design for the outer position loop is presented in Section 4. The design implementation of IDI control for tracking the servo angle command is given in Section 5. The detailed stability analysis that will guarantee semi-global asymptotic stability was illustrated in Section 6. Controller validation through computer simulations and experimental findings are reported in Sections 7 and 8. Finally, the concluding remarks are given in Section 9.

Quanser developed BBS, a benchmark laboratory test bench [_{b} _{b} _{b}

Similarly, the relation between the beam angle _{l}_{beam} _{arm}

The second major component of the ball and beam setup is the rotary servo base unit which comprises a DC motor with a gearbox, as shown in _{b} _{m} _{l} _{m}_{eq}

and the equivalent damping term _{eqv}

In the expressions of the equivalent moment of inertia and damping, _{g} _{m} _{t,} _{g} _{m}

The numerical value of the BBS parameters is listed in

Parameters | Value | Unit |
---|---|---|

A cascade control architecture based on the time separation principle is proposed to control the ball's position on the beam, as shown in _{ld} _{ld}

Given the desired and actual ball's position, i.e., _{d} _{ld} _{ld} _{p} _{d} _{ld} _{d} _{f}

This section discussed the design and implementation of IDI control for attitude control of the rotary servo unit of BBS. The proposed non-square inversion-based intelligent controller proves to be an efficient and effective control strategy to deal with complex nonlinear systems. The control law comprises two major components, i.e., an equivalent or baseline control and a switching control component. The general form of IDI control law is expressed as follows

The first component _{eq} _{rbt}

The equations of motion of rotary servo unit angular rate dynamics given by

The baseline or equivalent controlled voltage

A suitable discontinuous control element is determined to solve the reachability problem, which ensures robustness against parametric uncertainties, system nonlinearities, and external disturbances. The resultant GDI-SMC or IDI control law is constructed to be of the following form

Moreover,

The time derivatives of the sliding surfaces are evaluated as

It has been observed that the asymptotic convergence of the sliding surfaces

To prove the asymptotic convergence of the error dynamics, place the control law

Consider the following positive definite Lyapunov candidate function

The time derivative of

The 2-(induced) norms of

This section investigates the performance of IDI control law through computer simulations on the dynamical simulator of BBS developed by considering the model of ball and beam unit given by

To investigate the real-time performance of balancing the ball on a beam, the IDI control law developed in Section 5 is implemented on Quanser's ball and beam real hardware, as shown in

The experimental setup comprised the power module VoltPAQ-X1 for supplying voltage to the rotary servo motor module. The servo motor is placed at the right corner of the beam to move the beam in an upward and downward direction. The ball placed on the beam rolls on the track accordingly by changing the slope of the beam. The ball position and the angular position feedback commands are sent to the computer through Quanser's Q8-USB data acquisition card. The communication between Simulink/MATLAB and the experimental hardware is accomplished through the QUARC software. The sampling time is chosen to be

In this analysis, the core objective is to present the comparative analysis of IDI control with classical Linear Quadratic Regulator (LQR) and Fractional Order Control (FOC) strategies. The ball is commanded to follow the step input profile in this setup. The initial position of system states are

Furthermore, the FOC suffers from high overshoot, whereas many steady-state errors were observed in the LQR response. On the other hand, the IDI controller demonstrates very stable time-domain performance near reference input with less overshoot and steady-state error. The time evolution of desired and actual angular profiles and the time histories of the controlled voltages generated in response to the step input command are shown in

This section highlighted the disturbance rejection capability of the IDI controller when the ball is subjected to some manual disturbances. The ball position on the beam in a steady state is stimulated by tapping the ball at different time instants. The ball position plot under the effect of disturbances is shown in

In this plot, the controllers’ capability is further analyzed by commanding a reference square wave profile with an amplitude of

While examining the different behaviors of the controller response in the simulation and experimental results, it is claimed that the proposed controller proves itself a reasonable control approach for stabilizing the under-actuated electro-mechanical systems.

This paper investigates the performance of IDI control for a standard nonlinear ball and beam unstable system. A two-loops structured control is implemented in which PD control is responsible for generating the desired servo angle command based on the positional error of the ball placed on the beam. Furthermore, IDI control is engaged in the inner loop for precise attitude tracking. The results are verified through computer simulations and experimental studies in a MATLAB environment. Furthermore, a comparative analysis is presented, which exhibits the superior performance of IDI control over LQR and FOC in terms of faster error convergence with lesser steady-state error. Hence the proposed methodology is quite appealing and feasible for practical electro-mechanical systems.

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the Project Number (IFPRC-023-135-2020) and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.