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 [1]. The tilt angle of the beam is the most significant action through which the ball position is controlled systematically by varying the tilt angle. It is an under-actuated open-loop unstable system having two degrees of freedom. The fundamental idea of BBS can be applied to a wide range of applications such as stabilizing the aircraft in its horizontal position while landing, tanker on-road carrying liquid, aircraft yaw roll control, goods handling problem by industrial robots, etc.

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 [2]. Therefore, several control techniques are proposed in the literature to investigate the tracking and stabilization problem of BBS [3–5].

A feedback linearization method is applied for BBS, which produces singularity and does not produce good results [3]. A Fuzzy-based controller is investigated in [6], showing the tracking performance numerically without implementing it on a practical testbed. The Fuzzy and Neural Networks based algorithms are presented in [7,8]. The robust control technique based on the sliding mode approach is also investigated for the ball balancing problem, see [9,10]. A Backstepping controller is designed in [11] to simulate the tracking problem of BBS. An anti-windup compensator (AWC) control scheme is also demonstrated in [12]. Similarly, the optimal performance of Model Predictive Control (MPC) is investigated to evaluate the tracking characteristics for ball and beam equipment [5].

Recent work was proposed designing a controller comprising two embedded loops based on linear fractional-order calculus [2]. The outer-loop and inner-loop control the ball's location on the beam and angle of servo gear, respectively. The prime focus of this method of control is to demonstrate a fractional-order control scheme to enhance efficiency, which is reported in several journals [13–17]. However, the performance is unsatisfactory because an integer order traditional controller is related to the non-integer order integrator, which is used in the outer loop.

In addition to that, Nonlinear Dynamic Inversion (NDI) was also implemented for this ball and beam underactuated system, see [18]. However, this control method requires the exact model of the plant dynamics.

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 [19]. The control objectives of GDI are defined in the form of differential state constraints which are calculated along the solution paths of the system dynamics. The control law is obtained by inverting the suggested constraints using Moore-Penrose Generalized Inverse (MPGI) [20]. Further, the GDI algorithm is made intelligent by augmenting the switching term based on the concept of classical SMC principle to ensure robustness against parametric uncertainties and un-modeled dynamics while guaranteeing semi-global asymptotic stability conditions [21–25].

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 [1], to validate control algorithms that are defined relative degrees with nonlinear attributes. The two significant ball and beam setup components are the ball and beam unit and rotary servo unit. One end of the finite length beam is fixed by a shaft, whereas the other end is connected to the rotary servo unit, as shown in the schematic diagram in Fig. 1. Therefore, this unstable architecture is well known as an open-loop system in control technology. The core objective is to orient and stabilize the ball placed on the beam according to the commanded reference input profile. In Fig. 1, x is the linear displacement ball, m_b is the mass, r_b is the beam's radius, and J_b is the ball's momentum, respectively. The relative angle of the beam concerning the horizontal plane is denoted by α, whereas the gravitational acceleration is mentioned by g. The expression of the linear acceleration of the ball is given as Eq. (1). (1)x¨(t)=mbgsin⁡α(t)rb2mbrb2+Jb

Similarly, the relation between the beam angle α and servo angle θ_l(t) is expressed as (2)sin⁡α(t)=sin⁡θl(t)rarmLbeamwhere L_beam is the length of the beam and r_arm is the gap between the rotary servo output gear and the coupled joint. To relate the linear displacement x with the rotary servo angle θ, place the expression of sinα(t) in the linear acceleration equation given by Eq. (2), it implies (3)x¨(t)=mbgθl(t)rarmrb2Lbeam(mbrb2+Jb)

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 Fig. 2. The subscript m represents the motor attributes in the given schematic, whereas l characterizes the load shaft variables. In the rotary servo unit, the resistance is denoted by R, L is the inductance, ω is the angular speed, τ is the applied torque, J is the moment of inertia, and B stands for the viscous friction. The variable e_b represents the back-electromotive voltage, where k_m is the back-emf constant. To realize the dynamical model of the rotary servo unit, the mechanical and electrical systems are combined to form the relationship between the angular speed of the motor load shaft ω_l and the applied controlled voltage V_m. The dynamical system is represented by the following equations (4)θ˙l=ωl (5)ω˙lJeq+Beqvωl=AmVm(t)where the equivalent moment of inertia J_eq is expressed as (6)Jeq=ηgkg2Jm+Jl

and the equivalent damping term B_eqv is given by (7)Beqv=ηgkg2ηmktkm+BeqRmRm

In the expressions of the equivalent moment of inertia and damping, η_g and η_m represent the efficiency of the gearbox and motor, respectively. The motor torque constant is denoted by k_t, and k_g is the gear ratio. The actuator gain A_m is computed as (8)Am=ηgkgηmktRm

The numerical value of the BBS parameters is listed in Tab. 1, see [1,26].

A cascade control architecture based on the time separation principle is proposed to control the ball's position on the beam, as shown in Fig. 3. Based on the measured ball position x, the outer positional loop computes the desired angle θ_ld of the servo load shaft by engaging the classical PD control. The inner fast attitude loop incorporates IDI control to track the desired angle θ_ld by generating the corresponding motor controlled voltage V such that the desired ball position on the beam is achieved in minimal time.

Given the desired and actual ball's position, i.e., x_d and x, the classical PD controller is engaged in the outer loop to generate the desired servo angle command θ_ld for the inner loop to minimize the positional error of the ball placed on the beam. The asymptotically stable virtual (desired) dynamics to compute θ_ld based on the positional error of the ball in the horizontal plane is expressed as (9)kd(x˙−x˙d)+kp(x−xd)=θldwhere k_p and k_d are controller gains chosen to calculate the suitable value of θ_ld such that x approached x_d in minimum time. It is noteworthy to mention here that the position feedback of the ball is acquired from an analog sensor with its inherent noise. By taking its derivative, it will generate an amplified high-frequency signal transmitted to the inner loop via feedback which causes the grinding noise in the motor. The derivative is replaced by a first-order filter governed following dynamics to compensate for this effect. (10)H(s)=ωfss+ωfwhere ω_f represents the cutoff frequency, whose value is set to be 6.28 rad/s.

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 (11)uIDI=ueq+urbt

The first component u_eq represents the baseline controller, which enforces the prescribed constraint dynamics based on the attitude deviation function for stable attitude tracking. On the other hand, u_rbt represents the discontinuous control element whose design principle is based on the classical SMC approach to address robustness against uncertain dynamics, parametric variations, and external disturbances.

The time derivatives of the sliding surfaces are evaluated as (21)s˙θl=ξ¨θl+c1ξ˙θl+c2ξθl (22)s˙ωl=ξ˙ωl+c3ξωl

It has been observed that the asymptotic convergence of the sliding surfaces sθl and sωl to zero implies the asymptotic realization of the constraint dynamics given by the Eqs. (13) and (14), and its equivalent algebraic form given by Eq. (15). Hence, the sliding vector s is defined as (23)s˙=AVm−B

To prove the asymptotic convergence of the error dynamics, place the control law Vm given by Eq. (18) in sliding mode dynamics given by Eq. (23), resulting in s˙=A{A+B−A+Ks∥s∥}−B (24)={AA+−I2×2}B−A+Ks∥s∥

Consider the following positive definite Lyapunov candidate function (25)V=12sTs

The time derivative of V is computed as V˙=sTs˙=sT{(AA+−I2×2)B−AA+Ks∥s∥} (26)=sT{AA+−I2×2)}B−sTAA+Ks∥s∥

The 2-(induced) norms of AA+ and AA+−I2×2 are unity. It follows that V˙≤0 is guaranteed if σmin(K)>∥B(x(0),0)∥, where σmin is the minimum singular value function. Moreover, because V˙=0 occurs only at s=02, it follows Lassale's principle [27] that s=02 is attractive. Moreover, the discontinuous term in the right-hand side of Eq. (26) implies that this condition on K guarantees finite-time convergence of s and s˙ to the zero vectors, which implies asymptotic convergence of eθl and eωl to zero. Finally, because it is assumed that the matrix gain K can be increased arbitrarily such that the stability condition is satisfied for any initial state condition, it follows that the design guarantees semi-global asymptotic stability of (eθl,e˙θl)=(eωl,e˙ωl)=(0,0).

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 Eq. (3) and the model of rotary servo unit given by Eqs. (4) and (5). The simulation environment is created in Simulink/Matlab, in which the closed-loop performance is evaluated by commanding the square wave profile having an amplitude of ±20m with 0.05Hz frequency, assuming nominal system parameters. The initial condition of the system state vector is set as [xx˙θlθ˙lωlω˙l]=[000000] The time histories of the position tracking curve of the ball is shown in Fig. 4, which is quite appealing. The inner loop rotary servo angle tracking performance using IDI control is depicted in Fig. 5, which demonstrates faster convergence towards the angular reference command. Finally, the servo motor controlled voltage well within the saturation limit is displayed in Fig. 6.

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 Fig. 7, which proves to be a benchmark testbed for testing and validating control techniques.

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 2ms for all the experiments.

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.