Discrete-Inverse Optimal Control Applied to the Ball and Beam Dynamical System: A Passivity-Based Control Approach

Montoya, Oscar Danilo; Gil-González, Walter; Ramírez-Vanegas, Carlos

doi:10.3390/sym12081359

Cited by 11 publications

(5 citation statements)

References 21 publications

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“…with ν = 2 − µ and 1 − γ ≥ µν, which make each of the four additive terms of q(t, x t ) in (34), from (31), non-negative, as seen as follows concerning the first one:…”

Section: Theorem 4 Assume Thatmentioning

confidence: 99%

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On the Stabilization through Linear Output Feedback of a Class of Linear Hybrid Time-Varying Systems with Coupled Continuous/Discrete and Delayed Dynamics with Eventually Unbounded Delay

Sen

2022

Mathematics

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This research studies a class of linear, hybrid, time-varying, continuous time-systems with time-varying delayed dynamics and non-necessarily bounded, time-varying, time-differentiable delay. The considered class of systems also involves a contribution to the whole delayed dynamics with respect to the last preceding sampled values of the solution according to a prefixed constant sampling period. Such systems are also subject to linear output-feedback time-varying control, which picks-up combined information on the output at the current time instant, the delayed one, and its discretized value at the preceding sampling instant. Closed-loop asymptotic stabilization is addressed through the analysis of two “ad hoc” Krasovskii–Lyapunov-type functional candidates, which involve quadratic forms of the state solution at the current time instant together with an integral-type contribution of the state solution along a time-varying previous time interval associated with the time-varying delay. An analytic method is proposed to synthesize the stabilizing output-feedback time-varying controller from the solution of an associated algebraic system, which has the objective of tracking prescribed suited reference closed-loop dynamics. If this is not possible—in the event that the mentioned algebraic system is not compatible—then a best approximation of such targeted closed-loop dynamics is made in an error-norm sense minimization. Sufficiency-type conditions for asymptotic stability of the closed-loop system are also derived based on the two mentioned Krasovskii–Lyapunov functional candidates, which involve evaluations of the contributions of the delay-free and delayed dynamics.

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“…with ν = 2 − µ and 1 − γ ≥ µν, which make each of the four additive terms of q(t, x t ) in (34), from (31), non-negative, as seen as follows concerning the first one:…”

Section: Theorem 4 Assume Thatmentioning

confidence: 99%

“…In future works, it is planned to extend the results of this paper to the hyperstability and passivity theories, [32][33][34][35][36] by designing the controller gains so that "ad hoc" Popov stype inequalities be satisfied by a feedback control loop under generic nonlinear timevarying control laws.…”

Section: A CL (T)p(t) + P(t)mentioning

confidence: 99%

On the Stabilization through Linear Output Feedback of a Class of Linear Hybrid Time-Varying Systems with Coupled Continuous/Discrete and Delayed Dynamics with Eventually Unbounded Delay

Sen

2022

Mathematics

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“…The simulated results were compared with the proportional integration differentiation controller in which ADRC had a better performance than the integration differentiation controller. While Howimanporn et al [17] developed a nonlinear discrete optimal control technique for the regulation of all the state variables in the discrete mode of the BBS. The proposed controller showed passivity, stability, and optimality properties during closed-loop operation.…”

Section: Introductionmentioning

confidence: 99%

Performance Comparison of the Ball and Beam System using Linear Quadratic Regulator Controller

Umar¹,

Mu’azu²,

Haruna³

et al. 2023

PID Control for Linear and Nonlinear Industrial Processes

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This paper proposes the performance comparison of a linear quadratic regulator (LQR) controller for the ball and beam system (BBS). The BBS is a standard benchmark control system, which has two degree-of-freedom (2 DOF). It is an open loop and a highly nonlinear unstable system. This makes its parameter difficult to be estimated accurately, hence designing a controller for it is a challenging task. MatheThe BBS was modelled using Euler–Lagrange modeling technique, while the LQR controller was used for the stabilization of the ball on the beam. Simulation was done in MATLAB/Simulink 2022b environment, and the results simulated showed that for the two weighting matrices (Q and R), the state weighting matrix had a higher penalty on the ball displacement, ball velocity, beam angle, and beam angular velocity at lower values of Q. For the state weighting matrix had a better effect of penalty performance on the BBS with lower values. Also, as the diagonal element of the state weighting matrix Q increases from 0.1 to 20, the values of the optimal controller K increase, the reduced Ricatti matrix P increases, and the estimated eigenvalues E reduce. This implies that the ball displacement, ball velocity, beam angle, and beam angular velocity are better at lower values of Q.

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“…This approach however necessitates the utilization of sophisticated tools and hardware, potentially limiting its applicability to smaller or less complex control systems. Previous studies have also explored the use of nonlinear controllers such as neural networks [22]- [25], fuzzy logic [26]- [29], backstepping [30], deep reinforcement learning [31], and passivity-based control [32], as alternative strategies for stabilizing the BnB system.…”

Section: Introductionmentioning

confidence: 99%

Modeling and Hybrid PSO-WOA-Based Intelligent PID and State-Feedback Control for Ball and Beam Systems

Ahmad

2023

IEEE Access

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The ball and beam (BnB) system serves as a benchmark in control engineering as it provides a foundational concept applicable to addressing stabilization challenges of various underactuated nonlinear systems. This includes tasks like maintaining the balance of goods carried by mobile robots and controlling the attitude of unmanned aerial vehicles. In this study, the focus is on enhancing control optimization strategies for BnB systems that take into account inherent nonlinearities arising from actuator constraints and state measurements. The work introduces a novel intelligent control approach, termed hybrid PSO-WOA, which combines Particle Swarm Optimization (PSO) and Whale Optimization Algorithm (WOA) to automate optimal parameter search for proportional-integral-derivative (PID) and state feedback (SF) controllers. The collaborative technique between PSO and WOA is formulated to strike a balance between exploration and exploitation phases, and to mitigate premature convergence risks due to the system's complexities. Additionally, three control schemes, namely cascade PID-PID, cascade PID-SF, and cascade PID-observer are introduced, each with tailored cost functions for optimization through the hybrid PSO-WOA algorithm, accommodating both measurable and unmeasurable state scenarios. Simulation results consistently demonstrate the superior performance of the hybrid approach compared to individual PSO and WOA methods, as well as conventional PID and linear quadratic regulator approaches. Notable, the hybrid approach exhibits a significant improvement in error metrics, reducing integral-time absolute error by 18.99%, integral squared error by 35.37%, and steady-state error by 92.86%. This substantial enhancement suggests promising directions for future research in automated control parameter tuning for underactuated nonlinear systems.

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Discrete-Inverse Optimal Control Applied to the Ball and Beam Dynamical System: A Passivity-Based Control Approach

Cited by 11 publications

References 21 publications

On the Stabilization through Linear Output Feedback of a Class of Linear Hybrid Time-Varying Systems with Coupled Continuous/Discrete and Delayed Dynamics with Eventually Unbounded Delay

On the Stabilization through Linear Output Feedback of a Class of Linear Hybrid Time-Varying Systems with Coupled Continuous/Discrete and Delayed Dynamics with Eventually Unbounded Delay

Performance Comparison of the Ball and Beam System using Linear Quadratic Regulator Controller

Modeling and Hybrid PSO-WOA-Based Intelligent PID and State-Feedback Control for Ball and Beam Systems

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