Stabilization control of inverted pendulum systems by fractional order PD controller based on D-decomposition technique
Abstract
Many systems in nature are inherently under-actuated, with fewer actuators than degrees of freedom. However, even with reduced number of actuators, these systems are able to produce complex movements. To be capable of performing such motions, complex control algorithms must be implemented. Classical benchmark examples for studying problems of this kind include inverted pendulum systems. This paper deals with stability problem of two types of inverted pendulum controlled by a fractional order PD controller. Rotational and cart inverted pendulum are highly nonlinear mechanical systems with one control input and two degrees of freedom.
Detailed mathematical model of both pendulums are derived using the Rodriguez method. Stabilization of pendulum around its unstable equilibrium point is achieved by using the fractional order PDα controller, in combination with partial feedback linearization technique. There are several methods for determining stability region of a closed loop system, a...nd D-decomposition is one of them. Herein, D-decomposition method is applied to the inverted pendulum case, and determining its stability regions in parameters space of a fractional order PD controller is presented. D-decomposition for linear fractional systems is investigated, and for the case of linear
parameters dependence. Fractional order control laws are represented by a transfer functions which are not rational, which gives rise to a problem of practical implementation of the corresponding control algorithms. A method for rational approximation of linear fractional order systems used in this paper is computationally efficient, accurate, and relies on the interpolation of the frequency characteristics of the system on a predefined set of target frequencies. The performance of the proposed method is demonstrated with experimental verification of the stabilization control of the cart pendulum system.
Keywords:
Inverted Pendulum / Fractional Order PID Control / Asymptotic Stability / D-decomposition Technique / Robust Control / Rational ApproximationsSource:
Booklet of Abstracts Mini-symposium “ Fractional Calculus with applications in problems of diffusion, control and dynamics of complex systems”, July 13, 2016, 2016, 28, 2016, 28-28Publisher:
- Faculty of Mechanical Engineering, Belgrade
- University of Belgrade Mathematical Institute, Serbian Academy of Sciences and Arts
Funding / projects:
- Sustainability and improvement of mechanical systems in energetic, material handling and conveying by using forensic engineering, environmental and robust design (RS-MESTD-Technological Development (TD or TR)-35006)
- Intelligent Control Systems of the Air-conditioning for the Purpose of Achieving Energy Efficient Exploitation Regimes in the Complex Operating Conditions (RS-MESTD-Technological Development (TD or TR)-33047)
- Energy efficiency Improvement of Hydro and Thermal power plants in EPS by development and implementation of power electronics based regulation and automation equipment (RS-MESTD-Technological Development (TD or TR)-33020)
Collections
Institution/Community
Mašinski fakultetTY - CONF AU - Mandić, Petar AU - Lazarević, Mihailo AU - Šekara, Tomislav PY - 2016 UR - https://machinery.mas.bg.ac.rs/handle/123456789/6566 AB - Many systems in nature are inherently under-actuated, with fewer actuators than degrees of freedom. However, even with reduced number of actuators, these systems are able to produce complex movements. To be capable of performing such motions, complex control algorithms must be implemented. Classical benchmark examples for studying problems of this kind include inverted pendulum systems. This paper deals with stability problem of two types of inverted pendulum controlled by a fractional order PD controller. Rotational and cart inverted pendulum are highly nonlinear mechanical systems with one control input and two degrees of freedom. Detailed mathematical model of both pendulums are derived using the Rodriguez method. Stabilization of pendulum around its unstable equilibrium point is achieved by using the fractional order PDα controller, in combination with partial feedback linearization technique. There are several methods for determining stability region of a closed loop system, and D-decomposition is one of them. Herein, D-decomposition method is applied to the inverted pendulum case, and determining its stability regions in parameters space of a fractional order PD controller is presented. D-decomposition for linear fractional systems is investigated, and for the case of linear parameters dependence. Fractional order control laws are represented by a transfer functions which are not rational, which gives rise to a problem of practical implementation of the corresponding control algorithms. A method for rational approximation of linear fractional order systems used in this paper is computationally efficient, accurate, and relies on the interpolation of the frequency characteristics of the system on a predefined set of target frequencies. The performance of the proposed method is demonstrated with experimental verification of the stabilization control of the cart pendulum system. PB - Faculty of Mechanical Engineering, Belgrade PB - University of Belgrade Mathematical Institute, Serbian Academy of Sciences and Arts C3 - Booklet of Abstracts Mini-symposium “ Fractional Calculus with applications in problems of diffusion, control and dynamics of complex systems”, July 13, 2016, 2016, 28 T1 - Stabilization control of inverted pendulum systems by fractional order PD controller based on D-decomposition technique EP - 28 SP - 28 UR - https://hdl.handle.net/21.15107/rcub_machinery_6566 ER -
@conference{ author = "Mandić, Petar and Lazarević, Mihailo and Šekara, Tomislav", year = "2016", abstract = "Many systems in nature are inherently under-actuated, with fewer actuators than degrees of freedom. However, even with reduced number of actuators, these systems are able to produce complex movements. To be capable of performing such motions, complex control algorithms must be implemented. Classical benchmark examples for studying problems of this kind include inverted pendulum systems. This paper deals with stability problem of two types of inverted pendulum controlled by a fractional order PD controller. Rotational and cart inverted pendulum are highly nonlinear mechanical systems with one control input and two degrees of freedom. Detailed mathematical model of both pendulums are derived using the Rodriguez method. Stabilization of pendulum around its unstable equilibrium point is achieved by using the fractional order PDα controller, in combination with partial feedback linearization technique. There are several methods for determining stability region of a closed loop system, and D-decomposition is one of them. Herein, D-decomposition method is applied to the inverted pendulum case, and determining its stability regions in parameters space of a fractional order PD controller is presented. D-decomposition for linear fractional systems is investigated, and for the case of linear parameters dependence. Fractional order control laws are represented by a transfer functions which are not rational, which gives rise to a problem of practical implementation of the corresponding control algorithms. A method for rational approximation of linear fractional order systems used in this paper is computationally efficient, accurate, and relies on the interpolation of the frequency characteristics of the system on a predefined set of target frequencies. The performance of the proposed method is demonstrated with experimental verification of the stabilization control of the cart pendulum system.", publisher = "Faculty of Mechanical Engineering, Belgrade, University of Belgrade Mathematical Institute, Serbian Academy of Sciences and Arts", journal = "Booklet of Abstracts Mini-symposium “ Fractional Calculus with applications in problems of diffusion, control and dynamics of complex systems”, July 13, 2016, 2016, 28", title = "Stabilization control of inverted pendulum systems by fractional order PD controller based on D-decomposition technique", pages = "28-28", url = "https://hdl.handle.net/21.15107/rcub_machinery_6566" }
Mandić, P., Lazarević, M.,& Šekara, T.. (2016). Stabilization control of inverted pendulum systems by fractional order PD controller based on D-decomposition technique. in Booklet of Abstracts Mini-symposium “ Fractional Calculus with applications in problems of diffusion, control and dynamics of complex systems”, July 13, 2016, 2016, 28 Faculty of Mechanical Engineering, Belgrade., 28-28. https://hdl.handle.net/21.15107/rcub_machinery_6566
Mandić P, Lazarević M, Šekara T. Stabilization control of inverted pendulum systems by fractional order PD controller based on D-decomposition technique. in Booklet of Abstracts Mini-symposium “ Fractional Calculus with applications in problems of diffusion, control and dynamics of complex systems”, July 13, 2016, 2016, 28. 2016;:28-28. https://hdl.handle.net/21.15107/rcub_machinery_6566 .
Mandić, Petar, Lazarević, Mihailo, Šekara, Tomislav, "Stabilization control of inverted pendulum systems by fractional order PD controller based on D-decomposition technique" in Booklet of Abstracts Mini-symposium “ Fractional Calculus with applications in problems of diffusion, control and dynamics of complex systems”, July 13, 2016, 2016, 28 (2016):28-28, https://hdl.handle.net/21.15107/rcub_machinery_6566 .