The Inverted Pendulum in Control Theory and Robotics: From theory to new innovations
2: Department of Control and Robotics, National University of Mexico, Mexico City, Mexico
The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms. It is also an area of active study, with many new innovations and applications - for example, the problem is solved in the technology of the Segway, a self-balancing transportation device. This book provides an overall picture of historical and current trends and the developments in nonlinear control theory, based on the simple structure and rich nonlinear model of the inverted pendulum. After an introduction to the system and open/current problems, the book covers the topic in four parts: applications of robust state estimation and control to pendulum-cart systems; controllers for under-actuated mechanical systems; nonlinear controllers for mobile inverted pendulum systems; and robust controllers based observers via Takagi-Sugeno or linear approaches. With contributions from international researchers in the field, The Inverted Pendulum in Control Theory and Robotics is essential reading for researchers, scientists, engineers and students in the field of control theory, robotics and nonlinear systems
Inspec keywords: robust control; state estimation; linear systems; pendulums; nonlinear control systems; robots
Other keywords: Takagi-Sugeno approach; underactuated mechanical systems; control theory; robotics; nonlinear controllers; inverted pendulum; robust state estimation; linear approach; robust control
Subjects: Linear control systems; Stability in control theory; Nonlinear control systems; Robotics; General and management topics
- Book DOI: 10.1049/PBCE111E
- Chapter DOI: 10.1049/PBCE111E
- ISBN: 9781785613203
- e-ISBN: 9781785613210
- Page count: 450
- Format: PDF
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Front Matter
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1 The inverted pendulum: history and survey of open and current problems in control theory and robotics
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The inverted pendulum is a classical problem in dynamics and control theory widely used as a fundamental system for testing emerging control algorithms. In spite of its very simple structure appropriate for developing real-time implementation, the inverted-pendulum model is considered the richest one among common robotic benchmarks. Not only it can describe many engineering problems but also it can explain several biological examples. Various mathematical models and experimental designs for the inverted pendulum exist offering an attractive tool for education and research. The objective of this survey is to present an overview of the available varieties of such system, highlighting the richness of its dynamics and then providing an overall picture of different control design approaches and trendy robotic problems related to its simple structure. In total, 300 references in the open literature, dating back to Galileo's first experiments written in 1602, are compiled to provide an overall picture of historical, current and challenging developments based on the stabilization principle of the inverted pendulum.
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Part I: Robust state estimation and control: application to pendulum-cart systems
2 State estimation and parameter identification via sliding-mode techniques: pendulum-cart system
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In this chapter, the problems of state estimation and parameter identification for the pendulum-cart system are addressed. Different high-order sliding-modes techniques are applied for such a mechanical system. The mathematical model is studied and a couple of high-order sliding-modes observers are proposed to estimate the state, in spite of disturbances; exactly and in a finite or fixed time, respectively. Then, using the exact state estimation, two parameter identification algorithms based on sliding-modes techniques are introduced to identify the unknown parameters of the system, i.e. the friction coefficients. Some experiments and comparisons are presented to illustrate the effectiveness of the presented algorithms. Finally, some concluding remarks are given at the end of the chapter.
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Part I. Robust state estimation and control: application to pendulum-cart systems
3 Higher order sliding-mode stabilization of inverted cart-pendulum
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Stabilization of a cart-pendulum system through a generalized higher order surface-controller design is presented. The singular Linear Quadratic (LQ) method presents a natural relationship between order of singularity of a given performance index and the order of sliding-mode controller. Thus, several arbitrary relative degree optimal sliding surfaces and its corresponding higher order sliding-mode controller can be specified for a given system. Continuous higher order sliding-mode controllers are obtained through a robustification method for arbitrary relative degree nominal controllers based on integral sliding modes and supertwisting algorithm. An agreement between the accuracy/complexity of the Continuous Higher Order Sliding Mode (CHOSM) controller and the limited accuracy offered by the system's sensors and actuators is obtained through several experiments.
4 Stabilization and tracking control of the inverted pendulum on a cart via a modified PSO fractional order PID controller
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In this chapter, a fractional-order proportional integral derivative controller, PIλDμ, based super-twisting observer for the cart inverted pendulum system is proposed. The mathematical model of the underactuated robotic system is derived using Lagrange equation and Grunwald-Letnikov fractional calculus with the physical parameters of a commercial device, the Googol Technology experimental Laboratory. PIλDμ parameters are optimized using a modified Particle Swarm intelligence optimization approach with the help of a multiobjective fitness function. A comparative analysis with the classical Particle Swarm Optimization algorithm shows the superiority of the proposed approach.
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Part II. Controllers for underactuated mechanical systems
5 Model-free control of the inertia wheel inverted pendulum with real-time experiments
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In this chapter, we propose to design a control scheme based on model-free technique to deal with underactuation in stable limit cycle generation. In order to achieve stable limit cycles on both coordinates of the inertia wheel inverted pendulum, we first design a family of p-parameterized periodic trajectories for the pendulum's angle. Those trajectories are then tracked using the control input thanks to a classical model-free controller. Since the system is underactuated and nonminimum phase, a second controller is needed to update the parameter p of the above trajectories in order to deal with the convergence of the internal dynamics of the system.To achieve this control, we propose a second model-free controller using actuated coordinate (inertia wheel) as output and trajectories' parameter p as control input. Note that this control scheme can be easily applied to the stabilization case by carefully choosing appropriate trajectories. Numerical simulations as well as real-time experiments are presented to show the effectiveness of the proposed control scheme and its ability for external disturbances rejection.
6 Output feedback second-order sliding-mode tracking control for perturbed inertia wheel pendulum
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This paper addresses the robust output feedback tracking control problem for an inertia wheel pendulum in the presence of uniformly bounded matched disturbances. The periodic motion of the pendulum will be at the upright position which corresponds to the unstable equilibrium point of the unforced system. A two-relay-controller-based reference model was developed for generating the desired trajectories to be tracked by the unactuated link of the inertia wheel pendulum and then design an output feedback robust tracking controller. The desired amplitude and frequency were tuned by choosing the two-relay control gains properly. A second-order sliding-mode tracking controller interconnected with a second-order sliding-mode observer was capable to track the prescribed reference trajectory rejecting matched external disturbances. Performance issues of the constructed controller-observer were illustrated in a numerical study.
7 Switched integral sliding mode control for robust generation of self-oscillation in pendulum systems
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Self-oscillations for pendulum systems generated by a two-relay controller are robustified by using a switched integral sliding mode control. The robustification is theoretically exact, assuring the frequency and amplitude of the oscillations remain despite the presence of matched uncertainties/perturbations. To illustrate the efficacy of the proposed robustifying strategy, an inertial wheel pendulum is considered along the chapter.
8 Finite-time stabilization of underactuated mechanical systems in the presence of uncertainties: application to the cart-pole system
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Underactuated mechanical systems [13,14,27] are those systems with less control inputs than generalized coordinates (called also degrees of freedom), i.e., they have unactuated generalized coordinates. For such systems, the unactuated generalized coordinates may indirectly be controlled by the actuated coordinates through the dynamic coupling, inherent to these systems [8]. This coupling is often nonlinear, resulting in generally nonintegrable dynamic constraints and therefore second-order nonholonomic.
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Part III. Nonlinear controllers for mobile inverted pendulum systems
9 Advances in robust control of mobile wheeled inverted pendulum
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There has been increasing interest in a type of underactuated mechanical systems, mobile-wheeled inverted-pendulum (MWIP) models, which are widely used in autonomous robotics and intelligent vehicles. To cope with the model uncertainties and external disturbances, several robust controllers are designed for the MWIP models. For the velocity-tracking problem of the MWIP systems, we proposed two sliding-mode-control (SMC) methods. There is still a steady tracking error when the first SMC method is used. By assuming a novel sliding surface, the second SMC method is designed to solve this problem. Using a coordinate transformation, the non-“Class-I” type underactuated MWIP system is presented as a semistrict feedback form which is convenient for controller design. A dynamic surface controller with a nonlinear disturbance observer (DSCNDO) is then designed to solve the balance control problem of the MWIP systems. The proposed DSCNDO can compensate the external disturbances and the model uncertainties to improve the system performance significantly. The stabilities of the closed-loop MWIP systems using the proposed methods are proved by Lyapunov theorem. The effectiveness of all the methods is verified by numerical simulations.
10 Case studies on non-linear control theory of the inverted pendulum
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This chapter deals with the control of inverted pendulums through non-linear control theory. Simulations of the proposed control method are carried out in MATLAB® environment with satisfactory results in controlling systems of single and double-link inverted pendulum installed in a cart, for a range of starting positions. The proposed control technique can be extended to the control of many other non-linear systems. Examples are proposed as referring to design and teaching applications. In particular, a single inverted pendulum system constructed using Lego bricks and controlled by Lego Mindstorms EV3 is used to prove the user-friendliness of the proposed control method as implemented by Master students at Sheffield Hallam University. Moreover, 3 degrees of freedom parallel manipulator Cassino Parallel Manipulator is investigated for replacing a cart and providing three dimensional motions to an inverted pendulum while minimising dynamics effects at design stage.
11 Bipedal-double-pendulum walking robot control using recurrent hybrid neural network
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This chapter presents neural control scheme ofa planar-like double-pendulum-bipedal robot. For simplicity, only a five-link planar system is considered. The system effectively acts as two dynamically interacting planar robot arms. The scheme employs a single neural controller for the whole biped. Recurrent networks have feedback connections and thus an inherent memory for dynamics which makes them suitable for dynamic system modeling. A feature of the networks adopted is their hybrid hidden layer which includes both linear and nonlinear neurons. The standard proportional derivative (PD) controller is also employed for comparison. The results presented show the superior ability of the proposed neural control scheme at adapting to changes in the dynamics parameters of the bipedal robot.
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Part IV. Robust controllers-based observers via Takagi-Sugeno or linear approaches
12 A survey on the polytopic Takagi-Sugeno approach: application to the inverted pendulum
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This chapter gives a general scope, states the main results obtained and methods used for the polytopic Takagi-Sugeno approach with a detailed application to the inverted pendulum. Modeling, observer and controller design will be considered.
13 Robust fault-tolerant control of nonlinear inverted pendulum and cart system with simultaneous actuator and sensor faults sliding-mode observer
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In this chapter, we propose a robust active fault-tolerant control (AFTC) scheme for a class of uncertain nonlinear systems with simultaneous actuator and sensor faults described via Takagi-Sugeno (T-S) multiple models. First, by transforming the sensor fault into pseudoactuator fault, a novel T-S sliding-mode observer (TS-SMO) with two discontinuous terms is developed to provide separate estimates of the actuator and sensor faults for the purpose of fault compensation. The robustness of the proposed observer against uncertainties has been taken into account via H∞ norm minimization. Second, we use obtained on-line fault estimation information to design dynamic output feedback controller (DOFC) for robustly compensating the effects of actuator and sensor faults from the system inputs and outputs and guarantee the stability of the overall closed-loop system. The stability proof with H∞ performances and D-stability constraints is formulated as a linear matrix inequalities (LMI) optimization problem. The effectiveness of the proposed robust AFTC approach to treat simultaneous actuator and sensor faults is illustrated using a nonlinear inverted pendulum with cart system.
14 LMI-based control design for balancing and attitude stabilization of inverted pendulums
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This chapter explores the dynamic modeling and feedback stabilization of different types of inverted pendulums (IPs). It contains a theoretical analysis of the system dynamics and control methods, as well as a summary of MATLAB®/Simulink® simulation results. There are two primary objectives of this chapter: 1. To provide technical results pertaining to robust control-design approaches using convex optimization methods over linear-matrix inequalities. The design approaches include linear quadratic regulator (LQR), linear quadratic regulator with integral gain (LQRI), model-predictive control, H2 control, H∞ control and multiobjective state feedback. 2. To demonstrate the application of these approaches to the balancing and attitude stabilization of IP systems including pendulums on cart, double and triple link systems on pivot, rotational double IP and IP-type assistant robot.
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Back Matter
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