Radial basis function (RBF) neural networks have the advantages of excellent ability for the learning of the processes and certain immunity to disturbances when using in control systems. The robust trajectory tracking control of complex underactuated mechanical systems is a difficult problem that requires effective approaches. In particular, adaptive RBF neural networks are a good candidate to deal with that type of problems. In this document, a new method to solve the problem of trajectory tracking of an underactuated control moment gyroscope is addressed. This work is focused on the approximation of the unknown function by using an adaptive neural network with RBF fully tuned. The stability of the proposed method is studied by showing that the trajectory tracking error converges to zero while the solutions of the internal dynamics are bounded for all time. Comparisons between the model-based controller, a cascade PID scheme, the adaptive regressor-based controller, and an adaptive neural network-based controller previously studied are performed by experiments with and without two kinds of disturbances in order to validate the proposed method.

@article{sadeghzadeh_2020,
title = {Affine Linear Parameter-Varying Embedding of Nonlinear Models with Improved Accuracy and Minimal Overbounding},
author = {Sadeghzadeh, A.; Sharif, B.; Toth, R.},
journal = {arXiv},
year = {2020},
institution = {Eindhoven University of Technology, Netherlands; Institute for Computer Science and Control, Hungary},
abstract = {In this paper, automated generation of linear parameter-varying (LPV) state-space models to embed the dynamical behavior of nonlinear systems is considered, focusing on the trade-off between scheduling complexity and model accuracy and on the minimization of the conservativeness of the resulting embedding. The LPV state-space model is synthesized with affine scheduling dependency, while the scheduling variables themselves are nonlinear functions of the state and input variables of the original system. The method allows to generate complete or approximative embedding of the nonlinear system model and also it can be used to minimize complexity of existing LPV embeddings. The capabilities of the method are demonstrated on simulation examples and also in an empirical case study where the first-principle motion model of a 3-DOF control moment gyroscope is converted by the proposed method to LPV model with low scheduling complexity. Using the resulting model, a gain-scheduled controller is designed and applied on the gyroscope, demonstrating the efficiency of the developed approach. },
language = {English}
}

In this paper, automated generation of linear parameter-varying (LPV) state-space models to embed the dynamical behavior of nonlinear systems is considered, focusing on the trade-off between scheduling complexity and model accuracy and on the minimization of the conservativeness of the resulting embedding. The LPV state-space model is synthesized with affine scheduling dependency, while the scheduling variables themselves are nonlinear functions of the state and input variables of the original system. The method allows to generate complete or approximative embedding of the nonlinear system model and also it can be used to minimize complexity of existing LPV embeddings. The capabilities of the method are demonstrated on simulation examples and also in an empirical case study where the first-principle motion model of a 3-DOF control moment gyroscope is converted by the proposed method to LPV model with low scheduling complexity. Using the resulting model, a gain-scheduled controller is designed and applied on the gyroscope, demonstrating the efficiency of the developed approach.

@article{moreno-valenzuela_2020,
title = {Robust trajectory tracking control of an underactuated control moment gyroscope via neural network–based feedback linearization},
author = {Moreno–Valenzuela, J.; Montoya–Cháirez, J.; Santibáñez, V.},
journal = {Neurocomputing},
year = {2020},
institution = {Instituto Politécnico Nacional–CITEDI, Mexico; Tecnológico Nacional de México/Instituto Tecnológico de La Laguna, Mexico},
abstract = {In this document, an underactuated two degrees–of–freedom control moment gyroscope (CMG) is studied. Specifically, the problem of trajectory tracking in the non-actuated joint is addressed. The feedback linearization technique is used to design a model based–controller. Then, an adaptive neural network–based scheme is designed to add robustness with respect to model uncertainties. Studies on the internal and output dynamics are presented. The introduced theory is validated by means of real–time experiments. The comparisons among a linear controller, a cascaded PID–PID scheme and a known adaptive neural network controller are presented to assess the performance of the novel robust controller given in this work. Better tracking accuracy is obtained with the introduced approach. },
keywords = {Control moment gyroscope, underactuated systems, trajectory tracking control, neural networks, real–time experiments},
language = {English},
publisher = {Elsevier B.V.}
}

In this document, an underactuated two degrees–of–freedom control moment gyroscope (CMG) is studied. Specifically, the problem of trajectory tracking in the non-actuated joint is addressed. The feedback linearization technique is used to design a model based–controller. Then, an adaptive neural network–based scheme is designed to add robustness with respect to model uncertainties. Studies on the internal and output dynamics are presented. The introduced theory is validated by means of real–time experiments. The comparisons among a linear controller, a cascaded PID–PID scheme and a known adaptive neural network controller are presented to assess the performance of the novel robust controller given in this work. Better tracking accuracy is obtained with the introduced approach.

@article{chairez_2019,
title = {Adaptive control schemes applied to a control moment gyroscope of 2 degrees of freedom},
author = {Montoya–Cháirez, J.; Santibáñez, V.; Moreno–Valenzuela, J.},
journal = {Mechatronics},
year = {2019},
month = {02},
volume = {57},
institution = {Instituto Politécnico Nacional–CITEDI, Mexico; Instituto Tecnológico de La Laguna, Mexico},
abstract = {Control of underactuated mechanical systems has been an important trend in mechatronics and nonlinear systems. Typical examples are the Furuta pendulum and inertia wheel pendulum. On the other hand, gyroscopes are important in aerial and space systems. In this paper, a two-degree of freedom underactuated control moment gyroscope (CMG) is studied. This system is highly coupled of one joint to the another and is difficult to control, which make it an important benchmark system. The problem addressed in this paper is to achieve robust motion control of the underactuated CMG. Firstly, a trajectory tracking controller is developed by using the feedback linearization technique. Secondly, two new adaptive algorithms are introduced, which correspond to an adaptive neural network algorithm and an adaptive model regressor scheme. A real–time experimental comparison is carried out among a linear PD control law, a cascade PID-PID controller and the introduced schemes. The real-time experimental study validates the introduced theory, where the performance of the controllers is evaluated with and without a periodic disturbance at the input. },
keywords = {Control moment gyroscope, Underactuated systems, Trajectory tracking control, Adaptive control, Real–time experiments},
language = {English},
publisher = {Elsevier Ltd.},
pages = {73-85}
}

Control of underactuated mechanical systems has been an important trend in mechatronics and nonlinear systems. Typical examples are the Furuta pendulum and inertia wheel pendulum. On the other hand, gyroscopes are important in aerial and space systems. In this paper, a two-degree of freedom underactuated control moment gyroscope (CMG) is studied. This system is highly coupled of one joint to the another and is difficult to control, which make it an important benchmark system. The problem addressed in this paper is to achieve robust motion control of the underactuated CMG. Firstly, a trajectory tracking controller is developed by using the feedback linearization technique. Secondly, two new adaptive algorithms are introduced, which correspond to an adaptive neural network algorithm and an adaptive model regressor scheme. A real–time experimental comparison is carried out among a linear PD control law, a cascade PID-PID controller and the introduced schemes. The real-time experimental study validates the introduced theory, where the performance of the controllers is evaluated with and without a periodic disturbance at the input.

@conference{pejcic_2019,
title = {Experimental Verification of Sum-Of-Squares-Based Controller Tuning Technique with Extension to Parallel Multimodel Uncertainty Processing},
author = {Pejcic, I.; Jones, C.N.},
booktitle = {2019 18th European Control Conference (ECC)},
year = {2019},
institution = {Ecole Polytechnique Federale de Lausanne, Switzerland},
abstract = {Many control schemes require significant tuning effort to achieve desired performance targets, causing a need for general tools that can perform controller tuning in an automated fashion. This paper represents a continuation of the previous work in which a tuning method capable of designing Explicit Model Predictive Controllers for control of nonlinear systems was developed. Besides demonstrating a broader applicability of the tuning method by applying it here to a non-optimization-based, nonlinear control policy, the primary purpose of this paper is to provide experimental validation of the method by its application to a physical system, as well as to extend the method’s practical computational capability in case of multimodel plant uncertainty. The experimental results consist of the method’s application to tuning of an anti-windup equipped PID controller which is designed to be robustly stable with respect to multimodel uncertainty in the considered mechanical experimental setup. },
language = {English},
publisher = {IEEE},
isbn = {978-1-7281-1314-2}
}

Many control schemes require significant tuning effort to achieve desired performance targets, causing a need for general tools that can perform controller tuning in an automated fashion. This paper represents a continuation of the previous work in which a tuning method capable of designing Explicit Model Predictive Controllers for control of nonlinear systems was developed. Besides demonstrating a broader applicability of the tuning method by applying it here to a non-optimization-based, nonlinear control policy, the primary purpose of this paper is to provide experimental validation of the method by its application to a physical system, as well as to extend the method’s practical computational capability in case of multimodel plant uncertainty. The experimental results consist of the method’s application to tuning of an anti-windup equipped PID controller which is designed to be robustly stable with respect to multimodel uncertainty in the considered mechanical experimental setup.

@article{kammer_2018,
title = {A Data-Driven Fixed-Structure Control Design Method with Application to a 2-DOF Gyroscope},
author = {Kammer, C.; Karimi, A.},
year = {2018},
institution = {Laboratoire d’Automatique, Ecole Polytechnique Federale de Lausanne, Switzerland},
abstract = {This paper presents the practical aspects and application of a novel data-driven, fixed-structure, robust control design method. Only the frequency response data of the system is needed for the design, and no parametric model is required. The method can be used to design fully parametrized continuous- or discrete-time matrix transfer function controllers. The control performance is specified as constraints on the H-infinity or H2 norm of weighted sensitivity functions, and a convex formulation of the robust design problem is proposed. An application of the presented method is explored on an experimental setup, where a multivariable controller for a gyroscope is designed based only on the measured frequency response of the system. },
language = {English}
}

This paper presents the practical aspects and application of a novel data-driven, fixed-structure, robust control design method. Only the frequency response data of the system is needed for the design, and no parametric model is required. The method can be used to design fully parametrized continuous- or discrete-time matrix transfer function controllers. The control performance is specified as constraints on the H-infinity or H2 norm of weighted sensitivity functions, and a convex formulation of the robust design problem is proposed. An application of the presented method is explored on an experimental setup, where a multivariable controller for a gyroscope is designed based only on the measured frequency response of the system.

@article{cordero_2018,
title = {Inteconnection and Damping Assignment Passivity-based Control of an Underactuated 2 DOF Gyroscope},
author = {Cordero, G.; Santibanez, V.; Dzul, A.; Sandoval, J.},
journal = {International Journal of Applied Mathematics and Computer Science},
year = {2018},
volume = {28},
number = {4},
institution = {Laguna Institute of Technology, Mexico; La Paz Institute of Technology, Mexico},
abstract = {In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated gyroscope. First, the equations of motion of the complete system (3-DOF) are presented in both Lagrangian and Hamiltonian formalisms. Moreover, the conditions to reduce the system from a 3-DOF to a 2-DOF gyroscope, by using Routh’s equations of motion, are shown. Next, the solutions of the partial differential equations involved in getting the proper controller are presented using a reduction method to handle them as ordinary differential equations. Besides, since the gyroscope has no potential energy, it presents the inconvenience that neither the desired potential energy function nor the desired Hamiltonian function has an isolated minimum, both being only positive semidefinite functions; however, by focusing on an open-loop nonholonomic constraint, it is possible to get the Hamiltonian of the closed-loop system as a positive definite function. Then, the Lyapunov direct method is used, in order to assure stability. Finally, by invoking LaSalle’s theorem, we arrive at the asymptotic stability of the desired equilibrium point. Experiments with an underactuated gyroscopic mechanical system show the effectiveness of the proposed scheme. },
keywords = {gyroscope device, gyroscopic forces, cyclic coordinates, generalized momenta, Routh’s equations, IDA-PBC},
language = {English},
pages = {661-677}
}

In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated gyroscope. First, the equations of motion of the complete system (3-DOF) are presented in both Lagrangian and Hamiltonian formalisms. Moreover, the conditions to reduce the system from a 3-DOF to a 2-DOF gyroscope, by using Routh’s equations of motion, are shown. Next, the solutions of the partial differential equations
involved in getting the proper controller are presented using a reduction method to handle them as ordinary differential equations. Besides, since the gyroscope has no potential energy, it presents the inconvenience that neither the desired potential energy function nor the desired Hamiltonian function has an isolated minimum, both being only positive semidefinite functions; however, by focusing on an open-loop nonholonomic constraint, it is possible to get the Hamiltonian of the closed-loop system as a positive definite function. Then, the Lyapunov direct method is used, in order to assure stability. Finally, by invoking LaSalle’s theorem, we arrive at the asymptotic stability of the desired equilibrium point. Experiments with an underactuated gyroscopic mechanical system show the effectiveness of the proposed scheme.

@article{karimi_2016,
title = {A Data-driven Approach to Robust Control of Multivariable Systems by Convex Optimization},
author = {Karimi, A.; Kammer, C.},
year = {2016},
institution = {Laboratoire d'Automatique, Ecole Polytechnique Federale de Lausanne (EPFL), Switzerland},
abstract = {The frequency-domain data of a multivariable system in different operating points is used to design a robust controller with respect to the measurement noise and multimodel uncertainty. The controller is fully parametrized in terms of matrix polynomial functions and can be formulated as a centralized, decentralized or distributed controller. All standard performance specifications like H2, H-infinity and loop shaping are considered in a unified framework for continuous- and discrete-time systems. The control problem is formulated as a convex-concave optimization problem and then convexified by linearization of the concave part around an initial controller. The performance criterion converges monotonically to a local optimal solution in an iterative algorithm. The effectiveness of the method is compared with fixed-structure controllers using non-smooth optimization and with full-order optimal controllers via simulation examples. Finally, the experimental data of a gyroscope is used to design a data-driven controller that is successfully applied on the real system. },
keywords = {Data-driven control, robust control, convex optimization},
language = {English}
}

The frequency-domain data of a multivariable system in different operating points is used to design a robust controller with respect to the measurement noise and multimodel uncertainty. The controller is fully parametrized in terms of matrix polynomial functions and can be formulated as a centralized, decentralized or distributed controller. All standard performance specifications like H2, H-infinity and loop shaping are considered in a unified framework for continuous- and discrete-time systems. The control problem is formulated as a convex-concave optimization problem and then convexified by linearization of the concave part around an initial controller. The performance criterion converges monotonically to a local optimal solution in an iterative algorithm. The effectiveness of the method is compared with fixed-structure controllers using non-smooth optimization and with full-order optimal controllers via simulation examples. Finally, the experimental data of a gyroscope is used to design a data-driven controller that is successfully applied on the real system.

@article{oliveira_2015,
title = {Global tracking for a class of uncertain nonlinear systems with unknown sign-switching control direction by output feedback},
author = {Oliveira, T.R.; Peixoto, A.J.; Hsu, L.},
journal = {International Journal of Control},
year = {2015},
volume = {88},
number = {9},
abstract = {This paper addresses the design of a sliding mode controller for a class of high-order uncertain nonlinear plants with unmatched state-dependent nonlinearities and unknown sign of the high frequency gain, i.e., the control direction is assumed unknown. Differently from most previous studies, the control direction is allowed to switch its sign. We show that it is possible to obtain global exact tracking using only output-feedback by coupling a relay periodic switching function with a norm state observer. One significant advantage of the new scheme is its robustness and improved transient response under arbitrary changes of the control direction which have been theoretically demonstrated for jump variations and successfully tested by simulations. The proposed controller is also evaluated with a DC motor control experiment. },
keywords = {uncertain nonlinear systems; unknown control direction; Nussbaum gain; sliding mode control; variable structure systems; output-feedback; global exact tracking},
language = {English},
publisher = {Taylor & Francis},
pages = {1895-1910}
}

This paper addresses the design of a sliding mode controller for a class of high-order uncertain nonlinear plants with unmatched state-dependent nonlinearities and unknown sign of the high frequency gain, i.e., the control direction is assumed unknown. Differently from most previous studies, the control direction is allowed to switch its sign. We show that it is possible to obtain global exact tracking using only output-feedback by coupling a relay periodic switching function with a norm state observer. One significant advantage of the new scheme is its robustness and improved transient response under arbitrary changes of the control direction which have been theoretically demonstrated for jump variations and successfully tested by simulations. The proposed controller is also evaluated with a DC motor control experiment.