Global stabilisation of underactuated mechanical systems via PID passivity-based control

Romero, José Guadalupe; Donaire, Alejandro; Ortega, Roméo; Borja, Pablo

doi:10.1016/j.automatica.2018.06.040

Cited by 62 publications

(34 citation statements)

References 18 publications

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“…The PID control has the form of

u_{t} = K_{p} {[x_{c}, θ]}^{⊤} + K_{i} \int_{0}^{t} {[x_{c}, θ]}^{⊤} d τ + K_{d} {[{\dot{x}}_{c}, \dot{θ}]}^{⊤},

where K p , K i , and K d stands to the proportional, integral, and derivative gains. They require suitable tuning procedure . We use the Matlab control toolbox to tune the PID gains for the ideal control case, obtaining the following values K p =[5.12,20.34] ⊤ , K i =[1.54,0.57] ⊤ , K d =[1.51,1.56] ⊤ .…”

Section: Simulation and Experimentsmentioning

confidence: 99%

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Robust control under worst‐case uncertainty for unknown nonlinear systems using modified reinforcement learning

Perrusquía

2020

Intl J Robust & Nonlinear

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Reinforcement learning (RL) is an effective method for the design of robust controllers of unknown nonlinear systems. Normal RLs for robust control, such as actor-critic (AC) algorithms, depend on the estimation accuracy. Uncertainty in the worst case requires a large state-action space, this causes overestimation and computational problems. In this article, the RL method is modified with the k-nearest neighbor and the double Q-learning algorithm. The modified RL does not need the neural estimator as AC and can stabilize the unknown nonlinear system under the worst-case uncertainty. The convergence property of the proposed RL method is analyzed. The simulations and the experimental results show that our modified RLs are much more robust compared with the classic controllers, such as the proportional-integral-derivative, the sliding mode, and the optimal linear quadratic regulator controllers. K E Y W O R D Sk-nearest neighbors, double estimator, overestimation, robust reward, state-action space, worst-case uncertainty INTRODUCTIONThe objective of robust control is to achieve robust performance in presence of disturbances. Most robust controllers are inspired by the optimal control theory, such as  2 control, 1,2 which minimizes a certain cost function to find an optimal controller. The most popular  2 controller is the linear quadratic regulator (LQR). 3 It does not work well in presence of disturbances. The  ∞ control can find a robust controller when the system has disturbances. Its performance is poor compared with the  2 control. 4 The combination of  2 and  ∞ , called  2 ∕ ∞ control, has both advantages, that is., it has optimal performance with bounded disturbances. 5 The  2 ∕ ∞ controller design needs a complete knowledge of the system dynamics. 6 These controllers are model-based. The time-varying quadratic optimization can be calculated by the zeroing neural network. It can simultaneously achieve the finite-time convergence and inherent noise tolerance. 7 However, it requires prefect activation functions. Model-free controllers, like the proportional-integral-derivative (PID) control, 8,9 the sliding mode control (SMC), 10 neural control, 11-16 among others, do not require dynamic knowledge of the system. However, parameter tuning and some prior knowledge of the disturbances prevent these model-free controllers to perform optimally.Reinforcement learning (RL) 17,18 is another effective method without models. It is designed in the sense of  2 control. 19 The temporal difference (TD) rule, such as Q-learning, is applied to find an optimal solution for Markov decision processes. 20 The advantage of RL over the other model-free methods is that it can reach optimal performances. Recent results show that RL methods can learn  2 and  ∞ controllers without system dynamics. 21 2920The main objective of RL for  2 and  ∞ is to minimize the total accumulative reward. For a robust controller, the reward is designed in the sense of control problems  2 and  ∞ . This robust reward can be static or dynam...

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“…The PID control has the form of

u_{t} = K_{p} {[x_{c}, θ]}^{⊤} + K_{i} \int_{0}^{t} {[x_{c}, θ]}^{⊤} d τ + K_{d} {[{\dot{x}}_{c}, \dot{θ}]}^{⊤},

Section: Simulation and Experimentsmentioning

confidence: 99%

“…Model‐free controllers, like the proportional‐integral‐derivative (PID) control, the sliding mode control (SMC), neural control, among others, do not require dynamic knowledge of the system. However, parameter tuning and some prior knowledge of the disturbances prevent these model‐free controllers to perform optimally.…”

Section: Introductionmentioning

confidence: 99%

Robust control under worst‐case uncertainty for unknown nonlinear systems using modified reinforcement learning

Perrusquía

2020

Intl J Robust & Nonlinear

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“…In order to simplify the PDE problem Viola et al (2007) have introduced a change of coordinates and a modification of the target dynamics. With the objective to completely avoid PDEs, the following leading methods have been proposed: constructive procedures (Donaire et al, 2016a;Borja et al, 2016;Romero et al, 2017), implicit port-Hamiltonian representation (Macchelli, 2014;Castaños and Gromov, 2016) and an algebraic approach Batlle et al, 2007;Nunna et al, 2015). In addition, it has been shown in (Batlle et al, 2007;Donaire et al, 2016b) that a two step IDA-PBC may be restrictive in some cases, thus introducing a single step procedure (SIDA-PBC).…”

Section: Introductionmentioning

confidence: 99%

Algebraic IDA-PBC for Polynomial Systems with Input Saturation: An SOS-based Approach

Cieza

Reger

2019

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The necessity to deal with partial differential equations (PDEs) and the dissipation condition are the mainadversities in the application of Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC). Recently,an algebraic solution of IDA-PBC has been explored for a class of affine polynomial systems by using sum of squares(SOS) and semidefinite programming (SDP). In this work, we extend the previous method by incorporating actuatorsaturation (AS) and two minimization objectives in the SDP. Our results are validated on two polynomial systems.

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“…Gómez-Estern pendubot robot (Sandoval et al, 2008), among others. It is worth remarking that, in recent years, some authors have invented a way to shape the energy of mechanical systems without solving PDEs (Donaire et al, 2015), and even they were able to propose PID passivity-based control to reach global stabilization of underactuated mechanical systems (Romero et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

Interconnection and Damping Assignment Passivity–Based Control of an Underactuated 2–DOF Gyroscope

Cordero

Santibáñez

Dzul

et al. 2018

International Journal of Applied Mathematics and Computer Science

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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.

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Global stabilisation of underactuated mechanical systems via PID passivity-based control

Cited by 62 publications

References 18 publications

Robust control under worst‐case uncertainty for unknown nonlinear systems using modified reinforcement learning

Robust control under worst‐case uncertainty for unknown nonlinear systems using modified reinforcement learning

Algebraic IDA-PBC for Polynomial Systems with Input Saturation: An SOS-based Approach

Interconnection and Damping Assignment Passivity–Based Control of an Underactuated 2–DOF Gyroscope

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