Controller design for 2-DOF underactuated mechanical systems based on controlled Lagrangians and application to Acrobot control

On the basis of controlled Lagrangians,a controller design is proposed for underactuated mechanical systems with two degrees of freedom.A new kinetic energy equation(K-equation)independent of the gyroscopic forces is found due to the use of their property.As a result,the necessary and sufficient matching condition comprises the new K-equation and the potential energy equation(P-equation)cascaded,the regular condition,and the explicit gyroscopic forces.Further,for two classes of input decoupled systems that cover the main benchmark examples,the new K-equation,respectively,degenerates from a quasilinear partial differential equation(PDE)into an ordinary differential equation(ODE)under some choice and into a homogeneous linear PDE with two kinds of explicit general solutions.Benefiting from one of the general solutions,the obtained smooth state feedback controller for the Acrobots is of a more general form.Specifically,a constant fixed in a related paper by the system parameters is converted into a controller parameter ranging over an open interval along with some new nonlinear terms involved.Unlike what is mentioned in the related paper,some categories of the Acrobots cannot be stabilized with the existing interconnection and damping assignment passivity based control(IDA-PBC)method.As a contribution,the system can be locally asymptotically stabilized by the selection of the new controller parameter except for only one special case.

The discontinuous Petrov-Galerkin method for one-dimensional compressible Euler equations in the Lagrangian coordinate

In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume （RKCV） discontinuous finite ele- ment method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discon- tinuous Galerkin （RKDG） method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.

Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

In[14],Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates.These schemes use a node-based discretization of the numerical fluxes.The control volume version has several distinguished properties,including the conservation of mass,momentum and total energy and compatibility with the geometric conservation law(GCL).However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow.An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry,at the price of sacrificing conservation of momentum.In this paper,we apply the methodology proposed in our recent work[8]to the first order control volume scheme of Maire in[14]to obtain the spherical symmetry property.The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid,andmeanwhile itmaintains its original good properties such as conservation and GCL.Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry,non-oscillation and robustness properties.