Hybrid solution method for dynamic programming equations for MDOF stochastic systems
The authors consider a randomly excited controlled system with \(n\) degrees of freedom, governed by the following matrix equation of motion \[ M\ddot{X}+KX=U(t)+B(t)\zeta (t), \] where \(M\) and \(K\) are positively definite mass and stiffness matrices respectively, \(U(t)\) and \(B(t)\) are vectors of displacement and control forces respectively, \(\zeta (t)\) is a vector of independent Gaussian white noises. They investigate the problem of optimal control for such a multi-degree-of-freedom (MDOF) system. The problem is to minimise the expected response energy within a given time interval with the help of vector control force with given bounds on magnitudes of its components. This problem is governed by a Hamilton-Jacobi-Bellman (HJB) partial differential equation. The ''hybrid'' approach is applied by the authors to their MDOF systems using common transformations to modal coordinates. The multidimensional HJB equation is solved explicitly for the corresponding domain and the problem is reduced to a set of numerical solutions within bounded inner domains. So, the problem of bounded optimal control is solved completely, as long as the necessary modal control forces are implemented in the available actuators. If the control forces can be applied to the original generalised coordinates only, the resulting optimal control law may become unfeasible. A semioptimal control law is proposed for this case. It is based on projecting boundary points of the domain of the admissible transformed control forces onto boundaries of the domain of the original control forces. A case of a control with a single actuator is considered, when transformed control forces are dependent, and the solution of the HJB equation is derived.
scientific article; zbMATH DE number 1501675