Application of vibrational control to linear-quadratic control problems.
The application of the theory of vibration control to linear-quadratic control problems is presented. The general linear-quadratic problem for a vibration is stated as follows: \[ {dx\over dt}= Ax+{1\over \varepsilon}\sin{t\over \varepsilon} Kx+ Bu,\quad y= Cx\quad\text{and}\quad x(0)= x_0. \] Here, \(x\in \mathbb{R}^n\) is the state vector, \(y\in \mathbb{R}^m\) \((m\leq n)\) is the observed output, \(u\in \mathbb{R}^r\) is the control input, \(A\), \(K\), \(C\), \(B\) are constants matrices of proper dimensions, and \(\varepsilon\) is a small parameter. It is required to find the control input \(u\) which minimizes the following cost functional: \[ J(u)= y^T(T) Qy(T)+ \int^T_0 [y^T (t) Vy(t)+ u^T(s) Su(t)]\,dt, \] where \(S= S^T> 0\), \(Q= Q^T\geq 0\) and \(V= V^T\geq 0\). The control time interval \(T\) is assumed to be a multiple of \(2\pi\varepsilon\). Main result: This paper extends the theory of vibration control to linear-quadratic control problems in systems with vibration. The solution of the matrix Riccati differential equation (RDE) and the optimality trajectory are found approximately as power series of a small parameter. The region of attraction of the maximal solution of RDE for the case of a Hamiltonian matrix with imaginary axis eigenvalues is found. The application of vibration to the linear-quadratic problem of stabilization with respect to a part of the variables and transfer of the other variables to a given position is presented. Finally, a problem of vibration stabilization and optimal control of a carriage with an inverted pendulum is solved as an example.
Hamiltonian matrix with imaginary axis eigenvalues
scientific article; zbMATH DE number 2006009
scientific article