Synthesis of a robust optimal system for stabilizing a plant with failing elements (Q5951311)

The paper deals with the minimization of a given functional \(e\) by selecting the optimal structure of a feedback controller using the Wiener-Kolmogorov method. It takes into account equations of a given plant, the required matrix of transfer functions of the controller \(W\), the sensitivity function \(\Phi\), and a certain coupling relation. A model of the plant is described by a Fourier-transformed system of ODEs \(Px=Mu+\psi\), where \(x, u, \psi \) are appropriately dimensioned vectors of output, control, centered random disturbance, resp. \(P, M\) are square matrices of polynomials in \(s=j\omega\), and \(\text{det } P\) satisfies the Hurwitz condition. Let \(\zeta =(E, P)(\psi ', \varphi ')'\), \(F_{\nu}^{\zeta},\) and \(F_u^{\zeta}\), respectively denote a generalized disturbance, the matrix of transfer functions of the closed-loop system from \(\zeta\) to the output of the plant \(\nu\), and the matrix of transfer functions of the closed-loop system from \(\zeta\) to \(u\). Here \(\varphi\) is a measurement noise, and \(E\) is a unit matrix. Then, \(W=F_u^{\zeta}(F_{\nu}^{\zeta})^{-1}\) and \(\Phi=F_{\nu}^{\zeta}P=P^{-1}(MF_u^{\zeta}+E)P\) considering that the relation \(PF_{\nu}^{\zeta}-MF_u^{\zeta}=E\) holds. The performance functional has the form \(e=\langle \xi '(t)\Lambda\zeta (t)\rangle+\langle \zeta '(t)\Lambda\xi (t)\rangle+\langle u'(t)Cu(t)\rangle\), where \(\xi=\Phi\zeta\), \(\Lambda\) and \(C\) are symmetric negative definite weighting matrices, and \(\langle\;\rangle\) denotes a mathematical expectation.