Linear time-varying system control based on the inversion transformation (Q1361330)

This paper proposes a new nonlinear state feedback control design for linear time-varying systems (*): \(\dot x=A(t)x +B(t)u\), \(x(0)= x_0\), where \(x(t)\in R^n\) is the system state, \(u(t)\in R^{n\times m}\) is the control input, and \(A(t)\in R^{n\times m}\), \(B(t)\in R^{n \times m}\) are system matrices whose elements are bounded continuous or piecewise continuous functions of time. The new design is based on a nonlinear state transformation. Such a transformation results in a new real control (**): \(u= \gamma B^T(t) T(t)P^{-1} (t)x\), \(\gamma \geq 1/2\), where \(T(x)= I-2xx^T/x^Tx\) and \(P(t)\) is defined by \[ P(t)= \int^t_{t-\Delta} \psi(t,\tau) \overline B\bigl(x (\tau),\tau \bigr) \overline B^T \bigl(x(\tau), \tau\bigr) \psi^T(t,\tau) e^{\alpha (\tau-t)} d\tau, \] in which \(\Delta\) and \(\alpha\in R^+\), \(\psi (t,\tau) \in R^{n \times m}\) is defined by \[ {\partial \psi(t,\tau) \over\partial t} =A(x,t) \psi(t,\tau), \psi^{-1} (t,\tau) =\psi (\tau,t), \psi(t,t) =I, \] and \(A(x,t)\), \(\overline B(x,t)\) are given by \(\overline A(x,t)= T(x)A(t) T^T(x)\), \(\overline B(x,t)= T(x)B(t)\). The authors prove that global exponential stability of the controlled system is established if the open-loop system (*) is uniformly controllable. Compared with previous controls for linear time-varying systems (*), the proposed control (**) has the advantage that it requires neither forcast of future information of the system matrices nor time derivatives of the system matrices.