On the robustness of linear systems with nonlinear uncertain parameters (Q1294971)

The authors study the stability robustness of linear uncertain state space systems of the form (after the control law is given by \(u=Kx\)): \[ x'= (I +\Delta_m)[A +\Delta_a]x,\tag{L} \] where \(x\) denotes the state vector, \(x'\) is the derivative of \(x\), \(I\) is an identity matrix, and \[ \Delta_m=\sum^m_{j=1}k_jG_j,\quad \Delta_a=\sum^l_{i=1}r_iE_i,\quad \underline{k}_j\leq k_j\leq \overline{k}_j,\quad \underline{r}_i\leq r_i\leq \overline{r}_i, \] where \(k_j\), \(r_i\) are uncertain parameters, \(G_j\), \(E_i\) are known constant metrices, \((\overline *)\) and \((\underline{*})\) represent the upper and the lower bounds of \((*)\), respectively. It is supposed that the nominal system of (L) with \(\Delta_m = 0\), \(\Delta_a = 0\), is asymptotically stable. It is shown in the paper that the above model has a strong practical background. By applying the Lyapunov theorem, a new robustness criterion is established for System (L), which yields three different sufficient conditions for the existence of a positive-definite matrix \(P\) such that \(V = x^T Px\) is the desired Lyapunov function. The results obtained extended the existing approaches on stability robustness analysis to those systems with a class of second-order nonlinear uncertain parameters. An example is also given to convey the advantages of the proposed method.