Robust decentralized \(H_\infty\) control for nonlinear interconnected systems with uncertainty (Q2705745)

The authors consider a smooth uncertain nonlinear interconnected system modeled by equations of the form: NEWLINE\[NEWLINE\dot x_i= f_i(x_i) + \Delta f_i(x_i) + g_i(x_i)u_i + \Delta g_i(x_i)u_i + \sum_{\substack{ j\neq i\\ j=1}}^N r_{ij}g_i(x_i) Q_{ij}(x_j) + g_{i0}(x_i)\omega_i,NEWLINE\]NEWLINE NEWLINE\[NEWLINEz_i= h_{i1}(x_i) + d_{i1}(x_i)u_i,\qquad y_i= h_{i2}(x_i) + \Delta h_i(x_i) + d_{i2}(x_i)\omega_i,NEWLINE\]NEWLINE where \(i = 1,2,\dots,N\), \(x_i\in\mathbb R^n\) is the state, \(y_i \in\mathbb R^m\) is the measured output, \(u_i \in \mathbb R^r\) is the control input, \(\omega_i\in\mathbb R^m\) is the exogenous input noise, \(z_i\in\mathbb R^q\) is the signal to be controlled, \(f_i(x_i)\), \(g_i(x_i)\), \(g_{i0}(x_i)\), \(h_{i1}(x_i)\), \(h_{i2}(x_i)\), \(d_{i1}(x_i)\), \(d_{i2}(x_i)\), and \(Q_{ij}(x_i)\), are known matrix functions with appropriate dimensions, \(f_i(0) = 0\), \(h_{i1}(0)=0\), \(h_{i2}(0)=0\), \(\Delta f_{i}(x_i)\), \(\Delta g_{i}(x_i)\) and \(\Delta h_{i}(x_i)\) represent the uncertainties in the system and satisfy some intrinsic assumptions. The problem consists in the design of a state feedback controller and a dynamic output feedback controller such that the closed-loop system is Lyapunov stable and achieves a prescribed level of disturbance attenuation for all admissible uncertainties. The main results are connections between the robust decentralized \(H_\infty\) control problem and the nonlinear \(H_\infty\) control problem for system without uncertainty. This allows to solve the robust decentralized \(H_\infty\) control problem via existing nonlinear \(H_\infty\) control techniques. These results also provide a solution to the robust decentralized \(H_\infty\) control problem in terms of a set of scaled Hamilton-Jacobi inequalities.