On robust stability of dynamic interval systems (Q2726018)

The authors discuss the problem of robust stability for dynamic interval continuous systems and discrete interval systems. By using the Lyapunov method and a Riccati equation approach, some simple sufficient conditions for robust stability of dynamic continuous and discrete interval systems are obtained, respectively. These conditions can be tested by solving the Riccati inequalities only: NEWLINE\[NEWLINEXA_0+A_0^TX+ \lambda^{-1}XEE^TX +\lambda F^TF<0\text{ and }\|F(sI-A_0)^{-1} E\|<1,NEWLINE\]NEWLINE where \(A_0=(P+Q)/2\), \(H=(h_{ij})= (Q-P)/2\), with the \(n\times n^2\)-matrix NEWLINE\[NEWLINEE=[\sqrt{h_{11}} e_1\dots\sqrt {h_{1n}} e_1\dots \sqrt{h_{n1}}e_n\dots \sqrt{h_{nn} }e_n],NEWLINE\]NEWLINE and the \(n^2\times n\)-matrix NEWLINE\[NEWLINEF=[\sqrt{h_{11}} e_1\dots\sqrt{h_{1n}} e_n\dots\sqrt {h_{n1}}e_1\dots \sqrt {h_{nn}} e_n]^T,NEWLINE\]NEWLINE where \(e_i\) is the \(i\)-column vector of the unit \(n\times n\)-matrix, \(P=(p_{ij})\), \(Q=(q_{ij})\), \(N(P,Q)=\{A\in \mathbb{R}^{n\times n};p_{ij}\leq a_{ij}\leq q_{ij}\), \(i,j=1,2,\dots,n\}\), \(p_{ij}\) and \(q_{ij}\) are lower and upper bounds of \(a_{ij}\), respectively; the interval matrix \(A=(a_{ij})\in N(P,Q)\) is the coefficient matrix of the dynamic continuous and discrete interval systems: \(x(t)=Ax(t)\), \(x(k+1)=Ax(k)\). The criteria for robust stability of these dynamic systems are discussed and obtained.