Negative Imaginary Systems Theory for Nonlinear Systems: A Dissipativity Approach

The paper’s Theorem 2 states that, under Assumptions 1–4, the positive-feedback interconnection of an NNI system H1 (zero-state observable) and a WS-NNI system H2 has an asymptotically stable equilibrium at z=0, using the Lyapunov candidate W(x1,x2)=V1+V2−h1^T h2 and LaSalle’s invariance principle . While the lemma establishing W’s positive definiteness under the sector bound is provided (Lemma 8) , the proof of Theorem 2 omits key steps: it does not construct a compact positively invariant sublevel set needed for LaSalle, it reverses the logical order surrounding u2→0 and x2→0, and it states lim u2(t)=c instead of 0 despite Definition 4 (MS-NNI) implying the latter . By contrast, the model’s solution supplies the missing compactness/invariance argument, uses Assumption 4 to preclude nonzero equilibria, applies LaSalle correctly on a compact set, invokes MS-NNI to conclude u2→0 on the invariant set, and then uses standard asymptotically autonomous arguments to obtain x2→0, closing the loop to show the largest invariant set is {0}. Hence the theorem is (very likely) correct, but the paper’s proof sketch is incomplete; the model’s proof is coherent and fills the gaps.