STABILIZATION OF TWO-DIMENSIONAL NONLINEAR SYSTEMS DESCRIBED BY FORNASINI-MARCHESINI AND ROESSER MODELS

The paper considers nonlinear two-dimensional systems described by the Fornasini-Marchesini or Roesser state-space models. Conditions for such systems to have a physically motivated exponential stability property are derived using vector Lyapunov functions. A form of passivity, termed exponential passivity, is introduced and used, together with a vector storage function, to develop a feedback based control law that guarantees exponential stability of the controlled system. For cases where noise is present, stochastic dissipativity in the second moment is defined and then a particular case of this property, termed passivity in the mean square, is used, together with a vector storage function, to develop a feedback based control law such that the controlled system also has this property. Two physically motivated particular cases, a system with nonlinear actuator dynamics and additive noise and a linear system with state-dependent noise, respectively, are also considered to demonstrate the effectiveness of the new results.