Nonlinear Control of Multivariable Systems

Abstract

It is usual to think that there exist only a few control problems that can be posed directly in the frame of the input-state representation of a control system. They are presented by the well developed basic problems of stabilization about the state equilibrium point x = x* and tracking of the state reference trajectory x*(t) generated by a dynamical exosystem (reference model). In the meanwhile, we can easily point to a wide class of somewhat more sophisticated but not less familiar problems that can be formulated (directly or after a certain transformation) by using a description of non-trivial regular geometric objects in the system state space ℝ n . The most evident ones arise as a result of solving various problems of qualitative and optimal control [14, 54, 132, 192, 230, 269, 284], where the desired performance of the resulting system is often provided if its trajectories x(t,x 0) belong to some submanifolds (curves and surfaces) of the including state space ℝ n (see Section 1.2, Example 1.7). These geometric objects are usually constructed during the system preliminary analysis (optimization) and can be written in the implicit form .