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 ℝⁿ. 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 ℝⁿ (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

EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaai % ikaiaadIhacaGGPaGaeyypa0JaaGimaaaa!3BC5!EquationSource$$ \[\varphi (x) = 0\] $$

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