Operator Approach to Linear Control Systems
The idea of optimization runs through most parts of control theory. The simplest optimal controls are preplanned (programmed) ones. The problem of constructing optimal preplanned controls has been extensively worked out in literature (see, e. g. , the Pontrjagin maximum principle giving necessary conditions of preplanned control optimality). However, the concept of op timality itself has a restrictive character: it is limited by what one means under optimality in each separate case. The internal contradictoriness of the preplanned control optimality ("the better is the enemy of the good") yields that the practical significance of optimal preplanned controls proves to be not great: such controls are usually sensitive to unregistered disturbances (includ ing the round-off errors which are inevitable when computer devices are used for forming controls), as there is the effect of disturbance accumulation in the control process which makes controls to be of little use on large time inter vals. This gap is mainly provoked by oversimplified settings of optimization problems. The outstanding result of control theory established in the end of the first half of our century is that controls in feedback form ensure the weak sensitivity of closed loop systems with respect to "small" unregistered internal and external disturbances acting in them (here we do not need to discuss performance indexes, since the considered phenomenon is of general nature). But by far not all optimal preplanned controls can be represented in a feedback form.
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Introduction to systems theory
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admissible control admits algorithms analytical arbitrary assumed bounded operator called causal operators closed loop system coefficients component consider constructing control plant control system corresponding defined depend eigenvalues extended resolution space external disturbance F°-weak filter finite finite-dimensional follows formula Fourier transform frequency given half-plane Hilbert resolution space Hilbert space inequality infimum initial data inner product input invertible Kalman-Bucy filter Lemma linear causal linear functional linear operator Lur'e equation method minimal non-negative norm optimal control optimal feedback optimization problem output parameter parameterization Parseval equality performance index polynomial positive definite preplanned control properties relation representation respect Riccati equation satisfying the condition sequence set F SLQP solution solvable solving space H spectral factorization square form stabilizing stochastic element subspace system operator systems theory takes the form time-invariant transfer function transfer operator uniquely unit disk values variables vector vector-function Wiener estimations Wiener problem zero