Synthesis Control in the Systems with Distributed and Concentrated Parameters
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They consider a controlled system with distributed and lumped parameters, the perturbed state of which is described by linear differential equations in partial and ordinary derivatives. The system consists of one distributed link with finite-dimensional links at its both ends (for example, the systems containing electrical circuits, sufficiently long elastic shafts, the pipelines in which it is necessary to take into account the distributed nature of the fluid flow, etc.). The paper studies the problem of a lumped optimal control development applied to finite-dimensional links and/or to the end of a distributed link that ensures the stable system operation. Two tasks are solved to develop such control. First, the method of Lyapunov functions is used to determine the set of controls that ensure the asymptotic stability of
the closed-loop system; then, the Lagrange multiplier method is used on this set to develop an optimal control with the smallest value of the norm at each moment of time. At that, the original equations in private higher-order partial differential equations are represented as a system of evolutionary equations and first-order partial differential equations by introducing additional variables. The constraint equations are the equations without time derivatives. A
modified method of Lagrange multipliers is used to take such equations into account when calculating the derivative of the Lyapunov function in view of the system under consideration. The transition to partial differential equations of the first order, together with the development of ordinary differential equations in the standard Cauchy form, allows to construct the Lyapunov function as the sum of integral and ordinary quadratic forms constructively using specific equations, the sign-definiteness of which is checked by the Sylvester criterion, to ensure the stability of the closedloop system and obtain simpler control laws as the linear functions of phase coordinates. In the case of onedimensional distributed systems, these controls can be quite simply implemented in the form of lumped boundary
controls that require measuring the system state only at the boundary points, which is of great practical importance. The results obtained in this work significantly expand the possibilities of Lyapunov functions method use when they solve applied problems of control synthesis in engineering objects with distributed and lumped parameters. As an example, we consider the problem of determining the minimum boost pressure of the fuel tank, which ensures stable operation of the heating furnace.
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