Optimal Control Under Stochastic Uncertainty

Kurt Marti
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Kurt Marti: University of Bundeswehr Munich

Chapter Chapter 1 in Optimization Under Stochastic Uncertainty, 2020, pp 3-32 from Springer

Abstract: Abstract Optimal control and regulator problems that arise in many concrete applications (mechanical, electrical, thermodynamical, chemical, etc.) are modeled by dynamical control systems obtained from physical measurements and/or known physical (a priori) laws. The basic control system (input–output system) is mathematically represented by a system of first order differential equations with random parameters: ż ( t ) = g ( t , ω , z ( t ) , u ( t ) ) , t 0 ≤ t ≤ t f , ω ∈ Ω z ( t 0 ) = z 0 ( ω ) . $$\displaystyle \begin {array}{rcl} \dot z (t) & = & g \Big ( t, \omega , z (t) , u (t) \Big ), t_0 \leq t \leq t_f , ~ \omega \in \Omega \\ z (t_0) & = & z_0 ( \omega ). \end {array} $$ Here, ω is the basic random element taking values in a probability space ( Ω , A , P ) $$(\Omega , \mathcal {A}, P)$$ , Probability space and describing the random variations of model parameters or the influence of noise terms. The probability space ( Ω , A , P ) $$(\Omega , \mathcal {A}, P)$$ consists of the sample space or set of elementary events Ω, the σ-algebra A $$\mathcal {A}$$ of events and the probability measure P. The plant state vector z = z(t, ω) is an m-vector involving direct or indirect measurable/observable quantities like displacements, stresses, voltage, current, pressure, concentrations, etc., and their time derivatives (velocities), z 0(ω) is the random initial state. The plant control or control input u(t) is a deterministic or stochastic n-vector denoting system inputs like external forces or moments, voltages, field current, thrust program, fuel consumption, production rate, etc. Furthermore, ż $$\dot z$$ denotes the derivative with respect to the time t.

Date: 2020
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DOI: 10.1007/978-3-030-55662-4_1

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