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1
Content available Extremals in the problem of minimum time obstacle avoidance for a 2D double integrator system
Osmolovskii N. P., Figura A., Koska M., Wojtowicz M.
Control and Cybernetics
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2015
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Vol. 44, no. 2
185--209
EN
This paper provides an analysis of time optimal control problem of motion of a material point in the plane outside the given circle, without friction. The point is controlled by a force whose absolute value is limited by one. The closure of exterior of the circle plays the role of the state constraint. The analysis of the problem is based on the minimum principle.
2
Content available Application of Digital Control Techniques for Satellite Medium Power DC-DC Converters
Skup K. R., Grudziński P., Orleański P.
International Journal of Electronics and Telecommunications
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2011
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Vol. 57, No. 1
77-83
EN
The objective of this paper is to present a work concerning a digital control loop system for satellite medium power DC-DC converters that is done in Space Research Centre. The whole control process of a described power converter is based on a high speed digital signal processing. The paper presents a development of a FPGA digital controller for voltage and current mode stabilization that was implemented using VHDL. The described controllers are based on a classical digital PID controller. The converter used for testing is a 200 kHz, 750W buck converter with 50V/15A output. A high resolution digital PWM approach is presented. Additionally a simple and effective solution of filtering of an analog-to-digital converter output is presented.
3
Content available Regularization and discretization of linear-quadratic control problems
Alt W., Seydenschwanz M.
Control and Cybernetics
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2011
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Vol. 40, no 4
903-920
EN
We analyze regularizations of a class of linear-quadratic optimal control problems with control appearing linearly. It is shown that if the optimal control is bang-bang or if a coercivity condition for the state variables is satisfied, the solutions are continuous functions of the regularization parameter. Combining error estimates for Euler discretizations of the regularized problems with those for the regularization error, we choose the regularization parameter in dependence of the meshsize to obtain optimal convergence rates for the discrete solutions. Numerical experiments confirm the theoretical findings.
4
Content available remote Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour
Felgenhauer U., Poggiolini L., Stefani G.
Control and Cybernetics
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2009
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Vol. 38, no 4B
1305-1325
EN
The paper considers parametric optimal control problems with bang-bang control vector function. For this problem we give regularity and second-order optimality conditions at the nominal solution which are sufficient to: (i) existence and local uniqueness of extremals, (ii) local structure stability, (iii) strong local optimality, under parameter perturbations. Here "local" means in a L∞ neighbourhood of the nominal trajectory, regardless of the control values. Stability results were obtained by the first author using the shooting approach, while optimality results were obtained by the other authors, using the Hamiltonian approach. The paper, combining both approaches, allows to unify the assumptions and to close some gaps between optimality and stability results.
5
Content available remote The shooting approach in analyzing bang-bang extremals with simultaneous control switches
Felgenhauer U.
Control and Cybernetics
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2008
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Vol. 37, no 2
307-327
EN
The paper is devoted to stability investigation of optimal structure and switching points position for parametric bang-bang control problem with special focus on simultaneous switches of two control components. In contrast to problems where only simple switches occur, the switching points in general are no longer differentiable functions of input parameters. Conditions for Lipschitz stability are found which generalize known sufficient optimality conditions to nonsmooth situation. The analysis makes use of backward shooting representation of extremals, and of generalized implicit function theorems. The Lipschitz properties are illustrated for an example by constructing backward parameterized family of extremals and providing first-order switching points prediction.
6
Content available remote Equivalence of second order optimality conditions for bang-bang control problems. Part 2 : Proofs, variational derivatives and representations
Osmolovskii N. P., Maurer H.
Control and Cybernetics
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2007
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Vol. 36, no 1
5-45
EN
In Part 1 of this paper (Osmolovskii and Maurer, 2005), we have summarized the main results on the equivalence of two quadratic forms from which second order necessary and sufficient conditions can be derived for optimal bang-bang control problems. Here, in Part 2, we give detailed proofs and elaborate explicit relations between Lagrange multipliers and elements of the critical cones in both approaches. The main analysis concerns the derivation of formulas for the first and second order derivatives of trajectories with respect to variations of switching times, initial and final time and initial point. This leads to explicit representations of the second order derivatives of the Lagrangian for the induced optimization problem. Based on a suitable transformation, we obtain the elements of the Hessian of the Lagrangian in a form which involves only first order variations of the nominal trajectory. Finally, a careful regrouping of all terms allows us to find the desired equivalence of the two quadratic forms.
7
Content available Sliding mode control for active magnetic bearing system
Gosiewski Z., Żokowski M.
Mechanics / AGH University of Science and Technology
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2005
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Vol. 24, no. 2
68-72
EN
Sliding mode control of a single-axis type bang-bang megnetic bearing actuator is reported in this paper. The two electromagnets are driven by switching between a positive and a negative constant voltage. Sliding mode control using these switching surfaces in turn is shown to be possible. The sliding mode turns out to take place in a subregion of state space defined by s1(x)s2(x) lessim 0 rather than on a surface defined by s(x) = 0 as in most standard cases.
PL
Praca przedstawia ideę sterowania ślizgowego jednoosiowego siłownika aktywnego łożyska magnetycznego z symulowanym sterowaniem typu przekaźnikowego (bang-bang). Dwa elektromagnesy zasilane są poprzez przełączanie z dużą częstotliwością na przemian ujemnym i dodatnim napięciem stałym. Sterowanie ślizgowe opiera się na płaszczyźnie fazowej, która charakteryzuje fazę ruchu symulowanego (badanego) układu. Ruch ślizgowy odbywa się w obszarze płaszczyzny fazowej zdefiniowanej jajo s1(x) . s2(x) lessim 0, a nie jak w większości przypadków, gdzie płaszczyzna fazowa jest zdefiniowana w następujący sposób: s(x) = 0.
8
Content available remote Optimality properties of controls with bang-bang components in problems with semilinear state equation
Felgenhauer U.
Control and Cybernetics
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2005
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Vol. 34, no 3
763-785
EN
In this paper we study optimal control problems with bang-bang solution behavior for a special class of semilinear dynamics. Generalizing a former result for linear systems, optimlity conditions are derived by a duality based approach. The results apply for scalar as well as for vector control functions and, in particular, for the case of the so-called multiple switches, too. Further, an iterative procedure for determining switching points is proposed, and convergence results are provided.
9
Content available remote Equivalence of second order optimality conditions for bang-bang control problems. Part 1: Main results
Osmolovskii N. P., Maurer H.
Control and Cybernetics
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2005
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Vol. 34, no 3
927-950
EN
Second order optimality conditions have been derived in the literature in two different forms. Osmolovskii (1988a, 1995, 2000, 2004) obtained second order necessary and sufficient conditions requiring that, a certain quadratic form be positive (semi)-definite on a critical cone. Agrachev, Stefani, Zezza (2002) first, reduced the bang-bang control problem to finite-dimensional optimization and then show that well-known sufficient optimality conditions for this optimization problem supplemented by the strict bang-bang property furnish sufficient conditions for the bang-bang control problem. In this paper, we establish the equivalence of both forms of sufficient conditions and give explicit relations between corresponding Lagrange multipliers and elements of critical cones. Part 1 summarizes the main results while detailed proofs will be given in Part 2.
10
Content available remote Optimality and sensitivity for semilinear bang-bang type optimal control problems
Felgenhauer U.
International Journal of Applied Mathematics and Computer Science
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2004
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Vol. 14, no 4
447-454
EN
In optimal control problems with quadratic terminal cost functionals and systems dynamics linear with respect to control, the solution often has a bang-bang character. Our aim is to investigate structural solution stability when the problem data are subject to perturbations. Throughout the paper, we assume that the problem has a (possibly local) optimum such that the control is piecewise constant and almost everywhere takes extremal values. The points of discontinuity are the switching points. In particular, we will exclude the so-called singular control arcs, see Assumptions 1 and 2, Section 2. It is known from the results by Agrachev et al. (2002) stating that regularity assumptions, together with a certain strict second-order condition for the optimization problem formulated in switching points, are sufficient for strong local optimality of a state-control solution pair. This finite-dimensional problem is analyzed in Section 3 and optimality conditions are formulated (Lemma 2). Using well-known results concerning solution sensitivity for mathematical programs in Rn (Fiacco, 1983) one may further conclude that, under parameter changes in the problem data, the switching points will change Lipschitz continuously. The last section completes these qualitative statements by calculating sensitivity differentials (Theorem 2, Lemma 6). The method requires a simultaneous solution of certain linearized multipoint boundary value problems.
11
Content available remote Second order optimality conditions for bang-bang control problems
Maurer H., Osmolovskii N. P.
Control and Cybernetics
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2003
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Vol. 32, no 3
555-584
EN
Second order necessary and sufficient optimality conditions for bang-bang control problems have been studied in Milyutin, Osmolovskii (1998). These conditions amount to testing the positive (semi-)definiteness of a quadratic form on a critical cone. The assumptions are appropriate for numerical verification only in some special cases. In this paper, we study various transformations of the quadratic form and the critical cone which will be tailored to different types of control problems in practice. In particular, by means of a solution to a linear matrix differential equation, the quadratic form can be converted to perfect squares. We demonstrate by three practical examples that the conditions obtained can be verified numerically.
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