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AUTOMATIC CONTROL THEORY II

Slovak University of Technology Faculty of Material Science and Technology in Trnava. AUTOMATIC CONTROL THEORY II. Optimal Control of Hybrid Systems. Discretization utilizing a computer for a specific control problem a straight forward approach

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AUTOMATIC CONTROL THEORY II

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  1. Slovak University of Technology Faculty of Material Science and Technology in Trnava AUTOMATIC CONTROL THEORY II

  2. Optimal Control of Hybrid Systems • Discretization • utilizing a computer for a specific control problem • a straight forward approach • is to grid the state space to require the inequalities to be met at a set of evenly distributed points in X

  3. Optimal Control of Hybrid Systems • this approximation will not guarantee a lower bound on the optimal cost • unless the nature of fqand Vq between the grid points is taken into consideration • in the case of a two-dimensional continuous state space, introduce the next notation

  4. Optimal Control of Hybrid Systems

  5. Optimal Control of Hybrid Systems • e1 and e2 are unit vectors along the coordinate axes • h is the grid size • illustration of Xjkand X ˆjk

  6. Optimal Control of Hybrid Systems • introduce new vector variablesfor (j, k, q) such that • the inequalities can then be replaced byand next

  7. Optimal Control of Hybrid Systems • fordefine the interpolating function

  8. Optimal Control of Hybrid Systems • DISCRETIZATION IN R2 • if • Vjkqsatisfy for all q єQ • and for all grid points • such that Xjkintersects X • then • for every (x0, q0), Vq0 (x0) is a lower bound of J(x0, q0)

  9. Optimal Control of Hybrid Systems • any function that meet the constraints • even the trivial choice Vq(x) = 0 is a lower bound on the true cost • to yield useful bounds, Vq(x) need to be maximized • the maximizationcould be carried out • in either one point (x0, q0) • or several points, (x, q)єX x Q

  10. Optimal Control of Hybrid Systems • For the original, non-discretized problem, the result of a maximization of Vq(x) is always identical to the optimal cost • regardless if • the maximization is done at a particular initial state • or by summing the values at several initial states

  11. Optimal Control of Hybrid Systems • for the discretized problem, different choices of maximization criteria may lead to different results • the difference between the results ofa single-point and a multi-point maximization is often small • it’s possible to compute the valuefunction in a large subset of X xQ solving one LP

  12. Optimal Control of Hybrid Systems • the restriction x(t) єX in the optimal control problem is essential • the maximization of Vq0 (x0) can lead to arbitrarily large values • the theorem DISKRETIZATION in R2 is easily extended to Rn

  13. Optimal Control of Hybrid Systems • Define j = (j1, j2, . . . , jn)and exchange jk for the new multi-index j in the previous inequalities • the limits of all summations and enumerations should also be adjusted • assume that

  14. Optimal Control of Hybrid Systems • Noting thatthe inequalities taken at grid pointsjk, j(k+1), (j+1)k, and (j+1)(k+1) give

  15. Optimal Control of Hybrid Systems • The gradient of Vqis given by

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