Optimal Control with Engineering Applications Episode 1 pps
Hans P. Geering Optimal Control with Engineering Applications pot
... Exercises 222 Optimal Control 232.1 Optimal Control Problems with a Fixed Final State 242.1.1TheOptimalControlProblemofTypeA 242.1.2Pontryagin’sMinimumPrinciple 252.1.3Proof 252.1.4 Time -Optimal, ... has been solved in [15]. Hans P. Geering, Ph.D.Professor of Automatic Control and MechatronicsMeasurement and Control LaboratoryDepartment of Mechanical and Process Engineering ETH ZurichSonneggstrasse ... nicely reveals that the solution of an optimal control prob-lem always is “as bad” as the considered formulation of the optimal control problem. This optimal control problem lacks any sustainability...
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Optimal Control with Engineering Applications With 12 Figures Hans P. Geering potx
... gameproblem:State variables: x1and x2. Control variables: u ∈ [−1, +1] and v ∈ [−π, π]. Hans P. Geering Optimal Control with Engineering Applications With 12 Figures 123 2 List of SymbolsIntegersi, ... 113Solutions to Exercises 117References 129 Index 131 Hans P. Geering Optimal Control with Engineering Applications 26 2 Optimal Control Introducing the Hamiltonian functionH(x(t),u(t),λ(t),λ0,t)=λ0L(x(t),u(t),t)+λ(t)Tf(x(t),u(t),t)and ... 412.4 Optimal Control Problems with aPartially Constrained Final State 432.4.1TheOptimalControlProblemofTypeB 432.4.2Pontryagin’sMinimumPrinciple 432.4.3Proof 442.4.4Energy-OptimalControl...
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Optimal Control with Engineering Applications Hans P. Geering pptx
... been solved in [15]. ...
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Optimal Control with Engineering Applications Episode 12 ppsx
... minimizing control of the resulting optimal control problem ofType C.1 and, conversely, when the state feedback law u(t)=ku(x(t),t)is applied, vo(.) is a globally maximizing control of ... the optimal costate λo(.) is free.Remark: The calculus of variations only requires the local minimization ofthe Hamiltonian H with respect to the control u and a local maximizationof H with ... maximized with respect to v andthen minimized with respect to u or vice versa. The result is the same inboth cases. 4.2 LQ Differential Game 111The resulting optimal state-feedback control laws...
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Optimal Control with Engineering Applications Episode 1 pps
... 923.3.4LQQSpeedControl 963.4 Exercises 994 Differential Games 10 34 .1 Theory 10 34 .1. 1 Problem Statement 10 44 .1. 2TheNash-PontryaginMinimaxPrinciple 10 54 .1. 3Proof 10 64 .1. 4Hamilton-Jacobi-IsaacsTheory 10 74.2 ... TheLQDifferentialGameProblem 10 94.2 .1 Solved with the Nash-Pontryagin Minimax Principle 10 94.2.2 Solved with the Hamilton-Jacobi-Isaacs Theory . . 11 14.3 H∞-ControlviaDifferentialGames 11 3Solutions to Exercises 11 7References ... Contents2.3 Optimal Control Problems with a Free Final State 382.3.1TheOptimalControlProblemofTypeC 382.3.2Pontryagin’sMinimumPrinciple 382.3.3Proof 392.3.4 The LQ Regulator Problem 41 2.4 Optimal Control...
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Optimal Control with Engineering Applications Episode 2 potx
... are fixed. 2) This problem nicely reveals that the solution of an optimal control prob-lem always is “as bad” as the considered formulation of the optimal control problem. This optimal control ... problem is often named “time -optimal control ofthe double integrator”. It is analyzed in detail in Chapter 2. 1.4.Problem 2: Time -optimal, horizontal motion of a mass with viscous frictionThis ... dynamic system⎡⎣˙x1(t)˙x 2 (t)˙x3(t)⎤⎦=⎡⎣x 2 (t)1x3(t)u(t) −1 2 Aρcwx 2 2(t)−αu(t)⎤⎦is transferred from the initial state⎡⎣x1(0)x 2 (0)x3(0)⎤⎦=⎡⎣savama⎤⎦to...
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Optimal Control with Engineering Applications Episode 3 pdf
... 0andλ2=0∂F∂λ 3 = x1+ x2− 4= 0 and λ 3 ≥ 0< 0andλ 3 =0The optimal solution is:xo1=1xo2=1.5λo1=0.5λo2 =3 λo 3 =0.The third constraint is inactive. 14 1 Introduction control ... corresponding necessary con-ditions for the optimality of a solution of an optimal control problem. Thegeneralization from constrained static optimization to optimal control is verystraightforward, indeed. ... λ0=1:F (x1,x2,λ1,λ2,λ 3 )=x21+ x22+ λ1(1−x1)+λ2(2−0.5x1−x2)+λ 3 (x1+x2−4)∂F∂x1=2x1− λ1− 0.5λ2+ λ 3 =0∂F∂x2=2x2− λ2+ λ 3 =0∂F∂λ1=1− x1= 0 and...
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Optimal Control with Engineering Applications Episode 4 ppt
... Introduction1 .4 Exercises1. In all of the optimal control problems stated in this chapter, the control constraint Ω is required to be a time-invariant set in the control spaceRm.For the control ... equalityconstraints x − y = 2 and x2+ y2 =4. 2 Optimal Control In this chapter, a set of necessary conditions for the optimality of a solution ofan optimal control problem is derived using the calculus ... xo(Ã)ando(Ã).ã Eliminate locally optimal solutions which are not globally optimal. ã If possible, convert the resulting optimal open-loop control uo(t)intoan optimal closed-loop control uo(xo(t),t)...
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Optimal Control with Engineering Applications Episode 6 pdf
... textbookson linear optimal control, such as [1], [11], [ 16] , and [25]. 50 2 Optimal Control Theorem DIf the control uo:[ta,tb] → Ω is optimal (in the non-singular case with λo0=1), then ... > 0.Using the optimal open-loop control lawuo(t)=−bλo(t)=−be−atλo(0) , 46 2 Optimal Control 2) If the optimal control uo(t)attimet lies in the interior of the control constraint ... restriction is justified in Chapter 2.2.1.2.4.4 Energy -Optimal Control Statement of the optimal control problem:Consider the following energy -optimal control problem for an unstable sys-tem: Find u...
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Optimal Control with Engineering Applications Episode 8 potx
... — Thus, Theorem 2 covers ourtime -optimal and fuel -optimal control problems as well. 2 .8 Non-Scalar-Valued Cost Functional 672 .8 Optimal Control Problems with a Non-Scalar-Valued Cost FunctionalUp ... functional isJ(u)=x3(tb) , 66 2 Optimal Control Theorem 1. The following optimal control problem has a globally optimal solution:Find an unconstrained optimal control u :[ta,tb] → Rm, such ... Rm. — The following optimal control problem has aglobally optimal solution:Find an optimal control u :[ta,tb] → Ω ⊂ Rm, such that the dynamic system˙x(t)=f(x(t),u(t)) with the continuously...
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Optimal Control with Engineering Applications Episode 9 ppt
... 3 Optimal State Feedback Control Proof3) Otherwise, combining the optimal solution of the modified antecedent optimal control problem with the succedent part of the solution of theoriginal optimal ... coincides with the succedent part of the optimal solution of the original problem.2) The optimal solution of the antecedent optimal control problem coincides with the antecedent part of the optimal ... 76 3 Optimal State Feedback Control Suppose that we have found the unique globally optimal solution with the optimal control trajectory uo:[ta,tb] → Ω ⊆ Rmand the correspondingoptimal...
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Optimal Control with Engineering Applications Episode 10 pot
... time-invariant optimal control problem with an infinitehorizon for a nonlinear dynamic system with a non-quadratic cost functional,which is structured as follows:Find a time-invariant optimal state ... Hamilton-Jacobi-Bellman Theory 833.2.4 The Time-Invariant Case with Infinite HorizonIn this section, time-invariant optimal control problems with the uncon-strained control vector u(t) ∈ Rm, an infinite horizon, ... nonlinear optimal control law is depicted for the examplewhere a =1,b =1,q =8,andβ = 1 with the LQ regulator gain g = 4 andthe large-signal gain g∞= −43. 3.3 Approximatively Optimal Control...
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Optimal Control with Engineering Applications Episode 11 ppt
... −1012345−150−100−5050100150Fig. 3.3. Approximatively optimal controller for a =3, q = 100. 92 3 Optimal State Feedback Control 3.3.3 Controller with a Progressive CharacteristicFor a linear time-invariant ... following optimal control problem of Type B.1 where thereis an additional state constraint x(t1)∈ S1⊂Rnat the fixed time t1withinthetimeinterval[ta,tb]:Find a piecewise continuous control ... thrustF0=12cwArρv20is needed.We want to augment the obvious open-loop control strategy F (t) ≡ F0 with a feedback control such that the velocity v(t) is controlled more precisely,should any discrepancy occur...
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Plastics Engineering 3E Episode 1 pps
... 45 48 61 66 71 74 81 84 95 95 10 3 11 0 11 6 11 9 12 1 12 1 12 7 13 1 13 4 13 6 13 7 13 8 14 0 14 2 14 2 14 3 14 5 14 5 14 7 14 8 15 0 15 2 15 2 15 4 16 8 16 8 16 8 xiv Preface to the ... G 0.97- 1. 34 1. 1 1 - 1. 21 -50 -13 0 -40 -13 0 40D to 72D 30A to 45D 43 to62 60 to 70 21 to 45 25 to 45 E E E F G P E PIE 1. 17 -1. 25 0.93 -1. 0 -65- 13 0 -30- 12 0 40D to ... Avenue, Woburn, MA 018 01- 20 41 First published 19 81 Second edition 19 87 Reprinted with corrections 19 90. 19 92 Third edition 19 98 Reprinted 19 99.20 01, 2002 Copyright 0 19 87, 19 98 R.J. Crawford....
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