Optimal Control with Engineering Applications Episode 4 ppt

... Exercises 22 2 Optimal Control 23 2.1 Optimal Control Problems with a Fixed Final State 24 2.1.1TheOptimalControlProblemofTypeA 24 2.1.2Pontryagin’sMinimumPrinciple 25 2.1.3Proof 25 2.1.4 Time -Optimal, ... has been solved in [15]. Hans P. Geering, Ph.D. Professor of Automatic Control and Mechatronics Measurement and Control Laboratory Department of Mechanical and Process...

Ngày tải lên: 05/03/2014, 18:20

... Matlab Robust Control Toolbox. Finally, we obtain the feasible parameters 40 γ = and 1 0.02γ= , and a minimum attenuation level 2 0 0.93ρ= and a Robust Control, Theory and Applications 1 34 ... 2 0.85 84 -1. 344 2 -0.5699 -5.5768 Y ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ Robust Control, Theory and Applications 110 6. Discussion and conclusion In this study, we have de...

Ngày tải lên: 20/06/2014, 04:20

... game problem: State variables: x 1 and x 2 . Control variables: u ∈ [−1, +1] and v ∈ [−π, π]. Hans P. Geering Optimal Control with Engineering Applications With 12 Figures 123 2 List of Symbols Integers i, ... 113 Solutions to Exercises 117 References 129 Index 131 Hans P. Geering Optimal Control with Engineering Applications 26 2 Optimal Cont...

Ngày tải lên: 27/06/2014, 08:20

... been solved in [15].

Ngày tải lên: 27/06/2014, 18:20

... minimizing control of the resulting optimal control problem of Type C.1 and, conversely, when the state feedback law u(t)=k u (x(t),t) is applied, v o (.) is a globally maximizing control of ... the optimal costate λ o (.) is free. Remark: The calculus of variations only requires the local minimization of the Hamiltonian H with respect to the control u and a local maximization o...

Ngày tải lên: 12/08/2014, 16:21

... 92 3.3.4LQQSpeedControl 96 3.4 Exercises 99 4 Differential Games 10 3 4 .1 Theory 10 3 4 .1. 1 Problem Statement 10 4 4 .1. 2TheNash-PontryaginMinimaxPrinciple 10 5 4 .1. 3Proof 10 6 4 .1. 4Hamilton-Jacobi-IsaacsTheory 10 7 4.2 ... TheLQDifferentialGameProblem 10 9 4.2 .1 Solved with the Nash-Pontryagin Minimax Principle 10 9 4.2.2 Solved with the Hamilton-Jacobi-Isaacs Theory . ....

Ngày tải lên: 12/08/2014, 16:21

... 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 of the double integrator”. It is analyzed in detail in Chapter 2. 1.4. Problem 2: Time -optimal, horizontal motion of a mass with vi...

Ngày tải lên: 12/08/2014, 16:21

... 0andλ 2 =0 ∂F ∂λ 3 = x 1 + x 2 − 4  = 0 and λ 3 ≥ 0 < 0andλ 3 =0 The optimal solution is: x o 1 =1 x o 2 =1.5 λ o 1 =0.5 λ o 2 =3 λ o 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. The generalization from constrained static optimization to optimal...

Ngày tải lên: 12/08/2014, 16:21

... Introduction 1 .4 Exercises 1. 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 space R m . For the control ... equality constraints x − y = 2 and x 2 + y 2 =4. 2 Optimal Control In this chapter, a set of necessary conditions for the optimality of a solution of an optimal control...

Ngày tải lên: 12/08/2014, 16:21

... textbooks on linear optimal control, such as [1], [11], [ 16] , and [25]. 50 2 Optimal Control Theorem D If the control u o :[t a ,t b ] → Ω is optimal (in the non-singular case with λ o 0 =1), then ... > 0. Using the optimal open-loop control law u o (t)=−bλ o (t)=−be −at λ o (0) , 46 2 Optimal Control 2) If the optimal control u o (t)attimet lies in the interi...

Ngày tải lên: 12/08/2014, 16:21

... — Thus, Theorem 2 covers our time -optimal and fuel -optimal control problems as well. 2 .8 Non-Scalar-Valued Cost Functional 67 2 .8 Optimal Control Problems with a Non-Scalar-Valued Cost Functional Up ... functional is J(u)=x 3 (t b ) , 66 2 Optimal Control Theorem 1. The following optimal control problem has a globally optimal solution: Find an unconstrained optima...

Ngày tải lên: 12/08/2014, 16:21

... 3 Optimal State Feedback Control Proof 3) Otherwise, combining the optimal solution of the modified antecedent optimal control problem with the succedent part of the solution of the original 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...

Ngày tải lên: 12/08/2014, 16:21

... time-invariant optimal control problem with an infinite horizon 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 83 3.2.4 The Time-Invariant Case with Infinite Horizon In this section, time-invariant optimal control problems with the uncon- strained contr...

Ngày tải lên: 12/08/2014, 16:21

... −1012345 −150 −100 −50 50 100 150 Fig. 3.3. Approximatively optimal controller for a =3, q = 100. 92 3 Optimal State Feedback Control 3.3.3 Controller with a Progressive Characteristic For a linear time-invariant ... following optimal control problem of Type B.1 where there is an additional state constraint x(t 1 )∈ S 1 ⊂R n at the fixed time t 1 within thetimeinterval[t a ,t b ]:...

Ngày tải lên: 12/08/2014, 16:21