Model Predictive Control

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180 10 Constrained Optimal <strong>Control</strong> J ∗ 0 (x(0)) = minU0 N−1 J0(x(0), U0) = PxN p + Qxkp + Rukp 10.5.1 Batch Approach k=0 subj. to xk+1 = Axk + Buk, k = 0, . . .,N − 1 xk ∈ X, uk ∈ U, k = 0, . . .,N − 1 xN ∈ Xf x0 = x(0) (10.73) The optimal control problem (10.73) with p = ∞ can be rewritten as a linear program by using the standard approach (see e.g. [68]) presented in Section 9.1. Therefore, problem (10.73) can be reformulated as the following LP problem
10.5 State Feedback Solution, 1-Norm and ∞-Norm Case 181 min z0 subj. to −1nε x ⎡ k ≤ ±Q ⎣A k k−1 x0 + ε x 0 + . . . + εx N + εu 0 + . . . + εu N−1 −1rε x N ⎡ ≤ ±P ⎣A N x0 + j=0 N−1 j=0 A j Buk−1−j ⎤ A j BuN−1−j (10.74a) ⎦, (10.74b) ⎤ ⎦, (10.74c) −1mε u k ≤ ±Ruk, (10.74d) A k k−1 x0 + A j Buk−1−j ∈ X, uk ∈ U, (10.74e) j=0 A N N−1 x0 + A j BuN−1−j ∈ Xf, (10.74f) j=0 k = 0, . . .,N − 1 x0 = x(0) (10.74g) where constraints (10.74b)–(10.74f) are componentwise, and ± means that the constraint appears once with each sign. Problem (10.74) can be rewritten in the more compact form min c z0 ′ 0z0 subj. to G0z0 ¯ ≤ ¯ W0 + ¯ (10.75) S0x(0) where z0 {εx 0 , . . . , εxN , εu0 , . . .,εu N−1 , u′ 0 , . . . , u′ N−1 } ∈ Rs , s (m + 1)N + N + 1 and Gε ¯G0 = 0 0 G0 , ¯ Sε S0 = S0 , ¯ Wε W0 = W0 (10.76) Vector c0 and the submatrices Gǫ, Wǫ, Sǫ associated with the constraints (10.74b)- (10.74d) are defined in (9.10). The matrices G0, W0 and E0 are defined in (10.41) for i = 0. As in the 2-norm case, by treating x(0) as a vector of parameters, Problem (10.75) becomes a multiparametric linear program (mp-LP) that can be solved as described in Section 6.2. Once the multiparametric problem (10.74) has been solved, the explicit solution z ∗ 0 (x(0)) of (10.75) is available as a piecewise affine function of x(0), and the optimal control law u ∗ (0) is also available explicitly, as the optimal input u ∗ (0) consists simply of m components of z ∗ 0 (x(0)) u ∗ (0) = [0 . . .0 Im 0 . . . 0]z ∗ 0(x(0)). (10.77) Theorem 6.4 states that there always exists a continuous PPWA solution z ∗ 0(x) of the mp-LP problem (10.75). Clearly the same properties are inherited by the controller. The following Corollaries of Theorem 6.4 summarize the analytical properties of the optimal control law and of the value function.
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Predictive Control for linear and h
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ii Preface Dynamic optimization has
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iv book, in Chapter 3 we introduce
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vi Acknowledgements Large part of t
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viii Contents III Optimal Control 1
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x Contents Symbols and Acronyms Log
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xii Contents Dynamical Systems x(k)
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Chapter 1 Main Concepts In this Cha
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1.1 Optimization Problems 5 Active,
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1.2 Convexity 7 Operations preservi
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Chapter 2 Optimality Conditions 2.1
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2.2 Lagrange Duality Theory 11 wher
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2.3 Complementary Slackness 13 Note
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2.4 Karush-Kuhn-Tucker Conditions 1
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2.4 Karush-Kuhn-Tucker Conditions 1
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Chapter 3 Polyhedra, Polytopes and
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3.2 Polyhedra Definitions and Repre
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3.2 Polyhedra Definitions and Repre
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3.3 Polytopal Complexes 25 Definiti
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3.4 Basic Operations on Polytopes 2
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3.5 Operations on P-collections 37
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Chapter 4 Linear and Quadratic Opti
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4.1 Linear Programming 47 4.1.2 Dua
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4.1 Linear Programming 49 x2 c =
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4.1 Linear Programming 51 J(z) 6 5
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4.2 Quadratic Programming 53 0.5z
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4.3 Mixed-Integer Optimization 55 A
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4.3 Mixed-Integer Optimization 57 4
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Part II Multiparametric Programming
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62 5 General Results for Multiparam
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74 6 Multiparametric Programming: a
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100 6 Multiparametric Programming:
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Chapter 7 General Formulation and D
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7.2 Solution via Batch Approach 119
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7.3 Solution via Recursive Approach
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7.4 Optimal Control Problem with In
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7.4 Optimal Control Problem with In
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7.5 Lyapunov Stability 127 - asympt
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230 11 Receding Horizon Control x 2
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232 11 Receding Horizon Control
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234 12 Constrained Robust Optimal C
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268 13 On-line Control Computation
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Chapter 14 Models of Hybrid Systems
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14.2 Piecewise Affine Systems 285 X
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14.2 Piecewise Affine Systems 287 x
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14.2 Piecewise Affine Systems 289 k
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14.3 Discrete Hybrid Automata 291 A
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14.3 Discrete Hybrid Automata 293 e
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14.3 Discrete Hybrid Automata 295
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14.4 Logic and Mixed-Integer Inequa
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14.5 Mixed Logical Dynamical System
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14.7 The HYSDEL Modeling Language 3
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14.8 Literature Review 307 systems
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14.8 Literature Review 309 and allo
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Chapter 15 Optimal Control of Hybri
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15.1 Problem Formulation 313 Consid
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15.2 Properties of the State Feedba
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15.4 Computation of the Optimal Con
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15.6 State Feedback Solution via Re
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15.8 Receding Horizon Control 335 U
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15.8 Receding Horizon Control 337 x
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15.8 Receding Horizon Control 339 x
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References [1] L. Chisci A. Bempora
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References 343 [25] A. Bemporad. Re
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References 345 [52] F. Blanchini an
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References 347 [81] B. De Schutter
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References 349 [110] E. G. Gilbert
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References 351 [138] J.N. Hooker. L
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References 353 [164] M. Lazar, D. M
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References 355 [194] K. R. Muske an
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References 357 [222] M. Schechter.
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References 359 [249] R. Vidal, S. S
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Model Predictive Control

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