Robust Model Predictive Control

Robust Model Predictive Control Saˇsa V. Raković, Ph.D. DIC

Université Catholique de Louvain, Louvain–la–Neuve, January 16, 2018

Outline

Opening Model Predictive Control (MPC) Robust Model Predictive Control (RMPC) Tube Model Predictive Control (TMPC) Trends & Directions Closing

Saˇsa V. Rakovi´ c, Ph.D. DIC

Robust Model Predictive Control

UCL, Louvain, January 16, 2018

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Theme: MPC under uncertainty

Repetitive decision making process point of view. Technical aspects. Computational issues.




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MPC Analogy Jean Piaget (1896 – 1980) Cognitive Psychology Children learning and environment controlling

1. Image

2. Aim


3. Action


4. Collation


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MPC Analogy Jacques Richalet (1936 – ) Predictive Functional Control Credits for Brilliant Analogy

1. Image

2. Aim

3. Action

4. Collation

1. Model

2. Reference

3. Control

4. Feedback




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MPC Paradigm Goals: Constraint satisfaction, Stability, and Optimized performance. Tool: Model predictive control.

Model predictive control (MPC): Repetitive decision making process (DMP). Basic DMP is finite horizon optimal control. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Basic DMP = Finite Horizon Optimal Control Given an integer N ∈ N and a state x ∈ X select predicted sequences of control actions uN−1 := {u0 , u1 , . . . , uN−1 }, and controlled states xN := {x0 , x1 , . . . , xN−1 , xN },

which, for each k ∈ NN−1 := {0, 1, . . . , N − 1}, satisfy xk+1 = f (xk , uk ) with x0 = x, xk ∈ X,

uk ∈ U, and x N ∈ Xf ,

and which minimize

PN−1 k=0

`(xk , uk )+ Vf (xN ) .

(Hereafter, Xf and Vf are terminal constraint set and cost function.) Saˇsa V. Rakovi´ c, Ph.D. DIC



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Finite Horizon Optimal Control The set of admissible control sequences UN : UN (x) := {uN−1 : ∀k ∈ NN−1 , xk+1 = f (xk , uk ), with x0 = x,

xk ∈ X, uk ∈ U, and xN ∈ Xf }.

The cost VN : VN (x, uN−1 ) :=

(P N−1 k=0

`(xk , uk ) + Vf (xN )

∞

if uN−1 ∈ UN (x) if uN−1 6∈ UN (x).

The finite horizon optimal control problem PN : VN0 (x) := inf VN (x, uN−1 ), and, uN−1

u0N−1 (x)

:= arg min VN (x, uN−1 ). uN−1

The N–step controllability set XN : XN := {x ∈ Rn : UN (x) 6= ∅}. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Prototype Algorithm and Principal Components Prototype Algorithm: At any state x and time t in the control process. Solve basic DMP to evaluate control u00 (x) and value function VN0 (x). Implement control action u00 (x). Principal Components: Control law u00 . (Possibly set–valued) Feedback implicitly evaluated at current state. Closed–loop controlled dynamics x + ∈ f (x, u00 (x)). (Possibly set–valued) Implicitly evaluated at encountered states. Value function VN0 . Lyapunov certificate for closed–loop controlled dynamics. Controllability set, the domain of the value function, XN . Positively invariant set for closed–loop controlled dynamics. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Key Facts Main properties: MPC law u00 is feedback implicitly evaluated at current state. Predictions and optimized predictions are, however, open–loop. Consistently improving and stabilizing (under mild assumptions). Theoretical implementation: Mathematical (nonlinear) programming in general case. Strictly convex programming in most frequent cases. Practical implementation: Online optimization. Offline parameteric optimization and online look–up tables. Combinations of the online and offline parameteric optimization. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Main Properties Well–posedness. (Under fair assumptions VN0 , u0N−1 , and,XN are well–defined.) Positive invariance–inducing. (Initial feasibility ⇒ ad–infinitum feasibility.) Stabilizing. (Ensures stabilization of the desired equilibrium.) Consistently improving. (Increasing N decreases VN0 and enlarges XN , etc.) Optimizing. (Enhances performance of the induced control process.) Computationally plausible/effective. (Systematic implementation via standard optimization techniques.)




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Main Properties: Existence and Regularity Fair assumptions: f : Rn × Rm → Rn continuous.

X ⊆ Rn , U ⊆ Rm , and Xf ⊆ X closed.

` : Rn × Rm → [0, ∞), and Vf : Rn → [0, ∞) continuous.

ψ(u) ≤ `(x, u) for some radially unbounded ψ : Rm → [0, ∞). (Or U is bounded.) 0 = f (0, 0), 0 ∈ Xf ⊆ X, and 0 ∈ U. Existence and Regularity (Goebel and Raković): The value function VN0 : Rn → [0, ∞] is proper and lower semicontinuous. The value function VN0 (x) is finite if and only if x ∈ XN .

The set of minimizers u0N−1 (x) is nonempty and compact for every x ∈ XN . Saˇsa V. Rakovi´ c, Ph.D. DIC



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Main Properties: Continuity and Convexity Continuity (Goebel and Raković): If x¯ ∈ XN and UN (¯ x ) is continuous, VN0 (¯ x ) is continuous.

If VN0 (¯ x ) is continuous, u0N−1 (¯ x ) is outer semicontinuous, and thus continuous when it is single-valued. Convexity (Goebel and Raković): If, in addition, f is affine, X, U, and Xf are convex, and ` and Vf are convex. The controllability set XN is convex, the value function VN0 is convex, and thus continuous on the interior of XN . The set of minimizers u0N−1 has convex values.




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Main Properties: Strong Positive Invariance

Control invariance of Xf : ∀x ∈ Xf ⊆ X, ∃u ∈ U : f (x, u) ∈ Xf . w w Strong positive invariance of XN : ∀x ∈ XN ⊆ X, ∀u ∈ u00 (x) ⊆ U : f (x, u) ∈ XN .




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Main Properties: Strong Lyapunov Decrease

Control invariance of, and Lyapunov decrease of Vf over, Xf : ∀x ∈ Xf , ∃u ∈ U : f (x, u) ∈ Xf , and Vf (f (x, u)) ≤ Vf (x) − `(x, u). w w Strong positive invariance of, and Lyapunov decrease of VN0 over, XN : ∀x ∈ XN , ∀u ∈ u00 (x) : f (x, u) ∈ XN , and VN0 (f (x, u)) ≤ VN0 (x)−`(x, u).




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Main Properties: Consistent Improvement

Control invariance of, and Lyapunov decrease of Vf over, Xf : ∀x ∈ Xf , ∃u ∈ U : f (x, u) ∈ Xf , and Vf (f (x, u)) ≤ Vf (x) − `(x, u). w w Consistent Improvement of MPC, i.e., ∀N ∈ N: Xf =: X0 ⊆ XN ⊆ XN+1 , and 0 (x) ≤ V 0 (x) ≤ V 0 (x) := ∀x ∈ XN+1 , VN+1 Xf Vf (x). 0 N




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Main Properties Well–posedness. (Under fair assumptions VN0 , u0N−1 , and,XN are well–defined.) Positive invariance–inducing. (Initial feasibility ⇒ ad–infinitum feasibility.) Stabilizing. (Ensures stabilization of the desired equilibrium.) Consistently improving. (Increasing N decreases VN0 and enlarges XN , etc.) Optimizing. (Enhances performance of the induced control process.) Computationally plausible/effective. (Systematic implementation via standard optimization techniques.)




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A Few Points MPC is the best thing since sliced bread! High impact on diverse parts of academic community. Versatile applicability and beneficial utility over range of industries.

MPC’s evolution has followed an interesting path. Many misconceptions, and considerable time to overcome these. Relatively mature, but still offering considerable challenges.

MPC’s feedback nature and robustness. A form of feedback due to repetition of open–loop optimal control. Open–loop optimal control might be fragile (non–robust), and it is not the most adequate tool for control synthesis under uncertainty.




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Uncertainty Sources and Types Uncertainty Sources: Modeling. (Approximation, imprecision, etc.) Sensing. (Imperfection, estimation, etc.) Computing. (Cyber realm, software limitations, etc.) Implementing. (Physical realm, hardware limitations, etc.) ...

Uncertainty Types: Set–membership. Probabilistic. Combined.

Systematic analysis of uncertainty and its forms is a topic in its own right! Saˇsa V. Rakovi´ c, Ph.D. DIC



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Why Bother? Academic Aspect A simple, elegant and natural paradigm. A systematic, optimization based, synthesis method. A rich set of underlying theoretical problems/issues.

Applied Aspect The rare advanced control methodology replacing, and superseding, conventional controllers. Versatile applicability with thousands of applications across range of industries (petrochemical, energy, automotive, aerospace, ...). Ultimately, it is about optimizing performance and, hence, increasing profit or properties of interest (e.g. safety, quality, resilience, etc.). Saˇsa V. Rakovi´ c, Ph.D. DIC



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Part





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Robust Model Predictive Control Paradigm Goals: Robust constraint satisfaction, Robust stability, Optimized robust performance, and Computational practicability. Tool: Robust model predictive control. Robust model predictive control (RMPC): Repetitive decision making process (DMP). Basic DMP is finite horizon robust optimal control. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Pivotal Concerns Convoluted interaction of uncertainty with: System evolution, Constraints, and Performance.

Fragility (non–robustness) of conventional MPC, Intractability of exact, closed–loop robust MPC, Insensitivity of naive, open–loop robust MPC, Conflicting demands: Strong structural properties and Online computational tractability. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Value of Information Scalar system: Constraints: Target Set:

x+ = x + u + w, X = [−1, 1], U = [−3, 3] and W = [−2, 2], Xf = {0}

Objective I

Minimax stabilization of Xf . (Minimax control functions u : X → U.)

Objective II

Maximin stabilization of Xf . (Maximin control functions u : X × W → U.)

Observations:

Minimax stabilization not possible, as Xf ⊆ X ⊆ W. Maximin stabilization possible, take u(x, w ) = −x − w .

Explanation:

Inequality supa∈A inf b∈B c(a, b) ≤ inf b∈B supa∈A c(a, b). Our focus is on minimax informational setting.




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Predicting Without Uncertainty (x + = x + u) States xk depend on: initial state x0 , and controls u0 , u1 , . . . , uk−1 .

Controls uk depend on: initial state x0 .

Open–loop optimal control = Closed–loop optimal control. Can use open–loop control sequences!




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Predicting Without Uncertainty (x + = x + u)

In charge of selecting states {x0 , x1 , . . . , xN }, and

controls {u0 , u1 , . . . , uN−1 }.

These are open–loop, and: dynamically consistent, constraint admissible, and cost optimal.




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Predicting Under Uncertainty (x + = x + u + w ) States xk depend on: initial state x0 , controls u0 , u1 , . . . , uk−1 , and disturbances w0 , w1 . . . , wk−1 .

Controls uk depend on: current state xk .

Disturbances wk are independent. Open–loop optimal control (often ≺) Closed–loop optimal control. Should use closed–loop control policies! Saˇsa V. Rakovi´ c, Ph.D. DIC



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Predicting Under Uncertainty (x + = x + u + w ) Not in charge of selecting states {x0 , x1 , . . . , xN }, and

controls {u0 , u1 , . . . , uN−1 }.

These should be closed–loop, and worst–case: dynamically consistent, constraint admissible, and cost optimal.

In charge of selecting sets of possible states {X0 , X1 , . . . , XN }, and

controls {U0 , U1 , . . . , UN−1 }. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Robustness of Conventional MPC Necessary to consider robustness properties w.r.t. both cost and constraints. MPC is robust in some sense for some classes of problems. But, MPC is fragile because of discontinuity of the value function and its optimizer. The absence of nominal robustness is “predicted” by the celebrated Artstein’s Theorem, and “illustrated” by the famous Artstein’s circles. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Fragility of Conventional MPC

Key issues (mostly due to state, including terminal, constraints): Asymptotic stability needs not be a robust property (Teel), Optimal control of a continuous control system might induce a discontinuous controlled dynamics, and Optimal control might be a fragile process itself (Raković).

Message: There’s no thing such as a free lunch. Ensure robustness by design rather than hoping to get it for free.




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Computational Complexity of Exact Robust MPC

Exact robust MPC employs exact robust optimal control. Exact robust optimal control is an infinite–dimensional minimaximization problem taking generic form: minΠN−1 ∈ΠN−1 (x) maxwN−1 ∈WN VN (x, ΠN−1 , wN−1 ). Solvable via dynamic programming.




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Dynamic Programming Based Robust MPC Richard E. Bellman (1920 – 1984) Dynamic Programming Curse of Dimensionality

Minimax DP Recursion

(with boundary conditions V0 := Vf and X0 := Xf )

Max value functions Jk : ∀(x, u) ∈ Xk × U, Jk (x, u) = maxw {`(x, u, w ) + Vk−1 (f (x, u, w )) : w ∈ W}. Minimax value functions Vk : ∀x ∈ Xk , Vk (x) = minu {Jk (x, u) : u ∈ U ∧ ∀w ∈ W, f (x, u, w ) ∈ Xk−1 }. Minimax optimal control laws uk are the optimizers of the minimax value functions: ∀x ∈ Xk , uk (x) = arg minu {Jk (x, u) : u ∈ U ∧ ∀w ∈ W, f (x, u, w ) ∈ Xk−1 }. Domains of the minimax value functions Xk are the minimax controllability sets: Xk = F −1 (Xk−1 ) where F −1 (X ) = {x ∈ X : ∃u ∈ U, ∀w ∈ W, f (x, u, w ) ∈ X }. For DP based RMPC, one makes (repetitive) use of (a selection of) uN (·) and VN (·) over XN . Saˇsa V. Rakovi´ c, Ph.D. DIC



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Closed–Loop Robust MPC

System: x+ = x + u + w Uncertainty: w ∈ [−1, 1] Prediction horizon: N=4

Closed–Loop (or brute force scenarios based) RMPC is clearly intractable! Saˇsa V. Rakovi´ c, Ph.D. DIC



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Open–Loop Robust MPC

System: x+ = x + u + w Uncertainty: w ∈ [−1, 1] Prediction horizon: N=4

Open–Loop (or a careless man crossing street) RMPC is clearly insensitive! Saˇsa V. Rakovi´ c, Ph.D. DIC



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Breaking Down Complexity and Robust MPC Consensus What robust MPC a rational and intelligent man should be happy with? Improved computability w.r.t. DP based and closed–loop RPMC. Improved sensibility w.r.t. conventional and open–loop RMPC. And anything else on top of that as a bonus. Need for generalized: Control and state predictions, Constraint satisfaction, Stability notions, and Performance criteria, with the aim to: retain structural properties and ensure computational practicability. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Separable State Feedback (SSF) RMPC Parameterization via partial states P xk = kj=0 x(j,k)

controls P uk = kj=0 u(j,k) dynamics

x(j,k+1) = Ax(j,k) + Bu(j,k) initial conditions

x+ = x + u + w, w ∈ [−1, 1], N = 4.

x(0,0) = x ∧ x(k,k) = wk−1

Employ separable RMPC as compatible with the superposition principle! Saˇsa V. Rakovi´ c, Ph.D. DIC



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Separable prediction structure (x, u)–part time 0 1 0 x(0,0) = x x(0,1) 0 u(0,0) u(0,1) 1 x(1,1) = w0 1 u(1,1) 2 2

2 x(0,2) u(0,2) x(1,2) u(1,2) x(2,2) = w1 u(2,2)

... ... ... ... ... ... ... ... ...

k k

N−1 N−1 N N total total

k x(0,k) u(0,k) x(1,k) u(1,k) x(2,k) u(2,k)

... ... ... ... ... ... ...

N−1 x(0,N−1) u(0,N−1) x(1,N−1) u(1,N−1) x(2,N−1) u(2,N−1)

N x(0,N) x(1,N) x(2,N)

x(k,k) = wk−1 . . . x(k,N−1) x(k,N) u(k,k) . . . u(k,N−1) ... ... x(N−1,N−1) = wN−2 x(N−1,N) u(N−1,N−1) x(N,N) = wN−1 x(0,0)

P1

P2

u(0,0)

P1

P2

j=0 x(j,1) j=0

u(j,1)

...

Pk

u(j,2) . . .

Pk

j=0 x(j,2) j=0

j=0 x(j,k) j=0

u(j,k)

... ...

PN−1 Pj=0 N−1 j=0

x(j,N−1)

PN

j=0 x(j,N)

u(j,N−1)

The x–rows dynamics x(j,k+1) = Ax(j,k) + Bu(j,k) are deterministic. For worst case cost use column–wise the k th –(x,u)–rows. For worst case constraints use row–wise the k th –(x,u)–columns. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Main Existing Parameterizations in RMPC RMPC Method NO–RMPC OL–RMPC TIASF–RMPC TVASF–RMPC APDF–RMPC SSF–RMPC CL–RMPC DP–Based–RMPC

Parameterized States P xk = kj=0 x(j,k) P xk = kj=0 x(j,k) P xk = kj=0 x(j,k) P xk = kj=0 x(j,k) P xk = kj=0 x(j,k) P xk = kj=0 x(j,k) xk xk

Parameterized Controls P uk (xk , x) = K kj=0 x(j,k) uk (xk , x) = u(0,k) P uk (xk , x) = u(0,k) + K kj=1 x(j,k) Pk uk (xk , x) = u(0,k) + j=1 K(j,k) x(j,k) P uk (xk , x) = u(0,k) + kj=1 M(j,k) x(j,j) P uk (xk , x) = u(0,k) + kj=1 u(j,k) (x(j,k) ) uk (xk , x) = uk (xk ) uk (xk , x) = uk (xk )

Prediction Structure Separable Separable Separable Separable Separable Separable Aggregated Aggregated (∞)

OL–RMPC (Blanchini; Lee and Yu;...), TIASF–RMPC (Chisci and Zappa; Kouvaritakis and Cannon;...), TVASF–RMPC (...;Löfberg;...), APDF–RMPC (van Hessem and Bosgra; L¨ ofberg; Kerrigan;...), SSF–RMPC (Raković; Raković, Kouvaritakis, Cannon and Panos), CL–RMPC (Bertsekas; Scoekert and Mayne; ...), and DP–Based–RMPC (Bellman; Bertsekas; Mayne;...). Saˇsa V. Rakovi´ c, Ph.D. DIC



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RMPC Personal References Encyclopedia of Systems and Control DOI 10.1007/978-1-4471-5102-9_2-1 © Springer-Verlag London 2014

Robust Model-Predictive Control Saša Raković Member of the Senior Common Room at St. Edmund Hall, Oxford University, Oxford, UK

Invention of Prediction Structures and Categorization of Robust MPC Syntheses ⋆ Saˇ sa V. Rakovi´ c∗ ∗

Abstract Model-predictive control (MPC) is indisputably one of the rare modern control techniques that has significantly affected control engineering practice due to its unique ability to systematically handle constraints and optimize performance. Robust MPC (RMPC) is an improved form of the nominal MPC that is intrinsically robust in the face of uncertainty. The main objective of RMPC is to devise an optimization-based control synthesis method that accounts for the intricate interactions of the uncertainty with the system, constraints, and performance criteria in a theoretically rigorous and computationally tractable way. RMPC has become an area of theoretical relevance and practical importance but still offers the fundamental challenge of reaching a meaningful compromise between the quality of structural properties and the computational complexity. Keywords Model-predictive control • Robust model-predictive control • Robust optimal control • Robust stability

Keywords: Robust Model Predictive Control, Prediction Structures, Parameterized Tubes. 1. INTRODUCTION

Introduction RMPC is an optimization-based approach to the synthesis of robust control laws for constrained control systems subject to bounded uncertainty. RMPC synthesis can be seen as an adequately defined repetitive decision-making process, in which the underlying decision-making process is a suitably formulated robust optimal control (ROC) problem. The underlying ROC problem is specified in such a way so as to ensure that all possible predictions of the controlled state and corresponding control actions sequences satisfy constraints and that the “worst-case” cost is minimized. The decision variable in the corresponding ROC problem is a control policy (i.e., a sequence of control laws) ensuring that different control actions are allowed at different predicted states, while the uncertainty takes on a role of the adversary. RMPC utilizes recursively the solution to the associated ROC problem in order to implement the feedback control law that is, in fact, equal to the first control law of an optimal control policy. A theoretically rigorous approach to RMPC synthesis can be obtained either by employing, in a repetitive fashion, the dynamic programming solution of the corresponding ROC problem or by solving online, in a recursive manner, an infinite-dimensional optimization problem (Rawlings and Mayne 2009). In either case, the associated computational complexity renders the exact RMPC synthesis hardly ever tractable. This computational impracticability of the theoretically exact RMPC, in conjunction with the convoluted interactions of the uncertainty with the evolution of the �

St Edmund Hall, Oxford University, UK

Abstract: This plenary paper is concerned with robust model predictive control (MPC) synthesis. In particular, the novel notion of prediction structures is introduced, and then utilized to derive a precise and compact overview of the existing robust MPC (RMPC) syntheses as well as to indicate direct improvements of their system theoretic properties. The prediction structures paradigm allows for a systematic, implementation and examples independent, comparison and classification of the existing RMPC syntheses. The corresponding categorization of the currently available RMPC syntheses is derived by: (i) introducing the adequate indices as measures of structural (including topological and system theoretic) properties and computational complexity, and (ii) analysing the trade–off between the guaranteed structural properties and the necessary computational complexity. The associated analysis is based on the classical game and utility theory notions; for simplicity, it is delivered by deploying the aggregated structural property and computational complexity indices; furthermore, it is complemented with an unambiguous analysis of the ”holy–grail” trade–off between the quality of structural properties and the degree of computational complexity.

E-mail: svr@sasavRaković.com

Page 1 of 11

Robust model predictive control is an area of theoretical relevance and practical importance (Rawlings and Mayne, 2009). The field has attracted significant research attention over the last few decades. Nevertheless, the area still offers the fundamental challenge of reaching a reasonable compromise between computational practicability and quality of both topological and system theoretic properties. From a theoretical point of view, RMPC synthesis is an adequately defined repetitive decision making process, in which the underlying decision making process is a suitably formulated robust optimal control (ROC) problem. Thus, theoretically exact RMPC synthesis can be attained by employing, in a repetitive fashion, the dynamic programming (Bellman, 1957) solution of the underlying ROC problem (Mayne et al., 2000; Bertsekas, 2007). Unfortunately, the associated computational complexity is, in general, impracticable. It is, hence, rather tempting to aim to develop the methods which yield directly approximate solutions (Jones et al., 2007) to the underlying exact ROC problem. However, this approach is unwieldy, since the inherited complexity of the traditionally employed minimax optimization (Scokaert and Mayne, 1998; Kerrigan and Maciejowski, 2003) is bound to induce a reasonably high degree of computational complexity to any approximation based methodology that would guarantee reasonable system theoretic properties. All in all, the prohibitive computational complexity of the exact RMPC synthesis (or its direct minimax optimization based approximations) has motivated the development of alternative approaches.

The existing approaches to RMPC can be divided broadly into two categories depending on the treatment of the uncertainty and its interactions with evolution of the system, constraints and performance criteria. The first category of alternatives includes the methods that aim to employ the inherent robustness, when plausible, of deterministic MPC. These methods employ essentially deterministic MPC synthesis, albeit applied to a suitably modified description of the system, constraints and performance criteria, and utilize the notion of input–to–state stability. This category of the alternative approaches to RMPC results potentially in computationally practicable methods. However, the presence of uncertainty is taken care of in an indirect way, and furthermore such approaches seem to be limited since the deterministic MPC is itself an inherently fragile (non–robust) process. In this sense, it is known that the stability property of deterministic MPC is non– robust (Grimm et al., 2004), but the situation is, in fact, even worse since the optimal control of constrained discrete time systems is a fragile process itself (Raković, 2009). A more detailed overview of this category of RMPC synthesis methods can be found in (Raimondo et al., 2009). The second category of alternative approaches to RMPC includes the methods that account for the effects of the uncertainty directly, but aim to exploit suitable parameterization of the underlying control policy in order to reduce the corresponding computational complexity (Blanchini, 1990; Chisci et al., 2001; Löfberg, 2003b,a; Langson et al., 2004; Mayne et al., 2005; S. V. Raković, 2005; Goulart et al., 2006; Raković et al., 2012d). This category of approaches has been developed compatibly with a commonly accepted consensus: there is a need for simplifying approximations of the underlying control policy and for cost functions that

⋆ Email: [email protected].

What delayed the use of the superposition principle for design of RMPC?




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An Appraisal David Q. Mayne (1929 – ) Control Theory Model Predictive Control

..



. . . . . . .. .. .. .. ..


38

Part





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TMPC Paradigm The state tubes are sequences of sets of possible states. The control tubes are sequences of sets of possible controls. State and control tubes play role of state and control sequences. Tubes are induced from the dynamics, uncertainty and control policy. Optimal tubes are obtained via tube optimal control. TMPC is repetitive online utilization of related tube optimal control. All RMPC methods result in tubes. Parameterization of tubes and control policy is of major importance. Saˇsa V. Rakovi´ c, Ph.D. DIC



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TMPC Illustration

Controlled Inner and Outer Guaranteed Tubes Saˇsa V. Rakovi´ c, Ph.D. DIC



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TMPC Methods

Rigid TMPC (2002 – 2006). Homothetic TMPC (2007 – 2009). Elastic TMPC (2015 – 2016). SSF TMPC (2007 – 2010).




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Rigid TMPC Key Features Parameterizations (with S(1) := {x : Cx ≤ 1}) States xk = zk + sk .

Controls uk = vk + Ksk . State cross–sections Xk = zk ⊕ S(1).

Control cross–sections Uk = vk ⊕ KS(1).

Dynamics Set–dynamics Azk + Bvk ⊕ (A + BK )S(1) ⊕ W ⊆ zk+1 ⊕ S(1). “Nominal” dynamics zk+1 = Azk + Bvk .

“Local” dynamics (A + BK )S(1) ⊕ W ⊆ S(1).

Constraints State Xk ⊆ X ⇔ zk ∈ Z,

Control Uk ⊆ U ⇔ vk ∈ V, Saˇsa V. Rakovi´ c, Ph.D. DIC

Z := X S(1).

V := U KS(1).



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Rigid Tube MPC: Basic DMP/FHOC Given an integer N ∈ N and a state x ∈ X select finite sequences of control tubes centers vN−1 := {v0 , v1 , . . . , vN−1 }, and state tubes centers zN := {z0 , z1 , . . . , zN−1 , zN }, which, for each k ∈ {0, 1, . . . , N − 1}, satisfy zk+1 = Azk + Bvk , x − z0 ∈ S(1), zk ∈ Z, vk ∈ V, and

and which minimize Saˇsa V. Rakovi´ c, Ph.D. DIC

PN−1 k=0

zN ∈ Zf , `(zk , vk ) + Vf (zN ). Robust Model Predictive Control


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Rigid TMPC References State and, observer based, output feedback rigid TMPC. Automatica 45 (2009) 2082–2087

Contents lists available at ScienceDirect Automatica 41 (2005) 219 – 224

Automatica

Automatica 42 (2006) 1217 – 1222 www.elsevier.com/locate/automatica

www.elsevier.com/locate/automatica

Brief paper

Brief paper

Robust model predictive control of constrained linear systems with bounded disturbances�

Robust output feedback model predictive control of constrained linear systems�

D.Q. Maynea,∗ , M.M. Seronb , S.V. Rakovića

D.Q. Mayne a , S.V. Raković a,∗ , R. Findeisen b , F. Allgöwer b

a Department of Electrical and Electronic Engineering, Imperial College, London, SW72BT, UK b School of Electrical Engineering and Computer Science, University of Newcastle, New South Wales, Australia

a Department of Electrical and Electronic Engineering, Imperial College London b Institute for Systems Theory and Automatic Control (IST), University of Stuttgart, Germany

Received 9 February 2004; received in revised form 14 July 2004; accepted 23 August 2004 Available online 8 December 2004

Received 13 August 2005; received in revised form 22 November 2005; accepted 10 March 2006 Available online 4 May 2006

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Robust output feedback model predictive control of constrained linear systems: Time varying case✩ D.Q. Mayne a , S.V. Raković b,c,∗,1 , R. Findeisen c , F. Allgöwer d a

Abstract

Abstract

This paper provides a novel solution to the problem of robust model predictive control of constrained, linear, discrete-time systems in the presence of bounded disturbances. The optimal control problem that is solved online includes, uniquely, the initial state of the model employed in the problem as a decision variable. The associated value function is zero in a disturbance invariant set that serves as the ‘origin’ when bounded disturbances are present, and permits a strong stability result, namely robust exponential stability of the disturbance invariant set for the controlled system with bounded disturbances, to be obtained. The resultant online algorithm is a quadratic program of similar complexity to that required in conventional model predictive control. � 2004 Elsevier Ltd. All rights reserved.

This paper provides a solution to the problem of robust output feedback model predictive control of constrained, linear, discrete-time systems in the presence of bounded state and output disturbances. The proposed output feedback controller consists of a simple, stable Luenberger state estimator and a recently developed, robustly stabilizing, tube-based, model predictive controller. The state estimation error is bounded by an invariant set. The tube-based controller ensures that all possible realizations of the state trajectory lie in a simple uncertainty tube the ‘center’ of which is the solution of a nominal (disturbance-free) system and the ‘cross-section’ of which is also invariant. Satisfaction of the state and input constraints for the original system is guaranteed by employing tighter constraint sets for the nominal system. The complexity of the resultant controller is similar to that required for nominal model predictive control. � 2006 Elsevier Ltd. All rights reserved.

Keywords: Robust model predictive control; Robustness; Bounded disturbances Keywords: Output feedback; Robust model predictive control; Observer; Robust control invariant tubes

Department of Electrical and Electronic Engineering, Imperial College London, UK

b

Centre for Process Systems Engineering, Imperial College London, UK

c

Otto-von-Guericke-Universität, Institute for Automative Engineering, Magdeburg, Germany

d

Institute for Systems Theory and Automatic Control, University of Stuttgart, Stuttgart, Germany

article

info

Article history: Received 30 June 2008 Received in revised form 2 February 2009 Accepted 12 May 2009 Available online 27 June 2009 Keywords: Output feedback Robust model predictive control State observer

abstract The problem of output feedback model predictive control of discrete time systems in the presence of additive but bounded state and output disturbances is considered. The overall controller consists of two components, a stable state estimator and a tube based, robustly stabilizing model predictive controller. Earlier results are extended by allowing the estimator to be time varying. The proposed robust output feedback controller requires the online solution of a standard quadratic program. The closed loop system renders a specified invariant set robustly exponentially stable. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction

1. Introduction Model predictive control is widely employed for the control of constrained systems and an extensive literature on the subject exists some of which is reviewed in Bemporad and Morari (1999); Allgöwer, Badgwell, Qin, Rawlings, and Wright (1999); Mayne, Rawlings, Rao, and Scokaert (2000). Several methods for achieving robustness have been considered. The simplest is to ignore the disturbance and rely on the inherent robustness of deterministic model predictive control applied to the nominal system (Scokaert & Rawlings, 1995; Marruedo, Álamo, & Camacho, 2002). Open-loop model predictive control that determines the current control action by solving on-line an optimal control problem in which the decision variable is a sequence � This paper was not presented at any IFAC Meeting. This paper was recommended for publication in revised form by Associate Editor M. Guay under the direction of Editor F. Allgower. Research supported by the Engineering and Physical Sciences Research Council, UK. ∗ Corresponding author. E-mail address: [email protected] (D.Q. Mayne).

{u0 , u1 , . . . , uN−1 } of control actions was proposed in Zheng and Morari (1993); this method cannot contain the ‘spread’ of predicted trajectories resulting from disturbances. Hence feedback model predictive control in which the decision variable is a policy �, which is a sequence {�0 (·), �1 (·), . . . , �N−1 (·)} of control laws, was advocated in, for example, Mayne (1995); Kothare, Balakrishnan, and Morari (1996); Lee and Yu (1997); Scokaert and Mayne (1998); De Nicolao, Magni, and Scattolini (2000); Magni, Nijmeijer, and van der Schaft (2001); Magni, De Nicolao, Scattolini, and Allgöwer (2003). Determination of a control policy is usually prohibitively difficult so research has concentrated on various simplifying approximations (Mayne, 1995; Kothare et al., 1996; Scokaert & Mayne, 1998; Park & Kwon, 1999; Kouvaritakis, Rossiter, & Schuurmans, 2000; Schuurmans & Rossiter, 2000; Lee & Kouvaritakis, 2000; Mayne & Langson, 2001; Chisci, Rossiter, & Zappa, 2001; Lee, Kouvaritakis, & Cannon, 2002; Langson, Chryssochoos, Raković, & Mayne, 2004).In many papers, following (Lee & Kouvaritakis, 2000), a sub-optimal control policy in which �i (x) = vi + Kx is employed.

0005-1098/$ - see front matter � 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.08.019

1. Introduction Model predictive control has received considerable attention driven largely by its ability to handle hard constraints as well as nonlinearity. An inherent problem is that model predictive control normally requires full knowledge of the state (Findeisen, Imsland, Allgöwer, & Foss, 2003; Mayne, Rawlings, Rao, & Scokaert, 2000). Whereas, in many control problems, not all states can be measured exactly. In practice this problem is often overcome by employing ‘certainty equivalence’. For linear systems not subject to disturbances that employ an observer and linear control, stability of the closed-loop can be guaranteed by the separation principle. However, when state and output disturbances are present and the system or its controller is

� This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Franco Blanchini under the direction of Editor Roberto Tempo. Research supported by the Engineering and Physical Sciences Research Council, UK. ∗ Corresponding author. E-mail address: [email protected] (S.V. Raković).

nonlinear (as is the case with model predictive control of constrained systems), stability of the closed-loop cannot, in general, be ensured by simply combining a stable estimator with a stable state feedback controller (Atassi & Khalil, 1999; Teel & Praly, 1995). Robust stability is obtained if the nominal system is inherently robust and the estimation errors are sufficiently small; however, predictive controllers are not always inherently robust (Grimm, Messina, Teel, & Tuna, 2004; Scokaert, Rawlings, & Meadows, 1997). An appealing approach for overcoming this drawback is to use robust controller design methods that take the state estimation error directly into account. In the late sixties, Witsenhausen (1968a, 1968b) studied robust control synthesis and set-membership estimation of linear dynamic systems subject to bounded uncertainties; this work was followed by Schweppe (1968), Bertsekas and Rhodes (1971a,1971b), Glover and Schweppe (1971) and by related work on viability (Aubin, 1991; Kurzhanski & Vályi, 1997; Kurzhanski & Filippova, 1993). Further results include: a paper by Blanchini (1990) that deals with linear, constrained, uncertain, discrete-time systems, an anti-windup scheme that employs invariant sets in a similar fashion to their use in this

0005-1098/$ - see front matter � 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.03.005

Model predictive control has been widely applied, especially in the process industries, because of its ability to handle hard constraints. While output feedback is inevitably employed in practice, much of the literature deals with ‘deterministic’ model predictive control in which it is assumed that the system state is exactly known and that there is no model error or external disturbance. In the ‘deterministic’ case it is possible to achieve stabilizing control by solving an open loop optimal control problem online yielding implicit state feedback. When uncertainty is present in the form of state estimation error, model error or external disturbance, it is desirable that the decision variable in the optimal control problem solved online is a feedback policy.

✩ Research supported by the Engineering and Physical Sciences Research Council, UK. The material in this paper was partially presented at the 45th IEEE Conference on Decision and Control 2006, San Diego, CA, USA, December 2006. This paper was recommended for publication in revised form by Associate Editor Yasumasa Fujisaki under the direction of Editor Roberto Tempo. ∗ Corresponding address: Centre for Process Systems Engineering, Imperial College London, Roderic Hill Building, South Kensington Campus, SW7 2AZ London, UK. Tel.: +44 77 99775366; fax: +44 20 7594 6606. E-mail address: [email protected] (S.V. Raković). 1 S.V. Raković is currently a scientific associate at Otto-von-Guericke-Universität,

Magdeburg, Germany and an Honorary Research Associate at Imperial College London.

To avoid the consequent complexity, many schemes have been proposed for robust model predictive control in the face of uncertainty; see the review papers Findeisen, Imsland, Allgöwer and Foss (2003) and Mayne, Rawlings, Rao and Scokaert (2000). This paper extends the results on output feedback predictive control in Mayne, Raković, Findeisen and Allgöwer (2006) which may be consulted for a discussion of the relevant literature. In output feedback model predictive control, the state has to be estimated and this introduces a special type of uncertainty, state estimation error. Because optimality cannot be achieved in this case with a reasonable computational burden, compromises have to be made. Our main objective in this paper, and its predecessor (Mayne et al., 2006), is simplicity of the optimal control problem that is solved online. To achieve simplicity in state estimation, we employ a Luenberger observer to estimate the state instead of a moving horizon or set membership estimator; if the initial state estimation error and the disturbances are bounded, the state estimation error can be shown to be bounded. To achieve simplicity in control, we control the system and, hence, the observer, which is a dynamic system subject to an additive disturbance, using tube based model predictive control described, for the time invariant case, in Mayne, Seron, and Raković (2005). A novel nominal optimal control problem is defined whose solution yields both a central trajectory that satisfies constraints that are tighter than those specified for the original system and that converges to the origin as well as a control that maintains the observer trajectory within a known neighbourhood of the central trajectory. Combining this

0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.05.009

A few other papers supporting strongly the methodology.




45

Homothetic TMPC Key Features Parameterizations (with S(α) := {x : Cx ≤ α1} for α ∈ R≥0 ) States xk = zk + sk . Controls uk = vk + Ksk . State cross–sections Xk = zk ⊕ S(αk ).

Control cross–sections Uk = vk ⊕ KS(αk ).

Dynamics Set–dynamics Azk + Bvk ⊕ (A + BK )S(αk ) ⊕ W ⊆ zk+1 ⊕ S(αk+1 ). Equivalent constraints (zk , vk , αk , zk+1 , αk+1 ) ∈ F.

Constraints State Xk ⊆ X ⇔ (zk , αk ) ∈ G.

Control Uk ⊆ U ⇔ (vk , αk ) ∈ H. Saˇsa V. Rakovi´ c, Ph.D. DIC



46

Homothetic Tube MPC: Basic DMP/FHOC Given an integer N ∈ N and a state x ∈ X select finite sequences of control tubes centers vN−1 := {v0 , v1 , . . . , vN−1 },

state tubes centers zN := {z0 , z1 , . . . , zN−1 , zN }, and

tubes scalings aN := {α0 , α1 , . . . , αN−1 , αN }, which, for each k ∈ {0, 1, . . . , N − 1}, satisfy

(zk , vk , αk , zk+1 , αk+1 ) ∈ F, x − z0 ∈ S(α0 ),

(zk , αk ) ∈ G,

(vk , αk ) ∈ H, and

while minimizing

PN−1


k=0

(zN , αN ) ∈ Gf ,

`(zk , vk , αk ) + Vf (zN , αN ). Robust Model Predictive Control


47

Homothetic TMPC References

Approximate reachability and state feedback homothetic TMPC. Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008

Approximate Reachability Analysis for Linear Discrete Time Systems Using Homothety and Invariance

∗ Automatic Control Laboratory, ETH Z¨ urich, 8092 Z¨ urich, Switzerland, E-mail: [email protected] Department of Systems and Automatic, University of Seville, 41092 Seville, Spain, E-mail: [email protected]

Abstract: This paper introduces a method for approximate reachability, for linear discrete time systems, based on homothety and set invariance. The proposed method utilizes two particular families of sets, more precisely their members, and particular forms of the approximation maps to obtain simple inner and outer approximate reachable sets/tubes. The resulting set–dynamics, induced by the uncertainty set, the underlying dynamics in the state space and the approximation maps, are restricted to these particular families of sets and under standard assumptions yield bounded and convergent approximate reachable sets/tubes. A tractable computational procedure is suggested and a few illustrative examples are provided. c 2008 IFAC. Copyright Keywords: Reachability Analysis, Approximations, Homothety, Set Invariance. 1. INTRODUCTION Reachability analysis is one of the central research topics in the control theory due to its intimate relationship with optimal control, set invariance, set–membership state estimation, safety verification and control synthesis under uncertainty, see the monographs (Aubin, 1991; Kurzhanski and Vályi, 1997), the survey papers (Milanese and Vicino, 1991; Blanchini, 1999) and more recent references (Raković and Mayne, 2005; Artstein and Raković, 2008; Raković, 2007). Initial ideas related to reachability analysis and guaranteed state estimation can be traced back to the pioneering control literature (Witsenhausen, 1968; Hermes and Lasalle, 1969; Bertsekas and Rhodes, 1971; Schweppe, 1973; Kurzhanski, 1977). The research on reachability has recognized that the exact reachability is computationally demanding and has, therefore, focused on approximate reachability. Approximate reachability methods suggested in, for example, (Milanese and Vicino, 1991; Chernousko, 1994; Kurzhanski and Vályi, 1997; K¨ uhn, 1998a; Alamo et al., 2005; Kurzhanski and Varaiya, 2006) employ, essentially, ellipsoidal, polytopic or even zonotopic calculus to obtain guaranteed, possibly optimal with respect to a utilized criterion, inner and/or outer estimates of the exact reachable sets/tubes or sets of possible states consistent with acquired information, system dynamics and the uncertainty specification. Approximate reachability methods result in “optimal approximations” obtained, often, by somehow prioritizing “shape simplicity” and “geometric criteria” over “dynamical aspects” of the approximation procedures. The “dy1 Corresponding Author. 2 The second author acknowledges MCYT-Spain for funding this work DPI2007-66718-C04-01.

978-1-1234-7890-2/08/$20.00 © 2008 IFAC

namical aspects” of the approximation procedure can be addressed more directly using the set invariance theory. Recent advances in the set invariance theory include a theoretical framework for the examination of the minimality of invariant sets for the nonlinear–compact case (Artstein and Raković, 2008) and its specialization to the linear– convex case (Raković, 2007). Following ideas of (Artstein and Raković, 2008), we discuss a version of approximate reachability by utilizing homothety and set invariance. We analyze set–dynamics, induced by the uncertainty set, the underlying dynamics in the state space and the approximation maps, evolving within two particular families of sets whose members are homothetic copies of invariant inner and outer basic shape sets SY and SZ . We establish that, under modest assumptions, the resulting approximate reachable sets/tubes are bounded and convergent. Paper Structure: Section 2 presents necessary preliminaries. Sections 3 and 4 discuss the use of homothety and invariance in approximate reachability. Sections 5 and 6 provide computational remarks, examples and conclusions. Basic Nomenclature and Definitions: The sets of non–negative, positive integers and non–negative real numbers are denoted, respectively, by N , N+ and R+ , i.e. N := {0, 1, 2, . . .}, N+ := {1, 2, . . .} and R+ := {x ∈ R : x ≥ 0}. For two sets X ⊂ Rn and Y ⊂ Rn , the Minkowski set addition is defined by X ⊕ Y := {x + y : x ∈ X, y ∈ Y }. For a set X ⊂ Rn and a vector x ∈ Rn we write x ⊕ X instead of {x} ⊕ X. Given the sequence of sets {Xi ⊂ Rn }bi=a , a ∈ N, b ∈ N, b > a, we �b denote i=a Xi := Xa ⊕· · ·⊕Xb . Given a set X and a real matrix M of compatible dimensions (possibly a scalar) we define M X := {M x : x ∈ X}. Given a matrix M ∈ Rn×n , ρ(M ) denotes the largest absolute value of its eigenvalues.

15327

Systems & Control Letters 62 (2013) 209–217

Contents lists available at SciVerse ScienceDirect

Automatica

Systems & Control Letters

journal homepage: www.elsevier.com/locate/automatica

journal homepage: www.elsevier.com/locate/sysconle

Brief paper

Saˇ sa V. Rakovi´ c ∗,1 Mirko Fiacchini ∗∗,2

∗∗

Automatica 48 (2012) 1631–1638

Contents lists available at SciVerse ScienceDirect

10.3182/20080706-5-KR-1001.1233


Homothetic tube model predictive control✩ Saša V. Raković a,1 , Basil Kouvaritakis c , Rolf Findeisen b , Mark Cannon c a

Institute for Systems Research, University of Maryland, College Park, USA

b

Otto-von-Guericke-Universität, Institute for Automation Engineering, Magdeburg, Germany

c

Oxford University, Department of Engineering Sciences, Oxford, UK

article

info

Article history: Received 24 May 2010 Received in revised form 12 August 2011 Accepted 6 December 2011 Available online 23 June 2012 Keywords: Tube model predictive control Set invariance Set-dynamics

abstract

1. Introduction

✩ The material in this paper was partially presented at the 19th International Symposium on Mathematical Theory of Networks and Systems MTNS 2010, July 5–9, 2010, Budapest, Hungary. This paper was recommended for publication in revised form by Associate Editor Lalo Magni under the direction of Editor Frank Allgöwer. E-mail address: [email protected] (S.V. Raković). 1 Tel.: +1 202 621 4752; fax: +1 301 314 9218.

Saša V. Raković a,∗ , Basil Kouvaritakis b , Mark Cannon b a

Oxford University, St. Edmund Hall, Oxford, UK

b

Oxford University, Department of Engineering Science, Oxford, UK

article

The robust model predictive control for constrained linear discrete time systems is solved through the development of a homothetic tube model predictive control synthesis method. The method employs several novel features including a more general parameterization of the state and control tubes based on homothety and invariance, a more flexible form of the terminal constraint set and a relaxation of the controlled dynamics of the sets that define the state and control tubes. Under natural assumptions, the proposed method is computationally efficient and it induces strong system theoretic properties. © 2012 Elsevier Ltd. All rights reserved.

Tube model predictive control (TMPC) (Mayne, Raković, Findeisen, & Allgöwer, 2009; Mayne, Seron, & Raković, 2005; Raković, 2009) is a computationally tractable framework for robust model predictive control (RMPC). It is also mathematically sensible since tubes, which are sequences of sets, capture the totality of possible state and control sequences arising due to uncertainty. The effective use of tubes in RMPC is enabled through a parameterization of the control policy that allows for the direct handling of the disturbance and its interaction with the system dynamics, constraints and performance. In the case of linear systems with known models which are subject to bounded additive disturbances and convex constraint sets it is possible to use separable control policies in order to separate the evolution of the nominal system from that of the local uncertain system. The effect of disturbances can be accounted for by considering the local exact reachable tubes centered around the trajectories of the nominal system and invoking suitably modified constraints on the nominal variables as well as employing computationally more tractable performance objectives (Blanchini, 1990; Chisci, Rossiter, & Zappa, 2001; Lee, Kouvaritakis, & Cannon, 2002; Mayne et al., 2009, 2005; Raković, 2009).

Equi-normalization and exact scaling dynamics in homothetic tube model predictive control

info

Article history: Received 15 November 2011 Received in revised form 15 October 2012 Accepted 14 November 2012 Available online 3 January 2013

abstract This paper develops an equi-normalization process in order to verify the existence of, and characterize, the minimal robust positively invariant set generated by a number of a-priori, but suitably, selected linear inequalities. It also develops the exact scaling dynamics associated with this set. The paper then proposes an improved homothetic tube model predictive control synthesis for linear systems subject to additive disturbances and polytopic constraints. © 2012 Elsevier B.V. All rights reserved.

Keywords: Robust Optimal Constrained control Linear systems

Our main task is to utilize recent homothetic TMPC (HTMPC) concepts (Raković, Kouvaritakis, & Findeisen, 2009; Raković, Kouvaritakis, Findeisen, & Cannon, 2010), and develop an HTMPC synthesis for a more structured setting. The employed state and control tubes are sequences of the homothetic sets {Xk = zk ⊕ αk S }k∈NN and {Uk = vk ⊕ αk R}k∈NN −1 which are parameterized via the fixed shape sets S and R and the sequences of the centers, {zk }k∈NN and {vk }k∈NN −1 , and scalings {αk }k∈NN . The associated separable control policy {πk (·, Xk , Uk )}k∈NN −1 is a sequence of control laws parameterized via a local control function ν (·) and the sequences of the centers of the homothetic state and control tubes {zk }k∈NN and {vk }k∈NN −1 (so that πk (y, Xk , Uk ) = vk + ν(y − zk ) for all y ∈ Xk ). The tube basic shape sets S and R and the local control function ν (·) are designed off-line and satisfy reasonable assumptions, while the sequences {zk }k∈NN , {vk }k∈NN −1 and {αk }k∈NN are decision variables in the on-line optimization. In comparison to Blanchini (1990), Chisci et al. (2001), Lee et al. (2002), Mayne et al. (2005) and Mayne et al. (2009) our proposal introduces several novel features. First, our homothetic tubes are more general as is our separable control policy that makes use of a positively homogeneous local control function ν (·) (which includes linear local control functions as a special case). Second, we employ exact global set-dynamics of the underlying state and control tubes which allows for less conservative on-line constraint handling. Third, we introduce an improved terminal constraint set which is obtained by analyzing the local homothetic tube dynamics. The exact local set-dynamics is simplified by employing affine vector-valued dynamics describing the evolution of the centers and the scalings of the outer-bounding homothetic tubes; this, in turn, allows for the utilization of the classical set invariance

0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.05.003


1. Introduction Model predictive control (MPC) need not be robust [1–3], while min–max feedback MPC [4] is computationally intractable, hence the need for robust MPC synthesis that strikes a balance between optimality and computability. Tube MPC (TMPC) [2,3,5,6] forms a rigorous set theoretic approach to robust MPC synthesis, which results in computationally efficient handling of constraints and uncertainty. Early TMPC [3,5] used ‘‘rigid’’ tubes with fixed crosssection shape sets whose centers were optimized on-line. The deployment of the fixed tube cross-section shape sets results in a degree of conservatism. Latter ‘‘homothetic’’ tubes allowed for more flexibility: the cross-sections of homothetic state and control tubes are parameterized in terms of the sequences of associated centers and scalings [7]. These sequences form decision variables for the on-line optimization, and are subject to constraints that enforce robust constraint satisfaction. Global conditions describing the tube dynamics and ensuring robust constraint satisfaction were given in [7]. The dynamics of the local homothetic state and control tubes is characterized by the dynamics of the local state and control tubes centers and scalings. These dynamics plays a crucial role for derivation of the stabilizing conditions through the use of a terminal constraint set and a local Lyapunov function. In [7], an affine relationship between the tube scalings at consecutive

∗

Corresponding author. Tel.: +44 7799775366. E-mail addresses: [email protected], [email protected] (S.V. Raković).

times was used. The affine scalings dynamics represents an upper approximation of the exact dynamics and, as such, it induces a degree of conservatism. This article introduces an equi-normalization process, derives a description of the exact scaling dynamics and achieves a significant constraint relaxation. Using the proposed equi-normalization we establish the existence of, and characterize, the minimal robust positively invariant (RPI) set generated by a number of a-priori, but suitably, selected linear inequalities. The function inducing the exact scaling dynamics is shown to be a piecewise affine, convex, continuous, non-negative and monotonically nondecreasing function. Remarkably, this function is composed of, at most, two affine functions irrespective of the system dimension and the number of utilized linear inequalities. In particular, this function can be expressed as the maximum of two affine functions and yields the exact scaling dynamics (in contrast to the affine upper approximate scaling dynamics of [7]). The equi-normalized minimal RPI (mRPI) set together with the exact scaling function is used to analyze the dynamics of the local state and control tubes centers (z , Kz ) and scalings α . (The latter dynamics is determined completely by the (z , α)-dynamics.) This analysis enables us to ensure the maximality (w.r.t. both the volume and set inclusion) of the constraint set on the local state and control tubes centers (z , Kz ) and scalings α . (These constraints are characterized fully by constraints on the (z , α)-variable.) The equi-normalized mRPI set induces the minimal (w.r.t. set inclusion) homothetic tubes cross-section shape sets, while the exact scaling function induces the tightest (w.r.t. set inclusion) (z , α)-dynamics. In turn, this yields a significantly improved terminal constraint set, and

0167-6911/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2012.11.010


48

Elastic TMPC Key Features Parameterizations (with S(a) := {x : Cx ≤ a} for a ∈ Rq≥0 ) States xk = zk + sk . Controls uk = vk + K (ak )sk . State cross–sections Xk = zk ⊕ S(ak ).

Control cross–sections Uk = vk ⊕ K (ak )S(ak ).

Dynamics Set–dynamics Azk + Bvk ⊕ (A + BK (ak ))S(ak ) ⊕ W ⊆ zk+1 ⊕ S(ak+1 ). Guaranteed convexified constraints (zk , vk , ak , zk+1 , ak+1 ) ∈ F.

Guaranteed convexified constraints State Xk ⊆ X ⇐ (zk , ak ) ∈ G.

Control Uk ⊆ U ⇐ (vk , ak ) ∈ H. Saˇsa V. Rakovi´ c, Ph.D. DIC



49

Elastic Tube MPC: Basic DMP/FHOC Given an integer N ∈ N and a state x ∈ X select finite sequences of control tubes centers vN−1 := {v0 , v1 , . . . , vN−1 },

state tubes centers zN := {z0 , z1 , . . . , zN−1 , zN }, and

tubes elasticity parameters aN := {a0 , a1 , . . . , aN−1 , aN }, which, for each k ∈ {0, 1, . . . , N − 1}, satisfy (zk , vk , ak , zk+1 , ak+1 ) ∈ F, x − z0 ∈ S(a0 ),

(zk , ak ) ∈ G,

(vk , ak ) ∈ H, and

while minimizing

PN−1


k=0

(zN , aN ) ∈ Gf , `(zk , vk , ak ) + Vf (zN , aN ). Robust Model Predictive Control


50

Elastic TMPC References State feedback elastic TMPC. Elastic Tube Model Predictive Control Saˇsa V. Raković, William S. Levine and Behçet Açıkmes¸e

Abstract— This paper introduces elastic tube model predictive control (MPC) synthesis. The proposed framework is a natural generalization of the rigid and homothetic tube MPC design methods. The cross–sections of the employed state and control tubes are allowed to change more elastically, while the local component of the tubes control policy is permitted to take a more general form. The related stabilizing terminal conditions are also adequately generalized in order to take advantage of more flexible tubes and tubes control policy parameterizations. These novel features result in an improved tube MPC at a cost of manageable increase in computational complexity.

I. I NTRODUCTION Due to its unique ability to systematically handle constraints and optimize performance, MPC has become an elemental and contemporary research field that has seen important advances in underlying theory [1]–[3] as well as numerous industrial applications [4]. Robust MPC (RMPC) is an improved MPC variant providing structural properties (e.g. performance, invariance and stability) that are robust in face of the bounded uncertainty. A real intricacy in RMPC arises due to the facts that the closed loop RMPC [5] provides strong structural properties but it is computationally unwieldy, while the conventional MPC is not necessarily robust [6] even though it is computationally convenient. Being one of MPC’s fundamental subfields, RMPC has been the subject of intensive research investigations [5]–[17]. See also comprehensive monographs [2], [18], main survey papers [1], [3] and encyclopedic and plenary articles [19], [20] for an in–depth overview of the current state of affairs in RMPC and for a number of additional relevant references. Tube MPC (TMPC) has emerged as a dominant design framework for RMPC, since it addresses effectively the fundamental challenge of reaching a meaningful compromise between the quality of guaranteed structural properties and the associated computational complexity. TMPC considers predicted behaviour in terms of the sets of possible states and controls due to the spread of trajectories caused by the uncertainty. This results in state and control tubes that represent either the exact or outer bounding sequences of the sets of possible states and associated controls. The sensible parameterizations of the state and control tubes and related tubes control policy lead to computationally highly attractive TMPC that induces strong structural properties. TMPC is particularly effective for constrained linear systems subject to additive uncertainty [10], [11], [14]–[16]. Rigid TMPC (RTMPC) [11] is the first generation of TMPC. The cross–sections of the state and control tubes Saˇsa V. Raković and Behçet Açıkmes¸e are with the University of Texas at Austin, USA. William S. Levine is with the University of Maryland at College Park, USA.

in RTMPC are simple translations of the fixed cross–section basic shape sets S(1) and K(1)S(1) so that Xk = zk ⊕ S(1) and Uk = vk ⊕ K(1)S(1) where zk and vk are the centers of the state and control tubes. The related control laws take form πk (xk , zk , vk ) = vk + K(1)(xk − zk ). Homothetic TMPC (HTMPC) [14], [15] is the second generation of TMPC. The cross–sections of the state and control tubes in HTMPC are homothetic copies of the fixed cross–section basic shape sets S(1) and K(1)S(1) so that Xk = zk ⊕ αk S(1) and Uk = vk ⊕ αk K(1)S(1) where scalar αk is the scaling factor of the state and control tubes. Parameterized TMPC (PTMPC) [16], [19] is the third generation of TMPC. The cross–sections of the state and control tubes in PTMPC are expressed in terms of partial cross–sections so that Xk = X(0,k) ⊕ X(1,k) ⊕ . . . ⊕ X(k,k) and Uk = U(0,k) ⊕ U(1,k) ⊕ . . . ⊕ U(k,k) where sets X(j,k) and U(j,k) are paramaterized via finitely many points. The related control laws πk (xk , Xk , Uk ) are separable and nonlinear functions composed of partial control rules π(j,k) (x(j,k) , X(j,k) , U(j,k) ) �k so that πk (xk , Xk , Uk ) = j=0 π(j,k) (x(j,k) , X(j,k) , U(j,k) ) �k for xk = j=0 x(j,k) with x(j,k) ∈ X(j,k) . The online implementations of RTMPC, HTMPC and PTMPC reduce to convex optimization for which the number of decision variables and constraints scales linearly w.r.t. the prediction horizon in cases of RTMPC and HTMPC, and quadratically in case of PTMPC. RTMPC [11] induces strong system theoretic properties that are, however, weaker than those guaranteed by HTMPC [14], [15]. Both RTMPC and HTMPC are considerably outperformed by PTMPC [16], [19]. As a matter of fact, PTMPC outperforms all major methods for RMPC such as those based on the constraint tightening with or without prestabilization [7], [8] or on the use of affine–in–the–predicted–states and affine–in–the– past–disturbances control policies [9], [12].

S.V. Raković, W.S. Levine and B. A¸cıkme¸se

Elastic Tube Model Predictive Control Saˇsa V. Raković (1), William S. Levine (2), Beh¸cet A¸cıkme¸se (3), (1) The University of Texas, Austin, USA (2) The University of Maryland, College Park, USA (3) The University of Washington, Seattle, USA This paper introduces elastic tube model predictive control (MPC) synthesis. The proposed framework is a natural generalization of the rigid and homothetic tube MPC design methods. The cross–sections of the employed state and control tubes are allowed to change more elastically, while the local component of the tubes control policy is permitted to take a more general form. The related stabilizing terminal conditions are also adequately generalized in order to take advantage of more flexible tubes and tubes control policy parameterizations. These novel features result in an improved tube MPC at a cost of manageable increase in computational complexity.

The main aim of our proposal is to improve considerably RTMPC and HTMPC design methods at a cost of controlled increase in computational complexity. This goal is achieved by introducing more flexible parameterizations of the state and control tubes and related tubes control policy. The cross– sections Xk of the state tubes are parameterized in terms of the centers zk and vector–valued elasticity parameters ak as Xk = zk ⊕ S(ak ). Likewise, the cross–sections Uk of the control tubes are parameterized in terms of the centers vk and elasticity parameters ak as Uk = vk ⊕ K(ak )S(ak ). The related tubes control policy is formed from the control laws πk (xk , zk , vk , ak ) = vk + K(ak )(xk − zk ). In the spirit of RTMPC and HTMPC, the suitable set–valued function S (·), with values S(a), and matrix–valued function K (·),

1

Conference version out. Journal version to be submitted. Saˇsa V. Rakovi´ c, Ph.D. DIC



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Rigid, Homothetic and Elastic Tube MPCs: Comparison

Rigid TMPC Homothetic TMPC Elastic TMPC Saˇsa V. Rakovi´ c, Ph.D. DIC



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Recall: SSF RMPC

Parameterization via partial states P xk = kj=0 x(j,k)

controls P uk = kj=0 u(j,k) dynamics

x(j,k+1) = Ax(j,k) + Bu(j,k) initial conditions

x+ = x + u + w, w ∈ [−1, 1], N = 4.

x(0,0) = x ∧ x(k,k) = wk−1




53

Recall: separable prediction structure (x, u)–part time 0 1 2 0 x(0,0) = x x(0,1) x(0,2) 0 u(0,0) u(0,1) u(0,2) 1 x(1,1) = w0 x(1,2) 1 u(1,1) u(1,2) 2 x(2,2) = w1 2 u(2,2)

... ... ... ... ... ... ... ... ...

k k

k x(0,k) u(0,k) x(1,k) u(1,k) x(2,k) u(2,k)

... ... ... ... ... ... ...

N−1 x(0,N−1) u(0,N−1) x(1,N−1) u(1,N−1) x(2,N−1) u(2,N−1)

N x(0,N) x(1,N) x(2,N)

x(k,k) = wk−1 . . . x(k,N−1) x(k,N) u(k,k) . . . u(k,N−1) ... ... x(N−1,N−1) = wN−2 x(N−1,N) u(N−1,N−1) x(N,N) = wN−1

N−1 N−1 N N total

x(0,0)

P1

P2

total

u(0,0)

P1

P2

j=0 x(j,1) j=0 u(j,1)

...

Pk

...

j=0 u(j,2) . . .

Pk

...

j=0 x(j,2)

j=0 x(j,k) j=0 u(j,k)

PN−1 Pj=0 N−1

x(j,N−1)

PN

j=0 x(j,N)

u(j,N−1) j=0

Disturbance set W := convh({w ˜i : i ∈ I}). The deterministic “extreme” x–rows dynamics x(i,j,k+1) = Ax(i,j,k) + Bu(i,j,k) . Saˇsa V. Rakovi´ c, Ph.D. DIC



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SSF TMPC Key Features Parameterizations (with flexible, optimizable cross–section shapes) States xk =

Pk

j=0 x(j,k) .

Partial state cross–sections X(j,k) = convh({x(i,j,k) : i ∈ I}). Lk State cross–sections Xk = x(0,k) ⊕ j=1 X(j,k) . Pk Controls uk = j=0 u(j,k) .

Partial control cross–sections U(j,k) = convh({u(i,j,k) : i ∈ I}). Lk Control cross–sections Uk = u(0,k) ⊕ j=1 U(j,k) .

Dynamics

Partial dynamics x(0,k+1) = Ax(0,k) + Bu(0,k) ,

x(0,0) = x.

Partial extreme dynamics x(i,j,k+1) = Ax(i,j,k) + Bu(i,j,k) ,

Constraints State x(0,k) ⊕

Lk

Control u(0,k) ⊕ Saˇsa V. Rakovi´ c, Ph.D. DIC

j=1

Lk

x(i,j,j) = w ˜i .

convh({x(i,j,k) : i ∈ I}) ⊆ X.

j=1

convh({u(i,j,k) : i ∈ I}) ⊆ U.



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SSF Tube MPC: Basic DMP/FHOC Given an integer N ∈ N and a state x ∈ X select separable L N−1 control UN−1 := {u(0,k) ⊕ kj=1 convh({u(i,j,k) : i ∈ I})}k=0 , and Lk state XN := {x(0,k) ⊕ j=1 convh({x(i,j,k) : i ∈ I})}N k=0 ,

tubes which, for each k ∈ {0, 1, . . . , N − 1}, ensure that

x(0,k+1) = Ax(0,k) + Bu(0,k) , x(0,0) = x,

while minimizing

PN−1


k=0

x(i,j,k+1) = Ax(i,j,k) + Bu(i,j,k) , x(i,j,j) = w ˜i , Lk x(0,k) ⊕ j=1 convh({x(i,j,k) : i ∈ I}) ⊆ X, Lk u(0,k) ⊕ j=1 convh({u(i,j,k) : i ∈ I}) ⊆ U, and LN x(0,N) ⊕ j=1 convh({x(i,j,N) : i ∈ I}) ⊆ Xf ,

`(Xk , Uk ) + Vf (XN ).



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SSF TMPC References Robust control invariance and state feedback SSF TMPC. IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 7, JULY 2010

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Parameterized Robust Control Invariant Sets for Linear Systems: Theoretical Advances and Computational Remarks Saˇsa V. Rakovic´ and Miroslav Baric´

Abstract—We characterize a family of parametrized robust control invariant sets for linear discrete time systems subject to additive but bounded state disturbances. The existence of a member of the introduced family of parametrized robust control invariant sets can be verified by solving a tractable convex optimization problem in the linear convex case, which reduces to the standard linear or convex quadratic programme in the linear polytopic case. The developed method can also be utilized to detect and obtain an implicit representation of local control Lyapunov functions in the linear convex case from the solution of a single and tractable convex optimization problem. The offered method permits for the computation of polytopic robust control invariant sets and local control Lyapunov functions of indirectly controlled and limited complexity in the linear polytopic case. Index Terms—Control Lyapunov functions, linear polytopic case.

I. INTRODUCTION

T

HE research on the set invariance theory and its application to robust control synthesis and analysis problems is a topical one; see, for instance, the comprehensive monographs [1], [2], the survey paper [3] and references therein for a more detailed overview. Indeed, invariant sets permit for the synthesis of controllers guaranteeing robust constraint satisfaction, and under additional and relatively mild assumptions robust stability and convergence to an adequate set despite the presence of uncertainty and hard constraints on system variables. The main issues in set invariance are, for obvious reasons, the characterization and computation of the maximal and minimal invariant sets and their invariant approximations [1]–[12]. Though the maximal and minimal invariant sets are of particular interest, a direct inspection of the related literature suggests that any invariant set can be utilized for the corresponding control synthesis or analysis of the uncertain dynamics; see, for example,

Manuscript received November 17, 2008; revised September 11, 2009 and December 01, 2009. First published February 05, 2010; current version published July 08, 2010. Recommended by Associate Editor D. Arzelier. S. V. Rakovic´ is with the Institute for Automation Engineering, Otto-von-Guericke-Universität, Magdeburg 39106, Germany. He is also with the Department of Engineering Sciences, University of Oxford, Oxford OX1 2JD, U.K. (e-mail: [email protected]). M. Baric´ is with the Department of Mechanical Engineering, University of California, Berkeley, CA 94720 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2010.2042341

[13]–[19] and references therein. Consequently, computationally more tractable methods (compared to the standard set invariance computations) have been also considered [20]. The optimized robust control invariance approach [20] addresses the issues related to the finite time termination of the viability kernel based algorithms, and permits for an indirect control of the complexity of the resulting robust control invariant set. In this manuscript, we offer further advances and characterize a novel family of parametrized robust control invariant sets for linear discrete time systems subject to additive but bounded state disturbances. The reported results are more general than those of the optimized robust control invariance [20] and several other existing methods [8], [10], [12], since the considered construction utilizes more general feedback control laws. The theoretical developments lead to efficient computational procedures based on the standard optimization methods; in fact, we show that the existence of a member of the introduced family of parametrized robust control invariant sets for the constrained case can be verified by solving a tractable convex optimization problem whose size scales polynomially with the size of input data. A formulation of the corresponding convex optimization problem is discussed in more detail for a frequently encountered case – namely, the linear polytopic case. Apart from the characterization and implicit computation of robust control invariant sets, we demonstrate that the reported results can be utilized for the construction of indirectly controlled and limited complexity polytopic local control Lyapunov functions for constrained linear discrete time systems via an optimization procedure. Paper Structure: Section II provides necessary preliminaries and paper objectives. Section III introduces the underlying principle permitting for the finite time characterization of parameterized robust control invariant sets. Section IV presents a novel family of polytopic parameterized robust control invariant sets. Section V suggests the utilization of reported results in the stability analysis. Section VI discusses computational aspects. Sections VII and VIII present numerical examples and concluding remarks. Basic Nomenclature and Definitions: The sets of non–negative and positive integers are denoted, respectively, by and . Let for given and such that ; denotes for . Similarly, a set of non–negative real numbers is denoted by so that . The Minkowski set addition of two (nonempty) subsets of , say and , is denoted by ; in addition, if is a vector in (which could be a sum of vectors), we write to denote . Given a sequence of sets such that , with ,

0018-9286/$26.00 © 2010 IEEE

2746

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 11, NOVEMBER 2012

Parameterized Tube Model Predictive Control

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust. Nonlinear Control 2012; 22:1330–1361 Published online 7 May 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.2825

Saa V. Raković, Basil Kouvaritakis, Mark Cannon, Christos Panos, and Rolf Findeisen

Fully parameterized tube model predictive control‡ AbstractThis paper develops a parameterized tube model predictive control (MPC) synthesis method. The most relevant novel feature of our proposal is the online use of a single tractable linear program that optimizes parameterized, Minkowski decomposable, state and control tubes and an associated, fully separable, nonlinear, control policy. The induced control policy enjoys a higher degree of nonlinearity than existing tube MPC and robust MPC using disturbance af�ne control policy. Our proposal offers greater generality than the state of the art robust MPC methods. It is conjectured, and also established in three cases, that our proposal is equivalent, feasibility-wise, to dynamic programming (DP). It is also shown that, under natural assumptions, our method is computationally ef�cient while it possesses rather strong system theoretic properties. Index TermsConstrained control, convex programming, robust control, set-dynamics, tube model predictive control.

I. INTRODUCTION

R

OBUST model predictive control (RMPC) is an area of practical importance, which received a lot of research attention over the last two decades [1] but still offers the signi�cant challenge of reaching a reasonable compromise between computational tractability and degree of optimality. A theoretically rigorous way for RMPC synthesis is to employ, in a repetitive fashion, the DP solution [2][4] of the associated robust optimal control problem. However, prohibitive computational complexity motivated the development of alternatives. Early RMPC was based on open loop min-max optimal control problems and thus had limited control theoretic properties. An alternative proposed a reformulation of a DP-based solution and used closed loop min-max optimal control in the repetitive fashion [5], [6]. This is computationally unwieldy since the associated approach reduces, in general, to an in�nite dimensional optimization, or in more structured cases [5], [6], to a �nite-dimensional optimization for which the number of decision variables and constraints scales exponentially with the prediction horizon. The inadequacy of open loop min-max Manuscript received August 10, 2010; revised July 25, 2011; accepted February 20, 2012. Date of publication March 15, 2012; date of current version October 24, 2012. The �nalization of this work was supported by the University of Maryland and the University of Maryland Foundation. This research started while S. V. Raković was a Visiting Academic Fellow at the University of Oxford and a Scienti�c Associate at the at the Ottovon Guericke University of Magdeburg. Recommended by Associate Editor X. Chen. S. V. Raković is with the University of Maryland, College Park, MD 207423271 USA (e-mail: [email protected]; [email protected]). B. Kouvaritakis and M. Cannon are with the Department of Engineering Science, Oxford University, Oxford OX1 3PJ, U.K. (e-mail: basil. [email protected]; [email protected]). C. Panos is with the Department of Chemical Engineering, Imperial College London, London WC2 H9N, U.K. (e-mail: [email protected]). R. Findeisen is with the Systems Theory and Control Laboratory, Institute for Automation, Otto-von-Guericke-University of Magdeburg, 39016 Magdeburg, Germany (e-mail: rolf.�[email protected]). Digital Object Identi�er 10.1109/TAC.2012.2191174

RMPC and the impracticability of DP based and closed loop min-max RMPC has led to a consensus: there is a need for simplifying approximations of the underlying control policy and for cost functions that ensure computational tractability while retaining system theoretic rigor. An approach with this potential is tube model predictive control (TMPC) [7][11] which uses a parameterization of the partially separable policy that allows for the direct handling of uncertainty and its interaction with the system dynamics, constraints and performance. Early TMPC (e.g., [9] and [10]) employs state and control tubes with time varying cross sections and results in a reduction of the computational complexity. More recent proposals [7] employ tubes with constant cross sections, but allow for the optimization of the initial condition of the nominal system and guarantee robust stability and attractivity of the corresponding minimal robust positively invariant set. Recent advances of TMPC [11], termed homothetic tube model predictive control, employ: a more general parameterization of state and control tubes and associated control policy based on homothety, invariance and homogeneity; a more �exible form of the terminal constraint set; and a relaxation of the global tube dynamics. However, all existing TMPC methods utilize the of�ine design of the tube cross sections and the local feedback control law, and thus can be conservative. The RMPC with disturbance af�ne control policy [12][14] leads to improved system theoretic properties at a manageable computational cost, but the restriction that the dependence of control laws should be af�ne constitutes a serious limitation. This paper proposes a novel TMPC methodology which supersedes RMPC with the disturbance af�ne control policy [12][14] as it provides a more general nonlinear framework with which to achieve more optimal results at the same computational cost. The key to this is a new parameterization of state and control tubes for constrained linear systems which are subject to additive polytopic uncertainty. Our proposal, in contrast to the existing TMPC methods, allows uniquely for the simultaneous online optimization of the state and control tubes as well as the associated control policy. The new parameterized TMPC (PTMPC) offers, at the same computational cost, greater generality compared to RMPC with the disturbance af�ne control policy, while the associated parameterized tube optimal control (PTOC) problem overcomes the tractability problem of closed loop min-max RMPC [5], [6], by leading to a standard, �nite dimensional, linear program for which the number of decision variables and constraints scales quadratically with the prediction horizon. Combined use of linearity, separability, superposition and convexity allows a decomposition of the -steps-ahead into components state satisfying . Likewise, the -steps-ahead control rule is decomposed into components satisfying . Consequently, the state components evolve according to the deterministic recur, and hence the sion

Saša V. Raković1,*,† , Basil Kouvaritakis2 , Mark Cannon2 and Christos Panos3 1 Institute

for Systems Research, University of Maryland, College Park, MD, USA of Engineering Science, Oxford University, Oxford, UK for Process Systems Engineering, Imperial College London, London, UK

2 Department 3 Centre

SUMMARY The recently proposed parameterized tube model predictive control (MPC) exploited linearity to separate the treatment of future disturbances in robust model predictive control, thereby gaining significant computational advantages while superseding the state-of-the-art closed-loop strategies. The approach used an upper triangular parameterized tube prediction structure and a linear cost. The current paper proposes an extension of the tube prediction structure, which is fully parameterized and allows for the use of more general cost functions. The introduced parameterization generalizes the one associated with parameterized tube MPC (PTMPC), while it shares its strong system theoretic properties as well as its computational tractability. Copyright © 2012 John Wiley & Sons, Ltd. Received 9 August 2011; Revised 10 February 2012; Accepted 14 March 2012 KEY WORDS:

robust control; predictive control; constrained optimization

1. INTRODUCTION Despite the plethora of results on robust model predictive control (MPC) (e.g., [1, 2]), there remains the challenge of reaching an efficient compromise between computational complexity and degree of optimality. Optimality can be achieved via dynamic programming (DP) (e.g., [3]) and closedloop min–max MPC [4, 5], but the computation required by these approaches grows exponentially with the prediction horizon, N . Open-loop or semi-closed-loop strategies are computationally cheaper, and, among these, tube model predictive control (TMPC) provides an effective alternative. Tubes can be rigid with adjustable centers [6] or have variable centers and variable cross-sectional scalings [7, 8]. The so-called disturbance affine control policy [9, 10] provides a way of optimizing parameterized control policies, giving improved performance and larger regions of attraction at increased computational cost, which is nonetheless manageable because the numbers of decision variables and constraints depend quadratically (rather than exponentially) on N . However, the use of affine strategies can be restrictive, and this was circumvented in [11, 12] through the use of a state/control decomposition that separates the treatment of each future value of additive disturbances. This separation holds the key to preventing the exponential growth of computation: it does not require all extreme points of the regions in which all possible predicted states lie to be considered explicitly. More precisely, the decomposition replaces the single pair of state and control tubes used in [6–8, 13, 14] with a number of pairs of partial state and control tubes. Each partial state/control tube has as its initial cross section, either the current state/control or the polyhedral set, that contains term of the uncertain future state/control sequence. The parameterization employs an upper

*Correspondence to: Saša V. Raković, Institute for Systems Research, University of Maryland, College Park, MD, USA. † E-mail: [email protected] ‡ This paper honours Professor David W. Clarke’s early contributions to the model predictive control field. Copyright © 2012 John Wiley & Sons, Ltd.

0018-9286/$31.00 © 2012 IEEE

A “dual/polar” parameterization developed, and it is about to be reported.




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TMPC Methods in Books

Includes my work with D. Q. Mayne on Rigid state feedback TMPC. Rigid, observer based, output feedback TMPC. Time–varying, observer based, output feedback TMPC. And a few other results that I developed.




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TMPC Methods in Books

Includes my work on Rigid TMPC (with D. Q. Mayne). Homothetic TMPC. SSF TMPC. And a few other results that I developed.




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Part





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Present Trends

Scenarios based “quasi–robust” MPC. Real–time RMPC (Real–time TMPC). Stochastic MPC. Decentralized MPC under uncertainty. Networked MPC under uncertainty. Economic MPC under uncertainty. And many more, MPC has seen tremendous expansion.




61

Potential Directions Modelling uncertainty for MPC. Contemporary uncertainty in MPC. Resilient MPC. Fault–tolerant MPC. MPC for autonomous systems. Collaboratively adaptive MPC. And many more classical and contemporary directions, since MPC is applicable to a wide range of conventional and modern areas.




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A Big Picture in MPC: Integration Integration of identification and MPC (e.g., Adaptive MPC). Integration of uncertainty modelling and MPC (e.g., Flexible MPC under uncertainty). Integration of estimation and MPC (e.g., Output feedback MPC). Integration of fault tolerance and MPC (e.g., Reconfigurable and actively fault tolerant MPC). Integration of MPC’s general components and optimization (i.e., Integrated MPC synthesis). Make sure that the sum of parts is equal to the whole! Saˇsa V. Rakovi´ c, Ph.D. DIC



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A Big Picture in MPC: Artificial Intelligence Interconnections of MPC and AI!

Create bidirectionally beneficial symbiosis of MPC and AI!




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Part





65

The Handbook of Model Predictive Control To be published by Springer/Birkhäuser. Edited with W. S. Levine. Board composed of 7 world renowned scholars. 3 parts: Theory composed of 13 chapters; Optimization composed of 8 chapters; Applications composed of 7 chapters.

All in all 28 chapters by leading experts in MPC and related areas. Saˇsa V. Rakovi´ c, Ph.D. DIC



66

The Second Summer School on MPC Previous summer school on MPC held at UW Madison in July 2017. To be held at UM at Ann Arbour in July 2018. Organized with I. V. Kolmanovsky and J. B. Rawlings. 5 days, 8 themes and instructors style school. Essential MPC by J. B. Rawlings. Robust MPC by S. V. Raković. Stochastic MPC by I. V. Kolmanovsky. Hybrid MPC by R. Sanfelice. Convex Optimization for MPC by S. Wright. Optimization for MPC by L. Biegler. MPC in Practise by T. Badgwell. Aerospace/Automotive Applications of MPC by S. Di Cairano. Saˇsa V. Rakovi´ c, Ph.D. DIC



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This Occasion

More of a story telling and a different perspective!

Follow up discussion is, as always, welcome!




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