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Invariant Set Design for Constrained Discrete-Time Linear Systems with Bounded Matched Disturbance
9th IFAC Symposium on Robust Control Design 9th IFAC Symposium on Robust Control Design Florianopolis, Brazil, September 3-5, 2018Design 9th IFAC Symposium on Robust Control Florianopolis, Brazil, September 3-5, 2018 Available online at www.sciencedirect.com 9th IFAC Symposium on Robust Control Florianopolis, Brazil, September 3-5, 2018Design Florianopolis, Brazil, September 3-5, 2018
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IFAC PapersOnLine 51-25 (2018) 55–60
Invariant Set Design for Constrained Invariant Set Design for Constrained Invariant Set Design for Constrained Discrete-Time Linear Systems with Bounded Invariant Set Design for Constrained Discrete-Time Linear Systems with Bounded Discrete-Time Linear Systems with Bounded Matched Disturbance Discrete-Time LinearDisturbance Systems with Bounded Matched Matched Disturbance Matched Disturbance ∗∗ ∗ Nathan Michel ∗,∗∗ ∗,∗∗ Sorin Olaru ∗∗ Sylvain Bertrand ∗
Nathan Michel ∗,∗∗ Sorin Olaru ∗∗ Sylvain Bertrand ∗ ∗∗ ∗∗ NathanGiorgio MichelValmorbida Sorin Olaru Sylvain Bertrand Dumur ∗∗ Didier ∗∗ Giorgio Valmorbida Didier Dumur ∗,∗∗ ∗∗ ∗ ∗∗ ∗∗ NathanGiorgio MichelValmorbida Sorin Olaru Sylvain Bertrand Didier Dumur ∗∗ ∗∗ Giorgio Valmorbida Didier Dumur ∗ ∗ ONERA - The French Aerospace Lab, F-91123 Palaiseau, France ∗ ONERA - The French Aerospace Lab, F-91123 Palaiseau, France ONERA The French Aerospace Lab, F-91123 Palaiseau, France (e-mail: [email protected] ;; [email protected]) [email protected] [email protected]) ∗ (e-mail: ∗∗ ONERA The French Aerospace Lab, F-91123 Palaiseau, France (e-mail: [email protected] ; [email protected]) L2S, CentraleSupélec, CNRS, Univ. Paris-Sud, ∗∗ CNRS, ;Univ. Paris-Sud, Université Université ∗∗ L2S, CentraleSupélec, (e-mail: [email protected] [email protected]) L2S, CentraleSupélec, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 3 rue Joliot-Curie, Gif-Sur-Yvette 91192, France Paris-Saclay, 3 rue Joliot-Curie, Gif-Sur-Yvette 91192, France ∗∗ L2S, CentraleSupélec, CNRS, Gif-Sur-Yvette Univ. Paris-Sud, Université Paris-Saclay, 3 rue Joliot-Curie, 91192, France (e-mail: [email protected] ; (e-mail: [email protected] ; Paris-Saclay, [email protected] rue Joliot-Curie, Gif-Sur-Yvette 91192, ;France (e-mail: [email protected] ; [email protected]) [email protected] ; [email protected]) (e-mail: [email protected] ; [email protected] ; [email protected]) [email protected] ; [email protected]) Abstract: Invariant Invariant set set theory theory has has been been recognized recognized as as an an important important tool tool for for control control design design Abstract: Abstract: Invariant setsubject theory to hasdisturbances. been recognized as an important tool for control design of constrained constrained systems subject to disturbances. Indeed, for aa given control law, entering an of systems Indeed, for given control law, entering an Abstract: Invariant setsubject theory to has been recognized as an important toolin for control design of constrained systems disturbances. Indeed, for a given control law, entering an invariant set guarantees recursive state and input constraint satisfaction closed-loop. This invariant set guarantees recursive state and input constraint satisfaction in closed-loop. This of constrained systems subject to state disturbances. for asatisfaction given control law, disturbance. enteringThis an invariant set guarantees recursive and subject inputIndeed, constraint in closed-loop. paper focuses on discrete-time linear systems to bounded matched additive paper focuses on discrete-time linear systems subject to bounded matched additive disturbance. invariant set of guarantees recursive state andlaws input constraint satisfaction inthat closed-loop. This paper focuses on discrete-time linear systems subject to bounded matched additive disturbance. The problem the joint synthesis of control and associated invariant sets are optimized The problem ofonthe joint synthesis of control laws and associated invariant sets that are optimized paper focusesto discrete-time linear systems subject to bounded matched additive disturbance. The of the joint of control laws and An associated invariant sets that are optimized withproblem regards the statesynthesis constraints is investigated. An interpolation method is used used to enlarge with regards to the state constraints is investigated. interpolation method is to enlarge The problem of the joint synthesis of control laws and associated invariant sets that are optimized with regards to the state constraints is investigated. An interpolation method is used to enlarge the controllable region. the controllable region. with regards to the state constraints is investigated. An interpolation method is used to enlarge the controllable region. © 2018, IFAC (International the controllable region. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Invariant Invariant set, set, disturbance, disturbance, state state constraints, constraints, input input constraints, constraints, nonlinear nonlinear control control Keywords: Keywords: Invariant set, disturbance, state constraints, input constraints, nonlinear control Keywords: Invariant set, disturbance, state constraints, input constraints, nonlinear control invariant 1. INTRODUCTION INTRODUCTION invariant set set respecting respecting the the constraints, constraints, denoted denoted Maximal Maximal 1. invariant set respecting the constraints, denoted Maximal 1. INTRODUCTION Robustly Positively Invariant (MRPI) set. Robustly Positively Invariant (MRPI) set. invariant respecting the constraints, denoted Maximal 1. INTRODUCTION Robustly set Positively Invariant (MRPI) set. Previous work has focused on the joint synthesis of Robustly Positively Invariant (MRPI) set. Previous work has focused on the joint synthesis of control control Constrained control of dynamic systems in presence of Previous work has focused on the joint synthesis of laws and associated invariant sets tailored to state orcontrol input Constrained control of dynamic systems in presence of laws and associated invariant sets tailored to state or input Constrained control dynamic systems presence of Previous work has focused on(2014); the tailored joint synthesis oforcontrol disturbance faces faces twoofmain main challenges: theinimpact impact of the the laws and associated invariant sets to state input constraints (Corradini et al. Nguyen (2012); Tahir disturbance two challenges: the of constraints (Corradini et al. (2014); Nguyen (2012); Tahir Constrained control dynamic systems presence of laws disturbance twooflocal main challenges: theinimpact of the and associated invariant sets tailored to state or Tahir input disturbancesfaces on the the local behavior around nominal constraints (Corradini etRakovi` al. (2014); Nguyen (2012); and Jaimoukha (2015); cc et al. (2007); Michel et al. disturbances on behavior around aa nominal and Jaimoukha (2015); Rakovi` et al. (2007); Michel al. et disturbance faces two main challenges: the impact of the disturbances on the local behavior around a nominal constraints (Corradini et al. (2014); Nguyen (2012); Tahir equilibrium and the characterization of the controllable and Jaimoukha (2015); Rakovi` c et al. (2007); Michel et al. (2018)). In Rakovi` c et al. (2007), a characterization of equilibrium and the characterization of the controllable (2018)). In Rakovi` c et al. (2007), a characterization of disturbances on the characterization local behavior around ainnominal controllable equilibrium and the of the and Jaimoukha (2015); Rakovi` c et al. (2007); Michel et al. region. Both challenges have been addressed several (2018)). In Rakovi` c et al. (2007), a characterization of families of robust control invariant sets, based on outer in several region. Both challenges have been addressed families of robust control invariant sets, based on outer equilibrium and the characterization oftothe in several region. challenges haveaccording been addressed In robust Rakovi` c et al.invariant (2007), asets, characterization of control Both design frameworks according thecontrollable tools and (2018)). families of basedThis on outer approximations of the set, charcontrol design frameworks to the tools and approximations of control the mRPI mRPI set, is is proposed. proposed. This charregion. Both challenges have been addressed in several control design frameworks according to the tools and families of robust control invariant sets, based on outer modelling assumptions: set theoretic methods (Blanchini approximations of the mRPI set, is proposed. This characterization be used to optimized invariant modelling assumptions: set theoretic methods (Blanchini acterization can can of bethe used to establish establish optimizedThis invariant control frameworks according(Jaulin to the(2000)), tools and modelling assumptions: theoretic methods (Blanchini approximations mRPI set, constraints. is proposed. char(1999)), design interval based set approaches or acterization optimized invariant can bestate used to input establish sets regarding the and The control (1999)), interval based approaches (Jaulin (2000)), or sets regarding the state and input constraints. The control modelling assumptions: set theoretic methods (Blanchini (1999)), interval based approaches (Jaulin (2000)), or acterization can be used to establish optimized invariant robust Model Predictive Control (Mayne et al. (2005)). sets regarding the state and input constraints. The control design in Michel et al. (2018) considers bounded matched robust Model Predictive Control (Mayne et al. (2005)). design in Michel etstate al. (2018) considers bounded matched (1999)), interval based Control approaches (Jaulin robust Model Predictive (Mayne et al.(2000)), (2005)). or sets regarding theet and input Thestrategy. control design indisturbance Michel al.and (2018) considers bounded matched additive adopts aaconstraints. sliding mode Set theoretic methods require a set description of the additive disturbance and adopts sliding mode strategy. robust Model Predictive Control (Mayne et al. (2005)). Set theoretic methods require a set description of the design in Michel et al. (2018) considers bounded matched a sliding additive disturbance and adopts mode strategy. The control law feedback gain minSet theoretic A methods require description of the disturbances. A systematic way ato to set assess the influence influence of additive The resulting resulting control and law is is the the linear linear feedback gain mindisturbances. systematic way assess the of disturbance apredefined sliding mode strategy. gain minThe resulting control law adopts is the feedback imizing the mRPI projection in aalinear direction. Set theoretic isA methods require ato set description of the disturbances. systematic way assess the influence of disturbances to compute invariant sets. Indeed, invariimizing the mRPI projection in predefined direction. disturbances is to compute invariant sets. Indeed, invariThe resulting control law is the linear feedback gain mindisturbances. systematic way toconstraints assess influence of imizing the mRPI projection in a predefined direction. Indeed, invaridisturbances isA to compute invariant sets.the ant sets sets are are certificates certificates for robust robust satisfaction, In this paper we extend the linear control law and associant for constraints satisfaction, imizing the mRPI projection in a predefined direction. In this paper we extend the linear control law and associdisturbances is to compute invariant Indeed, invariant sets are certificates robust constraints satisfaction, recursive feasibility, andfor mitigation of sets. the disturbances for control this paper we linearof and associated invariant set design strategy Michel et (2018) to disturbances for In recursive feasibility, and mitigation of the ated invariant setextend design the strategy ofcontrol Michellaw et al. al. (2018) to ant setscontrol are certificates for robust constraints satisfaction, disturbances recursive feasibility, and mitigation of the for In this paper we extend the linearand law and associa given law (Mayne et al. (2000)). Such approaches ated invariant set design strategy of Michel et al. (2018) to aa larger class of state constraints, we account for input arecursive given control law (Mayne et al. (2000)). Such approaches larger class of state constraints, and we account for input feasibility, and mitigation of the disturbances for ahave given control law in (Mayne et al. (2000)). approaches invariant design strategy of Michel et al. (2018) to been studied in the context context of model modelSuch predictive con- ated aconstraints. larger classThe ofset state constraints, and we account for input impact of around the have been studied the of predictive conconstraints. The impact of the the disturbances disturbances around the a given control law (Mayne etrobust al. (2000)). Such approaches of model predictive con- constraints. have been studied in the context a larger class of state constraints, and we account for input trol (Mayne et al. (2005)), time-optimal control The impact of the disturbances around the nominal equilibrium is further mitigated by relaxing trol (Mayne et al.in(2005)), robust time-optimal control nominal equilibrium is further by relaxing the have been studied the context model control (Mayne et al. (2005)), robust time-optimal control The impact of themitigated disturbances around sets the (Mayne and Schroeder (1997)), orofdesign design ofpredictive reference gov- constraints. nominal equilibrium is further byinvariant relaxing the linear structure, allowing for (Mayne and Schroeder (1997)), or of reference govlinear control control structure, allowingmitigated for smaller smaller invariant sets trol (Mayne et et al.al.(2005)), robust time-optimal control (Mayne and Schroeder (1997)), or design of reference govnominal equilibrium is further mitigated by relaxing the ernors (Falcone (2009)). A series of results regarding linear structure, allowing for smaller invariant sets in direction of constraints. An interpolationinterpolationernors (Falcone et al. (2009)). Aor series of results regarding in the the control direction of the the state state constraints. An (Mayne and (1997)), design of reference gov- linear ernors (Falcone et been al. (2009)). A series of results regarding structure, allowing for smaller invariant sets invariant set Schroeder have been established for linear linear systems, with in the control direction of theisstate constraints. An interpolationbased control design then proposed to enlarge the invariant set have established for systems, with based control design isstate thenconstraints. proposed toAn enlarge the conconernors (Falcone et al. subject (2009)). A series of results regarding invariant set have been established for linear systems, with in the direction of the interpolationa linear control law, to additive bounded disturthe conbased control design is then proposed to enlarge trollable region. disturainvariant linear control law, to additive bounded trollable region. set beensubject established for and linear systems, with based disturabance, linearsee control law, subject to additive bounded control design is then proposed to enlarge the consee forhave example Kolmanovsky Gilbert (1998). trollable region. bance, for example Kolmanovsky and Gilbert (1998). aOflinear control law, subject to additive bounded disturThe paper is organized bance, see for example Kolmanovsky and Gilbert (1998). trollable region. particular interest is the so-called minimal Robust PosThe paper is organized as as follows. follows. Section Section 22 presents presents the the Of particular is Kolmanovsky the so-called minimal Robust Pos- The bance, see forinterest example and Gilbert (1998). paper is organized asand follows. Section 2 presentsSecthe class of system studied important definitions. Of particular interest is the so-called minimal Robust Positively Invariant (mRPI) set. It is defined as the smallest class of system studied and important definitions. Secitively Invariant (mRPI) set. It is defined as the smallest The paper is organized as follows. Section 2 presents the Of particular is the so-called minimal system studied important tion introduces results on design of invariant invariantSecset itively Invariant set. It is defined as Robust the smallest invariant set interest for (mRPI) given disturbance set (Rakovi` (Rakovi` etPosal. class tion 33of introduces resultsand on the the design definitions. of set invariant set for aa given disturbance set cc et al. class ofintroduces system studied and important definitions. Secitively Invariant (mRPI) set. It is defined as the smallest tion 3 results on the design of invariant set tailored to the state constraints. Section 4 proposes an invariant set for a given disturbance set (Rakovi` c et al. (2005)). It corresponds to the limit set of state trajectories tailored to the state constraints. Sectionof4 invariant proposes set an (2005)). Itset corresponds to the limit set of trajectories 3 introduces results on to theenlarge design tailored to thebased state constraints. Sectionthe 4 proposes an invariant for aof given disturbance setstate (Rakovi` c of et the al. tion interpolation method controllable (2005)). It corresponds to the limit set of state trajectories for any sequence disturbances. Characterization interpolation based method to enlarge the controllable for any sequence of disturbances. Characterization of the to the 5state constraints. Sectionthe 4ofproposes an (2005)). It corresponds to thebelimit set of state trajectories interpolation to enlarge region. illustrative examples the for any sequence disturbances. Characterization of the tailored controllable regionofcan can also addressed by set set theoretic region. Section Sectionbased 5 gives givesmethod illustrative examples of controllable the results. results. theoretic controllable region also be addressed by interpolation based method to enlarge theof controllable for any sequence of disturbances. Characterization of the region. Section 5 gives illustrative examples the results. Finally, Section 6 draws conclusion and discusses perspeccontrollable region can also be addressed by set theoretic methods (Blanchini (Blanchini (1999); (1999); Mayne Mayne et et al. al. (2005)). (2005)). The The set set Finally, Section 6 draws conclusion and discusses perspecmethods region. Section 56gives of theperspecresults. controllable region can also Mayne be addressed byisset theoretic Finally, drawsillustrative conclusionexamples and discusses tives. methods (Blanchini (1999); et al. (2005)). The set of interest for such controllability analysis the largest tives. Section of interest(Blanchini for such (1999); controllability analysis is the The largest Finally, Section 6 draws conclusion and discusses perspecmethods Mayne et al. (2005)). set tives. of interest for such controllability analysis is the largest of interest for such controllability analysis is the largest tives.
2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 91 Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2018 IFAC 91 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 91 10.1016/j.ifacol.2018.11.081 Copyright © 2018 IFAC 91
IFAC ROCOND 2018 56 Florianopolis, Brazil, September 3-5, 2018 Nathan Michel et al. / IFAC PapersOnLine 51-25 (2018) 55–60
Notation: For a positive integer p, define Ip = {1, ..., p}. For a vector h ∈ Rn , denote hi its ith element, de fine ||h||∞ = max|hi |, i ∈ In , and |h| = [|h1 | ... |hn |] . For two vectors x and y, x ≤ y (x < y) denotes the element-wise (strict) inequalities between their components. Define 1p the vector of ones of dimension p. The ith power of a matrix A is denoted Ai . The ith line of a matrix A is denoted Ai . Define the set of invertible matrices Gn = {A ∈ Rn×n | det(A) = 0}. The set of Schur matrices of dimension n is denoted Cn . For a matrix A ∈ Rm×n and a set X ⊆ Rn , define the set AX = {y ∈ Rm | y = Ax, x ∈ X }. Given two sets A, B, define A ⊕ B = {a + b | a ∈ A, b ∈ B} and A B = {x | {x} ⊕ B ⊆ A}. The boundary of a set A is denoted ∂A.
constraint set (XK , W) is defined as the RPI set containing all the RPI sets, denoted here O∞ (K). Definition 4. The minimal Robust Positively Invariant (mRPI) set for the system x+ = (A + BK)x + Bw and constraint set (XK , W) is defined as the RPI set contained in any closed RPI set. If A + BK is Schur, the mRPI for the system x+ = (A + BK)x + Bw and constraint set (Rn , W) exists, is unique, compact and contains the origin in its interior. Moreover, it is given by the following infinite Minkowski sum ∞ (A + BK)i BW. Z∞ (K) = i=0
An RPI for the system x+ = (A + BK)x + Bw and constraint set (Xν , W) exists if and only if Z∞ (K) ⊆ XK . Remark 2. In general, we do not have an explicit characterization of the set Z∞ (K). For computational purposes, polytopic outer approximations of this set are sought (Olaru et al. (2010); Rakovi`c et al. (2005)).
2. PRELIMINARIES Consider the following discrete-time linear time-invariant system (1) x+ = Ax + B(u + w), B = 0⊥ n−m,m Im where x ∈ Rn is the state, u ∈ Rm is the input, w ∈ W ⊆ Rm is an unknown bounded disturbance, and we assume ¯ + B(u ¯ + w) that m < n. Note that any system x+ = Ax ¯ with rank(B) = m can be written as (1) with a linear change of coordinates. The system (1) is subject to the state and input constraints (2) x ∈ X = {x ∈ Rn | |Hx| ≤ 1p } , u ∈ U, p×n , m ≤ p. The sets U and W are bounded where H ∈ R polytopes containing the origin in their interior. This paper focuses on the stabilization of (1) to a neighborhood of the origin, characterized in terms of invariant set, along with the characterization of the controllable region.
The local behavior around the origin can be characterized in terms of RCI sets, or RPI sets and their associated control law. In this paper, we want to design a local control law that mitigates the impact of the disturbances on the state constraints satisfaction. Hence, our goal is to design a control law and an associated RPI set, or an RCI set, that is minimal in the direction of the state constraints. To this local control strategy we add an interpolation-based control design to enlarge the controllable region. A measure to evaluate the minimality of invariant sets with regards to the state constraints is introduced in the following section.
The following definitions are based on invariant set theory (Blanchini (1999); Kolmanovsky and Gilbert (1998)). Definition 1. The set Z is said to be Robustly Controlled positively Invariant (RCI) for the system (1) and constraint set (X , U, W) if Z ⊆ X , and ∀x ∈ Z there exists u ∈ U such that Ax + B(u + w) ∈ Z, ∀w ∈ W.
3. INVARIANT SET DESIGN The criterion for the design of RCI sets Z is the minimization of (3) h(Z, H) = max ||Hx||∞ . x∈Z
Indeed, Z accounts for the impact of the disturbance, and H characterizes the direction of the state constraints. However, we do not have an explicit characterization of all the RCI sets for the system (1). The proposed approach is to first minimize (3) among mRPI obtained with linear control laws, which is a choice for the computation of RPI sets using existing methods (Olaru et al. (2010); Rakovi`c et al. (2005)). The linear control structure is then relaxed to construct a decreasing sequence of RCI sets starting from this mRPI.
In the following, an RCI set will refer to an RCI set for the system (1) and constraint set (X , U, W).
Given a state feedback control law ν : Rn → Rm , we define the set Xν = {x ∈ X | ν(x) ∈ U} . Definition 2. A set Z ⊆ Rn is said Robustly Positively Invariant (RPI) for the system x+ = Ax + B(ν(x) + w) and constraint set (Xν , W), if Z ⊆ Xν , and ∀x ∈ Z, Ax + B(ν(x) + w) ∈ Z, ∀w ∈ W. Remark 1. Note that an RPI set for the system x+ = Ax+ Bν(x) + Bw and constraint set (Xν , W) is an RCI set. Likewise, from any RCI set Z it is possible to define a state feedback ν : Z → U such that Z is an RPI set for the system x+ = Ax + Bν(x) + Bw and constraint set (Xν , W) (see Rakovi`c et al. (2007)).
The computation of a feedback gain leading to an mRPI minimizing (3) is presented below and a strategy to further improve the solution by relaxing the linear control law structure is presented in Section 3.2. 3.1 Invariant set design using linear state feedback
The following definitions consider a linear state feedback law ν(x) = Kx, and we define the polytopic set XK = {x ∈ X | Kx ∈ U} . Definition 3. The Maximal Robustly Positively Invariant (MRPI) set for the system x+ = (A + BK)x + Bw and
We now briefly recall the results in Michel et al. (2018) for the minimization of (3) with (1)-(2), p = m and U = Rm . The design method allows to compute the linear control law ν(x) = Kx whose mRPI set Z∞ (K) minimizes (3) 92
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which, thanks to (6), (7), and (8), satisfies ∀x ∈ Ω0 , Ax + Bν0 (x) ∈ Ω0 BW, ν0 (x) ∈ U. The above definition of ν0 seeks to minimize the scaling factor α in the direction of the state constraints defined by the matrix H as in (2). From the definition of an RCI set, the feasible domain of the above optimization problem is guaranteed to be non-empty. Note that the above optimization problem is convex (convex constraints and linear cost function).
under the assumptions that the matrix H = [HB ⊥ HB ] satisfies HB ∈ Gm . In this paper, we propose a strategy that extends the approach for the case p > m. p } Let us consider the matrices Hσi ∈ Rm×n , i ∈ {1, ..., m as the matrices obtained from the combination of m distinct rows out of the p rows of H. Consider the partition of those matrices, Hσi = [H define pσi ,B ⊥ Hσi ,B ], and the set H = Hσi , i ∈ {1, ..., m } | Hσi ,B ∈ Gm . From every matrix Hσi ∈ H we compute the linear control law νσi (x) = Kσi x with the method presented in Michel et al. (2018) that minimizes (4) h(Z, Hσi ) = max ||Hσi x||∞ .
We then define the set Ω1 as Ω1 = ConvexHull {Av + Bν0 (v), v ∈ V(Ω0 )} ⊕ BW. By construction, the set Ω1 is polytopic, satisfies Ω1 ⊆ Ω0 , and is an RPI set for the system x+ = Ax + Bν0 (x) + Bw and constraint set (Xν0 , W). Remark 3. Note that the image set, namely {z ∈ Rn | z = Ax + Bν0 (x) + Bw, x ∈ Ω0 , w ∈ W} might not be convex.
x∈Z
Define theset p | K = Kσi , i ∈ 1, ..., m
Hσi ∈ H,Kσi Z∞ (Kσi ) ⊆ U
This set contains the linear feedback gains Kσi obtained from the matrices Hσi ∈ H, and such that Z∞ (Kσi ) is an RPI set for the system (1) and constraint set (X , U, W). If the set K is non-empty, we chose the element minimizing (3), that is max ||Hx||∞ . (5) K = arg min Kσi ∈K
57
Likewise, we can define a sequence of polytopic RCI sets Ωi , i ∈ N, as Ωi+1 = ConvexHull {Av + Bνi (v), v ∈ V(Ωi )}⊕BW, i ∈ N, where
x∈Z∞ (Kσi )
νi (x) = arg minimize α
Using these elements we are able to state the following result. Proposition 1. If Z∞ (K) ⊆ X , then robust asymptotic stability of the set Z∞ (K) is achieved with a region of attraction O∞ (K). Additionally, the finite determination of O∞ (K) is guaranteed.
u
subject to Ax + Bu ∈ Ωi BW
H(Ax + Bu) ∈ αH(Ωi BW) u∈U
0 ≤ α ≤ 1.
The strategy presented here allows to take the state constraints into account in the design of a linear feedback gain and construct the associated mRPI in a direct manner by exploiting the matched properties of the disturbance. If the set K is empty, alternative design strategies based on the complete characterization of the RCI sets taking into account input constraints are to be sought (Nguyen (2012); Rakovi`c et al. (2007); Tahir and Jaimoukha (2015)).
By construction, we have ∀i ∈ N, ∀j ∈ N, Ωi+j ⊆ Ωi , thus the objective (3) decreases (not strictly) as i increases. Let k ∈ N, and let Ω = Ωk . Consider the following control law ν : Ω0 → U: (9a) ν(x) = νk (x), x ∈ Ωk , (9b) ν(x) = νi (x), x ∈ Ωi \Ωi+1 , i ∈ {0, ..., k − 1}, ν(x) = ν0 (x), x ∈ Ω0 . (9c)
In the next section we propose to improve the solution proposed in this section by relaxing the linear control structure and allowing for nonlinear control policies.
For all x ∈ Ω0 , finite time convergence to Ω is guaranteed. Indeed, if x ∈ Ωi , i ∈ Ik−1 , x+ = Ax + Bν(x) + Bw ∈ Ωi+1 , ∀w ∈ W.
3.2 Refinement of RCI sets with nonlinear control laws
We have presented a method to construct, starting from a polytopic RCI set, a decreasing sequence of polytopic RCI sets that aims at minimizing (3).
In the previous section we imposed a linear structure to the control law in the design of an RPI set. We now propose an optimization-based method to obtain a decreasing sequence of RCI sets with nonlinear control policies starting from a polytopic RCI set.
In the following section we define the control law ν(x) outside of the set Ω0 using interpolation-based methods to extend the controllable region.
Let Ω0 be a polytopic RCI set. We consider the control law ν0 : Ω0 → U :
4. ENLARGEMENT OF THE CONTROLLABLE REGION
ν0 (x) = arg minimize α H(Ax + Bu) ∈ αH(Ω0 BW)
The previous section focused on the behavior of the system in a neighborhood around the origin. The proposed control law is defined locally. We now propose a method to enlarge the controllable region using interpolation-based control.
0≤α≤1
Let K be the stabilizing gain as given in (5), Ω0 be a polytopic outer approximation of the mRPI Z∞ (K), and
u
subject to Ax + Bu ∈ Ω0 BW u∈U
(6) (7) (8) 93
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Ω = Ωk for a given k ∈ N, the k th element of the decreasing polytopic RCI sets sequence starting from Ω0 as defined in Section 3.2.
ν(x) =
Proof. Let x ∈ Xch and w ∈ W. We have ∀j ∈ IL , ∗ ∗ uj (x∗j ) ∈ U, x∗+ j = Axj + Buj (xj ) + Bw ∈ Xj . Moreover, x+ satisfies x+ =
L
∗ Proof. From (13), ∀w ∈ W, the pairs (λ∗j x∗+ j , λj ), j ∈ IL satisfy the constraints of the optimization problem (11) for the state x+ .
Hence, ∀x ∈ Xch , ∀w ∈ W, V (x+ ) ≤ 1 − λ∗1 (x) = V (x)
Let us introduce the following definition. Definition 5. (Robust Contractivity) A set S is α robustly contractive for the closed-loop system x+ = f (x, w), w ∈ W if there exists 0 ≤ α < 1 such that ∀x ∈ S, ∀w ∈ W, x+ ∈ αS.
The following assumption will be considered in view of the convergence analysis. Assumption 1. The sets Xj , j ∈ IL are respectively αj robustly contractive for the closed-loop systems x+ = Ax+ B(uj (x) + w), w ∈ W.
(11) x ˆj
Let us define α = max({αj , j ∈ IL }) in view of the following result. Proposition 4. The set Xch is α robustly contractive for the closed-loop system x+ = Ax + Bν(x) + Bw, w ∈ W with ν(x) defined in (12) and under the conditions of the Assumption 1.
j=1
x ˆj ∈ λj Xj , ∀j ∈ IL λj ≥ 0, ∀j ∈ IL L
j=1
In view of the stability analysis for the closed-loop system, we introduce the following positive definite function: V (x) = 1 − λ∗1 (x). Based on the constraints on the interpolation factors we have, ∀x ∈ Xch , 0 ≤ V (x) ≤ 1, and V (x) = 0 if and only if x ∈ X1 . Proposition 3. The control law (12) ensures that the closed-loop system is robustly stable in the sense of Lyapunov (non-increase along the system trajectories) for all initial conditions x ∈ Xch .
Let us now define Xch = Conv ({Xj , j ∈ IL }) . L Any point x ∈ Xch can be written x = j=1 λj xj , with L j=1 λj = 1, λj ≥ 0, xj ∈ Xj , ∀j ∈ IL . Remark 5. The above expression of x is not unique. L By denoting x ˆj = λj xj , we have x = j=1 x ˆj and x ˆj ∈ ˆj λj Xj . To perform a selection among the feasible λj and x at each time-step, we minimize online the following linear cost function subject to convex constraints
x=
L λ∗j Ax∗j + Buj (x∗j ) + Bw = λ∗j x∗+ j . (13)
Since the set U is convex, and by definition of the set Xch , we have ν(x) ∈ U, x+ ∈ Xch .
• an LMI-based method to compute the invariant ellipsoid E(x0 ), and the associated feedback gain u = Kx0 x, that contains the most important extension on a direction defined by a reference point (Nguyen (2012)). This is of particular interest to enlarge the basin of attraction in specific directions. • Tube-based Model Predictive Control (Mayne et al. (2006)).
subject to
L j=1
Several methods to design the pairs uj (x), Xj exist in the literature. Among those, we can cite
− λ1
(12)
Note that ∀x ∈ X1 , ν(x) = u1 (x). We have the following result regarding the above control law. Proposition 2. The set Xch is an RPI set for the system x+ = Ax + Bν(x) + Bw and constraint set (Xν , W).
The definition of the control law outside of O∞ (K) proposed here relies on the existence of L−1 stabilizing control laws uj (x), j ∈ {2, ..., L} and associated RPI sets Xj ⊆ X , j ∈ {2, ..., L}. That is ∀j ∈ {2, ..., L}, ∀x ∈ Xj , uj (x) ∈ U, Ax + B(uj (x) + w) ∈ Xj , ∀w ∈ W. We denote u1 (x) the control law given by (9) and (10), and X1 = O∞ (K). Remark 4. The sets Xj , j = {2, ..., L} are assumed to be convex and compact. This assumption is not restrictive provided Conv{Xj } is an admissible convex RPI set.
minimize
λ∗j uj (x∗j ).
j=1
A first step to enlarge the basin of attraction is to define the control law on the set O∞ (K)\Ω0 as (10) ν(x) = Kx, x ∈ O∞ (K)\Ω0 . Indeed, any element of O∞ (K)\Ω0 is robustly steered to Ω0 with the linear control law u = Kx.
λj ,ˆ xj ,j∈IL
L
λj = 1
j=1
Proof. Let x ∈ Xch . From Assumption 1, we have, ∀j ∈ IL , ∀w ∈ W, x∗+ = Ax∗j + B(uj (x∗j ) + w) ∈ αj Xj ⊆ αXj . j L Hence, x+ = j=1 λ∗j x∗+ j ∈ αXch .
Let us denote (ˆ x∗j (x), λ∗j (x)), j ∈ IL the solution of (11) for the state x, and define for all j ∈ IL , x ˆ∗j (x) x∗j (x) = ∗ , if λ∗j = 0, λj (x) x∗j (x) = 0, if λ∗j = 0. The dependency on x will be dropped for clarity purpose. We use this selection to define the control law ν : Xch → U,
Proposition 5. The control law (12) ensures that the set X1 is robustly asymptotically stable for all initial conditions x ∈ Xch .
Proof. First, let us prove that if V (x) = 1 then x ∈ ∂Xch . 94
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59
1 1 0 ,B = , 0 1 1 0.375 0.25 2 X = x ∈ R | |Hx| ≤ 13 , H = 0.1786 0.357 , −0.25 0.25 U = {u ∈ R | |u| ≤ 1.5} , W = {w ∈ R | |w| ≤ 0.5} . In this exemple, p = 3, n = 2, and m = 1. The 1 × 3 submatrices of H verifying H ∈ H are Hi , i = 1, 2, 3. We use the results in Michel et al. (2018) to obtain the associated gains K1 = [−0.9 −1.9] , K2 = [−0.5 −1.5] , K3 = [−1.5 −2.5] . In this scenario, it is possible to have an explicit representation of the sets Z∞ (Ki ) given the eigenstructure of the matrices A + BKi and the dimensions of the problem. We represent these sets and the state constraints in Figure 1. Note that Z∞ (Ki ) ⊆ X , for all i ∈ {1, 2, 3}. We have K1 Z∞ (K1 ) = [−1.2 ; 1.2] , K2 Z∞ (K2 ) = [−1 ; 1] , K3 Z∞ (K3 ) = [−3.5 ; 3.5] .
Let us assume that x ∈ ∂Xch . Then, ∃ > 0 such that (1+ )x ∈ Xch . Moreover, X1 has a non-empty interior, hence ∃δ > 0 such that δx ∈ X1 .
x+ = Ax + B(u + w), A =
Considering γ = min( , δ), we have concomitantly (1 + γ)x ∈ Xch , γx ∈ X1 .
If γ ≥ 1, then δ ≥ 1 and it follows x ∈ X1 , which leads to V (x) = 0. Else, if γ < 1 and we can rewrite x as x = (1 − γ)((1 + γ)x) + γ(γx), where (1 + γ)x ∈ Xch , γx ∈ X1 , 0 < (1 − γ) < 1 and 0 < γ < 1. Hence we get V (x) ≤ 1 − γ < 1. We conclude that if V (x) = 1 then x ∈ ∂Xch .
Let w ∈ W. To prove robust asymptotic stability of the set X1 , we consider three cases regarding the value of V (x). Case 1: V (x) = 1. From Proposition 4, we have x+ ∈ αXch . Thus, V (x+ ) < 1 = V (x).
Case 2: 0 < V (x) < 1. We have λ∗1 (x) > 0. Given ∈ α1 X1 . that the set X1 is α1 robustly contractive, x∗+ 1 Moreover, X1 has an non-empty interior, hence ∃ > 0 such ∗+ + + that x∗+ 1 + (x − x1 ) ∈ X1 . Note that if ≥ 1, x ∈ X1 + and thus V (x ) = 0. Else, we denote ∗+ + z1 = x∗+ (14) 1 + (x − x1 ) ∈ X1 . + z − x 1 This leads to x∗+ . We can rewrite x+ asx+ = 1 = 1− L z1 − x+ + ∗ λ∗1 + j=2 λ∗j x∗+ ) = j . Hence, x (1 + λ1 1− 1− z 1 L λ∗1 + j=2 λ∗j x∗+ j .This is equivalent to 1− L λ∗j λ∗1 z1 x+ = + x∗+ . 1 − (1 − λ∗1 ) j=2 (1 + λ∗ ) j 1 1− λ∗1 > 0. From 0 < < 1, we have λ∗1 > 1 − (1 − λ∗1 ) Moreover, we can show that L λ∗j λ∗1 + =1 ∗ 1 − (1 − λ1 ) j=2 (1 + λ∗ ) 1 1− λ∗j ≥ 0. This proves and ∀j ∈ IL , ) (1 + λ∗1 1− λ∗1 + < 1 − λ∗1 = V (x). V (x ) ≤ 1 − 1 − (1 − λ∗1 )
For Z∞ (Ki ) to be an RPI set, it has to satisfy Z∞ (Ki ) ⊆ X and Ki Z∞ (Ki ) ⊆ U. Hence, the gain K3 is not admissible. Moreover, we have h(Z∞ (K1 ), H) = 0.44, h(Z∞ (K2 ), H) = 0.5.
The set Z∞ (K1 ) minimizes (3). Due to the dimension of the problem and the proposed approach, the refinement proposed Section 3 does not reduce the invariant set (Ω1 = Ω0 = Z∞ (K1 )). The MRPI of the feedback gains K1 and K2 are presented in Figure 2. The method proposed in Section 4 to enlarge the controllable region O∞ (K1 ) = X1 of the controller u1 (x) = K1 x has been tested in simulation. The control laws uj , j = 2, 3, 4 and the associated RPI sets Xj , j = {2, 3, 4} have been obtained using Tube-Based MPC as presented in Mayne et al. (2005) with Ktube,1 = K2 , Ktube,2 = [−0.61 −1.61] , Ktube,3 = [−0.58 −1.55] , and a prediction horizon N = 10. The weight matrices Q, R, and P as defined in Mayne et al. (2005) do not impact the region of attraction of the tube-based MPC controllers. We also consider X5 = O∞ (K2 ) and u5 (x) = K2 x. The set Xch = Conv(Xi , i = 1, ..., 5) can be seen in Figure 3. Trajectories emanating from ∂Xch and converging to Z∞ (K1 ) are represented in Figure 4.
Case 3: V (x) = 0. The control action is given by ν(x) = u1 (x) and it guarantees x+ ∈ X1 . Hence, V (x+ ) = 0.
This proves robust asymptotic stability of the set X1 for any initial condition x ∈ Xch .
6. CONCLUSION
We have proposed a design strategy to enlarge the controllability of the local control law defined in the previous section. Convergence properties have been established with regards to different assumptions on the sets considered for the interpolation-based control strategy.
We have proposed a control law for discrete-time linear systems subject to matched disturbances and input and state constraints. The proposed method allows to mitigate the impact of the disturbances in the direction of the state constraints in a neighborhood of the origin. The effect of the disturbances is assessed through invariant set. A first approach that imposes a linear structure to the control law is proposed. The control structure is then relaxed to further mitigate the impact of the disturbance on state constraints satisfaction, leading to smaller invariant sets
5. SIMULATION RESULTS The results presented in Section 3 and 4 are now illustrated. Consider the following system, constraints and disturbance set 95
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in the constraints direction. To this local control strategy we add an interpolation-based control design to enlarge the controllable region. REFERENCES Blanchini, F. (1999). Set invariance in control. Automatica, 35(11), 1747 – 1767. Corradini, M.L., Cristofaro, A., and Orlando, G. (2014). Sliding-mode control of discrete-time linear plants with input saturation: application to a twin-rotor system. International Journal of Control, 87(8), 1523–1535. Falcone, P., Borrelli, F., Pekar, J., and Stewart, G. (2009). A reference governor approach for constrained piecewise affine systems. European Control Conference (ECC), 4223–4228. Jaulin, L. (2000). Interval constraint propagation with application to bounded-error estimation. Automatica, 36(10), 1547 – 1552. Kolmanovsky, I. and Gilbert, E.G. (1998). Theory and computation of disturbance invariant sets for discretetime linear systems. Mathematical problems in engineering, 4(4), 317–367. Mayne, D., Rakovi`c, S., Findeisen, R., and Allgöwer, F. (2006). Robust output feedback model predictive control of constrained linear systems. Automatica, 42(7), 1217 – 1222. Mayne, D., Rawlings, J., Rao, C., and Scokaert, P. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789 – 814. Mayne, D. and Schroeder, W. (1997). Robust time-optimal control of constrained linear systems. Automatica, 33(12), 2103 – 2118. Mayne, D., Seron, M., and Rakovi`c, S. (2005). Robust model predictive control of constrained linear systems with bounded disturbances. Automatica, 41(2), 219 – 224. Michel, N., Olaru, S., Valmorbida, G., Bertrand, S., and Dumur, D. (2018). Invariant sets for discrete-time constrained linear systems using sliding mode approach. European Control Conference, 2929–2934. Nguyen, H.N. (2012). Constrained control for uncertain systems: an interpolation based control approach. Ph.D. thesis. EDSTIC Supélec 2012. Olaru, S., De Doná, J., Seron, M., and Stoican, F. (2010). Positive invariant sets for fault tolerant multisensor control schemes. International Journal of Control, 83(12), 2622–2640. Rakovi`c, S.V., Kerrigan, E.C., Kouramas, K.I., and Mayne, D.Q. (2005). Invariant approximations of the minimal robust positively invariant set. IEEE Transactions on Automatic Control, 50(3), 406–410. Rakovi`c, S., Kerrigan, E., Mayne, D., and Kouramas, K. (2007). Optimized robust control invariance for linear discrete-time systems: Theoretical foundations. Automatica, 43(5), 831 – 841. Tahir, F. and Jaimoukha, I.M. (2015). Low-complexity polytopic invariant sets for linear systems subject to norm-bounded uncertainty. IEEE Transactions on Automatic Control, 60(5), 1416–1421.
Fig. 1. Z∞ (Ki ) for K1 (red), K2 (blue), and K3 (yellow), and the state constraints X (black).
Fig. 2. The MRPI O∞ (Ki ) and the mRPI Z∞ (Ki ) for K1 (left), and K2 (right).
Fig. 3. The enlarged controllable region Xch (grey) and the initial MRPI O∞ (K1 ) (red).
Fig. 4. The enlarged controllable region Xch (grey), the mRPI Z∞ (K1 ) (red), and trajectories initialized on several vertices of Xch . 96
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IFAC PapersOnLine 51-25 (2018) 55–60
Invariant Set Design for Constrained Invariant Set Design for Constrained Invariant Set Design for Constrained Discrete-Time Linear Systems with Bounded Invariant Set Design for Constrained Discrete-Time Linear Systems with Bounded Discrete-Time Linear Systems with Bounded Matched Disturbance Discrete-Time LinearDisturbance Systems with Bounded Matched Matched Disturbance Matched Disturbance ∗∗ ∗ Nathan Michel ∗,∗∗ ∗,∗∗ Sorin Olaru ∗∗ Sylvain Bertrand ∗
Nathan Michel ∗,∗∗ Sorin Olaru ∗∗ Sylvain Bertrand ∗ ∗∗ ∗∗ NathanGiorgio MichelValmorbida Sorin Olaru Sylvain Bertrand Dumur ∗∗ Didier ∗∗ Giorgio Valmorbida Didier Dumur ∗,∗∗ ∗∗ ∗ ∗∗ ∗∗ NathanGiorgio MichelValmorbida Sorin Olaru Sylvain Bertrand Didier Dumur ∗∗ ∗∗ Giorgio Valmorbida Didier Dumur ∗ ∗ ONERA - The French Aerospace Lab, F-91123 Palaiseau, France ∗ ONERA - The French Aerospace Lab, F-91123 Palaiseau, France ONERA The French Aerospace Lab, F-91123 Palaiseau, France (e-mail: [email protected] ;; [email protected]) [email protected] [email protected]) ∗ (e-mail: ∗∗ ONERA The French Aerospace Lab, F-91123 Palaiseau, France (e-mail: [email protected] ; [email protected]) L2S, CentraleSupélec, CNRS, Univ. Paris-Sud, ∗∗ CNRS, ;Univ. Paris-Sud, Université Université ∗∗ L2S, CentraleSupélec, (e-mail: [email protected] [email protected]) L2S, CentraleSupélec, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 3 rue Joliot-Curie, Gif-Sur-Yvette 91192, France Paris-Saclay, 3 rue Joliot-Curie, Gif-Sur-Yvette 91192, France ∗∗ L2S, CentraleSupélec, CNRS, Gif-Sur-Yvette Univ. Paris-Sud, Université Paris-Saclay, 3 rue Joliot-Curie, 91192, France (e-mail: [email protected] ; (e-mail: [email protected] ; Paris-Saclay, [email protected] rue Joliot-Curie, Gif-Sur-Yvette 91192, ;France (e-mail: [email protected] ; [email protected]) [email protected] ; [email protected]) (e-mail: [email protected] ; [email protected] ; [email protected]) [email protected] ; [email protected]) Abstract: Invariant Invariant set set theory theory has has been been recognized recognized as as an an important important tool tool for for control control design design Abstract: Abstract: Invariant setsubject theory to hasdisturbances. been recognized as an important tool for control design of constrained constrained systems subject to disturbances. Indeed, for aa given control law, entering an of systems Indeed, for given control law, entering an Abstract: Invariant setsubject theory to has been recognized as an important toolin for control design of constrained systems disturbances. Indeed, for a given control law, entering an invariant set guarantees recursive state and input constraint satisfaction closed-loop. This invariant set guarantees recursive state and input constraint satisfaction in closed-loop. This of constrained systems subject to state disturbances. for asatisfaction given control law, disturbance. enteringThis an invariant set guarantees recursive and subject inputIndeed, constraint in closed-loop. paper focuses on discrete-time linear systems to bounded matched additive paper focuses on discrete-time linear systems subject to bounded matched additive disturbance. invariant set of guarantees recursive state andlaws input constraint satisfaction inthat closed-loop. This paper focuses on discrete-time linear systems subject to bounded matched additive disturbance. The problem the joint synthesis of control and associated invariant sets are optimized The problem ofonthe joint synthesis of control laws and associated invariant sets that are optimized paper focusesto discrete-time linear systems subject to bounded matched additive disturbance. The of the joint of control laws and An associated invariant sets that are optimized withproblem regards the statesynthesis constraints is investigated. An interpolation method is used used to enlarge with regards to the state constraints is investigated. interpolation method is to enlarge The problem of the joint synthesis of control laws and associated invariant sets that are optimized with regards to the state constraints is investigated. An interpolation method is used to enlarge the controllable region. the controllable region. with regards to the state constraints is investigated. An interpolation method is used to enlarge the controllable region. © 2018, IFAC (International the controllable region. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Invariant Invariant set, set, disturbance, disturbance, state state constraints, constraints, input input constraints, constraints, nonlinear nonlinear control control Keywords: Keywords: Invariant set, disturbance, state constraints, input constraints, nonlinear control Keywords: Invariant set, disturbance, state constraints, input constraints, nonlinear control invariant 1. INTRODUCTION INTRODUCTION invariant set set respecting respecting the the constraints, constraints, denoted denoted Maximal Maximal 1. invariant set respecting the constraints, denoted Maximal 1. INTRODUCTION Robustly Positively Invariant (MRPI) set. Robustly Positively Invariant (MRPI) set. invariant respecting the constraints, denoted Maximal 1. INTRODUCTION Robustly set Positively Invariant (MRPI) set. Previous work has focused on the joint synthesis of Robustly Positively Invariant (MRPI) set. Previous work has focused on the joint synthesis of control control Constrained control of dynamic systems in presence of Previous work has focused on the joint synthesis of laws and associated invariant sets tailored to state orcontrol input Constrained control of dynamic systems in presence of laws and associated invariant sets tailored to state or input Constrained control dynamic systems presence of Previous work has focused on(2014); the tailored joint synthesis oforcontrol disturbance faces faces twoofmain main challenges: theinimpact impact of the the laws and associated invariant sets to state input constraints (Corradini et al. Nguyen (2012); Tahir disturbance two challenges: the of constraints (Corradini et al. (2014); Nguyen (2012); Tahir Constrained control dynamic systems presence of laws disturbance twooflocal main challenges: theinimpact of the and associated invariant sets tailored to state or Tahir input disturbancesfaces on the the local behavior around nominal constraints (Corradini etRakovi` al. (2014); Nguyen (2012); and Jaimoukha (2015); cc et al. (2007); Michel et al. disturbances on behavior around aa nominal and Jaimoukha (2015); Rakovi` et al. (2007); Michel al. et disturbance faces two main challenges: the impact of the disturbances on the local behavior around a nominal constraints (Corradini et al. (2014); Nguyen (2012); Tahir equilibrium and the characterization of the controllable and Jaimoukha (2015); Rakovi` c et al. (2007); Michel et al. (2018)). In Rakovi` c et al. (2007), a characterization of equilibrium and the characterization of the controllable (2018)). In Rakovi` c et al. (2007), a characterization of disturbances on the characterization local behavior around ainnominal controllable equilibrium and the of the and Jaimoukha (2015); Rakovi` c et al. (2007); Michel et al. region. Both challenges have been addressed several (2018)). In Rakovi` c et al. (2007), a characterization of families of robust control invariant sets, based on outer in several region. Both challenges have been addressed families of robust control invariant sets, based on outer equilibrium and the characterization oftothe in several region. challenges haveaccording been addressed In robust Rakovi` c et al.invariant (2007), asets, characterization of control Both design frameworks according thecontrollable tools and (2018)). families of basedThis on outer approximations of the set, charcontrol design frameworks to the tools and approximations of control the mRPI mRPI set, is is proposed. proposed. This charregion. Both challenges have been addressed in several control design frameworks according to the tools and families of robust control invariant sets, based on outer modelling assumptions: set theoretic methods (Blanchini approximations of the mRPI set, is proposed. This characterization be used to optimized invariant modelling assumptions: set theoretic methods (Blanchini acterization can can of bethe used to establish establish optimizedThis invariant control frameworks according(Jaulin to the(2000)), tools and modelling assumptions: theoretic methods (Blanchini approximations mRPI set, constraints. is proposed. char(1999)), design interval based set approaches or acterization optimized invariant can bestate used to input establish sets regarding the and The control (1999)), interval based approaches (Jaulin (2000)), or sets regarding the state and input constraints. The control modelling assumptions: set theoretic methods (Blanchini (1999)), interval based approaches (Jaulin (2000)), or acterization can be used to establish optimized invariant robust Model Predictive Control (Mayne et al. (2005)). sets regarding the state and input constraints. The control design in Michel et al. (2018) considers bounded matched robust Model Predictive Control (Mayne et al. (2005)). design in Michel etstate al. (2018) considers bounded matched (1999)), interval based Control approaches (Jaulin robust Model Predictive (Mayne et al.(2000)), (2005)). or sets regarding theet and input Thestrategy. control design indisturbance Michel al.and (2018) considers bounded matched additive adopts aaconstraints. sliding mode Set theoretic methods require a set description of the additive disturbance and adopts sliding mode strategy. robust Model Predictive Control (Mayne et al. (2005)). Set theoretic methods require a set description of the design in Michel et al. (2018) considers bounded matched a sliding additive disturbance and adopts mode strategy. The control law feedback gain minSet theoretic A methods require description of the disturbances. A systematic way ato to set assess the influence influence of additive The resulting resulting control and law is is the the linear linear feedback gain mindisturbances. systematic way assess the of disturbance apredefined sliding mode strategy. gain minThe resulting control law adopts is the feedback imizing the mRPI projection in aalinear direction. Set theoretic isA methods require ato set description of the disturbances. systematic way assess the influence of disturbances to compute invariant sets. Indeed, invariimizing the mRPI projection in predefined direction. disturbances is to compute invariant sets. Indeed, invariThe resulting control law is the linear feedback gain mindisturbances. systematic way toconstraints assess influence of imizing the mRPI projection in a predefined direction. Indeed, invaridisturbances isA to compute invariant sets.the ant sets sets are are certificates certificates for robust robust satisfaction, In this paper we extend the linear control law and associant for constraints satisfaction, imizing the mRPI projection in a predefined direction. In this paper we extend the linear control law and associdisturbances is to compute invariant Indeed, invariant sets are certificates robust constraints satisfaction, recursive feasibility, andfor mitigation of sets. the disturbances for control this paper we linearof and associated invariant set design strategy Michel et (2018) to disturbances for In recursive feasibility, and mitigation of the ated invariant setextend design the strategy ofcontrol Michellaw et al. al. (2018) to ant setscontrol are certificates for robust constraints satisfaction, disturbances recursive feasibility, and mitigation of the for In this paper we extend the linearand law and associa given law (Mayne et al. (2000)). Such approaches ated invariant set design strategy of Michel et al. (2018) to aa larger class of state constraints, we account for input arecursive given control law (Mayne et al. (2000)). Such approaches larger class of state constraints, and we account for input feasibility, and mitigation of the disturbances for ahave given control law in (Mayne et al. (2000)). approaches invariant design strategy of Michel et al. (2018) to been studied in the context context of model modelSuch predictive con- ated aconstraints. larger classThe ofset state constraints, and we account for input impact of around the have been studied the of predictive conconstraints. The impact of the the disturbances disturbances around the a given control law (Mayne etrobust al. (2000)). Such approaches of model predictive con- constraints. have been studied in the context a larger class of state constraints, and we account for input trol (Mayne et al. (2005)), time-optimal control The impact of the disturbances around the nominal equilibrium is further mitigated by relaxing trol (Mayne et al.in(2005)), robust time-optimal control nominal equilibrium is further by relaxing the have been studied the context model control (Mayne et al. (2005)), robust time-optimal control The impact of themitigated disturbances around sets the (Mayne and Schroeder (1997)), orofdesign design ofpredictive reference gov- constraints. nominal equilibrium is further byinvariant relaxing the linear structure, allowing for (Mayne and Schroeder (1997)), or of reference govlinear control control structure, allowingmitigated for smaller smaller invariant sets trol (Mayne et et al.al.(2005)), robust time-optimal control (Mayne and Schroeder (1997)), or design of reference govnominal equilibrium is further mitigated by relaxing the ernors (Falcone (2009)). A series of results regarding linear structure, allowing for smaller invariant sets in direction of constraints. An interpolationinterpolationernors (Falcone et al. (2009)). Aor series of results regarding in the the control direction of the the state state constraints. An (Mayne and (1997)), design of reference gov- linear ernors (Falcone et been al. (2009)). A series of results regarding structure, allowing for smaller invariant sets invariant set Schroeder have been established for linear linear systems, with in the control direction of theisstate constraints. An interpolationbased control design then proposed to enlarge the invariant set have established for systems, with based control design isstate thenconstraints. proposed toAn enlarge the conconernors (Falcone et al. subject (2009)). A series of results regarding invariant set have been established for linear systems, with in the direction of the interpolationa linear control law, to additive bounded disturthe conbased control design is then proposed to enlarge trollable region. disturainvariant linear control law, to additive bounded trollable region. set beensubject established for and linear systems, with based disturabance, linearsee control law, subject to additive bounded control design is then proposed to enlarge the consee forhave example Kolmanovsky Gilbert (1998). trollable region. bance, for example Kolmanovsky and Gilbert (1998). aOflinear control law, subject to additive bounded disturThe paper is organized bance, see for example Kolmanovsky and Gilbert (1998). trollable region. particular interest is the so-called minimal Robust PosThe paper is organized as as follows. follows. Section Section 22 presents presents the the Of particular is Kolmanovsky the so-called minimal Robust Pos- The bance, see forinterest example and Gilbert (1998). paper is organized asand follows. Section 2 presentsSecthe class of system studied important definitions. Of particular interest is the so-called minimal Robust Positively Invariant (mRPI) set. It is defined as the smallest class of system studied and important definitions. Secitively Invariant (mRPI) set. It is defined as the smallest The paper is organized as follows. Section 2 presents the Of particular is the so-called minimal system studied important tion introduces results on design of invariant invariantSecset itively Invariant set. It is defined as Robust the smallest invariant set interest for (mRPI) given disturbance set (Rakovi` (Rakovi` etPosal. class tion 33of introduces resultsand on the the design definitions. of set invariant set for aa given disturbance set cc et al. class ofintroduces system studied and important definitions. Secitively Invariant (mRPI) set. It is defined as the smallest tion 3 results on the design of invariant set tailored to the state constraints. Section 4 proposes an invariant set for a given disturbance set (Rakovi` c et al. (2005)). It corresponds to the limit set of state trajectories tailored to the state constraints. Sectionof4 invariant proposes set an (2005)). Itset corresponds to the limit set of trajectories 3 introduces results on to theenlarge design tailored to thebased state constraints. Sectionthe 4 proposes an invariant for aof given disturbance setstate (Rakovi` c of et the al. tion interpolation method controllable (2005)). It corresponds to the limit set of state trajectories for any sequence disturbances. Characterization interpolation based method to enlarge the controllable for any sequence of disturbances. Characterization of the to the 5state constraints. Sectionthe 4ofproposes an (2005)). It corresponds to thebelimit set of state trajectories interpolation to enlarge region. illustrative examples the for any sequence disturbances. Characterization of the tailored controllable regionofcan can also addressed by set set theoretic region. Section Sectionbased 5 gives givesmethod illustrative examples of controllable the results. results. theoretic controllable region also be addressed by interpolation based method to enlarge theof controllable for any sequence of disturbances. Characterization of the region. Section 5 gives illustrative examples the results. Finally, Section 6 draws conclusion and discusses perspeccontrollable region can also be addressed by set theoretic methods (Blanchini (Blanchini (1999); (1999); Mayne Mayne et et al. al. (2005)). (2005)). The The set set Finally, Section 6 draws conclusion and discusses perspecmethods region. Section 56gives of theperspecresults. controllable region can also Mayne be addressed byisset theoretic Finally, drawsillustrative conclusionexamples and discusses tives. methods (Blanchini (1999); et al. (2005)). The set of interest for such controllability analysis the largest tives. Section of interest(Blanchini for such (1999); controllability analysis is the The largest Finally, Section 6 draws conclusion and discusses perspecmethods Mayne et al. (2005)). set tives. of interest for such controllability analysis is the largest of interest for such controllability analysis is the largest tives.
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IFAC ROCOND 2018 56 Florianopolis, Brazil, September 3-5, 2018 Nathan Michel et al. / IFAC PapersOnLine 51-25 (2018) 55–60
Notation: For a positive integer p, define Ip = {1, ..., p}. For a vector h ∈ Rn , denote hi its ith element, de fine ||h||∞ = max|hi |, i ∈ In , and |h| = [|h1 | ... |hn |] . For two vectors x and y, x ≤ y (x < y) denotes the element-wise (strict) inequalities between their components. Define 1p the vector of ones of dimension p. The ith power of a matrix A is denoted Ai . The ith line of a matrix A is denoted Ai . Define the set of invertible matrices Gn = {A ∈ Rn×n | det(A) = 0}. The set of Schur matrices of dimension n is denoted Cn . For a matrix A ∈ Rm×n and a set X ⊆ Rn , define the set AX = {y ∈ Rm | y = Ax, x ∈ X }. Given two sets A, B, define A ⊕ B = {a + b | a ∈ A, b ∈ B} and A B = {x | {x} ⊕ B ⊆ A}. The boundary of a set A is denoted ∂A.
constraint set (XK , W) is defined as the RPI set containing all the RPI sets, denoted here O∞ (K). Definition 4. The minimal Robust Positively Invariant (mRPI) set for the system x+ = (A + BK)x + Bw and constraint set (XK , W) is defined as the RPI set contained in any closed RPI set. If A + BK is Schur, the mRPI for the system x+ = (A + BK)x + Bw and constraint set (Rn , W) exists, is unique, compact and contains the origin in its interior. Moreover, it is given by the following infinite Minkowski sum ∞ (A + BK)i BW. Z∞ (K) = i=0
An RPI for the system x+ = (A + BK)x + Bw and constraint set (Xν , W) exists if and only if Z∞ (K) ⊆ XK . Remark 2. In general, we do not have an explicit characterization of the set Z∞ (K). For computational purposes, polytopic outer approximations of this set are sought (Olaru et al. (2010); Rakovi`c et al. (2005)).
2. PRELIMINARIES Consider the following discrete-time linear time-invariant system (1) x+ = Ax + B(u + w), B = 0⊥ n−m,m Im where x ∈ Rn is the state, u ∈ Rm is the input, w ∈ W ⊆ Rm is an unknown bounded disturbance, and we assume ¯ + B(u ¯ + w) that m < n. Note that any system x+ = Ax ¯ with rank(B) = m can be written as (1) with a linear change of coordinates. The system (1) is subject to the state and input constraints (2) x ∈ X = {x ∈ Rn | |Hx| ≤ 1p } , u ∈ U, p×n , m ≤ p. The sets U and W are bounded where H ∈ R polytopes containing the origin in their interior. This paper focuses on the stabilization of (1) to a neighborhood of the origin, characterized in terms of invariant set, along with the characterization of the controllable region.
The local behavior around the origin can be characterized in terms of RCI sets, or RPI sets and their associated control law. In this paper, we want to design a local control law that mitigates the impact of the disturbances on the state constraints satisfaction. Hence, our goal is to design a control law and an associated RPI set, or an RCI set, that is minimal in the direction of the state constraints. To this local control strategy we add an interpolation-based control design to enlarge the controllable region. A measure to evaluate the minimality of invariant sets with regards to the state constraints is introduced in the following section.
The following definitions are based on invariant set theory (Blanchini (1999); Kolmanovsky and Gilbert (1998)). Definition 1. The set Z is said to be Robustly Controlled positively Invariant (RCI) for the system (1) and constraint set (X , U, W) if Z ⊆ X , and ∀x ∈ Z there exists u ∈ U such that Ax + B(u + w) ∈ Z, ∀w ∈ W.
3. INVARIANT SET DESIGN The criterion for the design of RCI sets Z is the minimization of (3) h(Z, H) = max ||Hx||∞ . x∈Z
Indeed, Z accounts for the impact of the disturbance, and H characterizes the direction of the state constraints. However, we do not have an explicit characterization of all the RCI sets for the system (1). The proposed approach is to first minimize (3) among mRPI obtained with linear control laws, which is a choice for the computation of RPI sets using existing methods (Olaru et al. (2010); Rakovi`c et al. (2005)). The linear control structure is then relaxed to construct a decreasing sequence of RCI sets starting from this mRPI.
In the following, an RCI set will refer to an RCI set for the system (1) and constraint set (X , U, W).
Given a state feedback control law ν : Rn → Rm , we define the set Xν = {x ∈ X | ν(x) ∈ U} . Definition 2. A set Z ⊆ Rn is said Robustly Positively Invariant (RPI) for the system x+ = Ax + B(ν(x) + w) and constraint set (Xν , W), if Z ⊆ Xν , and ∀x ∈ Z, Ax + B(ν(x) + w) ∈ Z, ∀w ∈ W. Remark 1. Note that an RPI set for the system x+ = Ax+ Bν(x) + Bw and constraint set (Xν , W) is an RCI set. Likewise, from any RCI set Z it is possible to define a state feedback ν : Z → U such that Z is an RPI set for the system x+ = Ax + Bν(x) + Bw and constraint set (Xν , W) (see Rakovi`c et al. (2007)).
The computation of a feedback gain leading to an mRPI minimizing (3) is presented below and a strategy to further improve the solution by relaxing the linear control law structure is presented in Section 3.2. 3.1 Invariant set design using linear state feedback
The following definitions consider a linear state feedback law ν(x) = Kx, and we define the polytopic set XK = {x ∈ X | Kx ∈ U} . Definition 3. The Maximal Robustly Positively Invariant (MRPI) set for the system x+ = (A + BK)x + Bw and
We now briefly recall the results in Michel et al. (2018) for the minimization of (3) with (1)-(2), p = m and U = Rm . The design method allows to compute the linear control law ν(x) = Kx whose mRPI set Z∞ (K) minimizes (3) 92
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which, thanks to (6), (7), and (8), satisfies ∀x ∈ Ω0 , Ax + Bν0 (x) ∈ Ω0 BW, ν0 (x) ∈ U. The above definition of ν0 seeks to minimize the scaling factor α in the direction of the state constraints defined by the matrix H as in (2). From the definition of an RCI set, the feasible domain of the above optimization problem is guaranteed to be non-empty. Note that the above optimization problem is convex (convex constraints and linear cost function).
under the assumptions that the matrix H = [HB ⊥ HB ] satisfies HB ∈ Gm . In this paper, we propose a strategy that extends the approach for the case p > m. p } Let us consider the matrices Hσi ∈ Rm×n , i ∈ {1, ..., m as the matrices obtained from the combination of m distinct rows out of the p rows of H. Consider the partition of those matrices, Hσi = [H define pσi ,B ⊥ Hσi ,B ], and the set H = Hσi , i ∈ {1, ..., m } | Hσi ,B ∈ Gm . From every matrix Hσi ∈ H we compute the linear control law νσi (x) = Kσi x with the method presented in Michel et al. (2018) that minimizes (4) h(Z, Hσi ) = max ||Hσi x||∞ .
We then define the set Ω1 as Ω1 = ConvexHull {Av + Bν0 (v), v ∈ V(Ω0 )} ⊕ BW. By construction, the set Ω1 is polytopic, satisfies Ω1 ⊆ Ω0 , and is an RPI set for the system x+ = Ax + Bν0 (x) + Bw and constraint set (Xν0 , W). Remark 3. Note that the image set, namely {z ∈ Rn | z = Ax + Bν0 (x) + Bw, x ∈ Ω0 , w ∈ W} might not be convex.
x∈Z
Define theset p | K = Kσi , i ∈ 1, ..., m
Hσi ∈ H,Kσi Z∞ (Kσi ) ⊆ U
This set contains the linear feedback gains Kσi obtained from the matrices Hσi ∈ H, and such that Z∞ (Kσi ) is an RPI set for the system (1) and constraint set (X , U, W). If the set K is non-empty, we chose the element minimizing (3), that is max ||Hx||∞ . (5) K = arg min Kσi ∈K
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Likewise, we can define a sequence of polytopic RCI sets Ωi , i ∈ N, as Ωi+1 = ConvexHull {Av + Bνi (v), v ∈ V(Ωi )}⊕BW, i ∈ N, where
x∈Z∞ (Kσi )
νi (x) = arg minimize α
Using these elements we are able to state the following result. Proposition 1. If Z∞ (K) ⊆ X , then robust asymptotic stability of the set Z∞ (K) is achieved with a region of attraction O∞ (K). Additionally, the finite determination of O∞ (K) is guaranteed.
u
subject to Ax + Bu ∈ Ωi BW
H(Ax + Bu) ∈ αH(Ωi BW) u∈U
0 ≤ α ≤ 1.
The strategy presented here allows to take the state constraints into account in the design of a linear feedback gain and construct the associated mRPI in a direct manner by exploiting the matched properties of the disturbance. If the set K is empty, alternative design strategies based on the complete characterization of the RCI sets taking into account input constraints are to be sought (Nguyen (2012); Rakovi`c et al. (2007); Tahir and Jaimoukha (2015)).
By construction, we have ∀i ∈ N, ∀j ∈ N, Ωi+j ⊆ Ωi , thus the objective (3) decreases (not strictly) as i increases. Let k ∈ N, and let Ω = Ωk . Consider the following control law ν : Ω0 → U: (9a) ν(x) = νk (x), x ∈ Ωk , (9b) ν(x) = νi (x), x ∈ Ωi \Ωi+1 , i ∈ {0, ..., k − 1}, ν(x) = ν0 (x), x ∈ Ω0 . (9c)
In the next section we propose to improve the solution proposed in this section by relaxing the linear control structure and allowing for nonlinear control policies.
For all x ∈ Ω0 , finite time convergence to Ω is guaranteed. Indeed, if x ∈ Ωi , i ∈ Ik−1 , x+ = Ax + Bν(x) + Bw ∈ Ωi+1 , ∀w ∈ W.
3.2 Refinement of RCI sets with nonlinear control laws
We have presented a method to construct, starting from a polytopic RCI set, a decreasing sequence of polytopic RCI sets that aims at minimizing (3).
In the previous section we imposed a linear structure to the control law in the design of an RPI set. We now propose an optimization-based method to obtain a decreasing sequence of RCI sets with nonlinear control policies starting from a polytopic RCI set.
In the following section we define the control law ν(x) outside of the set Ω0 using interpolation-based methods to extend the controllable region.
Let Ω0 be a polytopic RCI set. We consider the control law ν0 : Ω0 → U :
4. ENLARGEMENT OF THE CONTROLLABLE REGION
ν0 (x) = arg minimize α H(Ax + Bu) ∈ αH(Ω0 BW)
The previous section focused on the behavior of the system in a neighborhood around the origin. The proposed control law is defined locally. We now propose a method to enlarge the controllable region using interpolation-based control.
0≤α≤1
Let K be the stabilizing gain as given in (5), Ω0 be a polytopic outer approximation of the mRPI Z∞ (K), and
u
subject to Ax + Bu ∈ Ω0 BW u∈U
(6) (7) (8) 93
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Ω = Ωk for a given k ∈ N, the k th element of the decreasing polytopic RCI sets sequence starting from Ω0 as defined in Section 3.2.
ν(x) =
Proof. Let x ∈ Xch and w ∈ W. We have ∀j ∈ IL , ∗ ∗ uj (x∗j ) ∈ U, x∗+ j = Axj + Buj (xj ) + Bw ∈ Xj . Moreover, x+ satisfies x+ =
L
∗ Proof. From (13), ∀w ∈ W, the pairs (λ∗j x∗+ j , λj ), j ∈ IL satisfy the constraints of the optimization problem (11) for the state x+ .
Hence, ∀x ∈ Xch , ∀w ∈ W, V (x+ ) ≤ 1 − λ∗1 (x) = V (x)
Let us introduce the following definition. Definition 5. (Robust Contractivity) A set S is α robustly contractive for the closed-loop system x+ = f (x, w), w ∈ W if there exists 0 ≤ α < 1 such that ∀x ∈ S, ∀w ∈ W, x+ ∈ αS.
The following assumption will be considered in view of the convergence analysis. Assumption 1. The sets Xj , j ∈ IL are respectively αj robustly contractive for the closed-loop systems x+ = Ax+ B(uj (x) + w), w ∈ W.
(11) x ˆj
Let us define α = max({αj , j ∈ IL }) in view of the following result. Proposition 4. The set Xch is α robustly contractive for the closed-loop system x+ = Ax + Bν(x) + Bw, w ∈ W with ν(x) defined in (12) and under the conditions of the Assumption 1.
j=1
x ˆj ∈ λj Xj , ∀j ∈ IL λj ≥ 0, ∀j ∈ IL L
j=1
In view of the stability analysis for the closed-loop system, we introduce the following positive definite function: V (x) = 1 − λ∗1 (x). Based on the constraints on the interpolation factors we have, ∀x ∈ Xch , 0 ≤ V (x) ≤ 1, and V (x) = 0 if and only if x ∈ X1 . Proposition 3. The control law (12) ensures that the closed-loop system is robustly stable in the sense of Lyapunov (non-increase along the system trajectories) for all initial conditions x ∈ Xch .
Let us now define Xch = Conv ({Xj , j ∈ IL }) . L Any point x ∈ Xch can be written x = j=1 λj xj , with L j=1 λj = 1, λj ≥ 0, xj ∈ Xj , ∀j ∈ IL . Remark 5. The above expression of x is not unique. L By denoting x ˆj = λj xj , we have x = j=1 x ˆj and x ˆj ∈ ˆj λj Xj . To perform a selection among the feasible λj and x at each time-step, we minimize online the following linear cost function subject to convex constraints
x=
L λ∗j Ax∗j + Buj (x∗j ) + Bw = λ∗j x∗+ j . (13)
Since the set U is convex, and by definition of the set Xch , we have ν(x) ∈ U, x+ ∈ Xch .
• an LMI-based method to compute the invariant ellipsoid E(x0 ), and the associated feedback gain u = Kx0 x, that contains the most important extension on a direction defined by a reference point (Nguyen (2012)). This is of particular interest to enlarge the basin of attraction in specific directions. • Tube-based Model Predictive Control (Mayne et al. (2006)).
subject to
L j=1
Several methods to design the pairs uj (x), Xj exist in the literature. Among those, we can cite
− λ1
(12)
Note that ∀x ∈ X1 , ν(x) = u1 (x). We have the following result regarding the above control law. Proposition 2. The set Xch is an RPI set for the system x+ = Ax + Bν(x) + Bw and constraint set (Xν , W).
The definition of the control law outside of O∞ (K) proposed here relies on the existence of L−1 stabilizing control laws uj (x), j ∈ {2, ..., L} and associated RPI sets Xj ⊆ X , j ∈ {2, ..., L}. That is ∀j ∈ {2, ..., L}, ∀x ∈ Xj , uj (x) ∈ U, Ax + B(uj (x) + w) ∈ Xj , ∀w ∈ W. We denote u1 (x) the control law given by (9) and (10), and X1 = O∞ (K). Remark 4. The sets Xj , j = {2, ..., L} are assumed to be convex and compact. This assumption is not restrictive provided Conv{Xj } is an admissible convex RPI set.
minimize
λ∗j uj (x∗j ).
j=1
A first step to enlarge the basin of attraction is to define the control law on the set O∞ (K)\Ω0 as (10) ν(x) = Kx, x ∈ O∞ (K)\Ω0 . Indeed, any element of O∞ (K)\Ω0 is robustly steered to Ω0 with the linear control law u = Kx.
λj ,ˆ xj ,j∈IL
L
λj = 1
j=1
Proof. Let x ∈ Xch . From Assumption 1, we have, ∀j ∈ IL , ∀w ∈ W, x∗+ = Ax∗j + B(uj (x∗j ) + w) ∈ αj Xj ⊆ αXj . j L Hence, x+ = j=1 λ∗j x∗+ j ∈ αXch .
Let us denote (ˆ x∗j (x), λ∗j (x)), j ∈ IL the solution of (11) for the state x, and define for all j ∈ IL , x ˆ∗j (x) x∗j (x) = ∗ , if λ∗j = 0, λj (x) x∗j (x) = 0, if λ∗j = 0. The dependency on x will be dropped for clarity purpose. We use this selection to define the control law ν : Xch → U,
Proposition 5. The control law (12) ensures that the set X1 is robustly asymptotically stable for all initial conditions x ∈ Xch .
Proof. First, let us prove that if V (x) = 1 then x ∈ ∂Xch . 94
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1 1 0 ,B = , 0 1 1 0.375 0.25 2 X = x ∈ R | |Hx| ≤ 13 , H = 0.1786 0.357 , −0.25 0.25 U = {u ∈ R | |u| ≤ 1.5} , W = {w ∈ R | |w| ≤ 0.5} . In this exemple, p = 3, n = 2, and m = 1. The 1 × 3 submatrices of H verifying H ∈ H are Hi , i = 1, 2, 3. We use the results in Michel et al. (2018) to obtain the associated gains K1 = [−0.9 −1.9] , K2 = [−0.5 −1.5] , K3 = [−1.5 −2.5] . In this scenario, it is possible to have an explicit representation of the sets Z∞ (Ki ) given the eigenstructure of the matrices A + BKi and the dimensions of the problem. We represent these sets and the state constraints in Figure 1. Note that Z∞ (Ki ) ⊆ X , for all i ∈ {1, 2, 3}. We have K1 Z∞ (K1 ) = [−1.2 ; 1.2] , K2 Z∞ (K2 ) = [−1 ; 1] , K3 Z∞ (K3 ) = [−3.5 ; 3.5] .
Let us assume that x ∈ ∂Xch . Then, ∃ > 0 such that (1+ )x ∈ Xch . Moreover, X1 has a non-empty interior, hence ∃δ > 0 such that δx ∈ X1 .
x+ = Ax + B(u + w), A =
Considering γ = min( , δ), we have concomitantly (1 + γ)x ∈ Xch , γx ∈ X1 .
If γ ≥ 1, then δ ≥ 1 and it follows x ∈ X1 , which leads to V (x) = 0. Else, if γ < 1 and we can rewrite x as x = (1 − γ)((1 + γ)x) + γ(γx), where (1 + γ)x ∈ Xch , γx ∈ X1 , 0 < (1 − γ) < 1 and 0 < γ < 1. Hence we get V (x) ≤ 1 − γ < 1. We conclude that if V (x) = 1 then x ∈ ∂Xch .
Let w ∈ W. To prove robust asymptotic stability of the set X1 , we consider three cases regarding the value of V (x). Case 1: V (x) = 1. From Proposition 4, we have x+ ∈ αXch . Thus, V (x+ ) < 1 = V (x).
Case 2: 0 < V (x) < 1. We have λ∗1 (x) > 0. Given ∈ α1 X1 . that the set X1 is α1 robustly contractive, x∗+ 1 Moreover, X1 has an non-empty interior, hence ∃ > 0 such ∗+ + + that x∗+ 1 + (x − x1 ) ∈ X1 . Note that if ≥ 1, x ∈ X1 + and thus V (x ) = 0. Else, we denote ∗+ + z1 = x∗+ (14) 1 + (x − x1 ) ∈ X1 . + z − x 1 This leads to x∗+ . We can rewrite x+ asx+ = 1 = 1− L z1 − x+ + ∗ λ∗1 + j=2 λ∗j x∗+ ) = j . Hence, x (1 + λ1 1− 1− z 1 L λ∗1 + j=2 λ∗j x∗+ j .This is equivalent to 1− L λ∗j λ∗1 z1 x+ = + x∗+ . 1 − (1 − λ∗1 ) j=2 (1 + λ∗ ) j 1 1− λ∗1 > 0. From 0 < < 1, we have λ∗1 > 1 − (1 − λ∗1 ) Moreover, we can show that L λ∗j λ∗1 + =1 ∗ 1 − (1 − λ1 ) j=2 (1 + λ∗ ) 1 1− λ∗j ≥ 0. This proves and ∀j ∈ IL , ) (1 + λ∗1 1− λ∗1 + < 1 − λ∗1 = V (x). V (x ) ≤ 1 − 1 − (1 − λ∗1 )
For Z∞ (Ki ) to be an RPI set, it has to satisfy Z∞ (Ki ) ⊆ X and Ki Z∞ (Ki ) ⊆ U. Hence, the gain K3 is not admissible. Moreover, we have h(Z∞ (K1 ), H) = 0.44, h(Z∞ (K2 ), H) = 0.5.
The set Z∞ (K1 ) minimizes (3). Due to the dimension of the problem and the proposed approach, the refinement proposed Section 3 does not reduce the invariant set (Ω1 = Ω0 = Z∞ (K1 )). The MRPI of the feedback gains K1 and K2 are presented in Figure 2. The method proposed in Section 4 to enlarge the controllable region O∞ (K1 ) = X1 of the controller u1 (x) = K1 x has been tested in simulation. The control laws uj , j = 2, 3, 4 and the associated RPI sets Xj , j = {2, 3, 4} have been obtained using Tube-Based MPC as presented in Mayne et al. (2005) with Ktube,1 = K2 , Ktube,2 = [−0.61 −1.61] , Ktube,3 = [−0.58 −1.55] , and a prediction horizon N = 10. The weight matrices Q, R, and P as defined in Mayne et al. (2005) do not impact the region of attraction of the tube-based MPC controllers. We also consider X5 = O∞ (K2 ) and u5 (x) = K2 x. The set Xch = Conv(Xi , i = 1, ..., 5) can be seen in Figure 3. Trajectories emanating from ∂Xch and converging to Z∞ (K1 ) are represented in Figure 4.
Case 3: V (x) = 0. The control action is given by ν(x) = u1 (x) and it guarantees x+ ∈ X1 . Hence, V (x+ ) = 0.
This proves robust asymptotic stability of the set X1 for any initial condition x ∈ Xch .
6. CONCLUSION
We have proposed a design strategy to enlarge the controllability of the local control law defined in the previous section. Convergence properties have been established with regards to different assumptions on the sets considered for the interpolation-based control strategy.
We have proposed a control law for discrete-time linear systems subject to matched disturbances and input and state constraints. The proposed method allows to mitigate the impact of the disturbances in the direction of the state constraints in a neighborhood of the origin. The effect of the disturbances is assessed through invariant set. A first approach that imposes a linear structure to the control law is proposed. The control structure is then relaxed to further mitigate the impact of the disturbance on state constraints satisfaction, leading to smaller invariant sets
5. SIMULATION RESULTS The results presented in Section 3 and 4 are now illustrated. Consider the following system, constraints and disturbance set 95
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in the constraints direction. To this local control strategy we add an interpolation-based control design to enlarge the controllable region. REFERENCES Blanchini, F. (1999). Set invariance in control. Automatica, 35(11), 1747 – 1767. Corradini, M.L., Cristofaro, A., and Orlando, G. (2014). Sliding-mode control of discrete-time linear plants with input saturation: application to a twin-rotor system. International Journal of Control, 87(8), 1523–1535. Falcone, P., Borrelli, F., Pekar, J., and Stewart, G. (2009). A reference governor approach for constrained piecewise affine systems. European Control Conference (ECC), 4223–4228. Jaulin, L. (2000). Interval constraint propagation with application to bounded-error estimation. Automatica, 36(10), 1547 – 1552. Kolmanovsky, I. and Gilbert, E.G. (1998). Theory and computation of disturbance invariant sets for discretetime linear systems. Mathematical problems in engineering, 4(4), 317–367. Mayne, D., Rakovi`c, S., Findeisen, R., and Allgöwer, F. (2006). Robust output feedback model predictive control of constrained linear systems. Automatica, 42(7), 1217 – 1222. Mayne, D., Rawlings, J., Rao, C., and Scokaert, P. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789 – 814. Mayne, D. and Schroeder, W. (1997). Robust time-optimal control of constrained linear systems. Automatica, 33(12), 2103 – 2118. Mayne, D., Seron, M., and Rakovi`c, S. (2005). Robust model predictive control of constrained linear systems with bounded disturbances. Automatica, 41(2), 219 – 224. Michel, N., Olaru, S., Valmorbida, G., Bertrand, S., and Dumur, D. (2018). Invariant sets for discrete-time constrained linear systems using sliding mode approach. European Control Conference, 2929–2934. Nguyen, H.N. (2012). Constrained control for uncertain systems: an interpolation based control approach. Ph.D. thesis. EDSTIC Supélec 2012. Olaru, S., De Doná, J., Seron, M., and Stoican, F. (2010). Positive invariant sets for fault tolerant multisensor control schemes. International Journal of Control, 83(12), 2622–2640. Rakovi`c, S.V., Kerrigan, E.C., Kouramas, K.I., and Mayne, D.Q. (2005). Invariant approximations of the minimal robust positively invariant set. IEEE Transactions on Automatic Control, 50(3), 406–410. Rakovi`c, S., Kerrigan, E., Mayne, D., and Kouramas, K. (2007). Optimized robust control invariance for linear discrete-time systems: Theoretical foundations. Automatica, 43(5), 831 – 841. Tahir, F. and Jaimoukha, I.M. (2015). Low-complexity polytopic invariant sets for linear systems subject to norm-bounded uncertainty. IEEE Transactions on Automatic Control, 60(5), 1416–1421.
Fig. 1. Z∞ (Ki ) for K1 (red), K2 (blue), and K3 (yellow), and the state constraints X (black).
Fig. 2. The MRPI O∞ (Ki ) and the mRPI Z∞ (Ki ) for K1 (left), and K2 (right).
Fig. 3. The enlarged controllable region Xch (grey) and the initial MRPI O∞ (K1 ) (red).
Fig. 4. The enlarged controllable region Xch (grey), the mRPI Z∞ (K1 ) (red), and trajectories initialized on several vertices of Xch . 96