- Email: [email protected]
Symmetric Constrained Optimal Control
Control Control Preprints, 5th IFAC2015. Conference on Nonlinear Model Predictive Control September 17-20, Seville, Spain September 17-20, Seville, on Spain Preprints, IFAC2015. Conference Nonlinear Model Predictive Control September5th 17-20, 2015. Seville, Spain Available online at www.sciencedirect.com Control September 17-20, 2015. Seville, Spain September 17-20, 2015. Seville, Spain
ScienceDirect
Symmetric Constrained Optimal Control IFAC-PapersOnLine 48-23 (2015) 366–371 Symmetric Constrained Optimal Control Symmetric Constrained Optimal Control ∗∗ Symmetric Optimal Control Claus Constrained Danielson ∗∗∗ Francesco Francesco Borrelli ∗∗ Claus Danielson Borrelli ∗∗
Claus Danielson ∗ Francesco Borrelli ∗∗ Claus Danielson ∗∗ Francesco Borrelli ∗∗ ∗∗ Claus Danielson Borrelli Mitsubishi Electric Research Laboratories, Cambridge MA Mitsubishi Electric ResearchFrancesco Laboratories, Cambridge MA ∗∗ Mitsubishi Electric Research Laboratories, Cambridge MA ∗∗ University of California, Berkeley CA of California, Berkeley CA ∗∗ University Mitsubishi Electric Research Laboratories, Cambridge MA University of California, Berkeley CA ∗∗ Electric Research Laboratories, Cambridge MA Mitsubishi University of California, Berkeley CA ∗∗ ∗∗ University ofresults California, Berkeley CAstrictly convex linear model Abstract: Abstract: This This paper paper extends extends previous previous results on on symmetry symmetry in in strictly convex linear model Abstract: This paper extends previous results on symmetry strictly convex linear predictive control to non-strictly non-strictly convex results and nonlinear nonlinear modelin predictive control. We model define predictive control to convex and model predictive control. We define Abstract: This paper extends previous on symmetry strictly convex linear model predictive for control to non-strictly convex and and nonlinear modelinpredictive control. We define symmetry constrained systems, controllers, model predictive control problems. We show symmetry for constrained systems, controllers, and model predictive control problems. We show Abstract: This paper extends previous results on symmetry in strictly convex linear model predictive for control to non-strictly convex and and nonlinear model predictive control. We symmetry constrained systems, controllers, model predictive control problems. Wedefine show that symmetric model predictive control problems produce symmetric controllers. We show that predictive control to non-strictly convex and nonlinear model predictive control. We define that symmetric model predictive control problems produce symmetriccontrol controllers. We show that symmetry for constrained systems, controllers, and model predictive problems. We show that symmetric model predictive control problems produce symmetric controllers. We show that the previously established methods of memory reduction can be applied to non-strictly convex the previously established methods of memory reduction can be applied to non-strictly convex symmetry for constrained systems, controllers, and model predictive control problems. We show that symmetric model predictive control problems produce symmetric controllers. We show that the previously established methods of memory reduction can be applied to non-strictly convex problems. We established apply these memory reduction techniques to be theapplied battery balancing problem. problems. We apply memory reduction techniques to the battery balancing problem. that symmetric modelthese predictive control problems producecan symmetric controllers. We show that the previously methods of memory reduction to non-strictly convex problems. We apply these memory reductionmemory techniques to theand battery balancing problem. Exploiting symmetry leads to an exponential reduction simple, intuitive optimal Exploiting symmetry leads to an exponential memory reduction and simple, intuitive optimal the previously established methods of memory reduction can be applied to non-strictly convex problems. We apply these memory reductionmemory techniques to theand battery balancing Exploiting symmetry leads to an exponential reduction simple, intuitiveproblem. optimal controllers. problems. apply these memory reductionmemory techniques to theand battery balancing controllers. Exploiting We symmetry leads to an exponential reduction simple, intuitiveproblem. optimal controllers. Exploiting symmetry leads to an exponential simple, intuitive optimal controllers. © 2015, IFAC (International Federation of Automaticmemory Control) reduction Hosting by and Elsevier Ltd. All rights reserved. 1. INTRODUCTION INTRODUCTION large scale systems can be determined controllers. 1. large scale systems can be determined by by using using symmetry symmetry 1. INTRODUCTION large scale systems can bedecoupled determined by using symmetry and checking the smaller subsystems. In Cogill Cogill and checking the smaller decoupled subsystems. In 1. INTRODUCTION large scale systems can be determined by using symmetry andal.checking the smaller decoupled subsystems. In design Cogill In (2015) et (2008) symmetry was used to simplify the INTRODUCTION scale systems can be determined by using symmetry In Danielson Danielson and and1.Borrelli Borrelli (2015) we we studied studied symmetry symmetry large et al. (2008) symmetry was used to simplify the design andal.checking the smallerwas decoupled subsystems. In design Cogill In Danielson and Borrellicontrol (2015) (mpc). we studied symmetry et (2008) symmetry used to simplify the in linear model predictive Symmetries are of H and H controllers. Using symmetry adapted basis and checking the smaller decoupled subsystems. In Cogill in linear model predictive control (mpc). Symmetries are ∞symmetry of H∞ controllers. Using symmetry adapted basis In linear Danielson and Borrellicontrol (2015) (mpc). we studied symmetry et H al.222 and (2008) was used to simplify the design in model predictive Symmetries are of H and H controllers. Using symmetry adapted basis ∞ patterns in the dynamics and constraints of a system. More In Danielson and Borrelli (2015) we studied symmetry the authors designed H and H controllers for each of et al. (2008) symmetry was used to simplify the design patterns in the dynamics and constraints of a system. More 2 ∞ the authors designed H and H controllers for each of in linear model predictive control (mpc). Symmetries are 2 ∞ of H and H controllers. Using symmetry adapted basis 2 ∞ patterns in the dynamics and constraints of a system. More the authors designed H and H controllers for each of 2 ∞ formally, symmetries are transformations of the system in linear model predictive control (mpc). Symmetries are the decoupled subsystems. The squared H norm of the of H formally, symmetries are transformations of the system 2 the decoupled subsystems. The squared H norm of the and H controllers. Using symmetry adapted basis 2 ∞ patterns insymmetries the dynamics and constraints of aofsystem. More the 2 2 ∞ authors designed H2 and H∞ controllers for each of formally, aredo transformations the system the decoupled subsystems. The squared H norm of the 2 states and inputs that not change the dynamics or patterns in the dynamics and constraints of a system. More full system is the sum of the squared H norms of the states andsymmetries inputs thataredotransformations not change theofdynamics or the 2 norms full authors system is the sumH22ofand the HH of the designed Hsquared controllers for each of ∞ formally, the system 2 ∞ the decoupled subsystems. The squared norm of the 2 states and inputs that do not change the dynamics or full system is the sum of the squared H norms of the 2 full constraints. We showed showed that these patterns appear in the the formally, symmetries are transformations ofappear the system decoupled subsystems. The H norm of the system is the decoupled subsystems. The squared H norm of the the constraints. We that these patterns in ∞squared decoupled subsystems. The H norm of the full system is 2norms states and inputs that do not change the dynamics or ∞ 2 full system is the sum of the H of 2 full system is constraints. We showed that these patterns appear in the the decoupled subsystems. TheofHthe of the ∞ norm optimal control-law and can be exploited to reduce states and inputs that do not change the dynamics or the max of the H norm decoupled subsystems. optimal control-law and can be exploited to reduce the ∞ the max of the H norm of the decoupled subsystems. full system is the sum of the squared H norms of the 2 constraints. We showed that these patterns appear in the ∞ 2 decoupled subsystems. The H norm of the full system ∞ optimal control-law andthat can be exploited reduce max of the H∞ norm of the decoupled subsystems. is complexity of the controller. constraints. We patterns to appear in the the the complexity of theshowed controller. decoupled The Hthe of thesubsystems. full system is ∞ optimal control-law and can these be exploited to reduce the ∞ norm the ofsubsystems. the H∞ norm decoupled This paper is as In complexity of the controller. Thismax paper is organized organized asoffollows. follows. In Section Section 22 we we define define optimal control-law and can be exploited to reduce the the max of the H norm of the decoupled subsystems. ∞ This paper is organized as follows. In Section 2 we define complexity of the controller. ∞ An explicit explicit model model predictive predictive controller controller is is a a closed-form closed-form symmetry symmetry for constrained systemsInand and controllers, and An constrained systems controllers, and This paper for is organized as follows. Section 2 we define complexity ofmodel the controller. An explicit predictive finite-time controller isoptimal a closed-form for constrained systemsofand controllers, and solution to the control analyze the invariance properties symmetric systems. solution to model the constrained constrained finite-time optimal control symmetry analyze the invariance properties of symmetric systems. This paper is organized as follows. In Section 2 we define An explicit predictive controller is a closed-form for constrained systemsofand controllers, and solution to the constrained finite-time optimal control symmetry analyze the invariance properties symmetric systems. problem. For linear model predictive control, the explicit An explicit model predictive controller is a closed-form In Section 3 we use symmetry to analyze model predicproblem. Forthe linear model predictive control, the explicit In Section 3 invariance weconstrained use symmetry to of analyze modelsystems. predicsystems and controllers, and solution to constrained finite-time optimal control symmetry analyze thefor properties symmetric problem. For linear model predictive control, the explicit In Section 3 we use symmetry to analyze model prediccontroller takes the form ofpredictive a piecewise piecewise affine on polytope solution to the constrained finite-time optimal control tive controllers. We show that symmetric model predicanalyze the3 invariance properties of symmetric systems. controller takes the form of a affine on polytope tive controllers. We show that symmetric model predicproblem. For linear model control, the explicit In Section we use symmetry to analyze model prediccontroller takes the form a piecewise affine on controllers. We show that symmetric symmetric model model predicfunction. In Danielson andof Borrelli (2015) we used used symmeproblem. For linear model predictive control, thepolytope explicit tive tive control3 problems problems produce predicfunction. Danielson and (2015) we symmetive control produce In Section we We use show symmetry to analyze model controllerIn takes the form ofBorrelli a piecewise affine on polytope tive controllers. that symmetric symmetric model predicpredicfunction. In Danielson andof Borrelli (2015) weexplicit used symmetive control problems produce symmetric model predictry to reduce the memory requirements of model controller takes the form a piecewise affine on polytope tive controllers. In Section 4 we review the symmetry try to reduce the memory requirements ofweexplicit model tive controllers. In Section 4 we review the symmetry controllers. We show that symmetric model predicfunction. In Danielson and Borrelli (2015) used symmecontrol problems produce symmetric model predictry to reduce the memory requirements ofwe explicit model tive tive controllers. In Section 4 we review the symmetry predictive controllers. Symmetry was used to find relationfunction. In Danielson and Borrelli (2015) used symmeexploiting controller designs presented in Danielson and predictive controllers. Symmetry was usedoftoexplicit find relationexploiting controller designs in Danielson and tive control problems produce symmetric model predictry to reduce the memory requirements model tive controllers. In Section 4presented we review the symmetry predictive controllers. Symmetry was used to find relationexploiting controller designs presented in Danielson and ship between pieces of the controller. Using these relationtry to reduce the memory requirements of explicit model Borrelli (2015). Finally in Section 5 we apply the results tive controllers. In Section 4 we review the symmetry ship between pieces of the controller. Using these relationBorrelli (2015). Finally in Section 5 we apply the results predictive controllers. Symmetry was used to find relationexploiting controller designs presented in Danielson and ship between pieces of the controller. Using these relationBorrelli (2015). Finally in Section 5 we apply the results ships, redundant pieces ofcontroller. the controller controller were discarded predictive controllers. was Using used to finddiscarded relation- exploiting of this paper paper to Finally the battery battery balancing problem. Weresults show ships, redundant pieces the were of this to the balancing problem. We show controller designs presented in Danielson and ship between pieces of Symmetry theof these relationBorrelli (2015). in Section 5 we apply the ships, redundant pieces of the controller were discarded this paper tosymmetry the battery balancing problem. We show reducing the to the The ship between pieces ofneeded theofcontroller. Using thesediscarded relationthat exploiting leads to simple, intuitive optimal reducing the memory memory needed to store store the control-law. control-law. The of that exploiting symmetry leads to simple, intuitive optimal Borrelli (2015). Finally in Section 5 we apply the results ships, redundant pieces the controller were of this paper tosymmetry the battery balancing problem. We show reducing the memory needed tobe store the control-law. The that exploiting leads to simple, intuitive optimal discarded controller pieces efficiently reconstructed ships, redundant pieces ofcan the controller were discarded Where the results of are of this paper tosymmetry theapplicable, battery balancing problem. We show discarded controller pieces can efficiently reconstructed controllers. Where applicable, the results of this this paper paper are reducing the memory needed tobe store the control-law. The controllers. that exploiting leads to simple, intuitive optimal discarded controller pieces can be efficiently reconstructed controllers. Where applicable, the results of this paper are online using the symmetry. reducing the memory needed to store the control-law. The proved for the general non-linear case. that exploiting symmetry leads to simple, intuitive optimal online using the symmetry. proved for the general non-linear case. discarded controller pieces can be efficiently reconstructed controllers. Where applicable, the results of this paper are online using the symmetry. proved for the general non-linear case. discarded controller pieces can be efficiently reconstructed controllers. Where applicable, the results of this paper are online using the symmetry. proved for the general non-linear case. The results of Danielson and Borrelli (2015) focused on The results Borrelli (2015) focused on 1.1 Notation and Definitions online using of theDanielson symmetry.and 1.1 Notation and Definitions The results of Danielson and Borrelli (2015) focused on proved for the general non-linear case. linear systems with polytopic polytopic constraints andfocused quadratic linear systems with constraints and quadratic The results of Danielson and Borrelli (2015) on 1.1 Notation and Definitions linear systems with polytopic constraints and quadratic 1.1 Notation and Definitions cost functions. In this paper we extend these results to linThe results of Danielson and Borrelli (2015) focused on + cost functions. In this paper we extend these results to linlinearfunctions. systems In with polytopic constraints and quadratic Consider the discrete-time discrete-time nonlinear system system x x+ = ff (x, (x, u) u) Consider the nonlinear = Notation and Definitions cost thispolytopic paper wepredictive extend these results to lin- 1.1 ear programming based model control problems. linear systems with constraints and quadratic Consider the discrete-time nonlinear system=x+ = :f (x, u) ear programming based model predictive control problems. cost functions. In this paper we extend these results to linwith x ∈ X and u ∈ U . The set Pre(S) ∃u ∈ +{x with x ∈ X and u ∈ U . The set Pre(S) = {x : ∃u ∈ ear programming based model predictive control problems. Consider discrete-time nonlinear system=x+ = :f (x, For non-strictly convex model predictive control problems, cost functions. Inconvex this paper wepredictive extend these results to lin- with xf (x, ∈the X and uis ∈ Uset . The set Pre(S) ∃u u) ∈ For non-strictly model control problems, +{x ear programming based model predictive control problems. U s.t. u) ∈ S} the of states x that can be driven Consider the discrete-time nonlinear system x = f (x, u) U s.t. f (x, u) ∈ S} is the set of states x that can be driven For non-strictly convex model predictive control problems, with x ∈ X and u ∈ U . The set Pre(S) = {x : ∃u ∈ the optimal control-law is not necessarily unique and some ear programming based model predictive control problems. n U s.t. f (x, u) ∈ S} is the set of states x that can be driven the optimal control-law is not necessarily unique and some n in one-step. The set Reach(S) = For optimal non-strictly convex model predictive control problems, into the set S∈ ⊆ uR R in one-step. The set Reach(S) = into the set S ⊆ with x ∈ X and ∈ U . The set Pre(S) = {x : ∃u ∈ the control-law is not necessarily unique and some n U s.t. f (x, u) S} is the set of states x that can be driven of the control-laws may be non-symmetry. However, we For non-strictly convex predictive control problems, one-step. The set Reach(S) = the set S∈ ⊆ Rn in of control-laws maymodel However, we into thethe optimal control-law is be notnon-symmetry. necessarily unique and some {f (x, u) : ∃u U and x ∈ S} is the set of state f (x, u) {f (x, u) : ∃u ∈ U and x ∈ S} is the set of state f (x, u) U s.t. f (x, u) ∈ S} is the set of states x that can be driven of the control-laws may be non-symmetry. However, we one-step. The Reach(S) = into Rnn in show that at least is For this the optimal control-law notnon-symmetry. necessarily unique and {f (x,the u) :set ∃u Sreached ∈⊆ U and x ∈ S} is the setset of from state f (x,set u) show that at least one one controller is symmetric. symmetric. Forsome this of the control-laws mayiscontroller be However, we that can be one-step starting the that can be in one-step starting the in one-step. The Reach(S) = into R in show that at least one controller is symmetric. For this {f (x,the u)nn :set ∃u Sreached ∈⊆ U and x ∈ S} is the setset of from state f (x,set u) controller, the previously established memory reduction of the control-laws may be non-symmetry. However, we that can be reached in one-step starting from the set controller, the previously established memory reduction show that at least one controller is symmetric. For this S ⊆ R A C ⊆ X called control invariant if for {f (x, u)n ..: be ∃uset ∈ U and x is ∈ S} is the set of state the f (x, u) S ⊆ R A set C ⊆ X is called control invariant if for controller, the previously established memory reduction that can reached in one-step starting from set techniques can be readily applied. show that at one controller is symmetric. For this S RnC. there A setexists C ⊆ uX∈ isU called control for techniques can be readily applied. controller, theleast previously established memory reduction all ⊆ x∈ ∈ such that that (x, invariant u)from ∈ C. C. theif all x exists such ff (x, u) ∈ that can be reached in∈ isU one-step starting techniques can be readily applied. S ⊆ RnnC . there A set C ⊆u X called control invariant if set for controller, the previously established memory reduction all x ∈ C there exists u ∈ U such that f (x, u) ∈ C. techniques can readily applied. Symmetry has be a long long history of being being used used to to analyze analyze S ⊆ RC. (G, A ◦) setexists Ca ⊆set X∈ isUalong called control invariant if for Symmetry has a history of all x ∈ there u such that f (x, u) ∈ C. A group is G with a binary operator techniques can be readily applied. group (G, ◦) is a set G along with a binary operator ◦ ◦ Symmetry hascontrol a long design history(see of being used to (1981)). analyze A and simplify Tannenbaum all group x ∈the C (G, there u ∈ suchwith that ∈operator C. under◦ A ◦) exists is a isset G Ualong af (x, binary and simplify control design (see Tannenbaum (1981)). Symmetry has a long history of being used to analyze where operator associative, the set set G u) is closed closed where the operator is associative, the G is under and simplify control design (see Tannenbaum (1981)). A group (G, ◦) is a set G along with a binary operator ◦ Using group representation theory, a basis for the stateSymmetry a long design history of being used analyze the operator isset associative, theidentity set G is element, closed under Using grouphas representation theory, a basis fortothe state- where and simplify control (see Tannenbaum (1981)). the operation ◦,isthe the includes an and the operation set an identity element, and A group (G, ◦)◦, a isset Gincludes along with a binary operator ◦ Using group representation theory, a basis forfound the statewhere the operator associative, the set G is closed under space, input-space, and output-space can be that and simplify control design (see Tannenbaum (1981)). the operation ◦, theelement. set includes an identity simplicity element, and space, input-space, and output-space can be that the Using group representation theory, a basis forfound the stateinverse of each For notational we the inverse of each element. For notational simplicity we where the operator is associative, the set G is closed under space, input-space, and output-space can be found that the operation ◦, the set includes an identity element, and decomposes the system into smaller decoupled subsystems Using representation theory, decoupled a basis forsubsystems the stateinverse of each For gan notational we decomposes the system smaller space, group input-space, andinto output-space can be found that the will drop and write gh ◦◦ h. the theelement. set includes identity simplicity element, and will operation drop the theof◦◦◦, and write gh for for h. decomposes the system into smaller decoupled subsystems the inverse each element. For ggnotational simplicity we (see F¨ a ssler and Stiefel (1992)). This basis is called the space, input-space, and output-space can be found that will drop the ◦ and write gh for ◦ h. (see F¨ a ssler and Stiefel (1992)). This basis is called the decomposes the system into smaller decoupled subsystems the inverse of each element. For notational simplicity we (see F¨assleradapted andsystem Stiefel (1992)). This basis issubsystems called the A will drop the ◦ and write gh for g ◦ h. symmetry basis. A numerical method for finding group G acts on a set X if its elements g ∈ G index decomposes the into smaller decoupled symmetry basis.(1992)). A numerical finding group G acts on a set X if its elements g ∈ G index (see F¨assleradapted and Stiefel This method basis is for called the A will drop the ◦ and write gh for g ◦ h. symmetry adapted basis. A numerical method for finding A group G acts on a set X if its elements g ∈ G index this basis was presented in Danielson and (2015). bijective X satisfy law (see F¨assler and Stiefel This method basisBauer is for called the A this basis was presented in Danielson and Bauer (2015). bijective functions X → → Xif that that satisfy the the group law symmetry adapted basis.(1992)). A numerical finding group functions G acts onfffggga ::: set XX its elements g ∈group G index this basis was presented in Danielson and Bauer (2015). bijective functions Xh→ XGthat satisfy the group law In Fagnani and Willems (1991); Hazewinkel and Martin f ◦ f = f for all g, ∈ where ◦ denote function symmetry adapted basis. A numerical method for finding g h gh In Fagnani and Willems (1991); Hazewinkel and Martin f ◦ f = f for all g, h ∈ G where ◦ denote function A group G acts on a set X if its elements g ∈ G index g h functions gh this basis was presented in Danielson and Bauer (2015). bijective f : X → X that satisfy the group law g In Fagnani and Willems (1991); Hazewinkel and Martin f ◦ f = f for all g, h ∈ G where ◦ denote function g h gh (1983) it was was shown thatin controllability and stability in bijective composition. The orbit Gx := {f (x)satisfy ∈ G} of of a point point this basis was presented Danielson andand Bauer (2015). g (x) (1983) it shown that controllability stability in composition. :: ◦gg ∈ G} a fall XGx XG{f that the group law g : g, g g In Fagnani and Willems (1991); Hazewinkel and Martin fg ◦ fh =functions fghThe for orbit h→∈:= where denote function (1983) it was shown that controllability and stability in composition. The orbit Gx := {f (x) : g ∈ G} of a point g In Fagnani and Willems Hazewinkel Martin fgg ◦ fhh = fgh for orbit all g,Gx h ∈:=G{fwhere ◦g ∈ denote function ghThe (1983) it was shown that(1991); controllability and and stability in composition. (x) : G} of a point g (1983) it © Copyright ©was 2015shown IFAC that controllability and stability in366 366 composition. The orbit Gx := {fgg (x) : g ∈ G} of a point Copyright 2015 IFAC ∗ ∗ ∗ ∗ ∗ ∗
Copyright © 2015 IFAC 366 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 IFAC 366 Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2015 responsibility IFAC 366Control. 10.1016/j.ifacol.2015.11.307
2015 IFAC NMPC September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
x ∈ X is the set of images fg (x) of that point x under every element of the group g ∈ G. 2. SYMMETRIC SYSTEMS AND CONTROLLERS In this section we define symmetry for constrained systems and controllers. We analyze the reachable and invariant sets for symmetric systems. 2.1 Symmetries of Constrained Systems Consider the discrete-time nonlinear system x(t + 1) = f x(t), u(t) (1) subject to constraints on the state and input x(t) ∈ X (2a) u(t) ∈ U (2b) for all t ∈ N where X ⊆ Rn and U ⊆ Rm are convex, full-dimensional sets. A symmetry of the constrained system (1) and (2) is a state-space Θ and input-space Ω transformation that preserves the dynamics and constraints. Definition 1. The pair of invertible matrices Θ ∈ Rn×n and Ω ∈ Rm×m are a (linear) symmetry of the system (1) subject to the constraints (2) if they satisfy f (Θx, Ωu) = Θf (x, u) (3) for all x ∈ X and u ∈ U and ΘX = X (4a) ΩU = U. (4b) For an autonomous system, (3) says that the vector-field f (x) = f (x, 0) is invariant under the state-space transformation Θ i.e. f (x) = Θ−1 f (Θx). For a non-autonomous system (3) says that if the state and input trajectories ∞ {x(t)}∞ t=0 and {u(t)}t=0 satisfy the dynamics (1) then so does the state and input trajectories {Θx(t)}∞ t=0 and {Ωu(t)}∞ . t=0
The set of all pairs Θ and Ω that satisfy (3) is a group denoted by Aut(Σ). In Danielson (2014) it was shown how to efficiently identify the symmetric group of a linear system with polytopic constraints. Proposition 1. The set Aut(Σ) of all matrix pairs Θ and Ω that satisfy (3) is a group under pairwise matrix multiplication.
Next we study the reachable and invariant set of symmetric systems. The following proposition shows that the set of states that are driven into a symmetric set is symmetric. Likewise the set of states that can be reached from a symmetric set is symmetric. Proposition 2. Let S ⊆ Rn be a symmetric set ΘS = S for all Θ ∈ Aut(Σ). Then the backward Pre(S) and forward Reach(S) reachable sets for the system (1) subject to the constraints (2) are symmetric ΘPre(S) = Pre(S) ΘReach(S) = Reach(S) for all Θ ∈ Aut(Σ). The following proposition shows that the maximal control invariant set of a symmetric constrained system is symmetric. 367
367
Proposition 3. Let C∞ ⊆ X be the maximal control invariant set for the system (1) subject to the constraints (2). Then the set C∞ is symmetry with respect to the group Aut(Σ) C∞ = ΘC∞ for all Θ ∈ Aut(Σ). The following corollary provides a method for computing a symmetric control invariant set from any control invariant set. Corollary 1. Let C ⊆ X be a control invariant set for the linear system x(t + 1) = Ax(t) + Bu(t) (5) subject to convex constraints (2). Then the set C¯ = conv{ΘC : Θ ∈ Aut(Σ)} ⊆ X is a symmetric control invariant set. This corollary can be used to construct a symmetric control invariant terminal set for a model predictive controller. 2.2 Symmetries of Controllers In this section we define symmetries of controllers. In particular we study the symmetries of piecewise affine on polytope controllers. Consider the nonlinear state-feedback control-law u(t) = κ x(t) . (6) where κ : X → U . A symmetry of the control-law (6) is a state-space Θ and input-space Ω transformation that preserves the control-law. Definition 2. The pair of invertible matrices Θ ∈ Rn×n and Ω ∈ Rm×m is a symmetry of the controller (6) if Ωκ(x) = κ(Θx) (7) for all x ∈ X . Definition 2 says that the control actions at point x1 and x2 = Θx1 are related by a linear transformation Ω. For a linear controller u = F x, definition (7) says that a symmetry is a pair of invertible matrices Θ and Ω that commute ΩF = F Θ with the feedback gain matrix F . The set of all symmetries of the control-law u = κ(x) is a group denoted by Aut(κ). Proposition 4. The set Aut(κ) of all invertible matrices (Θ, Ω) that satisfy (7) is a group under pairwise matrix multiplication. The following proposition shows that the maximal positive invariant set O∞ is symmetric. Proposition 5. The maximal positive invariant set O∞ of the system (1) subject to the constraints (2) in closedloop with the controller (6) is symmetric ΘO∞ = O∞ with respect to the symmetry group Aut(Σ) ∩ Aut(κ). We are particularly interested in symmetries of piecewise affine on polytopes control-laws F1 x + G1 x ∈ R1 .. (8) κ(x) = . Fp x + G p x ∈ R p where Fi ∈ Rm×n and Gi ∈ Rm are the feedback and feedforward gains, and Ri are polytopes. The triples
2015 IFAC NMPC 368 September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
(Fi , Gi , Ri ) are called the controller pieces. The controller pieces are indexed by I = {1, . . . , p}. The regions Ri for i ∈ I partition the domain X of the controller κ : X → U.
We are interested in symmetries (Θ, Ω) ∈ Aut(κ) of the controller κ(x) that permute the controller pieces (Fi , Gi , Ri ) (9a) ΩFi = Fj Θ ΩGi = Gj (9b) (9c) ΘRi = Rj . Equation (9) says that the state-space transformation Θ maps the i-th region Ri ⊆ X to the j-th region Rj = ΘRi . Furthermore the symmetry (Θ, Ω) maps the control-law in region Ri to the control-law in region Rj κ(x) = Fj x + Gj = ΩFi Θ−1 x + ΩGi . We will denote the subgroup of symmetries that satisfy (9) by Aut(I).
Clearly symmetries (Θ, Ω) ∈ Aut(I) that permute the pieces I of the controller (8) are symmetries (Θ, Ω) ∈ Aut(κ) of the controller. However symmetries (Θ, Ω) ∈ Aut(κ) of the controller κ(x) do not necessarily permute the controller pieces I. The symmetries (Θ, Ω) ∈ Aut(I) that permute the controller pieces I depend on how the partition {Ri }i∈I is chosen. For instance we can break symmetry by splitting region Ri but not its image Rj = ΘRi . If we merge Ri ∪ Rj all regions Ri and Rj with identical feedback Fi = Fj and feedforward Gi = Gj gains then we obtain a symmetric controller as shown by the following theorem. The partition {Ri }i∈I is called minimal if all regions that can be merged have been merged. Theorem 1. Let {Ri }i∈I be the minimal partition of the domain of κ. Then Aut(I) = Aut(κ). The regions Ri of the minimal partition are not necessarily convex nor connected sets. However in the next section we will show that model predictive controllers defined on polytopic partitions are symmetric. 3. SYMMETRIC MODEL PREDICTIVE CONTROL In this section we use symmetry to analyzed model predictive controllers. 3.1 Model Predictive Control In mpc, the control input u(t) to the system (1) is obtained by solving the following constrained finite-time optimal control (cftoc) problem N −1 J (x) = minimize p(xN ) + k=0 q(xk , uk ) (10a) u0 ,...,uN −1
(10b) subject to xk+1 = f (xk , uk ) (10c) xk+1 ∈ X uk ∈ U xN ∈ XN (10d) (10e) x0 = x(t) ∈ X where x0 = x(t) is the current state of the system (1), xk is the predicted state under the control actions uk over the horizon N , and the functions p : X → R and q : X ×U → R are terminal-cost and stage-cost respectively.
368
The set of all initial conditions x0 ∈ X for which (10) is feasible is denoted by X0 ⊆ X . The model predictive controller u(t) = u 0 (x(t)) is the first-term u 0 : X0 → U of the open-loop control sequence u 0 , . . . , u N −1 which optimizes (10) as a function of the initial state x0 = x(t) ∈ X0 . See Rawlings and Mayne (2009) for details about closed-loop stability and persistent feasibility of model predictive control. In Bemporad et al. (2002b), it was shown that for linear systems with polytopic constraints and quadratic cost functions, the model predictive controller u 0 (x) is a piecewise affine on polytope function (8). The partitions Ri = CRi of the piecewise affine controller (8) are called critical regions CRi ⊆ X . This result was extended to linear system with polytopic constraints and linear cost functions in Bemporad et al. (2002a). 3.2 Symmetries of Model Predictive Controllers In this section we define symmetry for model predictive control problems. We show that the resulting model predictive controller is symmetric. A symmetry of the cost function (10a) is a state-space Θ and input-space Ω transformation that preserves the cost function. Definition 3. The pair of invertible matrices Θ and Ω is a symmetry of the cost function (10a) if it satisfies p(Θx) = p(x) (11a) q(Θx, Ωu) = q(x, u). (11b) Definition 3 says that the cost (10a) of the state trajectory N −1 {x(t)}N t=0 and input trajectory {u(t)}t=0 is the same as the cost (10a) of the state trajectory {Θx(t)}N t=0 and input −1 trajectory {Ωu(t)}N . The set of all symmetries of the t=0 cost function (10a) is a group denoted by Aut(J). In Proposition 3 we showed the maximal control invariant set C∞ of a constrained system is symmetric with respect to Aut(Σ). Furthermore in Corollary 1 we showed how to construct a symmetric control invariant set C¯ for a constrained linear system. Therefore we will assume the terminal set XN is symmetric with respect to the symmetries Aut(Σ) of the constrained system. Thus we will define the symmetry group Aut(mpc) = Aut(Σ) ∩ Aut(J) of the cftoc problem (10) as the intersection of the symmetry groups of the dynamics Aut(Σ) and cost function Aut(J). We will use the symmetry group Aut(mpc) to study the symmetry of the model predictive controllers. Our analysis will be based on the dynamic programming formulation of the cftoc problem (10). We define the terminal cost-to-go as JN (xN ) = p(xN ) where p(xN ) is the terminal cost of the model predictive control problem (10a). This cost JN (xN ) is defined on the terminal set XN . For each stage k = N − 1, . . . , 0 the costto-go is the solution to the one-step optimization problem Jk (x) = minimize q(x, u) + Jk+1 f (x, u) u∈U
subject to f (x, u) ∈ Xk+1 .
(12a) (12b)
2015 IFAC NMPC September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
and the cost function Jk (x) is defined on the feasible set Xk = x ∈ X : ∃u ∈ U, f (x, u) ∈ Xk+1 = Pre(Xk+1 ) ∩ X . The following theorem shows that the cost-to-go Jk (x) and feasible region Xk are symmetric with respect to Aut(mpc) at each stage k = 0, . . . , N . Furthermore the one-step optimization problem has a symmetric solution u k (x) at each stage k = 0, . . . , N . Theorem 2. Let the constrained finite-horizon optimal control problem (10) be convex. Then the cost-to-go Jk (x) and feasible region Xk are symmetric Jk (Θx) = Jk (x) ΘXk = Xk for all Θ ∈ Aut(mpc) and k = 0, . . . , N . The one-step optimization problem (12) has a symmetric solution u ¯ 0 (x) u ¯ k (Θx) = Ω¯ u k (x) for all Θ ∈ Aut(mpc) and k = 0, . . . , N − 1.
369
a symmetric piecewise affine on critical regions solution u ¯ 0 (x). Theorem 3. Suppose (10) can be posed as a multiparametric linear program. Then it has a symmetric piecewise affine control-law defined on the critical regions ¯ ¯ F1 x + G1 x ∈ CR1 u ¯ 0 (x) =
.. . ¯p F1 x + G
¯
x ∈ CRp .
The importance of this theorem is that it shows there exists a symmetric controller u ¯ 0 (x) with the same number of pieces |I| as the non-symmetric controllers u 0 (x). Thus we do not need to increase the complexity of the controller in order to apply the symmetric memory reductions discussed in the next section. 4. SYMMETRIC EXPLICIT CONTROLLERS
Theorem 2 says that every convex symmetric cftoc problem has at least one symmetric solution u ¯ 0 (x). However if there are multiple solutions then some of these solutions can be non-symmetric. The following corollary states that if the cftoc problem (10) is strictly convex then the model predictive controller u 0 (x) is symmetric. Corollary 2. Suppose (10) is strictly convex. Then the model predictive controller u 0 (x) is symmetric with respect to Aut(mpc).
In this section we review the fundamental domain controller and orbit controller introduced in Danielson and Borrelli (2015). These controllers replace storing controller pieces with storing the symmetry group. Typically the symmetry group has The symmetry group can be stored efficiently
If the cftoc problem (10) can be posed as a linear or quadratic program then the model predictive controller u 0 (x) is a piecewise affine on polytope function. According to Theorems 1 and 2, the symmetries (Θ, Ω) ∈ Aut(mpc) of (10) permute the pieces (Fi , Gi , Ri ) of the controller u 0 (x) when {Ri }i∈I is the minimal partition. In the following subsection we will show this holds when {Ri }i∈I = {CRi }i∈I is the critical region partition. This stronger result can be used to simplify the explicit model predictive controller.
4.1 Fundamental Domain Controller
3.3 Linear mpc with Linear Cost
By the definition of controller symmetry (7), the control inputs u = u 0 (x) at every point Θx ∈ Gx in the same orbit Gx are related u 0 (Θx) = Ωu 0 (x). Thus we only need determine the control-law u 0 for a single point x in each orbit Gx. In other words, we only need to define u 0 (x) on a fundamental domain Xˆ0 ⊆ X0 .
In this section we show that, if the cftoc problem (10) can be posed as a linear program then the controller pieces are permuted by the symmetries. The case where (10) can be posed as a quadratic program was handled in Danielson and Borrelli (2015). Suppose the cftoc problem (10) can can be posed as a multi-parametric linear program. Then the value function J (x) is a piecewise affine function J (x) = max c i x + di i∈I
where the critical regions CRi are (see Jones et al. (2008)) CRi = {x ∈ X0 : c i x + di ≥ c j x + dj , ∀j ∈ I}.
The symmetries Θ ∈ Aut(mpc) of the constrained finitehorizon optimal control problem (10) permute the value function pieces. Lemma 1. For each Θ ∈ Aut(mpc) and i ∈ I there exists j ∈ I such that Θ ci = cj and di = dj . One consequence of Lemma 1 is that, when (10) can be posed as a multi-parametric linear program, it has 369
exploit symmetry to reduce the memory needed to store the control-law.
A fundamental domain Xˆ ⊆ X is a subset of X that contains at least one representative from each orbit Gx = {Θx : Θ ∈ G} for x ∈ X . Equivalently a fundamental domain Xˆ can be defined as a subset of X that satisfies X = Θ∈G ΘXˆ .
See Danielson (2014) for details about constructing polytopic fundamental domains Xˆ .
Consider the cftoc problem (10) restricted to a fundamental domain Xˆ of the state-space X with respect to a subgroup of symmetries G ⊆ Aut(mpc) N −1 minimize p(xN ) + k=0 q(xk , uk ) (13a) u0 ,...,uN −1
subject to xk+1 = f (xk , uk ) xk+1 ∈ X uk ∈ U xN ∈ XN x0 = x(t) ∈ Xˆ .
(13b) (13c) (13d)
(13e) This problem is identical to the original problem (10) except that the initial state x0 = x(t) is restricted to a fundamental domain Xˆ . The feasible state-space Xˆ0 of (13) is a fundamental domain of the feasible set X0 of the original problem (10).
2015 IFAC NMPC 370 September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
Lemma 2. The set of initial conditions for which (13) is feasible Xˆ0 = X0 ∩ Xˆ is a minimal fundamental domain of X0 . Let u ˆ 0 , . . . , u ˆ N −1 be the optimal solution of problem (13). We call κ ˆ (x) = u ˆ 0 (x) the fundamental domain controller. We can extend this controller from Xˆ0 to the feasible statespace X0 of (10) using symmetry Ω κ ˆ (Θ−1 for x ∈ Θ1 Xˆ 1 x) 1 .. κ ¯ (x) = (14) . −1 Ω κ for x ∈ Θ|G| Xˆ . |G| ˆ (Θ|G| x)
The following theorem shows that κ ¯ (x) is a solution to the model predictive control problem (10). Theorem 4. The controller (14) is a solution to the model predictive control problem (10).
If the cftoc problem (10) can be posed as a linear or quadratic program then the fundamental domain controller u ˆ 0 is a piecewise affine on polytope function (8). Details about the synthesis and implementation of the fundamental domain controller can be found in Danielson and Borrelli (2015). 4.2 Orbit Controller In this section we review the orbit controller from Danielson and Borrelli (2015). We consider the case when (10) can be posed as a linear or quadratic multi-parametric program. In this case, the model predictive controller is piecewise affine on polytopes (8). According to Theorem 3, the pieces (Fi , Gi , CRi ) of the controller (8) are permuted (Fj , Gj , CRj ) = (ΩFi Θ−1 , ΩGi , ΘCRi ) by the symmetries (Θ, Ω) ∈ Aut(mpc). Thus we need only store one of these controller pieces i and j and the other piece can be recovered using the symmetry transformations Θ and Ω. This intuition is formalized and generalized using the concept of a controller orbit. Two controller pieces i, j ∈ I are symmetrically equivalent if there exists a state-space and input-space transformation pair (Θ, Ω) ∈ G ⊆ Aut(mpc) that maps controller piece i to piece j. The set of all controller pieces equivalent to the i-th piece is called a controller orbit G(i) = j ∈ I :(Fj , Gj , CRj ) = (ΩFi Θ−1 , ΩGi , ΘCRi ) for some Θ, Ω ∈ G where G = Aut(mpc). In terms of the controller orbits, the control-law u 0 (x) can be written as −1 ΩFi1 Θ x + ΩGir if x ∈ ΘCRi1 for some (Θ, Ω) ∈ G u 0 (x) =
.. . ΩFir Θ−1 x + ΩGir
if x ∈ ΘCRir for some (Θ, Ω) ∈ G
where {i1 , . . . , ir } = I/G (read as I modulo G) is a set containing one representative controller piece ij from each orbit G(ij ). The other controller pieces j ∈ G(i) can be recovered using the state-space and input-space symmetries (Θ, Ω) ∈ Aut(I) and the relation u 0 (x) = Fj x + Gj = ΩFi Θ−1 x + ΩGi . 370
for x ∈ Rj = ΘRi .
Details about the synthesis and implementation of the orbit controller, as well as the memory reduction it provides, can be found in Danielson and Borrelli (2015). 5. APPLICATION: BATTERY BALANCING In this section we apply the results of this paper to the battery balancing problem Danielson et al. (2013). We show how symmetry can be used to reduce the complexity of balancing controllers for two different battery balancing topologies. Consider a pack of n battery cells connected by a network of power electronics that allow the cells to exchange charge. The dynamics of the cells are described by x(t + 1) = x(t) + B∆τ u(t) (15) n where x ∈ R is the state-of-charge of the cells, u ∈ Rm is the current on the links, and ∆τ is the sample period of the controller. The matrix B ∈ Rn×m contains information about the network topology, charge capacity of the cells, and current capacity of the links. The state-of-charge x(t) and balancing current u(t) are subject to constraints (2). The state are restricted to the box X = [0, 1]n and the input constraints U ⊆ [−1, 1]m depend on the hardware topology. The objective of the controller is to drive the state-of-charge x(t) into a balanced state X¯ = x ∈ X : xi = xj , ∀i, j = 1, . . . , n . Further information about using model predictive control to stabilize a set of equilibria can be found in Danielson et al. (2013). The balancing current u(t) is given by v if τ ≥ ∆τ u(t) = τv if τ < ∆τ ∆τ where v and τ are the solution to the constrained optimal control problem (16a) minimize τ τ,v
subject to x + Bv ∈ X¯ (16b) v ∈ τU. (16c) This controller exploits the integrator dynamics (15) to compute total amount of time τ and charge v = τ u(t)dt needed to balance the cells x(τ ) ∈ X¯ . The 0 implemented current u(t) depends on whether the necessary charge transfer v can be accomplished in one sample period ∆τ . The stability and persistent feasibility of this controller are proven in Danielson (2014). A piecewise affine on polytope controller v (x) and τ (x) can be determined by solving the linear program (16) parametrically. In the next subsection we use the results of this paper to simplifying the piecewise affine expression for v and τ . 5.1 Storage Element Topology The storage element topology is shown in Figure 1. In this topology, charge is removed from one cell and distributed equally among all the cells in the stack. Likewise charge
2015 IFAC NMPC September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
4
10
371
Full Explicit Ctrl Orbit Ctrl
3
10
2
10
1
10
0
10
Fig. 1. Cell-to-Stack topology. Charge is removed from a cell and distributed equally among the stack or removed equally from the stack and added to a cell.
2
3
The topology matrix for the cell-to-stack topology is B = T C 1 n I (In − n 11 ) where 1 ∈ R is the all ones vector, n is the number of cells and m = n is the number of links. We use the sign convention ui > 0 to indicate charge flow from the stack to cell i and uj < 0 to indicate charge flowing from cell j to the stack. The input u is limited by peak current constraint of the links |ui | ≤ 1 for i = 1, 2, ..., m. Additionally, only one link can be active at each time instant u1 ≤ 1. Therefore the input constraint set is U = u ∈ Rn : −1 ≤ ui ≤ 1 and u1 ≤ 1 . (17)
We solved the optimal control problem (16) parametrically to obtain a piecewise affine on polytope control-law v for this topology. The number of controller pieces |I| grows exponentially with the size of the battery pack as shown in Figure 2. Using symmetry we can reduce the controller complexity by discarding symmetrically redundant controller pieces. The symmetry group Aut(mpc) of the storage element topology is the set Sn of all permutation matrices Π ∈ Rn×n . This symmetry groups says that we can arbitrarily permute the battery cells without change the dynamics or constraints. Thus we can assume the battery cells are sorted x1 ≥ x2 ≥ · · · ≥ xn by stateof-charge. The number of non-redundant controller pieces |I/G| depends on whether the number of cells n is odd or even. For battery packs with an odd-number of cells n, the orbit controller has one piece |I/G| = 1 given by n m−1 τ = CI x − x i i i=1 i=m+1 vi = − CI xi − xm
where xm is the median state-of-charge. This control-law says that the amount of charge vi that should be removed from cell i is proportional to the deviation of the cells state-of-charge xi from the median state-of-charge xm . The amount of time τ needed to implement this charge redistribution v is proportional m−1 to the difference between total amount of charge C i=1 xi in the cells n above the median xm and total amount of charge C i=m+1 xi in the cells below the median xm . For battery packs with an even-number of cells n the orbit controller has three piece |I/G| = 3 given by
371
5
6
7
8
9
10
Fig. 2. Number of pieces |I| for the full explicit controller u 0 (x) and number of pieces |I/G| for the orbit controller κo (x).
τ = can be drawn equally from all the cells and added to a single cell.
4
vi =
C I C I
n/2 i=1 xi
−
n
i=n/2+1 xi
−xi + xn/2 −x + xn/2+1 i −F3 x
for x ∈ CR1 for x ∈ CR2 for x ∈ CR3
where F3 ∈ Rn×n is the feedback matrix for region R3 . In region R1 , this controller has the cells track the uppermedian cell xn/2 . In region R2 , this controller has the cells track the lower-median cell xn/2+1 . In region R3 , the controller uses a more complex feedback law v = F3 x. REFERENCES Bemporad, A., Borrelli, F., and Morari, M. (2002a). Model predictive control based on linear programming - the explicit solution. Transactions on Automatic Control. Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E. (2002b). The explicit linear quadratic regulator for constrained systems. Automatica. Cogill, R., Lall, S., and Parrilo, P. (2008). Structured semidefinite programs for the control of symmetric systems. Automatica, 44(5), 1411–1417. Danielson, C. (2014). Symmetric Constrained Optimal Control: Theory, Algorithms, and Applications. Ph.D. thesis, University of California, Berkeley. Danielson, C. and Bauer, S. (2015). Numerical decomposition of symmetric linear systems. In Submitted to Conference on Decision and Control. Danielson, C. and Borrelli, F. (2015). Symmetric linear model predictive control. Transactions on Automatic Control. Danielson, C., Borrelli, F., Oliver, D., Anderson, D., and Phillips, T. (2013). Constrained flow control in storage networks: Capacity maximization and balancing. Automatica. Fagnani, F. and Willems, J. (1991). Representations of symmetric linear dynamical systems. In Conference on Decision and Control. F¨assler, A. and Stiefel, E. (1992). Group theoretical methods and their applications. Birkh¨ auser. Hazewinkel, M. and Martin, C. (1983). Symmetric linear systems: An application of algebraic systems theory. In International Journal of Control. Jones, C., Kerrigan, E., and Maciejowski, J. (2008). On polyhedral projection and parametric programming. Journal of Optimization Theory and Applications. Rawlings, J. and Mayne, D. (2009). Model Predictive Control: Theory and Design. Nob Hill Pub. Tannenbaum, A. (1981). Invariance and System Theory: Algebraic and Geometric Aspects. Springer-Verlag.
ScienceDirect
Symmetric Constrained Optimal Control IFAC-PapersOnLine 48-23 (2015) 366–371 Symmetric Constrained Optimal Control Symmetric Constrained Optimal Control ∗∗ Symmetric Optimal Control Claus Constrained Danielson ∗∗∗ Francesco Francesco Borrelli ∗∗ Claus Danielson Borrelli ∗∗
Claus Danielson ∗ Francesco Borrelli ∗∗ Claus Danielson ∗∗ Francesco Borrelli ∗∗ ∗∗ Claus Danielson Borrelli Mitsubishi Electric Research Laboratories, Cambridge MA Mitsubishi Electric ResearchFrancesco Laboratories, Cambridge MA ∗∗ Mitsubishi Electric Research Laboratories, Cambridge MA ∗∗ University of California, Berkeley CA of California, Berkeley CA ∗∗ University Mitsubishi Electric Research Laboratories, Cambridge MA University of California, Berkeley CA ∗∗ Electric Research Laboratories, Cambridge MA Mitsubishi University of California, Berkeley CA ∗∗ ∗∗ University ofresults California, Berkeley CAstrictly convex linear model Abstract: Abstract: This This paper paper extends extends previous previous results on on symmetry symmetry in in strictly convex linear model Abstract: This paper extends previous results on symmetry strictly convex linear predictive control to non-strictly non-strictly convex results and nonlinear nonlinear modelin predictive control. We model define predictive control to convex and model predictive control. We define Abstract: This paper extends previous on symmetry strictly convex linear model predictive for control to non-strictly convex and and nonlinear modelinpredictive control. We define symmetry constrained systems, controllers, model predictive control problems. We show symmetry for constrained systems, controllers, and model predictive control problems. We show Abstract: This paper extends previous results on symmetry in strictly convex linear model predictive for control to non-strictly convex and and nonlinear model predictive control. We symmetry constrained systems, controllers, model predictive control problems. Wedefine show that symmetric model predictive control problems produce symmetric controllers. We show that predictive control to non-strictly convex and nonlinear model predictive control. We define that symmetric model predictive control problems produce symmetriccontrol controllers. We show that symmetry for constrained systems, controllers, and model predictive problems. We show that symmetric model predictive control problems produce symmetric controllers. We show that the previously established methods of memory reduction can be applied to non-strictly convex the previously established methods of memory reduction can be applied to non-strictly convex symmetry for constrained systems, controllers, and model predictive control problems. We show that symmetric model predictive control problems produce symmetric controllers. We show that the previously established methods of memory reduction can be applied to non-strictly convex problems. We established apply these memory reduction techniques to be theapplied battery balancing problem. problems. We apply memory reduction techniques to the battery balancing problem. that symmetric modelthese predictive control problems producecan symmetric controllers. We show that the previously methods of memory reduction to non-strictly convex problems. We apply these memory reductionmemory techniques to theand battery balancing problem. Exploiting symmetry leads to an exponential reduction simple, intuitive optimal Exploiting symmetry leads to an exponential memory reduction and simple, intuitive optimal the previously established methods of memory reduction can be applied to non-strictly convex problems. We apply these memory reductionmemory techniques to theand battery balancing Exploiting symmetry leads to an exponential reduction simple, intuitiveproblem. optimal controllers. problems. apply these memory reductionmemory techniques to theand battery balancing controllers. Exploiting We symmetry leads to an exponential reduction simple, intuitiveproblem. optimal controllers. Exploiting symmetry leads to an exponential simple, intuitive optimal controllers. © 2015, IFAC (International Federation of Automaticmemory Control) reduction Hosting by and Elsevier Ltd. All rights reserved. 1. INTRODUCTION INTRODUCTION large scale systems can be determined controllers. 1. large scale systems can be determined by by using using symmetry symmetry 1. INTRODUCTION large scale systems can bedecoupled determined by using symmetry and checking the smaller subsystems. In Cogill Cogill and checking the smaller decoupled subsystems. In 1. INTRODUCTION large scale systems can be determined by using symmetry andal.checking the smaller decoupled subsystems. In design Cogill In (2015) et (2008) symmetry was used to simplify the INTRODUCTION scale systems can be determined by using symmetry In Danielson Danielson and and1.Borrelli Borrelli (2015) we we studied studied symmetry symmetry large et al. (2008) symmetry was used to simplify the design andal.checking the smallerwas decoupled subsystems. In design Cogill In Danielson and Borrellicontrol (2015) (mpc). we studied symmetry et (2008) symmetry used to simplify the in linear model predictive Symmetries are of H and H controllers. Using symmetry adapted basis and checking the smaller decoupled subsystems. In Cogill in linear model predictive control (mpc). Symmetries are ∞symmetry of H∞ controllers. Using symmetry adapted basis In linear Danielson and Borrellicontrol (2015) (mpc). we studied symmetry et H al.222 and (2008) was used to simplify the design in model predictive Symmetries are of H and H controllers. Using symmetry adapted basis ∞ patterns in the dynamics and constraints of a system. More In Danielson and Borrelli (2015) we studied symmetry the authors designed H and H controllers for each of et al. (2008) symmetry was used to simplify the design patterns in the dynamics and constraints of a system. More 2 ∞ the authors designed H and H controllers for each of in linear model predictive control (mpc). Symmetries are 2 ∞ of H and H controllers. Using symmetry adapted basis 2 ∞ patterns in the dynamics and constraints of a system. More the authors designed H and H controllers for each of 2 ∞ formally, symmetries are transformations of the system in linear model predictive control (mpc). Symmetries are the decoupled subsystems. The squared H norm of the of H formally, symmetries are transformations of the system 2 the decoupled subsystems. The squared H norm of the and H controllers. Using symmetry adapted basis 2 ∞ patterns insymmetries the dynamics and constraints of aofsystem. More the 2 2 ∞ authors designed H2 and H∞ controllers for each of formally, aredo transformations the system the decoupled subsystems. The squared H norm of the 2 states and inputs that not change the dynamics or patterns in the dynamics and constraints of a system. More full system is the sum of the squared H norms of the states andsymmetries inputs thataredotransformations not change theofdynamics or the 2 norms full authors system is the sumH22ofand the HH of the designed Hsquared controllers for each of ∞ formally, the system 2 ∞ the decoupled subsystems. The squared norm of the 2 states and inputs that do not change the dynamics or full system is the sum of the squared H norms of the 2 full constraints. We showed showed that these patterns appear in the the formally, symmetries are transformations ofappear the system decoupled subsystems. The H norm of the system is the decoupled subsystems. The squared H norm of the the constraints. We that these patterns in ∞squared decoupled subsystems. The H norm of the full system is 2norms states and inputs that do not change the dynamics or ∞ 2 full system is the sum of the H of 2 full system is constraints. We showed that these patterns appear in the the decoupled subsystems. TheofHthe of the ∞ norm optimal control-law and can be exploited to reduce states and inputs that do not change the dynamics or the max of the H norm decoupled subsystems. optimal control-law and can be exploited to reduce the ∞ the max of the H norm of the decoupled subsystems. full system is the sum of the squared H norms of the 2 constraints. We showed that these patterns appear in the ∞ 2 decoupled subsystems. The H norm of the full system ∞ optimal control-law andthat can be exploited reduce max of the H∞ norm of the decoupled subsystems. is complexity of the controller. constraints. We patterns to appear in the the the complexity of theshowed controller. decoupled The Hthe of thesubsystems. full system is ∞ optimal control-law and can these be exploited to reduce the ∞ norm the ofsubsystems. the H∞ norm decoupled This paper is as In complexity of the controller. Thismax paper is organized organized asoffollows. follows. In Section Section 22 we we define define optimal control-law and can be exploited to reduce the the max of the H norm of the decoupled subsystems. ∞ This paper is organized as follows. In Section 2 we define complexity of the controller. ∞ An explicit explicit model model predictive predictive controller controller is is a a closed-form closed-form symmetry symmetry for constrained systemsInand and controllers, and An constrained systems controllers, and This paper for is organized as follows. Section 2 we define complexity ofmodel the controller. An explicit predictive finite-time controller isoptimal a closed-form for constrained systemsofand controllers, and solution to the control analyze the invariance properties symmetric systems. solution to model the constrained constrained finite-time optimal control symmetry analyze the invariance properties of symmetric systems. This paper is organized as follows. In Section 2 we define An explicit predictive controller is a closed-form for constrained systemsofand controllers, and solution to the constrained finite-time optimal control symmetry analyze the invariance properties symmetric systems. problem. For linear model predictive control, the explicit An explicit model predictive controller is a closed-form In Section 3 we use symmetry to analyze model predicproblem. Forthe linear model predictive control, the explicit In Section 3 invariance weconstrained use symmetry to of analyze modelsystems. predicsystems and controllers, and solution to constrained finite-time optimal control symmetry analyze thefor properties symmetric problem. For linear model predictive control, the explicit In Section 3 we use symmetry to analyze model prediccontroller takes the form ofpredictive a piecewise piecewise affine on polytope solution to the constrained finite-time optimal control tive controllers. We show that symmetric model predicanalyze the3 invariance properties of symmetric systems. controller takes the form of a affine on polytope tive controllers. We show that symmetric model predicproblem. For linear model control, the explicit In Section we use symmetry to analyze model prediccontroller takes the form a piecewise affine on controllers. We show that symmetric symmetric model model predicfunction. In Danielson andof Borrelli (2015) we used used symmeproblem. For linear model predictive control, thepolytope explicit tive tive control3 problems problems produce predicfunction. Danielson and (2015) we symmetive control produce In Section we We use show symmetry to analyze model controllerIn takes the form ofBorrelli a piecewise affine on polytope tive controllers. that symmetric symmetric model predicpredicfunction. In Danielson andof Borrelli (2015) weexplicit used symmetive control problems produce symmetric model predictry to reduce the memory requirements of model controller takes the form a piecewise affine on polytope tive controllers. In Section 4 we review the symmetry try to reduce the memory requirements ofweexplicit model tive controllers. In Section 4 we review the symmetry controllers. We show that symmetric model predicfunction. In Danielson and Borrelli (2015) used symmecontrol problems produce symmetric model predictry to reduce the memory requirements ofwe explicit model tive tive controllers. In Section 4 we review the symmetry predictive controllers. Symmetry was used to find relationfunction. In Danielson and Borrelli (2015) used symmeexploiting controller designs presented in Danielson and predictive controllers. Symmetry was usedoftoexplicit find relationexploiting controller designs in Danielson and tive control problems produce symmetric model predictry to reduce the memory requirements model tive controllers. In Section 4presented we review the symmetry predictive controllers. Symmetry was used to find relationexploiting controller designs presented in Danielson and ship between pieces of the controller. Using these relationtry to reduce the memory requirements of explicit model Borrelli (2015). Finally in Section 5 we apply the results tive controllers. In Section 4 we review the symmetry ship between pieces of the controller. Using these relationBorrelli (2015). Finally in Section 5 we apply the results predictive controllers. Symmetry was used to find relationexploiting controller designs presented in Danielson and ship between pieces of the controller. Using these relationBorrelli (2015). Finally in Section 5 we apply the results ships, redundant pieces ofcontroller. the controller controller were discarded predictive controllers. was Using used to finddiscarded relation- exploiting of this paper paper to Finally the battery battery balancing problem. Weresults show ships, redundant pieces the were of this to the balancing problem. We show controller designs presented in Danielson and ship between pieces of Symmetry theof these relationBorrelli (2015). in Section 5 we apply the ships, redundant pieces of the controller were discarded this paper tosymmetry the battery balancing problem. We show reducing the to the The ship between pieces ofneeded theofcontroller. Using thesediscarded relationthat exploiting leads to simple, intuitive optimal reducing the memory memory needed to store store the control-law. control-law. The of that exploiting symmetry leads to simple, intuitive optimal Borrelli (2015). Finally in Section 5 we apply the results ships, redundant pieces the controller were of this paper tosymmetry the battery balancing problem. We show reducing the memory needed tobe store the control-law. The that exploiting leads to simple, intuitive optimal discarded controller pieces efficiently reconstructed ships, redundant pieces ofcan the controller were discarded Where the results of are of this paper tosymmetry theapplicable, battery balancing problem. We show discarded controller pieces can efficiently reconstructed controllers. Where applicable, the results of this this paper paper are reducing the memory needed tobe store the control-law. The controllers. that exploiting leads to simple, intuitive optimal discarded controller pieces can be efficiently reconstructed controllers. Where applicable, the results of this paper are online using the symmetry. reducing the memory needed to store the control-law. The proved for the general non-linear case. that exploiting symmetry leads to simple, intuitive optimal online using the symmetry. proved for the general non-linear case. discarded controller pieces can be efficiently reconstructed controllers. Where applicable, the results of this paper are online using the symmetry. proved for the general non-linear case. discarded controller pieces can be efficiently reconstructed controllers. Where applicable, the results of this paper are online using the symmetry. proved for the general non-linear case. The results of Danielson and Borrelli (2015) focused on The results Borrelli (2015) focused on 1.1 Notation and Definitions online using of theDanielson symmetry.and 1.1 Notation and Definitions The results of Danielson and Borrelli (2015) focused on proved for the general non-linear case. linear systems with polytopic polytopic constraints andfocused quadratic linear systems with constraints and quadratic The results of Danielson and Borrelli (2015) on 1.1 Notation and Definitions linear systems with polytopic constraints and quadratic 1.1 Notation and Definitions cost functions. In this paper we extend these results to linThe results of Danielson and Borrelli (2015) focused on + cost functions. In this paper we extend these results to linlinearfunctions. systems In with polytopic constraints and quadratic Consider the discrete-time discrete-time nonlinear system system x x+ = ff (x, (x, u) u) Consider the nonlinear = Notation and Definitions cost thispolytopic paper wepredictive extend these results to lin- 1.1 ear programming based model control problems. linear systems with constraints and quadratic Consider the discrete-time nonlinear system=x+ = :f (x, u) ear programming based model predictive control problems. cost functions. In this paper we extend these results to linwith x ∈ X and u ∈ U . The set Pre(S) ∃u ∈ +{x with x ∈ X and u ∈ U . The set Pre(S) = {x : ∃u ∈ ear programming based model predictive control problems. Consider discrete-time nonlinear system=x+ = :f (x, For non-strictly convex model predictive control problems, cost functions. Inconvex this paper wepredictive extend these results to lin- with xf (x, ∈the X and uis ∈ Uset . The set Pre(S) ∃u u) ∈ For non-strictly model control problems, +{x ear programming based model predictive control problems. U s.t. u) ∈ S} the of states x that can be driven Consider the discrete-time nonlinear system x = f (x, u) U s.t. f (x, u) ∈ S} is the set of states x that can be driven For non-strictly convex model predictive control problems, with x ∈ X and u ∈ U . The set Pre(S) = {x : ∃u ∈ the optimal control-law is not necessarily unique and some ear programming based model predictive control problems. n U s.t. f (x, u) ∈ S} is the set of states x that can be driven the optimal control-law is not necessarily unique and some n in one-step. The set Reach(S) = For optimal non-strictly convex model predictive control problems, into the set S∈ ⊆ uR R in one-step. The set Reach(S) = into the set S ⊆ with x ∈ X and ∈ U . The set Pre(S) = {x : ∃u ∈ the control-law is not necessarily unique and some n U s.t. f (x, u) S} is the set of states x that can be driven of the control-laws may be non-symmetry. However, we For non-strictly convex predictive control problems, one-step. The set Reach(S) = the set S∈ ⊆ Rn in of control-laws maymodel However, we into thethe optimal control-law is be notnon-symmetry. necessarily unique and some {f (x, u) : ∃u U and x ∈ S} is the set of state f (x, u) {f (x, u) : ∃u ∈ U and x ∈ S} is the set of state f (x, u) U s.t. f (x, u) ∈ S} is the set of states x that can be driven of the control-laws may be non-symmetry. However, we one-step. The Reach(S) = into Rnn in show that at least is For this the optimal control-law notnon-symmetry. necessarily unique and {f (x,the u) :set ∃u Sreached ∈⊆ U and x ∈ S} is the setset of from state f (x,set u) show that at least one one controller is symmetric. symmetric. Forsome this of the control-laws mayiscontroller be However, we that can be one-step starting the that can be in one-step starting the in one-step. The Reach(S) = into R in show that at least one controller is symmetric. For this {f (x,the u)nn :set ∃u Sreached ∈⊆ U and x ∈ S} is the setset of from state f (x,set u) controller, the previously established memory reduction of the control-laws may be non-symmetry. However, we that can be reached in one-step starting from the set controller, the previously established memory reduction show that at least one controller is symmetric. For this S ⊆ R A C ⊆ X called control invariant if for {f (x, u)n ..: be ∃uset ∈ U and x is ∈ S} is the set of state the f (x, u) S ⊆ R A set C ⊆ X is called control invariant if for controller, the previously established memory reduction that can reached in one-step starting from set techniques can be readily applied. show that at one controller is symmetric. For this S RnC. there A setexists C ⊆ uX∈ isU called control for techniques can be readily applied. controller, theleast previously established memory reduction all ⊆ x∈ ∈ such that that (x, invariant u)from ∈ C. C. theif all x exists such ff (x, u) ∈ that can be reached in∈ isU one-step starting techniques can be readily applied. S ⊆ RnnC . there A set C ⊆u X called control invariant if set for controller, the previously established memory reduction all x ∈ C there exists u ∈ U such that f (x, u) ∈ C. techniques can readily applied. Symmetry has be a long long history of being being used used to to analyze analyze S ⊆ RC. (G, A ◦) setexists Ca ⊆set X∈ isUalong called control invariant if for Symmetry has a history of all x ∈ there u such that f (x, u) ∈ C. A group is G with a binary operator techniques can be readily applied. group (G, ◦) is a set G along with a binary operator ◦ ◦ Symmetry hascontrol a long design history(see of being used to (1981)). analyze A and simplify Tannenbaum all group x ∈the C (G, there u ∈ suchwith that ∈operator C. under◦ A ◦) exists is a isset G Ualong af (x, binary and simplify control design (see Tannenbaum (1981)). Symmetry has a long history of being used to analyze where operator associative, the set set G u) is closed closed where the operator is associative, the G is under and simplify control design (see Tannenbaum (1981)). A group (G, ◦) is a set G along with a binary operator ◦ Using group representation theory, a basis for the stateSymmetry a long design history of being used analyze the operator isset associative, theidentity set G is element, closed under Using grouphas representation theory, a basis fortothe state- where and simplify control (see Tannenbaum (1981)). the operation ◦,isthe the includes an and the operation set an identity element, and A group (G, ◦)◦, a isset Gincludes along with a binary operator ◦ Using group representation theory, a basis forfound the statewhere the operator associative, the set G is closed under space, input-space, and output-space can be that and simplify control design (see Tannenbaum (1981)). the operation ◦, theelement. set includes an identity simplicity element, and space, input-space, and output-space can be that the Using group representation theory, a basis forfound the stateinverse of each For notational we the inverse of each element. For notational simplicity we where the operator is associative, the set G is closed under space, input-space, and output-space can be found that the operation ◦, the set includes an identity element, and decomposes the system into smaller decoupled subsystems Using representation theory, decoupled a basis forsubsystems the stateinverse of each For gan notational we decomposes the system smaller space, group input-space, andinto output-space can be found that the will drop and write gh ◦◦ h. the theelement. set includes identity simplicity element, and will operation drop the theof◦◦◦, and write gh for for h. decomposes the system into smaller decoupled subsystems the inverse each element. For ggnotational simplicity we (see F¨ a ssler and Stiefel (1992)). This basis is called the space, input-space, and output-space can be found that will drop the ◦ and write gh for ◦ h. (see F¨ a ssler and Stiefel (1992)). This basis is called the decomposes the system into smaller decoupled subsystems the inverse of each element. For notational simplicity we (see F¨assleradapted andsystem Stiefel (1992)). This basis issubsystems called the A will drop the ◦ and write gh for g ◦ h. symmetry basis. A numerical method for finding group G acts on a set X if its elements g ∈ G index decomposes the into smaller decoupled symmetry basis.(1992)). A numerical finding group G acts on a set X if its elements g ∈ G index (see F¨assleradapted and Stiefel This method basis is for called the A will drop the ◦ and write gh for g ◦ h. symmetry adapted basis. A numerical method for finding A group G acts on a set X if its elements g ∈ G index this basis was presented in Danielson and (2015). bijective X satisfy law (see F¨assler and Stiefel This method basisBauer is for called the A this basis was presented in Danielson and Bauer (2015). bijective functions X → → Xif that that satisfy the the group law symmetry adapted basis.(1992)). A numerical finding group functions G acts onfffggga ::: set XX its elements g ∈group G index this basis was presented in Danielson and Bauer (2015). bijective functions Xh→ XGthat satisfy the group law In Fagnani and Willems (1991); Hazewinkel and Martin f ◦ f = f for all g, ∈ where ◦ denote function symmetry adapted basis. A numerical method for finding g h gh In Fagnani and Willems (1991); Hazewinkel and Martin f ◦ f = f for all g, h ∈ G where ◦ denote function A group G acts on a set X if its elements g ∈ G index g h functions gh this basis was presented in Danielson and Bauer (2015). bijective f : X → X that satisfy the group law g In Fagnani and Willems (1991); Hazewinkel and Martin f ◦ f = f for all g, h ∈ G where ◦ denote function g h gh (1983) it was was shown thatin controllability and stability in bijective composition. The orbit Gx := {f (x)satisfy ∈ G} of of a point point this basis was presented Danielson andand Bauer (2015). g (x) (1983) it shown that controllability stability in composition. :: ◦gg ∈ G} a fall XGx XG{f that the group law g : g, g g In Fagnani and Willems (1991); Hazewinkel and Martin fg ◦ fh =functions fghThe for orbit h→∈:= where denote function (1983) it was shown that controllability and stability in composition. The orbit Gx := {f (x) : g ∈ G} of a point g In Fagnani and Willems Hazewinkel Martin fgg ◦ fhh = fgh for orbit all g,Gx h ∈:=G{fwhere ◦g ∈ denote function ghThe (1983) it was shown that(1991); controllability and and stability in composition. (x) : G} of a point g (1983) it © Copyright ©was 2015shown IFAC that controllability and stability in366 366 composition. The orbit Gx := {fgg (x) : g ∈ G} of a point Copyright 2015 IFAC ∗ ∗ ∗ ∗ ∗ ∗
Copyright © 2015 IFAC 366 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 IFAC 366 Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2015 responsibility IFAC 366Control. 10.1016/j.ifacol.2015.11.307
2015 IFAC NMPC September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
x ∈ X is the set of images fg (x) of that point x under every element of the group g ∈ G. 2. SYMMETRIC SYSTEMS AND CONTROLLERS In this section we define symmetry for constrained systems and controllers. We analyze the reachable and invariant sets for symmetric systems. 2.1 Symmetries of Constrained Systems Consider the discrete-time nonlinear system x(t + 1) = f x(t), u(t) (1) subject to constraints on the state and input x(t) ∈ X (2a) u(t) ∈ U (2b) for all t ∈ N where X ⊆ Rn and U ⊆ Rm are convex, full-dimensional sets. A symmetry of the constrained system (1) and (2) is a state-space Θ and input-space Ω transformation that preserves the dynamics and constraints. Definition 1. The pair of invertible matrices Θ ∈ Rn×n and Ω ∈ Rm×m are a (linear) symmetry of the system (1) subject to the constraints (2) if they satisfy f (Θx, Ωu) = Θf (x, u) (3) for all x ∈ X and u ∈ U and ΘX = X (4a) ΩU = U. (4b) For an autonomous system, (3) says that the vector-field f (x) = f (x, 0) is invariant under the state-space transformation Θ i.e. f (x) = Θ−1 f (Θx). For a non-autonomous system (3) says that if the state and input trajectories ∞ {x(t)}∞ t=0 and {u(t)}t=0 satisfy the dynamics (1) then so does the state and input trajectories {Θx(t)}∞ t=0 and {Ωu(t)}∞ . t=0
The set of all pairs Θ and Ω that satisfy (3) is a group denoted by Aut(Σ). In Danielson (2014) it was shown how to efficiently identify the symmetric group of a linear system with polytopic constraints. Proposition 1. The set Aut(Σ) of all matrix pairs Θ and Ω that satisfy (3) is a group under pairwise matrix multiplication.
Next we study the reachable and invariant set of symmetric systems. The following proposition shows that the set of states that are driven into a symmetric set is symmetric. Likewise the set of states that can be reached from a symmetric set is symmetric. Proposition 2. Let S ⊆ Rn be a symmetric set ΘS = S for all Θ ∈ Aut(Σ). Then the backward Pre(S) and forward Reach(S) reachable sets for the system (1) subject to the constraints (2) are symmetric ΘPre(S) = Pre(S) ΘReach(S) = Reach(S) for all Θ ∈ Aut(Σ). The following proposition shows that the maximal control invariant set of a symmetric constrained system is symmetric. 367
367
Proposition 3. Let C∞ ⊆ X be the maximal control invariant set for the system (1) subject to the constraints (2). Then the set C∞ is symmetry with respect to the group Aut(Σ) C∞ = ΘC∞ for all Θ ∈ Aut(Σ). The following corollary provides a method for computing a symmetric control invariant set from any control invariant set. Corollary 1. Let C ⊆ X be a control invariant set for the linear system x(t + 1) = Ax(t) + Bu(t) (5) subject to convex constraints (2). Then the set C¯ = conv{ΘC : Θ ∈ Aut(Σ)} ⊆ X is a symmetric control invariant set. This corollary can be used to construct a symmetric control invariant terminal set for a model predictive controller. 2.2 Symmetries of Controllers In this section we define symmetries of controllers. In particular we study the symmetries of piecewise affine on polytope controllers. Consider the nonlinear state-feedback control-law u(t) = κ x(t) . (6) where κ : X → U . A symmetry of the control-law (6) is a state-space Θ and input-space Ω transformation that preserves the control-law. Definition 2. The pair of invertible matrices Θ ∈ Rn×n and Ω ∈ Rm×m is a symmetry of the controller (6) if Ωκ(x) = κ(Θx) (7) for all x ∈ X . Definition 2 says that the control actions at point x1 and x2 = Θx1 are related by a linear transformation Ω. For a linear controller u = F x, definition (7) says that a symmetry is a pair of invertible matrices Θ and Ω that commute ΩF = F Θ with the feedback gain matrix F . The set of all symmetries of the control-law u = κ(x) is a group denoted by Aut(κ). Proposition 4. The set Aut(κ) of all invertible matrices (Θ, Ω) that satisfy (7) is a group under pairwise matrix multiplication. The following proposition shows that the maximal positive invariant set O∞ is symmetric. Proposition 5. The maximal positive invariant set O∞ of the system (1) subject to the constraints (2) in closedloop with the controller (6) is symmetric ΘO∞ = O∞ with respect to the symmetry group Aut(Σ) ∩ Aut(κ). We are particularly interested in symmetries of piecewise affine on polytopes control-laws F1 x + G1 x ∈ R1 .. (8) κ(x) = . Fp x + G p x ∈ R p where Fi ∈ Rm×n and Gi ∈ Rm are the feedback and feedforward gains, and Ri are polytopes. The triples
2015 IFAC NMPC 368 September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
(Fi , Gi , Ri ) are called the controller pieces. The controller pieces are indexed by I = {1, . . . , p}. The regions Ri for i ∈ I partition the domain X of the controller κ : X → U.
We are interested in symmetries (Θ, Ω) ∈ Aut(κ) of the controller κ(x) that permute the controller pieces (Fi , Gi , Ri ) (9a) ΩFi = Fj Θ ΩGi = Gj (9b) (9c) ΘRi = Rj . Equation (9) says that the state-space transformation Θ maps the i-th region Ri ⊆ X to the j-th region Rj = ΘRi . Furthermore the symmetry (Θ, Ω) maps the control-law in region Ri to the control-law in region Rj κ(x) = Fj x + Gj = ΩFi Θ−1 x + ΩGi . We will denote the subgroup of symmetries that satisfy (9) by Aut(I).
Clearly symmetries (Θ, Ω) ∈ Aut(I) that permute the pieces I of the controller (8) are symmetries (Θ, Ω) ∈ Aut(κ) of the controller. However symmetries (Θ, Ω) ∈ Aut(κ) of the controller κ(x) do not necessarily permute the controller pieces I. The symmetries (Θ, Ω) ∈ Aut(I) that permute the controller pieces I depend on how the partition {Ri }i∈I is chosen. For instance we can break symmetry by splitting region Ri but not its image Rj = ΘRi . If we merge Ri ∪ Rj all regions Ri and Rj with identical feedback Fi = Fj and feedforward Gi = Gj gains then we obtain a symmetric controller as shown by the following theorem. The partition {Ri }i∈I is called minimal if all regions that can be merged have been merged. Theorem 1. Let {Ri }i∈I be the minimal partition of the domain of κ. Then Aut(I) = Aut(κ). The regions Ri of the minimal partition are not necessarily convex nor connected sets. However in the next section we will show that model predictive controllers defined on polytopic partitions are symmetric. 3. SYMMETRIC MODEL PREDICTIVE CONTROL In this section we use symmetry to analyzed model predictive controllers. 3.1 Model Predictive Control In mpc, the control input u(t) to the system (1) is obtained by solving the following constrained finite-time optimal control (cftoc) problem N −1 J (x) = minimize p(xN ) + k=0 q(xk , uk ) (10a) u0 ,...,uN −1
(10b) subject to xk+1 = f (xk , uk ) (10c) xk+1 ∈ X uk ∈ U xN ∈ XN (10d) (10e) x0 = x(t) ∈ X where x0 = x(t) is the current state of the system (1), xk is the predicted state under the control actions uk over the horizon N , and the functions p : X → R and q : X ×U → R are terminal-cost and stage-cost respectively.
368
The set of all initial conditions x0 ∈ X for which (10) is feasible is denoted by X0 ⊆ X . The model predictive controller u(t) = u 0 (x(t)) is the first-term u 0 : X0 → U of the open-loop control sequence u 0 , . . . , u N −1 which optimizes (10) as a function of the initial state x0 = x(t) ∈ X0 . See Rawlings and Mayne (2009) for details about closed-loop stability and persistent feasibility of model predictive control. In Bemporad et al. (2002b), it was shown that for linear systems with polytopic constraints and quadratic cost functions, the model predictive controller u 0 (x) is a piecewise affine on polytope function (8). The partitions Ri = CRi of the piecewise affine controller (8) are called critical regions CRi ⊆ X . This result was extended to linear system with polytopic constraints and linear cost functions in Bemporad et al. (2002a). 3.2 Symmetries of Model Predictive Controllers In this section we define symmetry for model predictive control problems. We show that the resulting model predictive controller is symmetric. A symmetry of the cost function (10a) is a state-space Θ and input-space Ω transformation that preserves the cost function. Definition 3. The pair of invertible matrices Θ and Ω is a symmetry of the cost function (10a) if it satisfies p(Θx) = p(x) (11a) q(Θx, Ωu) = q(x, u). (11b) Definition 3 says that the cost (10a) of the state trajectory N −1 {x(t)}N t=0 and input trajectory {u(t)}t=0 is the same as the cost (10a) of the state trajectory {Θx(t)}N t=0 and input −1 trajectory {Ωu(t)}N . The set of all symmetries of the t=0 cost function (10a) is a group denoted by Aut(J). In Proposition 3 we showed the maximal control invariant set C∞ of a constrained system is symmetric with respect to Aut(Σ). Furthermore in Corollary 1 we showed how to construct a symmetric control invariant set C¯ for a constrained linear system. Therefore we will assume the terminal set XN is symmetric with respect to the symmetries Aut(Σ) of the constrained system. Thus we will define the symmetry group Aut(mpc) = Aut(Σ) ∩ Aut(J) of the cftoc problem (10) as the intersection of the symmetry groups of the dynamics Aut(Σ) and cost function Aut(J). We will use the symmetry group Aut(mpc) to study the symmetry of the model predictive controllers. Our analysis will be based on the dynamic programming formulation of the cftoc problem (10). We define the terminal cost-to-go as JN (xN ) = p(xN ) where p(xN ) is the terminal cost of the model predictive control problem (10a). This cost JN (xN ) is defined on the terminal set XN . For each stage k = N − 1, . . . , 0 the costto-go is the solution to the one-step optimization problem Jk (x) = minimize q(x, u) + Jk+1 f (x, u) u∈U
subject to f (x, u) ∈ Xk+1 .
(12a) (12b)
2015 IFAC NMPC September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
and the cost function Jk (x) is defined on the feasible set Xk = x ∈ X : ∃u ∈ U, f (x, u) ∈ Xk+1 = Pre(Xk+1 ) ∩ X . The following theorem shows that the cost-to-go Jk (x) and feasible region Xk are symmetric with respect to Aut(mpc) at each stage k = 0, . . . , N . Furthermore the one-step optimization problem has a symmetric solution u k (x) at each stage k = 0, . . . , N . Theorem 2. Let the constrained finite-horizon optimal control problem (10) be convex. Then the cost-to-go Jk (x) and feasible region Xk are symmetric Jk (Θx) = Jk (x) ΘXk = Xk for all Θ ∈ Aut(mpc) and k = 0, . . . , N . The one-step optimization problem (12) has a symmetric solution u ¯ 0 (x) u ¯ k (Θx) = Ω¯ u k (x) for all Θ ∈ Aut(mpc) and k = 0, . . . , N − 1.
369
a symmetric piecewise affine on critical regions solution u ¯ 0 (x). Theorem 3. Suppose (10) can be posed as a multiparametric linear program. Then it has a symmetric piecewise affine control-law defined on the critical regions ¯ ¯ F1 x + G1 x ∈ CR1 u ¯ 0 (x) =
.. . ¯p F1 x + G
¯
x ∈ CRp .
The importance of this theorem is that it shows there exists a symmetric controller u ¯ 0 (x) with the same number of pieces |I| as the non-symmetric controllers u 0 (x). Thus we do not need to increase the complexity of the controller in order to apply the symmetric memory reductions discussed in the next section. 4. SYMMETRIC EXPLICIT CONTROLLERS
Theorem 2 says that every convex symmetric cftoc problem has at least one symmetric solution u ¯ 0 (x). However if there are multiple solutions then some of these solutions can be non-symmetric. The following corollary states that if the cftoc problem (10) is strictly convex then the model predictive controller u 0 (x) is symmetric. Corollary 2. Suppose (10) is strictly convex. Then the model predictive controller u 0 (x) is symmetric with respect to Aut(mpc).
In this section we review the fundamental domain controller and orbit controller introduced in Danielson and Borrelli (2015). These controllers replace storing controller pieces with storing the symmetry group. Typically the symmetry group has The symmetry group can be stored efficiently
If the cftoc problem (10) can be posed as a linear or quadratic program then the model predictive controller u 0 (x) is a piecewise affine on polytope function. According to Theorems 1 and 2, the symmetries (Θ, Ω) ∈ Aut(mpc) of (10) permute the pieces (Fi , Gi , Ri ) of the controller u 0 (x) when {Ri }i∈I is the minimal partition. In the following subsection we will show this holds when {Ri }i∈I = {CRi }i∈I is the critical region partition. This stronger result can be used to simplify the explicit model predictive controller.
4.1 Fundamental Domain Controller
3.3 Linear mpc with Linear Cost
By the definition of controller symmetry (7), the control inputs u = u 0 (x) at every point Θx ∈ Gx in the same orbit Gx are related u 0 (Θx) = Ωu 0 (x). Thus we only need determine the control-law u 0 for a single point x in each orbit Gx. In other words, we only need to define u 0 (x) on a fundamental domain Xˆ0 ⊆ X0 .
In this section we show that, if the cftoc problem (10) can be posed as a linear program then the controller pieces are permuted by the symmetries. The case where (10) can be posed as a quadratic program was handled in Danielson and Borrelli (2015). Suppose the cftoc problem (10) can can be posed as a multi-parametric linear program. Then the value function J (x) is a piecewise affine function J (x) = max c i x + di i∈I
where the critical regions CRi are (see Jones et al. (2008)) CRi = {x ∈ X0 : c i x + di ≥ c j x + dj , ∀j ∈ I}.
The symmetries Θ ∈ Aut(mpc) of the constrained finitehorizon optimal control problem (10) permute the value function pieces. Lemma 1. For each Θ ∈ Aut(mpc) and i ∈ I there exists j ∈ I such that Θ ci = cj and di = dj . One consequence of Lemma 1 is that, when (10) can be posed as a multi-parametric linear program, it has 369
exploit symmetry to reduce the memory needed to store the control-law.
A fundamental domain Xˆ ⊆ X is a subset of X that contains at least one representative from each orbit Gx = {Θx : Θ ∈ G} for x ∈ X . Equivalently a fundamental domain Xˆ can be defined as a subset of X that satisfies X = Θ∈G ΘXˆ .
See Danielson (2014) for details about constructing polytopic fundamental domains Xˆ .
Consider the cftoc problem (10) restricted to a fundamental domain Xˆ of the state-space X with respect to a subgroup of symmetries G ⊆ Aut(mpc) N −1 minimize p(xN ) + k=0 q(xk , uk ) (13a) u0 ,...,uN −1
subject to xk+1 = f (xk , uk ) xk+1 ∈ X uk ∈ U xN ∈ XN x0 = x(t) ∈ Xˆ .
(13b) (13c) (13d)
(13e) This problem is identical to the original problem (10) except that the initial state x0 = x(t) is restricted to a fundamental domain Xˆ . The feasible state-space Xˆ0 of (13) is a fundamental domain of the feasible set X0 of the original problem (10).
2015 IFAC NMPC 370 September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
Lemma 2. The set of initial conditions for which (13) is feasible Xˆ0 = X0 ∩ Xˆ is a minimal fundamental domain of X0 . Let u ˆ 0 , . . . , u ˆ N −1 be the optimal solution of problem (13). We call κ ˆ (x) = u ˆ 0 (x) the fundamental domain controller. We can extend this controller from Xˆ0 to the feasible statespace X0 of (10) using symmetry Ω κ ˆ (Θ−1 for x ∈ Θ1 Xˆ 1 x) 1 .. κ ¯ (x) = (14) . −1 Ω κ for x ∈ Θ|G| Xˆ . |G| ˆ (Θ|G| x)
The following theorem shows that κ ¯ (x) is a solution to the model predictive control problem (10). Theorem 4. The controller (14) is a solution to the model predictive control problem (10).
If the cftoc problem (10) can be posed as a linear or quadratic program then the fundamental domain controller u ˆ 0 is a piecewise affine on polytope function (8). Details about the synthesis and implementation of the fundamental domain controller can be found in Danielson and Borrelli (2015). 4.2 Orbit Controller In this section we review the orbit controller from Danielson and Borrelli (2015). We consider the case when (10) can be posed as a linear or quadratic multi-parametric program. In this case, the model predictive controller is piecewise affine on polytopes (8). According to Theorem 3, the pieces (Fi , Gi , CRi ) of the controller (8) are permuted (Fj , Gj , CRj ) = (ΩFi Θ−1 , ΩGi , ΘCRi ) by the symmetries (Θ, Ω) ∈ Aut(mpc). Thus we need only store one of these controller pieces i and j and the other piece can be recovered using the symmetry transformations Θ and Ω. This intuition is formalized and generalized using the concept of a controller orbit. Two controller pieces i, j ∈ I are symmetrically equivalent if there exists a state-space and input-space transformation pair (Θ, Ω) ∈ G ⊆ Aut(mpc) that maps controller piece i to piece j. The set of all controller pieces equivalent to the i-th piece is called a controller orbit G(i) = j ∈ I :(Fj , Gj , CRj ) = (ΩFi Θ−1 , ΩGi , ΘCRi ) for some Θ, Ω ∈ G where G = Aut(mpc). In terms of the controller orbits, the control-law u 0 (x) can be written as −1 ΩFi1 Θ x + ΩGir if x ∈ ΘCRi1 for some (Θ, Ω) ∈ G u 0 (x) =
.. . ΩFir Θ−1 x + ΩGir
if x ∈ ΘCRir for some (Θ, Ω) ∈ G
where {i1 , . . . , ir } = I/G (read as I modulo G) is a set containing one representative controller piece ij from each orbit G(ij ). The other controller pieces j ∈ G(i) can be recovered using the state-space and input-space symmetries (Θ, Ω) ∈ Aut(I) and the relation u 0 (x) = Fj x + Gj = ΩFi Θ−1 x + ΩGi . 370
for x ∈ Rj = ΘRi .
Details about the synthesis and implementation of the orbit controller, as well as the memory reduction it provides, can be found in Danielson and Borrelli (2015). 5. APPLICATION: BATTERY BALANCING In this section we apply the results of this paper to the battery balancing problem Danielson et al. (2013). We show how symmetry can be used to reduce the complexity of balancing controllers for two different battery balancing topologies. Consider a pack of n battery cells connected by a network of power electronics that allow the cells to exchange charge. The dynamics of the cells are described by x(t + 1) = x(t) + B∆τ u(t) (15) n where x ∈ R is the state-of-charge of the cells, u ∈ Rm is the current on the links, and ∆τ is the sample period of the controller. The matrix B ∈ Rn×m contains information about the network topology, charge capacity of the cells, and current capacity of the links. The state-of-charge x(t) and balancing current u(t) are subject to constraints (2). The state are restricted to the box X = [0, 1]n and the input constraints U ⊆ [−1, 1]m depend on the hardware topology. The objective of the controller is to drive the state-of-charge x(t) into a balanced state X¯ = x ∈ X : xi = xj , ∀i, j = 1, . . . , n . Further information about using model predictive control to stabilize a set of equilibria can be found in Danielson et al. (2013). The balancing current u(t) is given by v if τ ≥ ∆τ u(t) = τv if τ < ∆τ ∆τ where v and τ are the solution to the constrained optimal control problem (16a) minimize τ τ,v
subject to x + Bv ∈ X¯ (16b) v ∈ τU. (16c) This controller exploits the integrator dynamics (15) to compute total amount of time τ and charge v = τ u(t)dt needed to balance the cells x(τ ) ∈ X¯ . The 0 implemented current u(t) depends on whether the necessary charge transfer v can be accomplished in one sample period ∆τ . The stability and persistent feasibility of this controller are proven in Danielson (2014). A piecewise affine on polytope controller v (x) and τ (x) can be determined by solving the linear program (16) parametrically. In the next subsection we use the results of this paper to simplifying the piecewise affine expression for v and τ . 5.1 Storage Element Topology The storage element topology is shown in Figure 1. In this topology, charge is removed from one cell and distributed equally among all the cells in the stack. Likewise charge
2015 IFAC NMPC September 17-20, 2015. Seville, Spain
Claus Danielson et al. / IFAC-PapersOnLine 48-23 (2015) 366–371
4
10
371
Full Explicit Ctrl Orbit Ctrl
3
10
2
10
1
10
0
10
Fig. 1. Cell-to-Stack topology. Charge is removed from a cell and distributed equally among the stack or removed equally from the stack and added to a cell.
2
3
The topology matrix for the cell-to-stack topology is B = T C 1 n I (In − n 11 ) where 1 ∈ R is the all ones vector, n is the number of cells and m = n is the number of links. We use the sign convention ui > 0 to indicate charge flow from the stack to cell i and uj < 0 to indicate charge flowing from cell j to the stack. The input u is limited by peak current constraint of the links |ui | ≤ 1 for i = 1, 2, ..., m. Additionally, only one link can be active at each time instant u1 ≤ 1. Therefore the input constraint set is U = u ∈ Rn : −1 ≤ ui ≤ 1 and u1 ≤ 1 . (17)
We solved the optimal control problem (16) parametrically to obtain a piecewise affine on polytope control-law v for this topology. The number of controller pieces |I| grows exponentially with the size of the battery pack as shown in Figure 2. Using symmetry we can reduce the controller complexity by discarding symmetrically redundant controller pieces. The symmetry group Aut(mpc) of the storage element topology is the set Sn of all permutation matrices Π ∈ Rn×n . This symmetry groups says that we can arbitrarily permute the battery cells without change the dynamics or constraints. Thus we can assume the battery cells are sorted x1 ≥ x2 ≥ · · · ≥ xn by stateof-charge. The number of non-redundant controller pieces |I/G| depends on whether the number of cells n is odd or even. For battery packs with an odd-number of cells n, the orbit controller has one piece |I/G| = 1 given by n m−1 τ = CI x − x i i i=1 i=m+1 vi = − CI xi − xm
where xm is the median state-of-charge. This control-law says that the amount of charge vi that should be removed from cell i is proportional to the deviation of the cells state-of-charge xi from the median state-of-charge xm . The amount of time τ needed to implement this charge redistribution v is proportional m−1 to the difference between total amount of charge C i=1 xi in the cells n above the median xm and total amount of charge C i=m+1 xi in the cells below the median xm . For battery packs with an even-number of cells n the orbit controller has three piece |I/G| = 3 given by
371
5
6
7
8
9
10
Fig. 2. Number of pieces |I| for the full explicit controller u 0 (x) and number of pieces |I/G| for the orbit controller κo (x).
τ = can be drawn equally from all the cells and added to a single cell.
4
vi =
C I C I
n/2 i=1 xi
−
n
i=n/2+1 xi
−xi + xn/2 −x + xn/2+1 i −F3 x
for x ∈ CR1 for x ∈ CR2 for x ∈ CR3
where F3 ∈ Rn×n is the feedback matrix for region R3 . In region R1 , this controller has the cells track the uppermedian cell xn/2 . In region R2 , this controller has the cells track the lower-median cell xn/2+1 . In region R3 , the controller uses a more complex feedback law v = F3 x. REFERENCES Bemporad, A., Borrelli, F., and Morari, M. (2002a). Model predictive control based on linear programming - the explicit solution. Transactions on Automatic Control. Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E. (2002b). The explicit linear quadratic regulator for constrained systems. Automatica. Cogill, R., Lall, S., and Parrilo, P. (2008). Structured semidefinite programs for the control of symmetric systems. Automatica, 44(5), 1411–1417. Danielson, C. (2014). Symmetric Constrained Optimal Control: Theory, Algorithms, and Applications. Ph.D. thesis, University of California, Berkeley. Danielson, C. and Bauer, S. (2015). Numerical decomposition of symmetric linear systems. In Submitted to Conference on Decision and Control. Danielson, C. and Borrelli, F. (2015). Symmetric linear model predictive control. Transactions on Automatic Control. Danielson, C., Borrelli, F., Oliver, D., Anderson, D., and Phillips, T. (2013). Constrained flow control in storage networks: Capacity maximization and balancing. Automatica. Fagnani, F. and Willems, J. (1991). Representations of symmetric linear dynamical systems. In Conference on Decision and Control. F¨assler, A. and Stiefel, E. (1992). Group theoretical methods and their applications. Birkh¨ auser. Hazewinkel, M. and Martin, C. (1983). Symmetric linear systems: An application of algebraic systems theory. In International Journal of Control. Jones, C., Kerrigan, E., and Maciejowski, J. (2008). On polyhedral projection and parametric programming. Journal of Optimization Theory and Applications. Rawlings, J. and Mayne, D. (2009). Model Predictive Control: Theory and Design. Nob Hill Pub. Tannenbaum, A. (1981). Invariance and System Theory: Algebraic and Geometric Aspects. Springer-Verlag.