In a conference paper titled "Geometric Properties of Gradient Projection Anti-windup Compensated Systems," two main results were presented. The first is the controller state-output consistency property of gradient projection anti-windup (GPAW) compensated controllers. The second is a geometric bounding condition relating the vector fields of the uncompensated and GPAW compensated closed-loop systems with respect to a star domain. While the controller state-output consistency property stands without modifications, the proof of the geometric bounding condition depends on two lemmas, the proofs of which were found to be faulty. In this report, we present a new proof of the geometric bounding condition using concepts from convex analysis, together with minor miscellaneous corrections.DSO National Laboratories, Singapore, AFOSR grant FA9550-08-1-008

How, Jonathan P.

Teo, Justin

DSpace@MIT

1Corrections to “Geometric Properties of GradientProjection Anti-windup Compensated Systems”Justin Teo and Jonathan P. HowTechnical Report ACL10–02Aerospace Controls LaboratoryDepartment of Aeronautics and AstronauticsMassachusetts Institute of TechnologyJune 29, 2010Abstract—In a conference paper titled “Geometric Propertiesof Gradient Projection Anti-windup Compensated Systems,” twomain results were presented. The first is the controller state-output consistency property of gradient projection anti-windup(GPAW) compensated controllers. The second is a geometricbounding condition relating the vector fields of the uncompen-sated and GPAW compensated closed-loop systems with respectto a star domain. While the controller state-output consistencyproperty stands without modifications, the proof of the geometricbounding condition depends on two lemmas, the proofs of whichwere found to be faulty. In this report, we present a new proofof the geometric bounding condition using concepts from convexanalysis, together with minor miscellaneous corrections.Index Terms—gradient projection anti-windup, geometricproperties, corrections, combinatorial optimization, quadraticprogram, convex analysis, projection on polyhedral cone.I. INTRODUCTIONIN the conference paper titled “Geometric Propertiesof Gradient Projection Anti-windup Compensated Sys-tems” [1], two main results were presented. The first is thecontroller state-output consistency property of gradient projec-tion anti-windup (GPAW) compensated controllers, presentedas Theorem 1 in [1]. The second is a geometric boundingcondition relating the vector fields of the uncompensatedand GPAW compensated closed-loop systems with respectto a star domain [1, Definition 2], which is presented asTheorem 2 in [1]. While the controller state-output consistencyproperty [1, Theorem 1] stands without modifications, theproof of the geometric bounding condition [1, Theorem 2]depends on two lemmas [1, Lemmas 1 and 2], the proofs ofwhich were found to be faulty. In this report, we present anew proof of the geometric bounding condition [1, Theorem 2]J. Teo is a graduate student with the Aerospace Controls Laboratory, Depart-ment of Aeronautics & Astronautics, Massachusetts Institute of Technology,Cambridge, MA 02139, USA (email: csteo@mit.edu).J. P. How is director of Aerospace Controls Laboratory and Professor inthe Department of Aeronautics & Astronautics, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USA (email: jhow@mit.edu).using concepts from convex analysis. The new proof is basedon geometry and does not need [1, Lemmas 1 and 2], whichis algebraic in nature.This report is meant to be read together with [1]. As such,we will not repeat extraneous material, and refer the readerto [1]. We highlight the faults in the existing proofs in thefollowing section. For consistency, we present some minorcorrections to [1] in Section III. This is followed by the newproof of [1, Theorem 2] in Section IV. Equation, theorem,lemma, and proposition numbers in the present report havebeen prefixed by “C” for clarity.II. FAULTS WITH EXISTING PROOFSIn [1], Theorem 2 depends on Lemma 2, which in turndepends on Lemma 1. It was found that both the proofs ofLemmas 1 and 2 in [1] are faulty, which invalidates Theorem 2.Note that Lemma 1 being faulty alone is sufficient to invalidateTheorem 2.In the last part of the proof of Lemma 1 in [1], it was statedthat when the columns of NIsat are linearly independent, thenno columns of NIsat\I∗ (I∗ being an optimal solution to thecombinatorial optimization subproblem (7) in [1]) can be inthe span of the columns of NI∗ . It was then claimed thatthis implies all columns of ΦNIsat\I∗ must lie entirely in thespan of the columns of M , where M is chosen such that itscolumns, together with the columns of ΦNI∗ forms a basis,and satisfy NTI∗ΦTM = 0. The last assertion is wrong, asseen by the following counterexample.Example 1: Set ΦNIsat = [ 1 10 1 ] and ΦNI∗ = [1, 0]T. ThenM = [0, α]T for some α 6= 0. Clearly, ΦNIsat\I∗ = [1, 1]Tmust have a non-zero component lying in the span of thecolumns of ΦNI∗ , ie. it is not contained entirely in the spanof M , which is the faulty claim. 2The preceding counterexample is sufficient to invalidateLemma 2 in [1]. However, it was found that even if Lemma 1holds, the proof of Lemma 2 is also faulty. In the latter part ofthe proof of Lemma 2 in [1], it was claimed that the cardinalityof solutions φ(Iˆ1) will be increased by the described construc-tion. This is not true in general and invalidates Lemma 2, evenif Lemma 1 holds.III. MINOR CORRECTIONSFor consistency, the following changes to [1] are needed.(i) Following the proof of Proposition 1, the material statingLemmas 1 and 2 is to be removed. The paragraphpreceding equation (9) should be changed to read: “Ateach fixed time (so that (xg, y, r˜(t)) is fixed), let I∗ bea solution to subproblem (7). The GPAW compensatedcontroller derived from (2) is then given by (4) with . . . ”(ii) The Appendix containing the (faulty) proofs of Lem-mas 1 and 2 is to be removed.IV. CORRECTION TO PROOF OF GEOMETRIC BOUNDINGCONDITIONHere, we present a new proof of Theorem 2 in [1]. The newproof is based on the underlying geometry, in contrast with theexisting attempt in [1], which is algebraic in nature. We firstshow that a vector defined by a solution to the combinatorialoptimization subproblem (7) in [1] must be the unique solutionto a convex optimization problem. Then geometric propertiesof the solution to the convex optimization problem is used toprove Theorem 2 in [1].Observe that the GPAW parameter Γ ∈ Rq×q is symmetricpositive definite, so that it can always be decomposed asΓ = ΦTΦ for some nonsingular matrix Φ ∈ Rq×q [2,Theorem 7.2.7, pp. 406]. For any I such that |I| = 0, or suchthat NI(xg) is full rank, the matrix RI(xg) is well-definedand given by (see its definition following (6) in [1])RI(xg) =I − ΓNI(NTI ΓNI)−1NTI (xg), if |I| > 0,I, otherwise.For any well-defined RI(xg), let the principal projectionmatrix R˜I(xg) and complementary projection matrix SI(xg)induced by RI(xg) beR˜I(xg) := Φ−TRI(xg)ΦT, SI(xg) := I − R˜I(xg), (C1)so thatRI(xg) = ΦTR˜I(xg)Φ−T = I − ΦTSI(xg)Φ−T. (C2)It can be verified that R˜I(xg) and SI(xg) take the explicitformsR˜I(xg) =I − N˜I(N˜TI N˜I)−1N˜TI (xg), if |I| > 0,I, otherwise,SI(xg) =N˜I(N˜TI N˜I)−1N˜TI (xg), if |I| > 0,0, otherwise, (C3)where N˜I(xg) = ΦNI(xg). From [3, Lemma 1, Theorem 1],it can be seen that both are projection matrices. Using (C2)and (C3), the objective function of subproblem (7) in [1] canbe written asJ(I) = fTc Γ−1RIfc = fTc Φ−1Φ−T(I − ΦTSIΦ−T)fc,= f˜Tc f˜c − f˜Tc SI f˜c = ‖f˜c‖2 − f˜Tc SISI f˜c,= ‖f˜c‖2 − f˜Tc STI SI f˜c = ‖f˜c‖2 − ‖SI f˜c‖2,where f˜c := Φ−Tfc = Φ−Tfc(xg, y, r˜(t)). Observe thatfor any I such that RI(xg) is well-defined, we haveNTI (xg)fI(xg, y, r˜(t)) = 0 (≤ 0), so that the solutions tosubproblem (7) in [1] remains unchanged by addition of theseredundant constraints. Defining N˜I(xg) := ΦNI(xg), it canbe seen that subproblem (7) in [1] can be rewritten as aminimization problem with |Isat| inequality constraints,minI∈J‖(I − R˜I)f˜c‖2, subject to rank(N˜I) = |I|,N˜TIsatR˜I f˜c ≤ 0,(C4)where all function arguments have been dropped, and Jremains unaltered as the set of all subsets of Isat withcardinality less than or equal to q.Now, consider the quadratic programming problemminx∈Rq‖f˜c − x‖2, subject to N˜TIsatx ≤ 0, (C5)which is a convex optimization problem with a unique solu-tion [4, pp. 214]. It can be seen that problem (C5) seeks theunique projection of f˜c onto the (polyhedral) polar cone K◦ [4,pp. 513] induced by the (finitely generated) cone K generatedby the columns of N˜Isat [4, pp. 143]. Together with the dualcone K∗ [4, pp. 149] (dual to K), these cones are defined byK = {x ∈ Rq | x = N˜Isatz,∀z ≥ 0 ∈ R|Isat|},K◦ = {x ∈ Rq | N˜TIsatx ≤ 0 ∈ R|Isat|},K∗ = {x ∈ Rq | N˜TIsatx ≥ 0 ∈ R|Isat|}.(C6)The projection of f˜c onto K◦, together with the cone K andits dual K∗, are illustrated in Fig. 1. Observe that the polarof K◦, ie. K◦◦, satisfy K◦◦ = K for every finitely generated3K◦K K∗f˜cx∗Fig. 1. The projection x∗ of f˜c onto the polyhedral cone K◦, together withK (= K◦◦) and its dual K∗ (= −K◦).cone K [5, Lemma 2.7.9, pp. 54]. Moreover, a theorem due toMinkowski states that every polyhedral cone (eg. K◦ in (C6))is finitely generated [5, Theorem 2.8.6, pp. 55], and a theoremdue to Weyl states that every finitely generated cone (eg. Kin (C6)) is a polyhedral cone [5, Theorem 2.8.8, pp. 56].These allow results applicable to finitely generated cones tobe applied to polyhedral cones, and vice versa.We will need the following result from [6].Proposition C1 ( [6, Proposition 2]): Let x∗ be the pro-jection of a vector y of Rn into a convex polyhedral coneK = K(S) that is generated by a set S = {s1, . . . , sk}. LetR be the set of vectors si of S orthogonal to y − x. Thenthe vector x∗ is equal to the projection of y into the subspaceL(R) generated by the vectors of R.Proposition C1 applied to the convex polyhedral cone K◦shows that the unique solution x∗ to problem (C5) is equalto the projection of f˜c onto a subspace containing a face ofK◦ that x∗ resides in (including possibly the face K◦). Thisleads to the next result, which is crucial to the new proof ofTheorem 2 in [1].Lemma C1: The unique solution x∗ to the convex optimiza-tion problem (C5) satisfies x∗ = R˜I∗ f˜c for any solution I∗to the combinatorial optimization problem (C4).Proof: When f˜c ∈ K◦, the unique optimal solution x∗of problem (C5) must be x∗ = f˜c. In this case, ∅ is afeasible solution to problem (C4), so that the objective functionsatisfies ‖f˜c − R˜I∗ f˜c‖2 = J˜(I∗) ≤ J˜(∅) = 0 for any optimalsolution I∗. This implies R˜I∗ f˜c = f˜c, and x∗ = R˜I∗ f˜c.When f˜c 6∈ K◦, Proposition C1 shows that x∗ is equal tothe projection of f˜c onto some (possibly non-unique) subspaceL containing a face of K◦ that x∗ resides in. This subspaceL must be orthogonal to the span of some collection ofcolumn vectors of N˜Isat . The unique projection of f˜c onto thesubspace orthogonal to the subspace spanned by the columnsof N˜I for some I ⊂ Isat (such that the columns of N˜Iare linearly independent) is given by R˜I f˜c [3, Theorem 1].Since the definition of the set of candidate solutions J isexhaustive, it must include a set of indices IL such thatthe columns of N˜IL are linearly independent, and L isorthogonal to the subspace spanned by the columns of N˜IL .Hence x∗ = R˜IL f˜c by Proposition C1 and J˜(IL) = ‖f˜c −R˜IL f˜c‖2 = ‖f˜c − x∗‖2. The minimization in problem (C4)yields ‖f˜c − R˜I∗ f˜c‖2 = J˜(I∗) ≤ J˜(IL) = ‖f˜c − x∗‖2.Uniqueness of the solution x∗ of problem (C5) [4, pp. 214]ensures that ‖f˜c − x∗‖2 < ‖f˜c − y‖2 for all y 6= x∗. This,together with ‖f˜c − R˜I∗ f˜c‖2 ≤ ‖f˜c − x∗‖2, shows thatx∗ = R˜I∗ f˜c, as desired.Remark 1: Observe that ∅ ∈ J by the definition of J , andthat the constraint rank(N˜I) = |I| in (C4) is to ensure linearindependence of the columns of N˜I , or that I = ∅. Note alsothat the second part of the proof of Lemma C1 includes thecase f˜c ∈ K◦, but is presented as such for clarity. Remark 2: Lemma C1 shows that even if problem (C4)has no unique solutions, the projection of f˜c defined by anysolution I∗, namely R˜I∗ f˜c, is unique. Since Φ is nonsingular,it implies uniqueness of RI∗fc = ΦTR˜I∗ f˜c as well. Remark 3: An implication of Lemma C1 is that the GPAWcompensated controller can be defined by (compare with (4)and (6) in [1])x˙g = ΦTx∗, xg(0) = xc0, u = gc(xg),where x∗ is obtained as a solution to the quadratic pro-gram (C5) at each fixed time. This realization will be usefulif the quadratic program proves to be more computationallyattractive than the posed combinatorial optimization problems.However, observe that most of the inherent structure of theGPAW compensated controller is then concealed by this rep-resentation, which renders it ill suited for further analysis. Remark 4: Much work has been devoted to developingalgorithms for projection onto polyhedral cones (see [6], [7]and the references therein). This suggests that the quadraticprogram formulation may be computationally inefficient. For the next result, observe that any solution I∗ of sub-problem (7) in [1] satisfies rank(NI∗(xg)) = |I∗|, so thateither I∗ = ∅ or NI∗(xg) is full rank. This ensures thatRI∗(xg, y, r˜(t)), R˜I∗(xg, y, r˜(t)), and SI∗(xg, y, r˜(t)) arewell-defined. Recall the definition of the unsaturated regionin [1] (preceding [1, Remark 9])K = {x ∈ Rq | hi(x) ≤ 0,∀i ∈ I2m} ⊂ Rq,where hi are the saturation constraint functions in (5) of [1].Lemma C2: Let I∗ be a solution to subproblem (7) in [1].If the unsaturated region K ⊂ Rq is a star domain, then(x− xker)TΦ−1SI∗(x, y, r˜(t))Φ−Tfc(x, y, r˜(t)) ≥ 0holds for any boundary point x ∈ ∂K and any xker ∈ ker(K).4Proof: For notational convenience, defineN˜I := ΦNI(x), N˜I∗ := ΦNI∗(x, y, r˜(t)),RI∗ := RI∗(x, y, r˜(t)), R˜I∗ := R˜I∗(x, y, r˜(t)),SI∗ := SI∗(x, y, r˜(t)), f˜c := Φ−Tfc(x, y, r˜(t)),bearing in mind that x ∈ ∂K and Γ = ΦTΦ [2, Theorem 7.2.7,pp. 406]. With x˜ := Φ−T(x− xker), we need to show thatx˜TSI∗ f˜c ≥ 0. (C7)Lemma C1 shows that x∗ = R˜I∗ f˜c where x∗ is the uniqueoptimal solution to problem (C5). Since x∗ is the projection off˜c onto K◦, it satisfies x∗ − f˜c ∈ K◦∗ [4, Theorem E.9.2.0.1,pp. 726] where K◦∗ is the dual [4, pp. 149] to K◦. Moreover,because the dual cone is the negative polar cone [4, foot-note 2.53, pp. 149], and K◦◦ = K [5, Lemma 2.7.9, pp. 54],we have K◦∗ = −K. Hence the relation x∗ − f˜c ∈ K◦∗ isequivalent to f˜c − x∗ ∈ K. Using (C1) and x∗ = R˜I∗ f˜c (byLemma C1), we havef˜c − x∗ = f˜c − R˜I∗ f˜c = SI∗ f˜c ∈ K. (C8)From [1, Lemma 3], we have 〈x−xker,∇hi(x)〉 ≥ 0 for alli ∈ Isat, where Isat is the set of indices of active saturationconstraints (see its definition preceding (7) in [1]). This meansNTIsat(x− xker) = NTIsatΦTΦ−T(x− xker) = N˜TIsat x˜ ≥ 0,which implies that x˜ is in the dual of K, ie. x˜ ∈ K∗ [4,pp. 149]. This, together with the definition of the dual cone [4,pp. 149] and (C8) shows that (C7) holds.The corrected proof of the geometric bounding condition,with a minor generalization, is presented next.Theorem C1 ( [1, Theorem 2]): If K ⊂ Rq is a star do-main, then for any z ∈ (Rn × K) and any zker ∈ (Rn ×ker(K)),〈z − zker, Γ˜−1fp(t, z)〉 ≤ 〈z − zker, Γ˜−1fn(t, z)〉,holds for all t ∈ R, where Γ˜ = [ Θ 00 Γ ] ∈ R(n+q)×(n+q) andΘ ∈ Rn×n is any nonsingular square n× n matrix.Proof: Let z = (e, x) ∈ (Rn ×K), zker = (e∞, x∞) ∈(Rn × ker(K)), e¯ = e − e∞ ∈ Rn, and x¯ = x − x∞ ∈ Rq .With reference to (3) and (11) in [1], we need to show thate¯TΘ−1(f−r˙(t))+x¯TΓ−1fI∗ ≤ e¯TΘ−1(f−r˙(t))+x¯TΓ−1fc,or equivalently, x¯TΓ−1fI∗ ≤ x¯TΓ−1fc, where the functionarguments have been dropped. Using the definition of fI∗(see (6) in [1]), Γ = ΦTΦ [2, Theorem 7.2.7, pp. 406],and (C2), the preceding can be reduced tox¯TΓ−1fc(x)− x¯TΓ−1RI∗(x)fc(x) ≥ 0,x¯TΓ−1fc(x)− x¯TΓ−1(I − ΦTSI∗(x)Φ−T)fc(x) ≥ 0,x¯TΦ−1SI∗(x, y, r˜(t))Φ−Tfc(x, y, r˜(t)) ≥ 0, (C9)where by an abuse of notation, we mean RI∗(x) :=RI∗(x, y, r˜(t)), SI∗(x) := SI∗(x, y, r˜(t)), and fc(x) :=fc(x, y, r˜(t)).If x is in the interior of K, then hi(x) < 0 for all i ∈ I2mand Isat = ∅. Since I∗ ⊂ Isat, we have I∗ = ∅. From thedefinition of SI∗ (C3) with I∗ = ∅, we see that (C9) holdswith equality. If x is on the boundary of K, ie. x ∈ ∂K,Lemma C2 yields (C9), and hence the conclusion.V. CONCLUSIONSFaults in the existing proof of the geometric boundingcondition in [1] were revealed. Necessary changes and a newproof of a marginally generalized result was presented, usingconcepts from convex analysis. An intermediate result is thatthe GPAW compensated controller can be defined by thesolution to a quadratic program with a unique optimal solution.The geometric bounding condition as originally stated standswith this new proof.ACKNOWLEDGMENTSThe first author gratefully acknowledges Prof. Jean-JacquesSlotine for the numerous enlightening discussions that led tothe development of the GPAW scheme. He also gratefullyacknowledges the support of DSO National Laboratories,Singapore. Research funded in part by AFOSR grant FA9550-08-1-0086.REFERENCES[1] J. Teo and J. P. How, “Geometric properties of gradient projectionanti-windup compensated systems,” in Proc. American Control Conf.,Baltimore, MD, Jun./Jul. 2010, pp. 5973 – 5978.[2] R. A. Horn and C. R. Johnson, Matrix Analysis. New York, NY:Cambridge University Press, 1985.[3] J. B. Rosen, “The gradient projection method for nonlinear programming.part I. linear constraints,” J. Soc. Ind. Appl. Math., vol. 8, no. 1, pp. 181– 217, Mar. 1960.[4] J. Dattorro, Convex Optimization & Euclidean Distance Geometry. PaloAlto, CA: Meboo, 2005, version 2010.02.04.[5] J. Stoer and C. Witzgall, Convexity and Optimization in Finite DimensionsI, ser. Die Grundlehren der mathematischen Wissenschaften. Berlin,Germany: Springer, 1970, vol. 163.[6] M. Tenenhaus, “Canonical analysis of two convex polyhedral cones andapplications,” Psychometrika, vol. 53, no. 4, pp. 503 – 524, Dec. 1988.[7] A. B. Ne´meth and S. Z. Ne´meth, “How to project onto an isotoneprojection cone,” Linear Alg. Appl., vol. 433, no. 1, pp. 41 – 51, Jul.2010.

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