Gradient Projection Anti-windup Scheme on Constrained Planar LTI Systems. Justin Teo and Jonathan P. How

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1 1 Gradient Projection Anti-windp Scheme on Constrained Planar LTI Systems Jstin Teo and Jonathan P. How Technical Report ACL1 1 Aerospace Controls Laboratory Department of Aeronatics and Astronatics Massachsetts Institte of Technology March Abstract The gradient projection anti-windp GPAW) scheme was recently proposed as an anti-windp method for nonlinear mlti-inpt-mlti-otpt systems/controllers the soltion of which was recognized as a largely open problem in a recent srvey paper. This report analyzes the properties of the GPAW scheme applied to an inpt constrained first order linear time invariant LTI) system driven by a first order LTI controller where the objective is to reglate the system state abot the origin. We show that the GPAW compensated system is in fact a projected dynamical system PDS) and se reslts in the PDS literatre to assert eistence and niqeness of its soltions. The main reslt is that the GPAW scheme can only maintain/enlarge the eact region of attraction of the ncompensated system. We illstrate the qalitative weaknesses of some reslts in establishing tre advantages of anti-windp methods and propose a new paradigm to address the anti-windp problem where reslts relative to the ncompensated system are soght. Inde Terms gradient projection anti-windp constrained planar LTI systems projected dynamical systems eqilibria region of attraction. I. INTRODUCTION THE gradient projection anti-windp GPAW) scheme was proposed in [1] as an anti-windp method for nonlinear mlti-inpt-mlti-otpt MIMO) systems/controllers. It was recognized in a recent srvey paper [2] that antiwindp compensation for nonlinear systems remains largely an open problem. To this end [3] and relevant references in [2] represent some recent advances. The GPAW scheme ses a continos-time etension of the gradient projection method of nonlinear programming [4] [5] to etend the stop integration heristic otlined in [6] to the case of nonlinear MIMO systems/controllers. Application of the GPAW scheme to some nominal controllers reslts in a hybrid GPAW compensated controller [1] and hence a hybrid closed loop system. J. Teo is a gradate stdent with the Aerospace Controls Laboratory Department of Aeronatics & Astronatics Massachsetts Institte of Technology Cambridge MA 2139 USA csteo@mit.ed). J. How is director of Aerospace Controls Laboratory and Professor in the Department of Aeronatics & Astronatics Massachsetts Institte of Technology Cambridge MA 2139 USA jhow@mit.ed). Here we apply the GPAW scheme to a first order linear time invariant LTI) system stabilized by a first order LTI controller where the objective is to reglate the system state abot the origin. This case is particlarly insightfl becase the closed loop system is a planar dynamical system whose vector field is easily visalized and is highly tractable becase there is a large body of relevant work eg. [7 Chapter 2] [8 Chapter 2] [9 Chapter 2] [1 Chapter 3] [11]. Related literatre on constrained planar systems inclde [12] [18]. After presenting the generalities in Section II we address the eistence and niqeness of soltions to the GPAW compensated system. De to discontinities of the governing vector field of the GPAW compensated system on the satration constraint bondaries classical eistence and niqeness reslts based on Lipschitz continity of vector fields [7] [1] do not apply directly. We show that the GPAW compensated system is in fact a projected dynamical system PDS) [19] [22] in Section III. Observe that PDS is a significant line of independent research that has attracted the attention of economists and mathematicians among others. The link to PDS ths enables cross tilization of ideas and methods as demonstrated in [23]. Using reslts from the PDS literatre eistence and niqeness of soltions to the GPAW compensated system can ths be easily established as shown in Section IV. In Section V eqilibria of the systems are characterized leading to the stdy of the associated region of attraction ROA). It is widely accepted as a rle that the performance of a control system can be enhanced by trading off its robstness [24 Section 9.1]. As sch we consider an antiwindp scheme to be valid only if it can provide performance enhancements withot redcing the system s ROA. The first qestion to be addressed is whether the GPAW scheme satisfy sch a criterion and is shown to be affirmative in Section VI.

2 2 Nmerical reslts frther illminate this property of GPAW compensated systems. In Section VII we illstrate some qalitative weaknesses of some reslts in the anti-windp literatre and propose a new paradigm in addressing the anti-windp problem in which reslts relative to the ncompensated system are soght. This is the case for the main reslt of this report Proposition 4. II. PRELIMINARIES Let the system to be controlled be described by ẋ = a + b sat) 1) where the satration fnction is defined by ma if ma sat) = if min < < ma min if min and R are the plant state and control inpt respectively a b min ma R are constant plant parameters with min ma satisfying min < < ma. Let the nominal controller be ẋ c = c c + d = ẽ c where c R are the controller state and otpt respectively R is the measrement of the plant state and c d ẽ R are controller gains chosen to globally stabilize the nconstrained system ie. when ma = min =. Remark 1: It is important that the otpt eqation of the nominal controller namely = ẽ c depends only on the controller state c and independent of measrement. That is if the otpt eqation is = ẽ c + f then we reqire f =. This property ensres that fll controller state-otpt consistency ie. sat) = can be maintained at almost all times stated more precisely as Fact 1 below) when applying the GPAW scheme. For general nominal controllers this reqirement and its conseqences on the GPAW compensated controller are detailed in [25] together with remedies when the nominal controller does not have the reqired strctre. A simple transformation of 2) yields the eqivalent controller realization 2) = c + d 3) with c := dẽ d := c. Applying the GPAW scheme [1] to the preceding transformed nominal controller 3) yields the GPAW compensated controller see Appendi A) if ma c + d > = if min c + d < c + d otherwise which is similar to the conditionally freeze integrator method [26]. This is epected since the GPAW scheme can be viewed as a generalization of this idea to MIMO nonlinear controllers. Observe that the first order GPAW compensated controller is independent of the GPAW tning parameter Γ introdced in [1] which is tre for all first order controllers. Frthermore inspection of 4) reveals the following. Fact 1 Controller State-Otpt Consistency): If for some T R the control signal of the GPAW compensated controller 4) at time T satisfies min T ) ma then min t) ma holds for all t T. That is the GPAW compensated controller maintains fll controller state-otpt consistency sat) = for all ftre times once it has been achieved for any time instant. In particlar if the controller state is initialized sch that sat)) = ) then satt)) = t) holds for all t. Remark 2: For nonlinear MIMO controllers whose otpt eqation depends only on the controller state the same reslt state-otpt consistency of GPAW compensated controller) holds as shown in [25 Theorem 1]. The nominal constrained closed-loop system Σ n is described by 1) and 3) { ẋ = a + b sat) Σ n : = c + d while the GPAW compensated closed-loop system Σ g is described by 1) and 4) ẋ = a + b sat) if Σ g : ma c + d > = if min c + d < c + d otherwise. Each of these systems can be epressed in the form ż = fz) with f : R 2 R 2. The representing fnctions vector fields) for systems Σ n and Σ g will be denoted by f n and f g respectively. The following will be assmed. Assmption 1: The controller parameters c d satisfy 4) a + d < 5) ad bc > 6)

3 3 and bc. The characteristic eqation of the nconstrained system ie. Σ n with ma = min = can be verified to be s 2 a + d)s + ad bc) = so that Assmption 1 ensres that the origin is a globally eponentially stable eqilibrim point for the nominal nconstrained system. The condition bc ensres that c d can be chosen to satisfy 5) and 6) and that Σ n is a feedback system. We will need the following sets K = { ) R 2 min < < ma } K + = { ) R 2 > ma } K = { ) R 2 < min } K + = { ) R 2 = ma } K = { ) R 2 = min } K +div = { ) R 2 > ma c + d = } K +in = { ) R 2 > ma c + d < } K +ot = { ) R 2 > ma c + d > } K div = { ) R 2 < min c + d = } K in = { ) R 2 < min c + d > } K ot = { ) R 2 < min c + d < } K +in = { ) R 2 = ma c + d ma < } K +ot = { ) R 2 = ma c + d ma > } K in = { ) R 2 = min c + d min > } K ot = { ) R 2 = min c + d min < } and the points K = K K + K z + = d c ma ma ) z = d c min min ). These sets and associated vector fields are illstrated in Fig. 1 for an open loop nstable plant and in Fig. 2 for an open loop stable plant. Observe that K + = K +in K +div K +ot and K + = K +in K +ot {z + } with analogos conterparts for K and K. Observe frther that on K +in and K in vector fields of systems Σ n and Σ g f n and f g respectively) point into K. On K +ot f n points into K + and f g points into K +. On K ot f n points into K and f g points into K. By inspection of the vector fields f n and f g from their definitions we have the following. Fact 2: The vector fields f n and f g coincide in K K +in K in K +div K div K +in K in {z + z }. That is they coincide in R 2 \ K +ot K ot K +ot K ot ). Fact 3: Any soltion of systems Σ n or Σ g can pass from K + to K if and only if it intersects the line segment K +in and analogosly with respect to K and K in. Fact 4: Any soltion of system Σ n can pass from K to K + if and only if it intersects the line segment K +ot and analogosly with respect to K and K ot. III. GPAW COMPENSATED CLOSED LOOP SYSTEM AS A PROJECTED DYNAMICAL SYSTEM Two of the most fndamental properties reqired for a meaningfl stdy of dynamic systems is the eistence and niqeness of their soltions. As evident from the definition of the GPAW compensated controller 4) the vector field of the GPAW compensated system f g is in general discontinos on the satration constraint bondaries K +ot K + ) and K ot K ). Classical reslts on the eistence and niqeness of soltions [7] [1] rely on Lipschitz continity of the governing vector fields and hence do not apply to GPAW compensated systems. While reslts in [27] can be sed to assert sch properties we will se reslts from the projected dynamical system PDS) [19] [22] literatre to assert the eistence and niqeness of soltions to GPAW compensated systems. First we show here that the GPAW compensated system Σ g is in fact a PDS. Observe that the set K is a closed conve set in fact a closed conve polyhedron). The interior and bondary of K are K and K + K respectively. Let P : R 2 K be the projection operator [19] defined for all y R 2 by P y) = arg min y z z K with as the Eclidean norm. It can be seen that for any ) R 2 P )) = sat)). Net for any y K v R 2 define the projection of vector v at y by [19] [2] P y + δv) y πy v) = lim. δ δ Note that the limit is one-sided in the above definition [2]. With f n being the vector field of Σ n written eplicitly as [ ] a + b f n ) = ) K c + d

4 4 3 3 f f n f 2 K +ot K +in f n 2 K +ot K +in f g f n f g K +div 1 K +div 1 K div K K div K +ot K K +ot K +in K +in K ot 1 K ot 1 K in K in 2 K in 2 K in K ot K ot Fig. 1: Closed loop vector fields f n f g ) of systems Σ n Σ g and the nconstrained system Σ f ) associated with an open loop nstable system plant and controller parameters: a = 1 b = 1 c = 3 d = 2 min = ma = 1). Vector fields of systems Σ n Σ g and Σ f n f g f ) are shown on the left while the vector field differences f n f f g f n ) are shown on the right. 3 3 f f n f 2 f n K +ot K +in 2 f g f n K +ot K +in f g K +div 1 K +div 1 K div K div K +ot K +ot K K +in K K +in K ot 1 K ot 1 K in K in 2 K ot 2 K ot K in K in Fig. 2: Closed loop vector fields f n f g ) of systems Σ n Σ g and the nconstrained system Σ f ) associated with an open loop stable system plant and controller parameters: a = 1 b = 1 c = 1 d =.5 min = ma = 1). Vector fields of systems Σ n Σ g and Σ f n f g f ) are shown on the left while the vector field differences f n f f g f n ) are shown on the right. we have the following the corollary of which is the desired reslt. Claim 1: For all ) K the vector field f g of the GPAW compensated closed loop system Σ g satisfy f g ) = π ) f n )). Proof: If ) K the reslt follows from [2 Lemma 2.1i)] and Fact 2. Net consider a bondary point ) K +in {z + }. On this segment we have = ma and c+d ma from definition of the set K +in {z + }. Since sat ma +δβ) = ma +δβ for β and a sfficiently small δ > we have [ ] + δa + b) P ) + δf n )) = sat + δc + d)) [ ] + δa + b) = + δc + d) so that P ) + δf n )) ) π ) f n )) = lim δ δ [ ] a + b = = f n ) = f g ) c + d for all ) K +in {z + } where the final eqality follows from Fact 2.

5 5 Finally consider a bondary point ) K +ot. On this segment we have = ma and c + d ma > from the definition of K +ot. Since sat ma + δβ) = ma for β > and a sfficiently small δ > we have [ ] + δa + b) P ) + δf n )) = sat + δc + d)) [ ] + δa + b) = so that P ) + δf n )) ) π ) f n )) = lim δ δ [ ] a + b = = f g ) for all ) K +ot. The above established the claim for all points on K \ K. The verification on the bondary K is similar to that for K +. Corollary 1: The GPAW compensated system Σ g is a projected dynamical system [19] governed by ż = f g z) = πz f n z)) where z = ). Corollary 1 will be sed in the net section to assert the eistence and niqeness of soltions to system Σ g. See [19] [21] for a detailed development of PDS and [23] for known relations to other system descriptions. IV. EXISTENCE AND UNIQUENESS OF SOLUTIONS Here we assert the eistence and niqeness of soltions to both the nominal constrained system and GPAW compensated system. Claim 2: The nominal system Σ n has a niqe soltion for all initial conditions t ) t )) R 2 and all t t. Proof: For all z := ) R 2 the vector field f n can be written as [ ] [ ] b a f n z) = Az + sat) where A =. c d It can be verified [8 Eample 3.2 pp ] that the satration fnction is globally Lipschitz with nity Lipschitz constant ie. satα) satβ) α β. Then global Lipschitz continity of f n for all t t follows from f n z) f n z) = Az z) + [b ] T sat) satũ)) Az z) + [b ] T sat) satũ)) = Az z) + b sat) satũ) A z z + b ũ A + b ) z z 7) for all z := ) R 2 z := ũ) R 2. By [8 Theorem 3.2 pp. 93] Σ n has a niqe soltion defined for all t t for all t ) t )) R 2. We will need the following assmption sed to assert the eistence and niqeness of soltions to PDS. Assmption 2 [19 Assmption 1]): There eists B < sch that the vector field f n : R k R k satisfies the following conditions f n z) B1 + z ) z K 8) f n z) f n z) z z B z z 2 z z K 9) where y denotes the dot prodct of and y. The following reslt is stated withot proof in the remark following [19 Assmption 1]. Claim 3: If f n is Lipschitz in K R k then Assmption 2 holds. Proof: Since f n is Lipschitz in K there eists an L < sch that f n z) f n z) L z z for all z z K. To show that 8) holds observe that f n z) = f n z) f n z) + f n z) f n z) f n z) + f n z) L z z + f n z) L z + L z + f n z) for all z z K. Fi any z K and define α := L z + f n z) < ) and B := ma{l α} < ) so that the preceding ineqality becomes f n z) L z + α B1 + z ) z K which proves 8). By the Cachy-Schwarz ineqality we have f n z) f n z) z z f n z) f n z) z z L z z 2 B z z 2 for all z z K which proves 9). Remark 3: Both Assmption 2 and Claim 3 are stated for general vector fields f n and regions K in R k bt will be specialized to vector fields and regions in R 2 in the seqel. The following is the main reslt of this section. Proposition 1: The GPAW compensated system Σ g has a niqe soltion for all initial conditions t ) t )) R 2 and all t t.

6 6 Proof: Since f n : R 2 R 2 is globally Lipschitz see 7)) it is Lipschitz in K R 2 so that Assmption 2 holds de to Claim 3. Since Σ g is a PDS see Corollary 1 and [19 Eqation 7)]) it follows from Assmption 2 and [19 Theorem 2] that Σ g has a niqe soltion defined for all t t whenever the initial condition satisfies t ) t )) K also recall Fact 1). To assert the eistence and niqeness of soltions for all initial conditions t ) t )) R 2 it is sfficient to establish this otside K and if the soltion enters K there will be a niqe contination in K for all ftre times from this reslt. Consider the region K + = K +in K +ot K +div. The proof for the region K is similar. For any z 1 z 2 K + there are three possible cases. Firstly in the region ˆK +ot := K +ot K +div we get from the definition of f g and ˆK +ot that f g z) = f g ) = a + b ma ). Clearly for any z 1 z 2 ˆK +ot we have f g z 1 ) f g z 2 ) L ot z 1 z 2 where L ot = a <. Secondly from Fact 2 f g and f n coincide in ˆK +in := K +in K +div so that f g is also Lipschitz in ˆK +in. For any z 1 z 2 ˆK +in we have f g z 1 ) f g z 2 ) L in z 1 z 2 where L in = A + b < see 7)). The last case corresponds to z 1 and z 2 being in different regions ˆK+in and ˆK +ot. Withot loss of generality let z 1 ˆK +in and z 2 ˆK +ot. The straight line in R 2 connecting z 1 and z 2 then contains a point z K +div with the property that z ˆK +in ˆK +ot z 1 z z 1 z 2 and z 2 z z 1 z 2. Then we have f g z 1 ) f g z 2 ) = f g z 1 ) f g z) + f g z) f g z 2 ) f g z 1 ) f g z) + f g z 2 ) f g z) L in z 1 z + L ot z 2 z L in + L ot ) z 1 z 2 which together with the first two cases shows that f g is Lipschitz in K +. By [9 Theorem 3.1 pp ] Σ g has a niqe soltion contained in K + whenever t ) t )) K +. If the soltion stays in K + for all t the claim holds. Otherwise by [9 Theorem 2.1 pp. 17] the soltion can be contined to the bondary of K + K + K. In this case the first part of the proof shows that there is a niqe contination in K for all t. Remark 4: Care is de when interpreting the eistence and niqeness reslts of Proposition 1. Let φ n t z ) be the niqe soltion of system Σ n starting from z R 2 at time t =. For system Σ n eistence and niqeness of soltion implies that no two different paths intersect [9 pp. 38] and φ n t φ n t z )) = z t R z R 2. That is proceeding forwards and then backwards in time by the same amont the soltion always reaches its starting point. This is not tre for system Σ g whenever the soltion intersects K +ot or K ot. Inspection of the vector field f g reveals that in this case all forward soltions either stay in K +ot or K ot for all ftre times or they eventally reach the points z + or z. Frthermore traversing backwards in time from any point of K +ot or K ot the soltion stays on these segments indefinitely. That is K +ot and K ot are negative invariant sets [9 pp. 47] for system Σ g. If a forward soltion of Σ g intersects K +ot or K ot starting from some interior point z K then traversing backwards in time the soltion will never reach z. Eistence and niqeness of soltions of system Σ g means that if two distinct trajectories φ g t z 1 ) φ g t z 2 ) intersect at some time then they will be identical for all ftre times ie. if φ g T 1 z 1 ) = φ g T 2 z 2 ) for some T 1 T 2 R then φ g t + T 1 z 1 ) = φ g t + T 2 z 2 ) for all t. Specifically they can never diverge into two distinct trajectories. V. EQUILIBRIUM POINTS In this section we characterize all eqilibria of systems Σ n and Σ g. Of primary importance is the origin stated below. Claim 4: The origin z eq := ) is the only eqilibrim point of systems Σ n and Σ g in K and it mst be either a stable node or stable focs. Proof: In K the vector fields f n and f g coincide see Fact 2) and can be written as f n z) = f g z) = Ãz where à = [ a b c d ]. It is clear that the origin is an eqilibrim point de to f n z eq ) = f g z eq ) = Ãz eq = R 2. From 6) the matri à is invertible and hence z eq mst be the only eqilibrim point in K. De to Assmption 1 z eq mst be either a stable node or a stable focs [7 Section pp ]. Additional eqilibria of the nominal system Σ n are characterized below. Claim 5: Apart from the origin z eq the nominal system Σ n admits two additional isolated eqilibrim points defined by z eq+ = b a ma bc ad ) ma zeq = b a min bc ad min) only when i) the open loop system is nstable a > ) or

7 7 ii) the open loop system is strictly stable a < ) and controller parameter satisfies d a). Moreover if z eq+ and z eq are eqilibria of Σ n they are saddle points and lie strictly in K + and K respectively ie. z eq+ z eq K + K ). Remark 5: When z eq+ and z eq are eqilibria of Σ n it can be verified that they mst lie in K +div respectively. and K div Proof: All eqilibria of Σ n are determined from the condition f n z) =. It can be verified from the conditions a + b sat) = from f n z) = ) and bc of Assmption 1 that whenever the open loop system is marginally stable ie. a = there can be no eqilibria apart from z eq. Similarly whenever d = the conditions c + d = from f n z) = ) bc and a + b sat) = implies that there can be no additional eqilibria apart from z eq. Together this means ad and z eq+ and z eq are well-defined. A simple comptation shows that apart from z eq the additional eqilibria are z eq+ and z eq provided z eq+ K + K + and z eq K K. These hold if and only if ad and bc ad 1. From 6) bc ad 1 holds if and only if ad < which reslts in the strict condition bc ad > 1. Therefore if z eq+ and z eq are indeed eqilibria of Σ n they mst lie in K + and K respectively ie. they cannot lie on K + or K. If the open loop system is nstable ie. a > then from 5) we mst have d < a < which implies ad < and Σ n indeed has z eq+ and z eq as eqilibria. If the open loop system is strictly stable ie. a < then ad < and 5) hold if and only if d a). It remains to show that z eq+ and z eq mst be saddle points [7 Section pp ] whenever they are eqilibria of Σ n. The Jacobian of f n at the isolated eqilibrim points z eq+ K + and z eq K are identical and given by [ ] f n z z eq+) = f n z z a eq ) = A =. c d Since its eigenvales are a d and ad < the eqilibria z eq+ and z eq mst be saddle points. The following characterizes additional eqilibria of the GPAW compensated system Σ g. Claim 6: Apart from the origin z eq the GPAW compensated system Σ g admits additional eqilibria only when i) the open loop system is nstable a > ). Additional eqilibria are all points in the two connected sets defined by Z eq+ = { ) R 2 = b a ma ma bc ad ma} K+ K + ) Z eq = { ) R 2 = b a min bc ad min min } K K ). ii) the open loop system is strictly stable a < ) and controller parameter satisfies d a). Additional eqilibria are all points in the two connected sets defined by Z eq+ = { ) R 2 = b a ma bc ad ma} K+ Z eq = { ) R 2 = b a min bc ad min} K. Remark 6: Observe that whenever Σ n has additional eqilibria other than z eq so does Σ g. The converse statement is also easily verified. Moreover observe that z eq+ and z eq belongs to and lies on the endpoints of the sets Z eq+ and Z eq respectively. Proof: All eqilibria of Σ g are determined from the condition f g z) =. It can be verified from the conditions a + b sat) = from f g z) = ) and bc of Assmption 1 that whenever the open loop system is marginally stable ie. a = there can be no eqilibria apart from z eq. Comptation shows that apart from z eq all points in the sets Z eq+ = { ) R 2 = b a ma Z eq = { ) R 2 = b a min ma d bc a ma} min d bc a min} are also eqilibria of Σ g provided these sets are non-empty. Considering the conditions ma and d bc a ma and their analogos conterparts) these sets are non-empty if and only if a) d > b) d = and bc a or c) d < and bc ad 1. Consider case a). From 5) this case d > ) is possible only when a < ie. the open loop system is strictly stable. To satisfy 5) and d > we mst restrict d a). Hence ad < and 6) implies bc ad > 1. The above sets Z eq+ and Z eq then simplifies to those stated in the claim for case ii). Now consider case b). With d = conditions 5) and 6) redces to a < and bc < respectively which implies bc a >. Therefore Assmption 1 ensres that this case in particlar bc a ) cannot occr. Finally consider case c). From 6) this case in particlar bc ad 1) is possible only when ad < which in trn implies

8 8 bc ad > 1 holds with strict ineqality. The condition ad < for this case in particlar d < ) implies a > ie. the open loop system is nstable. It is easily verified that the above sets Z eq+ and Z eq simplifies to those stated in the claim for case i). Remark 7: Observe that the presence of additional eqilibria precldes the possibility of the origin being a globally asymptotically stable eqilibrim point for both systems Σ n and Σ g. However note that a d and b c) are given fied parameters in the anti-windp contet. In smmary z eq is an isolated stable eqilibrim point of systems Σ n and Σ g for all a b c d R satisfying Assmption 1 and it is the only eqilibrim point in K. When the open loop system is marginally stable or strictly stable with d there cannot be additional eqilibria. When the open loop system is nstable or strictly stable with d a) Σ n has two more isolated eqilibrim points z eq+ and z eq which are saddle points and Σ g has a continm of eqilibria Z eq+ and Z eq. VI. REGION OF ATTRACTION The prpose of anti-windp schemes is to provide performance improvements only in the presence of control satration. It is widely accepted as a rle that the performance of a control system can be enhanced by trading off its robstness [24 Section 9.1]. To distingish anti-windp schemes from conventional control methods we consider an antiwindp scheme to be valid only if it can provide performance enhancements withot redcing the system s region of attraction ROA). We show in this section that GPAW compensation can only maintain/enlarge the ROA of the nominal system Σ n. In other words the ROA of system Σ n is contained within the ROA of Σ g. While there may eist mltiple eqilibria for systems Σ n and Σ g we are primarily interested in the ROA of the eqilibrim point at the origin z eq. A distingishing featre is that the reslts herein refers to the eact ROA in contrast to ROA estimates that is fond in a significant portion of the literatre on anti-windp compensation. For clarity of presentation we present the reslt in two parts where the ROA containment is shown for the nsatrated region K and satrated region R 2 \ K separately. Some nmerical eamples will illstrate typical ROAs and show that the said ROA containment can hold strictly for some systems. In the seqel we will state and prove reslts only for one side of the state space namely with respect to K + K +. The analogos reslts with respect to K K can be readily etended and will not be epressly stated. Let φ n t z ) and φ g t z ) be the niqe soltions of systems Σ n and Σ g respectively both starting at initial state z at time t =. The ROA of the origin z eq for systems Σ n and Σ g are then defined by [8 pp. 314] R n = {z R 2 φ n t z) z eq as t } R g = {z R 2 φ g t z) z eq as t } respectively. We recall the notion of transverse sections and ω limit sets. Definition 1 [7 pp. 46]): A transverse section σ to a vector field f : R 2 R 2 is a continos connected arc in R 2 sch that the dot prodct of the nit normal to σ and f is not zero and does not change sign on σ. In other words the vector field has no eqilibrim points on σ and is never tangent to σ [7 pp. 46]. It is clear from the definition of K +in and K in that both of these line segments are in fact transverse sections of f n and f g. Moreover K +ot and K ot are also transverse sections of f n. Definition 2 [7 Definition 2.11 pp. 44]): A point z R 2 is said to be an ω limit point of a trajectory φt z ) if there eists a seqence of times t n n { } sch that t n as n for which lim n φt n z ) = z. The set of all ω limit points of a trajectory is called the ω limit set of the trajectory. For convenience let the straight line connecting two points α β R 2 be denoted by lα β) = lβ α)) and defined by lα β) = {z R 2 z = θα + 1 θ)β θ 1)}. Observe that lα β) does not contain the endpoints α β ecept for the degenerate case of identical endpoints in which case lα α) = {α}. Net the ROA containment in the nsatrated and satrated regions are shown separately which combines to yield the desired reslt. A. ROA Containment in Unsatrated Region What follows is a series of intermediate claims to arrive at the main reslt of this sbsection Proposition 2. Let the straight lines connecting the origin to the points z + and z be σ + = lz eq z + ) {z + } σ = lz eq z ) {z } 1) respectively. Consider a point z K +in with the property that z R n and φ n t z ) K + for all t. In other

9 9 ηz ) z Dz ) z z eq t int K K z + ηz ) z Dz ) Fig. 3: Closed path ηz ) encloses region Dz ) K K. A case where the soltion enters K and also intersects σ + is shown on the left while a case where the soltion never enters K and never intersects σ + is shown on the right. K z z eq words z is in the ROA of system Σ n and its soltion stays in K K for all t. As a conseqence of Fact 4 φ n t z ) can never intersect K +ot for all t. Let t int = inf{t ) φ n t z ) σ + }. That is t int is the first time instant that the soltion starting from z at t = intersects σ + or if it does not intersect σ +. If t int < the path η int z ) = {z R 2 z = φ n t z ) t [ t int ]} is well defined. Otherwise the path lφ n t int z ) z + ) {z + } lz z + ) η z ) = {z R 2 z = φ n t z ) t } K {z eq } σ + lz z + ) is well defined. Now define the path ηz ) R 2 by η int z ) if t int < ηz ) = η z ) otherwise which can be verified to be closed and connected. Let the open bonded region enclosed by ηz ) be Dz ) and its closre be Dz ). The region Dz ) is illstrated in Fig. 3. The following reslt states that Dz ) is a positive invariant set [9 pp. 47] and it mst contain the origin z eq. Claim 7: If there eists a point z K +in sch that z R n and φ n t z ) K K for all t then Dz ) K K is a positive invariant set for system Σ n and it mst contain z eq ie. z eq Dz ). Remark 8: The claim states specifically that nder the assmptions it is not possible for φ n t z ) to intersect σ + withot having ηz ) enclose z eq a case not illstrated in Fig. 3. z + Proof: Let lφ n t int z ) z + ) {z + } if t int < σ + = σ + otherwise. We first show that σ + is a transverse section to f n and that f n always points into Dz ) on σ +. Let α { 1 +1} be chosen sch that α T z + z z + > where T z + = ma d c ma) z+ is orthogonal to z +. Then α T z is the nit normal of σ + + that points into Dz ). Hence σ + is a transverse section to f n and f n points into Dz ) on σ + if and only if α T z + f n z) > holds with strict ineqality for all z σ +. Since z K +in we have from the definition of K +in that z = ma ) for some that satisfies c +d ma <. Then z z + = + d c ma ). De to c +d ma < the condition α T z + z z + = α ma d c ma) + d c ma ) = α c mac + d ma ) > can hold only if α = sgnc). From the definition of σ + any z σ + has the form z = θ d c ma θ ma ) for some θ 1] so that f n z) = b ad c )θ ma ) on σ +. Using the definition of f n on σ + we have α T z + f n z) = α ma d c ma) b ad c )θ ma ) = sgnc) ) b ad c θ 2 ma = ad bc c θ 2 ma. Since θ 1] for any z σ + we have from 6) that α T z + f n z) > which shows that σ + is a transverse section to f n and that f n always points into Dz ) on σ +. It is clear that lz z + ) K +in is also a transverse section to f n and that f n always points into Dz ) on lz z + ). Both of these reslts show that any soltion originating in Dz ) cannot eit Dz ) throgh the line segments σ + or lz z + ). Frthermore since the soltion is niqe and no two different paths can intersect [9 pp. 38] the region Dz ) enclosed by ηz ) mst be a positive invariant set [9 pp. 47] for system Σ n. The assmption φ n t z ) K K for all t implies ηz ) K K. Hence we have Dz ) K K. Finally from the assmption z R n we have φ n t z ) z eq as t. Since Dz ) is a positive invariant set and z Dz ) we have φ n t z ) Dz ) for all t. The conclsion z eq Dz ) then follows from the fact that Dz ) is closed and hence contains all its limit points. Claim 8: If there eists a point z K +in sch that z R n and φ n t z ) K for all t then all points in Dz ) K also lie in the ROA of system Σ n ie. Dz ) R n.

10 1 Remark 9: Specifically the conclsion implies z + Dz ) R n. Proof: Since K K K ) the hypotheses of Claim 7 are satisfied. Claim 7 then shows that Dz ) is a positive invariant set. The condition φ n t z ) K for all t implies Dz ) K. It was shown in [28 Section 6.2 pp ] [9 Theorem 1.3 pp. 55] that for planar dynamic systems with only a contable nmber of eqilibria and with niqe soltions the ω limit set of any trajectory contained in any bonded region can only be of three types: eqilibrim points closed orbits or heteroclinic/homoclinic orbits [29 pp. 45] which are nions of saddle points and the trajectories connecting them. It follows from Claims 4 and 5 that the origin z eq is the only eqilibrim point of Σ n in K which mst be a stable node or stable focs. Hence the ω limit set of any trajectory contained in Dz ) K cannot be heteroclinic/homoclinic orbits. By Bendison s Criterion [8 Lemma 2.2 pp. 67] and 5) region Dz ) contains no closed orbits. As a reslt the ω limit sets mst consist of eqilibrim points only and it mst be z eq since it is the only eqilibrim point in K. The conclsion follows by observing that Dz ) is a positive invariant set and any trajectory starting in it mst converge to the ω limit set {z eq } de to [8 Lemma 4.1 pp. 127]. The points z + K + and z K defined by z + := b a ma ma ) z := b a min min ) and the line segments ξ + := l z + z + ) K + ξ := l z z ) K will be needed in the sbseqent development. Claim 9: If the open loop system is stable or marginally stable ie. a then f g points towards z + on K +ot ie. f g z) = αz + z) for some α = αz) > and for all z K +ot. If the open loop system is nstable ie. a > then f g points towards z + on ξ + f g z + ) = and f g points away from z + on K +ot \ ξ + { z + }). Proof: From the definition of K +ot any z K +ot has the form z = ma ) for some satisfying c + d ma >. For any z K +ot we have f g z) = a + b ma ) and z + z = + d c ma) ) where z = ma ) and c + d ma >. The condition f g z) = αz + z) is clearly eqivalent to a + b ma = α c c + d ma ). Since c + d ma > it follows that f g z) = αz + z) can hold with α > if and only if ca + b ma ) <. 11) If a = 6) redces to bc < and 11) follows. If a < we have from 6) and c + d ma > that ca + b ma ) < ac + ad ma = ac + d ma ) < and 11) holds. This proves the first statement of the claim. Finally consider the case a >. Then 11) is eqivalent to c < bc a ma and c + d ma > is eqivalent to c > d ma. Hence f g z) points towards z + on some z = ma ) K +ot if and only if satisfies d ma < c < bc a ma. 12) It can be verified that d ma < bc a ma de to 6). The above condition 12) can be decomposed and rewritten as d c ma < < b a ma if c > b a ma < < d c ma otherwise so that 12) is eqivalent to = θ d c 1 θ) a) b ma for some θ 1). In other words f g z) points towards z + if and only if z ξ +. The fact that f g z + ) = can be verified by sbstittion and the last statement of the claim follows. Remark 1: It is clear that when a > z + Z eq+ where Z eq+ is the set of eqilibria defined in Claim 6. Claim 1: If the open loop system is nstable ie. a > and z K +ot R n then z ξ +. Proof: We will show that if a > and z K +ot \ ξ + then z R n see Appendi B). If z R n we have φ n t z ) z eq as t. Since z eq K it is sfficient to show that if a > and z K +ot \ ξ + then φ n t z ) K for all t. Let z = ma ) K +ot so that c + d ma >. At the point z we have f n z ) = a + b ma c +d ma ). It follows that ) = c +d ma > at time t = and t) mst increase and hence satt)) = ma ) at least for some non-zero interval. The initial vale problem to be considered is ẋ = a + b ma ) = = c + d ) = ma whose soltion will coincide with the soltion of Σ n ie. φ n t z ) as long as it remains otside K. We will show that t) ma for all t so that φ n t z ) K for all t. If c > we have d c ma < b a ma from 6). If z = ma ) K +ot \ ξ + then satisfies b a ma

11 11 and hence ẋ) = a + b ma. Moreover becase a > t) is non-decreasing at least ntil t) < ma. Hence t) and ct) c dring this interval. If c < then d c ma > b a ma from 6). If z = ma ) K +ot \ ξ + then satisfies b a ma and hence ẋ) = a + b ma. Moreover becase a > t) is non-increasing at least ntil t) < ma. Hence t) and ct) c dring this interval. In either case we have = c + d c + d ) = ma as the differential ineqality governing t). To apply the Comparison Lemma [8 Lemma 3.4 pp ] define v = so that v dv c v) = ma. Applying the Comparison Lemma [8 Lemma 3.4 pp ] to the above differential ineqality yields vt) ma e dt c d e dt 1 ) and hence t) = vt) ma e dt + c d e dt 1 ) t. Since a > it follows from 5) that d < a < and hence e dt 1 ) for all t. Becase c + d ma > we have c d < ma and c d e dt 1 ) ma e dt 1 ). With these the above ineqality becomes for all t. t) ma e dt + c d e dt 1 ) ma e dt ma e dt 1 ) = ma The above reslts are smmarized below. Claim 11: If there eists a z K +ot R n then for every z lz z + ) {z } there eists a T z) ) sch that the soltion of system Σ g satisfies φ g T z) z) = z + and φ g t z) K +ot for all t [ T z)). Proof: If a the reslt is a direct conseqence of Claim 9 and the fact that K +ot {z + } contains no eqilibrim points of Σ g. If a > then the reslt follows from Claim 1 and Claim 9 and the fact that ξ + {z + } contains no eqilibrim points of Σ g. Remark 11: Observe that nder the assmptions the soltion of the GPAW compensated system φ g t z ) slides along the line segment K +ot or ξ + as appropriate) to reach z +. Note that Fact 1 corroborates this observation. Net we will show that a soltion of Σ n converging to the origin can intersect K +ot or K ot only in a specific way namely that sbseqent intersection points if any mst steadily approach z + or z. Claim 12: If z K +ot R n and there eists a T ) sch that φ n T z ) K +ot then φ n T z ) lz z + ). Proof: We will show that if φ n T z ) lz z + ) then z R n. Let z 1 := φ n T z ) and assme z 1 K +ot \ lz z + ). If z 1 = z then the soltion forms a closed orbit and de to niqeness of soltions φ n t z ) will stay on the orbit for all t and never approach z eq. Hence z R n. Otherwise we have z 1 K +ot \ lz z + ) {z }). Let the closed bonded region enclosed by the closed path be ηz ) = {z R 2 z = φ n t z ) t [ T ]} lz z 1 ) Dz ). Note that φ n t z ) mst necessarily intersect K +in and enter K before it can intersect K +ot at time T de to Fact 4. It can be seen that lz z 1 ) K +ot is a transverse section to f n with f n pointing ot of Dz ) on lz z 1 ). Hence Dz ) is a negative invariant set of system Σ n. If z eq Dz ) then there is no way for φ n t z ) to reach z eq which will prove the claim. We will show that z eq mst be contained in Dz ) sing inde theory [7 Section 2.4 pp ] [28 Section 5.8 pp. 3 35]. Noting that the inde [7 Definition 2.16 pp. 49] of a closed orbit is +1 [28 pp. 31] it can be shown that the inde of the closed path ηz ) formed by a section of a trajectory and a transverse section is also +1 [28 pp ]. The indices of a node focs and saddle are and 1 respectively [28 pp. 31]. Since the inde of ηz ) is the sm of all indices of eqilibria enclosed by ηz ) [28 pp. 31] and system Σ n has only one node or focs at the origin with possibly two additional saddle points the only way for ηz ) to have an inde of +1 is for it to enclose the origin z eq alone. That is z eq Dz ). Remark 12: The above proof is most evident by visalizing the vector field f n on the path ηz ). The following is the main reslt of this sbsection. The proof amonts to sing the soltion of Σ n soltion of Σ g. to bond the Proposition 2: The part of the ROA of the origin of system Σ n contained in K is itself contained within the ROA of the origin of system Σ g ie. R n K) R g. Remark 13: The distinction between the soltions of systems Σ n and Σ g namely φ n t z) and φ g t z) and their ROAs R n and R g shold be kept clear when eamining the proof below. Proof: The following argment will be sed repeatedly in

12 12 the present proof. If for some z K we have φ n t z) K for all t then Fact 4 implies that φ n t z) cannot intersect K +ot or K ot ie. φ n t z) K \ K +ot K ot ) for all t. Fact 2 shows that f n and f g coincide in K \ K +ot K ot ) which implies φ g t z) = φ n t z) for all t. If in addition we have lim t φ n t z) = z eq then lim t φ g t z) = lim t φ n t z) = z eq. In smmary if φ n t z) K for all t and z R n then z R g. For ease of reference we call this the coincidence argment. We need to show that if z R n K then z R g. Let z R n K so that φ n z ) = z K and φ n t z ) z eq as t. Consider the case where φ n t z ) stays in K for all t. It follows from the coincidence argment that z R g. Now we let the soltion φ n t z ) enter K + and consider all possible continations. De to Fact 4 φ n t z ) mst intersect K +ot at least once. If φ n t z ) intersects mltiple times it can only intersect it for finitely K +ot many times. Otherwise there is an infinite seqence of times t m m { } sch that t m as m for which φ n t m z ) K +ot. Since z R n it follows that φ n t m z ) K +ot R n for every m. As a conseqence of Claim 12 we have lim m φ n t m z ) = z + which shows that z + is an ω limit point of φ n t z ). Bt this is impossible becase lim t φ n t z ) = z eq z +. Similarly if φ n t z ) intersects K ot mltiple times it can only intersect it for finitely many times. Hence let T 1 and T 2 be the first and last times for which φ n t z ) intersects K +ot and let T 3 be the only) time after T 2 that φ n t z ) intersects K +in. Then we have T 1 T 2 < T 3 < and φ n t z ) K + for all t T 2 T 3 ) φ n T 1 z ) φ n T 2 z ) K +ot and φ n T 3 z ) K +in with behavior after T 3 to be specified. Let z 1 = φ n T 1 z ) K +ot z 2 = φ n T 2 z ) K +ot and z 3 = φ n T 3 z ) K +in. Since z R n we have z 1 z 2 K +ot R n and z 3 K +in R n. It is clear that φ g t z ) = φ n t z ) for all t [ T 1 ]. By Claim 11 there eist a T 1 < sch that φ g T 1 + T 1 z ) = φ g T 1 φ g T 1 z )) = φ g T 1 φ n T 1 z )) = φ g T 1 z 1 ) = z +. Becase φ n t z ) cannot intersect K +ot for all t > T 2 the only possible continations from time T 3 > T 2 ) onwards are i) φ n t z ) stays in K for all t T 3 or ii) φ n t z ) enters K at some finite time. Consider case i) which implies Dz 3 ) K. Claim 8 yields z + Dz 3 ) R n and Claim 7 shows that Dz3 ) is a positive invariant set for system Σ n. Then we have φ n t z + ) Dz 3 ) K for all t. It follows from the coincidence argment that z + R g. Becase φ g t z + ) = φ g t φ g T 1 + T 1 z )) for all t we have z R g as desired. Now consider case ii). De to Fact 4 φ n t z ) mst intersect K ot at least once. From the above discssion φ n t z ) can intersect K ot only finitely many times. Let T 4 be the first time after T 3 ) and T 5 be the last time for which φ n t z ) intersects K ot and let T 6 be the only) time after T 5 that φ n t z ) intersects K in. Then T 3 < T 4 T 5 < T 6 < and φ n t z ) K for all t T 5 T 6 ) φ n T 4 z ) φ n T 5 z ) K ot and φ n T 6 z ) K in. Let z 4 = φ n T 4 z ) K ot z 5 = φ n T 5 z ) K ot and z 6 = φ n T 6 z ) K in. Since z R n we have z 4 z 5 K ot R n and z 6 K in R n. Now the only possible contination after T 6 is for φ n t z ) K for all t T 6. Recall the definition of ηz) and Dz) for some z K +in R n as illstrated in Fig. 3. It is clear that z + Dz 3 ). Claim 7 shows that Dz 3 ) with a portion in K ) is a positive invariant set for system Σ n so that φ n t z + ) Dz 3 ) for all t. Recall also that φ g T 1 + T 1 z ) = z + and we want to show that z + R g. There are two possible ways for the soltion φ n t z + ) to contine. Either φ n t z + ) stays in Dz 3 ) K for all t or it enters Dz 3 ) K at some finite time. If φ n t z + ) Dz 3 ) K for all t then as in the proof of Claim 8 Bendison s Criterion [8 Lemma 2.2 pp. 67] and the absence of saddle points in Dz 3 ) K means that {z eq } is the ω limit set of φ n t z + ) and hence z + R n. By the coincidence argment we have z + R g. It follows from φ g t z + ) = φ g t φ g T 1 + T 1 z )) for all t that z R g. Finally consider when φ n t z + ) enters Dz 3 ) K at some finite time. By Fact 4 φ n t z + ) mst intersect K ot at least once. Let T 2 < be sch that φ n T 2 z + ) K ot and φ n t z + ) K for all t T 2 ) and let z 2 = φ n T 2 z + ) K ot. Becase the bondary of Dz 3 ) intersects K ot at z 4 and z 2 Dz 3 ) K ot we have that z 2 lz 4 z ). Since z 4 K ot R n we have by the analogos conterpart to) Claim 11 that there eists a T 3 < sch that φ g T 3 z 2 ) = z. Since z 6 K in R n it follows from the analogos conterparts to) Claims 8 and 7 that z Dz 6 ) R n Dz6 ) is a positive invariant set and φ n t z ) Dz 6 ) K for all t. The coincidence argment then yields z R g. Since φ n t z + ) K {z + } for all t [ T 2 ) Fact 2 implies that φ g t z + ) = φ n t z + ) for all t [ T 2 ]. We can trace back the path to z by observing that φ g t z ) = φ g t φ g T 3 z 2 )) = φ g t +

13 13 T 3 z 2 ) = φ g t + T 3 φ n T 2 z + )) = φ g t + T 3 φ g T 2 z + )) = φ g t + T 3 + T 2 z + ) = φ g t + T 3 + T 2 φ g T 1 + T 1 z )) for all t. Since z R g we have z R g as desired. In similar manner it can be shown that if z R n K and the soltion φ n t z ) enters K first then z R g. Observe that the partial reslt stated in Proposition 2 is practically meaningfl becase the controller state can sally be initialized in a manner sch that the system state is in the nsatrated region. B. ROA Containment in Satrated Region In this sbsection we show that the ROA containment also holds in the satrated region. What follows is a series of intermediate claims to arrive at the main reslt of this sbsection Proposition 3. Define the line segments σ +div := K +div { ) R 2 < bc ad ma} σ div := K div { ) R 2 > bc ad min} σ +div := K +div \ σ +div σ div := K div \ σ div. It can be verified that σ +div = lz + z eq+ ) σ div = lz z eq ) z eq+ σ +div and z eq σ div whenever ad <. Claim 13: If the open loop system is i) marginally stable a = ) or strictly stable with a stable controller a < and d ) then K +div is a transverse section to f n. ii) strictly stable with an nstable controller a < and d a)) or nstable a > ) then σ +div K +div ) is a transverse section to f n. Proof: Since σ +div K +div K + we only need to consider f n in K +. For any z K + we have f n z) = f n ) = a+b ma c+d). Let T z + = ma d c ma). z+ For case i) respectively ii)) it can be verified that T z is a + nit normal of K +div respectively σ +div ). We need to show that T z + f n z) for all z K +div respectively z σ +div ). Any z K +div can be epressed as z = ) = d c ) for some > ma. On any point z K +div direct comptation yields T z + f n z) = ma d c ma) a + bma c + d) for some > ma. = ma d c ma) ad c + b ma ) = ad c + b ma) ma = 1 c ad bc ma) ma For case i) we have ad so that ad ad ma > bc ma where the last ineqality is de to 6). Then ad bc ma > and we have T z + f n z) for all z K +div as desired. For case ii) it can be verified that ad < de in part to 5) d < a < when a > ). Then bc ad > 1 de to 6) and σ +div. On K +div T z + f n z) = can hold if and only if ad bc ma =. This is assred on any point z = ) σ +div K +div de to < bc ad ma. Remark 14: For case ii) the proof also shows that σ +div \ {z eq+ } is also a transverse section to f n. Claim 14: If the open loop system is i) strictly stable with an nstable controller a < and d a)) or ii) nstable a > ) and z R n then z σ +div. Proof: We will show that if z σ +div then z R n. If z R n we have φ n t z ) z eq as t. Since z eq K it is sfficient to show that if z σ +div then φ n t z ) K for all t. It can be verified that ad < de in part to 5) d < a < when a > ). Then 6) yields bc ad > 1. Let z = ) σ +div so that bc ad ma > ma and c + d =. Since z σ +div K + consider the initial vale problem ẋ = a + b ma ) = = c + d ) = whose soltion will coincide with φ n t z ) as long as it remains in K + K +. Solving for t) yields t) = e at + b a mae at 1) t. We will show that t) ma for all t so that φ n t z ) K for all t. Consider case i) a < and d a)). Define v = so that v = = c + d = dv + c + d ) and consider v = dv + c + d ) v) = ) =. Clearly if vt) = t) for all t then t) bc ad ma > ma for all t and the conclsion follows. Since d > a sfficient condition is for the inpt of the preceding ordinary differential eqation to satisfy ct) + d for all t. Using c + d = and the soltion of t) we have ct) + d = c e at + bc a mae at 1) + d

14 14 = d e at + bc a mae at 1) + d = d bc a ma) 1 e at ) = d bc ad ma) 1 e at ) for all t where the final ineqality is de to a < d > and bc ad ma. Now consider case ii) a > ). From 5) we have d < a <. In trn we have bc a ma d de to bc ad ma. Becase a > we have e at 1 for all t. The evoltion of ct) then satisfy ct) = c e at + b a mae at 1)) c e at + d e at 1) = d = c for all t de to c + d =. Then t) is governed by the differential ineqality = c + d c + d ) =. In similar manner as the proof of Claim 1 define ṽ = so that ṽ dṽ c ṽ) =. Applying the Comparison Lemma [8 Lemma 3.4 pp ] to the above differential ineqality yields ṽt) e dt c d e dt 1 ). Using c + d = we have for all t. t) = ṽt) e dt + c d e dt 1 ) d e dt + c e dt 1 )) = 1 d = c d = > ma Claim 15: If z K +ot R n then there eists a T g ) sch that the soltion of the GPAW compensated system satisfy φ g T g z ) lz + φ n T n z )) K +in where T n ) is sch that the soltion of the nominal system satisfy φ n T n z ) K +in and φ n t z ) K + for all t [ T n ). Moreover T g < T n < holds. Proof: Let z = ) K +ot R n so that > ma c + d > and φ n t z ) z eq as t. Since z eq K and z K +ot K + Fact 3 shows that φ n t z ) mst intersect K +in at some finite time. Let T n be the first time instant that φ n t z ) intersects K +in. Then T n ) φ n T n z ) K +in and φ n t z ) K + for all t [ T n ). It is clear that lz + φ n T n z )) K +in. The soltion of the nominal system φ n t z ) = n t) n t)) is governed by ẋ n = a n + b ma n ) = n = h n n n ) = c n + d n n ) = as long as n t) ma ie. for all t T n. The soltion of the GPAW compensated system φ g t z ) = g t) g t)) is governed by ẋ g = a g + b ma g ) = g = h g g g ) g ) =. where if c g + d g h g g g ) = c g + d g otherwise as long as g t) ma. We need to show that there eists a T g ) sch that T g < T n and φ g T g z ) lz + φ n T n z )). Solving the initial vale problem ẋ = a + b ma ) = yields + b ma t if a = t) = 13) e at + b a mae at 1) otherwise for all t. It can be seen that n t) = t) for all t sch that n t) ma and g t) = t) for all t sch that g t) ma. Let T g = inf{t ) g t) ma } ie. T g is the first time instant that φ g t z ) intersects K +in or if φ g t z ) never intersects K +in. With T := min{t n T g } the preceding relations yield n t) = g t) = t) for all t [ T ]. Hence n t) and g t) are well defined at least for all t [ T ] and we have h n t n t)) := h n n t) n t)) = ct) + d n t) h g t g t)) := h g g t) g t)) if ct) + d g t) = ct) + d g t) otherwise. Observe that whenever ct) + d g t) we have h n t g t)) h g t g t)). When ct) + d g t) < we have h n t g t)) = h g t g t)). Hence h g t g t)) h n t g t)) for all g t) ma for all t [ T ]. The soltion of g t) is clearly governed by the differential ineqality g t) = h g t g t)) h n t g t)) g ) = for all t [ T ]. By the Comparison Lemma [8 Lemma 3.4 pp ] we have g t) n t) for all t [ T ]. To obtain a strict ineqality observe that c) + d n ) =