The feasibility issue in trajectory tracking by means of regions-of-attraction-based gain scheduling

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(14) . Lustosa, Leandro and Defay, François and Moschetta, Jean-Marc The feasibility issue in trajectory tracking by means of regions-of-attraction-based gain scheduling. (2017) IFAC-PapersOnLine, vol. 50 (n° 1). pp. 11504-11508. ISSN 2405-8963. .

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(24) Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017. The feasibility issue in trajectory tracking The feasibility issue in trajectory tracking The feasibility issue in trajectory tracking The feasibility issue in trajectory tracking by means of regions-of-attraction-based by means of regions-of-attraction-based by means of regions-of-attraction-based by means ofgain regions-of-attraction-based scheduling gain scheduling gain scheduling gain scheduling ∗ ∗ ∗. Leandro c y Leandro R. R. Lustosa Lustosa ∗ Fran¸ Fran¸ cois ois Defa¨ Defa¨ y ∗∗ Jean-Marc Jean-Marc Moschetta Moschetta ∗ Leandro R. Lustosa ∗∗ Fran¸ cois Defa¨ y ∗ Jean-Marc Moschetta ∗∗ Leandro R. Lustosa Fran¸ cois Defa¨ y Jean-Marc Moschetta ∗ eerieur ∗ Institut Sup´ rieur de de l’A´ l’A´eeronautique ronautique et et de de l’Espace, l’Espace, Toulouse Toulouse 31400, 31400, ∗ Institut Sup´ Sup´ e rieur de l’A´ e ronautique et de l’Espace, Toulouse 31400, France (e-mail: {leandro-ribeiro.lustosa, francois.defay, ∗ Institut France (e-mail: {leandro-ribeiro.lustosa, francois.defay, Institut Sup´ e rieur de l’A´ e ronautique et de l’Espace, Toulouse France (e-mail: {leandro-ribeiro.lustosa, francois.defay, 31400, jean-marc.moschetta}@isae.fr). jean-marc.moschetta}@isae.fr). France (e-mail: {leandro-ribeiro.lustosa, francois.defay, jean-marc.moschetta}@isae.fr). jean-marc.moschetta}@isae.fr). Abstract: Abstract: Linear Linear control control theory theory has has been been long long established established and and aa myriad myriad of of techniques techniques Abstract: Linear control controllers theory has for been longsystems established andof aconflicting myriad ofperformance techniques are available for designing linear in view are available for designing controllers for linear systems in view of conflicting Abstract: Linear control theory has been long established and a myriad ofperformance techniques are available for for linear systems in view conflicting requirements. On the hand, control techniques are often tailored to requirements. On designing the other other controllers hand, nonlinear nonlinear control techniques areof tailoredperformance to specific specific are available for designing controllers for linear systems in view ofoften conflicting performance requirements. On the other hand, nonlinear control techniques are often tailored to specific applications and versatile nonlinear control frameworks are still on their infancy. A common applications and nonlinear control frameworks are still are on their requirements. Onversatile the other hand, nonlinear control techniques often infancy. tailored Atocommon specific applications and versatile nonlinear control frameworks are stillset-points on their over infancy. A common approach resort to linearized descriptions at aa given desired approach is is to to resort to local local linearized descriptions at desired desired given desired applications and versatile nonlinear control frameworks are stillset-points on their over infancy. A common approach is to resort to local linearized descriptions at desired set-points over a given desired trajectory and employ linear tools. Furthermore, to enforce stability when switching controllers, trajectory and employ linear tools. Furthermore, to enforce stability when switching controllers, approach is to resort to local linearized descriptions at desired set-points over a given desired trajectory and employ linear tools. Furthermore, to enforce stability when switching controllers, the approach has attention. This paper questions whether the regions-of-attraction regions-of-attraction approach has gained gained recent recent attention. This paper questions whether trajectory and employ linear tools. Furthermore, to enforce stability when switching controllers, the regions-of-attraction approach has gained recent attention. This paper questions such method – when applied to a well-posed smooth nonlinear controllable system – always suchregions-of-attraction method – when applied to a has well-posed smoothattention. nonlinearThis controllable system –whether always the approach gained recent paper questions whether such method – when applied to a well-posed smooth nonlinear controllable system – always yields a sequence of controllers that successfully tracks a given reference equilibrium trajectory, yieldsmethod a sequence of controllers that successfullysmooth tracks anonlinear given reference equilibrium such – when applied to a well-posed controllable systemtrajectory, – always yields sequencecounter-example of controllers that successfully a givendiscussed. reference Finally, equilibrium trajectory, and analytic is and thoroughly our study and an ana is provided provided andtracks thoroughly our case case study yields a analytic sequencecounter-example of controllers that successfully tracks a givendiscussed. reference Finally, equilibrium trajectory, and an analytic counter-example is provided and thoroughly discussed. Finally, our case study additionally shed light on how gain scheduling fails to track particular trajectories for certain additionally shedcounter-example light on how gain scheduling fails to track discussed. particular Finally, trajectories for certain and an analytic is provided and thoroughly our case study additionally shed light on how gain scheduling fails to track particular trajectories for certain globally systems. globally controllable controllable systems. additionally shed light on how gain scheduling fails to track particular trajectories for certain globally controllable systems. globally controllable systems. Keywords: Keywords: Tracking; Tracking; Switching Switching stability stability and and control; control; Stability Stability of of nonlinear nonlinear systems. systems. Keywords: Tracking; Switching stability and control; Stability of nonlinear systems. Keywords: Tracking; Switching stability and control; Stability of nonlinear systems. 1. Gain-scheduling 1. INTRODUCTION INTRODUCTION Gain-scheduling techniques, techniques, on on the the other other hand, hand, are are based based 1. INTRODUCTION Gain-scheduling techniques, on the other hand, based on linearization of nonlinear systems at desired operaton linearization techniques, of nonlinearonsystems at hand, desiredare operat1. INTRODUCTION Gain-scheduling the other are based on linearization of nonlinear systems at desired operating points thus profiting of well established and comRecent advances in avionics hardware technology allow ing linearization points thus of profiting of systems well established andoperatcomRecent advances in avionics hardware technology allow on nonlinear at desired points thus profiting of well established and comRecent advances in avionics hardware technology allow ing putationally inexpensive linear control techniques. The for novel unmanned aerial vehicles (UAVs) architectures inexpensive techniques. The for noveladvances unmanned aerial vehicles (UAVs) architectures ing points thus profiting linear of wellcontrol established and comRecent in avionics hardware technology allow putationally putationally inexpensive linear control techniques. manual sequential controller design is tedious but has for novel unmanned aerial vehicles (UAVs) architectures as well as increased maneuverability of traditional ones. sequential controller is tedious but The has as well increased maneuverability of traditional ones. manual putationally inexpensive lineardesign control techniques. The for novelasunmanned aerial vehicles (UAVs) architectures manual sequential controller design is tedious but has being increasingly replaced by H -based automated alas well as increased maneuverability of traditional ones. For instance, the increasingly popular convertible archi∞ being increasingly replaced bydesign H∞ -based automated alForwell instance, the increasingly popularofconvertible manual sequential controller is tedious but has as as increased maneuverability traditional archiones. gorithms being increasingly replaced by H∞(2011)). -based automated al(Gahinet and Apkarian Furthermore, For instance, the increasingly popular convertible architectures such as tilt-body and tilt-wing vehicles are capa(Gahinet replaced and Apkarian Furthermore, tectures such as tilt-body and tilt-wing vehicles are archicapa- gorithms being increasingly by H∞(2011)). -based automated alFor instance, the increasingly popular convertible gorithms (Gahinet and Apkarian (2011)). Furthermore, tectures such as tilt-body and tilt-wing vehicles are capaadvances in regions-of-attraction computation by means ble between helicopter and fixed-wing modes regions-of-attraction by means ble of of switching switching and vehicles fixed-wing modes gorithms in (Gahinet and Apkariancomputation (2011)). Furthermore, tectures such as between tilt-bodyhelicopter and tilt-wing aremodes capaadvances advances in regions-of-attraction computation by means of direct of functions sumble of switching between and fixed-wing during flight for hovering in longdirect computation computation of Lyapunov Lyapunov functions using using sumduring flight allowing allowing forhelicopter hovering capabilities capabilities in aamodes long- of advances in regions-of-attraction computation by means ble of switching between helicopter and fixed-wing of direct computation of Lyapunov functions using sumof-squares (SOS) optimization (Johansen (2000); Parrilo during flight allowing for hovering capabilities in a longendurance vehicle (Lustosa et al. (2015)). Such vehicles (SOS) optimization (Johansen (2000); Parrilo endurance vehicle (Lustosa et al. (2015)). Suchin vehicles of direct computation of Lyapunov functions using sumduring flight allowing for hovering capabilities a long- of-squares of-squares (SOS) optimization (Johansen (2000); Parrilo (2000)) allow for efficient stability guarantees while switchendurance vehicle (Lustosa et al. (2015)). Such vehicles exhibit wide heterogeneous flight envelopes which preclude (2000)) allow for efficient stability guarantees while switchexhibit widevehicle heterogeneous flight envelopes which preclude of-squares (SOS) optimization (Johansen (2000); Parrilo endurance (Lustosa et al. (2015)). Such vehicles (2000)) allow for efficient stability guarantees while switchexhibit wide heterogeneous flight envelopes which preclude ing scheduled controllers. This global controllers as and/or scheduled controllers. This regions-of-attraction-based regions-of-attraction-based global linear linear controllers employment employment as auto-pilots auto-pilots and/or ing (2000)) allow for efficient stability guarantees while switchexhibit wide heterogeneous flight envelopes which preclude ing scheduled controllers. This regions-of-attraction-based gain scheduling (RoA-GS) strategy was global linear controllers employment as auto-pilots and/or stability augmentation systems (SAS). Nonlinear control gainscheduled scheduling (RoA-GS)This strategy was recently recently validated validated stability augmentation (SAS). Nonlinear control controllers. regions-of-attraction-based global linear controllers systems employment as auto-pilots and/or ing gain scheduling (RoA-GS) strategy was in agile fixed-wing vehicles (Moore et al. (2014)) and stability augmentation systems (SAS). Nonlinear control techniques, however, are often tailored to specific applicaagile fixed-wing vehicles strategy (Moore et al.recently (2014)) validated and proprotechniques, however, aresystems often tailored specific applicagain scheduling (RoA-GS) was recently validated stability augmentation (SAS).toNonlinear control in in agile fixed-wing vehicles (Moore et al. (2014)) and promoted to a motion planning technique by means of a techniques, however, are often tailored to specific applications and general nonlinear control frameworks are still on a motion planning technique a sparse sparse tions and general nonlinear control frameworks areapplicastill on moted in agiletofixed-wing vehicles (Moore et by al. means (2014))ofand protechniques, however, are often tailored to specific moted to a motion planning technique by means of a sparse randomized tree of scheduled linear quadratic regulators tions and general nonlinear control frameworks are still on their tree ofplanning scheduled linear quadratic regulators their infancy. infancy. moted to a motion technique by means of a sparse tions and general nonlinear control frameworks are still on randomized randomized tree et of al. scheduled quadratic regulators (LQR) (Tedrake (2010)). their infancy. (LQR) (Tedrake (2010)).linear randomized tree et of al. scheduled linear quadratic regulators their infancy. feedback For For instance, instance, feedback linearization linearization methods methods render render aa nonnon- (LQR) (Tedrake et al. (2010)). (LQR) (Tedrake et contributes al. (2010)). to present paper For instance, methods render non- The linear system linear means state present paper contributes to the the RoA-GS RoA-GS framework framework linear system feedback linear by bylinearization means of of appropriate appropriate state aadiffeodiffeoFor instance, feedback linearization methods render non- The The present paper contributes to the RoA-GS framework by investigating whether the RoA-GS planning technique linear system linear by means of appropriate state diffeomorphism transformations. However, such transformation by investigating whether the RoA-GS planning technique morphism transformations. However, such transformation The present paper contributes to the RoA-GS framework linear system linear by means of appropriate state diffeo- by investigating whether the RoA-GS planning always yield a finite (or even countable infinite) sequence morphism transformations. However, such transformation exists only for fully actuated systems (generally absent yield a finite (or even countableplanning infinite)technique sequence exists onlytransformations. for fully actuated systems (generally absent always by investigating whether the RoA-GS technique morphism However, such transformation always yield a finite (or even countable infinite) exists only for fullyand, actuated systemscontroller (generally absent of controllers leading to a desired set-point given a in aerial robotics) additionally, design is controllers to even a desired set-point givensequence a wellwellin aerial robotics) additionally, controller design is of always yield aleading finite (or countable infinite) sequence exists only for fullyand, actuated systems (generally absent controllers leading to a desired set-point given a equiwelladditionally, controller design is of posed controllable nonlinear system and aa reference in aerial robotics) and, often challenging due to unstable zero dynamics. Sliding posed controllable nonlinear system and reference often challenging due to additionally, unstable zerocontroller dynamics.design Sliding of controllers leading to a desired set-point given a equiwellin aerial robotics)due and, is controllable nonlinear system and a reference equito unstable zero dynamics. Sliding posed librium trajectory. Section 2 revisits the RoA-GS method often challenging mode control, on the other hand, applies to underactuated librium trajectory. nonlinear Section 2 system revisits and the RoA-GS method mode challenging control, on the other hand, applies to underactuated posed controllable a reference equioften due to unstable zero dynamics. Sliding librium trajectory. Section 2 revisits the RoA-GS method on the uncertainties other hand, applies to underactuated and up Section 33 proposes aa well-posed mode control, systems but may to and sets setstrajectory. up notation. notation. Section proposes well-posed systems but model model uncertainties may lead lead to excessive excessive librium Section 2 revisits the RoA-GS method mode control, on the uncertainties other hand, applies to underactuated and sets up notation. Section 3 proposes a well-posed systems butand model may lead to excessive nonlinear system that is globally controllable and chattering excitation of unmodeled dynamics. Adsystem that isSection globally and fullyfullychattering excitation of unmodeled dynamics. Ad- nonlinear and sets up notation. 3 controllable proposes a well-posed systems butand model uncertainties may lead to excessive system everywhere, that is globally controllable actuated almost but fails to track chattering and excitation of unmodeled dynamics. Ad- nonlinear vances hardware and theory but RoA-GS RoA-GS failsand to fullytrack vances in in computing computing hardware and optimization optimization theory nonlinear almost system everywhere, that is globally controllable and fullychattering and excitation of unmodeled dynamics. Ad- actuated actuated almost everywhere, but RoA-GS fails to track aa given equilibrium trajectory. Finally, section 4 presents vances in computing hardware and optimization theory allow for successful model predictive control design but its given equilibrium trajectory. Finally, section 4 presents allow for successful model predictive control design but its actuated almost everywhere, but RoA-GS fails to track vances in computing hardware and optimization theory aopportunities given equilibrium trajectory. Finally, section 4 presents for future work and conclusion. allow for successful model predictive control design but its non-convex nature poses challenges for high dimensional opportunities for future work and conclusion. non-convex nature model poses challenges for highdesign dimensional a given equilibrium trajectory. Finally, section 4 presents allow for successful predictive control but its non-convex nature poses challenges for high dimensional opportunities for future work and conclusion. embedded applications. embedded real-time real-time applications. non-convex nature poses challenges for high dimensional opportunities for future work and conclusion. real-time applications. embedded This workreal-time is partiallyapplications. supported by the Conselho Nacional de embedded This work is partially supported by the Conselho Nacional de. Desenvolvimento Cient´ıficosupported e Tecnol´ ogico, CNPq (Brazilian National work is partially by the Conselho Nacional de This Desenvolvimento Cient´ıficosupported e Tecnol´ ogico, CNPq (Brazilian National This Foundation), work is partially Nacional de Science through the ”Ciˆ eby nciathe semConselho Fronteiras” program. Desenvolvimento Cient´ ıfico e Tecnol´ ogico, CNPq (Brazilian National Science Foundation), through the ”Ciˆ enciaCNPq sem Fronteiras” Desenvolvimento Cient´ ıfico e Tecnol´ ogico, (Brazilian program. National Science Foundation), through the ”Ciˆ encia sem Fronteiras” program. Science Foundation), through the ”Ciˆ encia sem Fronteiras” program. Copyright © 2017 IFAC 11996 Copyright © 2017 IFAC 11996 Copyright © 2017 IFAC 11996 Copyright © 2017 IFAC 11996 10.1016/j.ifacol.2017.08.1609.

(25) Proceedings of the 20th IFAC World Congress. Toulouse, France, July 9-14, 2017. 2. REGIONS-OF-ATTRACTION-BASED SWITCHING CONTROL REVISITED Consider a nonlinear globally controllable dynamical system f ∈ C ∞ of the form dx = f (x, u) (1) dt where x ∈ Rn and u ∈ Rm are, respectively, state-space coordinates and control inputs. Furthermore, consider a given equilibrium point (x0 , u0 ), i.e., (2) f (x0 , u0 ) = 0 at which a local linear controller is designed by means of linear control techniques applied at the linearized system d ∆x0 = A0 ∆x0 + B0 ∆u0 (3) dt where ∆x0 = x − x0 (4) (5) ∆u0 = u − u0 ∂f (6) A0 = ∂x (x0 ,u0 ) and ∂f B0 = (7) ∂u (x0 ,u0 ) Linear control design techniques (e.g., pole-placement, LQR, H∞ ) yield local stabilizing controllers of the form (8) ∆u0 = −K0 ∆x0 which regulate the error ∆x0 on a neighborhood of x0 . A set R0 ⊂ Rn of initial conditions x(0) that are regulated by this local controller K0 is called a region-of-attraction (RoA), basin of attraction or attractor for the given controller around x0 (Chiang et al. (1988)). RoA computation is challenging in all but exceedingly simple systems and often conservative estimates are numerically computed instead by means of polynomial Lyapunov functions (Tan and Packard (2008); Topcu and Packard (2009); Topcu et al. (2010)).. enlarged region-of-attraction R0:1 = R0 ∪ R1 . Finally, N iterations of the aforementioned steps yield a controller K0:N that potentially allows for wide regions-of-attraction N Ri (9) R0:N = i=0. as figure 2 illustrates. For instance, Tedrake et al. (2010) applies this concept to motion planning by means of a sparse randomized tree of scheduled LQR controllers. R1. R3. R0. R2 x1. x3. x0. x2. Fig. 2. Sequence of scheduled controllers K0:N and respective regions-of-attraction R0:N , for N = 3. Consider now a given desired quasi-static trajectory 1 parametrized by xd (s) : [0, 1] → Rn , that is, we want to steer the state towards xd (0) from xd (1) by closely tracking the curve xd (s). We chose s as parameter symbol to reinforce the absence of time performance requirements in the trajectory definition.. s0 = 0 s1 s0,1 s2. R0. s1,2. R1. R1 R2 x(0) x1 R0. s3 = 1. K1. Fig. 3. Desired quasi-static trajectory xd (s) and illustrative iterations of the regions-of-attraction based switching linear control algorithm.. x0 K0. Switch!. Fig. 1. To steer x from x(0) towards x0 , controller K1 is employed until x reaches R0 . Afterwards, controller K0 drives x towards the desired setpoint. Moreover, consider a second linear regulator K1 with associated operating point x1 = x0 such that x1 ∈ R0 and an initial state x(0) ∈ R1 \ R0 (see figure 1). Since x(0) ∈ / R0 , controller K0 does not guarantee x(t) regulation in x0 . However, x0 -convergence is guaranteed by initially applying controller K1 until x(t) ∈ R0 (which is guaranteed since x1 ∈ R0 ) and then switching control to K0 (we denote the time that this occurs as t1 ). This methodology yields a RoA-GS controller K0:1 with an. A gain-scheduling control design strategy for xd (s) tracking is proposed (Burridge et al. (1999)) as follows. Firstly, we design a controller K0 for xd (s0 ), where s0 = 0 (see figure 3), by means of any preferred linear control technique. Let s0,1 be the point where R0 intersects xd (s), i.e., (10) s0,1 sup{s ∈ [0, 1] : xd (s) ∈ R0 } and, secondly, we design a controller K1 at xd (s1 ) where (11) s1 = µ(s0,1 − s0 ) + s0 with 0 < µ < 1. Appropriate µ design is dependent on the particular xd (s) parametrization, K0 and system 1. A quasi-static trajectory is system-dependent and means herein that f (xd (s)) = 0 for all s ∈ [0, 1], i.e., it is a trajectory composed of equilibrium points.. 11997.

(26) Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017. dynamics. Independently, small µ yield small contribution to the region-of-attraction expansion whereas overly large µ might preclude robustness. The aforementioned RoAGS procedure is iterated until sn reaches 1 (see figure 3) and delivers a controller K0:N that follows the desired trajectory. An important question is whether the aforementioned algorithm terminates for N < ∞. The answer is key not only to avoid infinite loops in practical implementations but, more importantly, to better understand, as we shall soon discuss, when gain scheduling is not suitable. Figure 4 illustrates a conceptual case where the regions-of-attraction Ri are successfully advancing forward but never reaching the end goal. Is that a possibility for smooth controllable nonlinear systems? Indeed, the next section proves these phenomena possible and further discusses the matter.. and it is underactuated in U ⊂ R2 , where U is given by U = R2 \ A = {(x1 , x2 ) ∈ R2 : x1 = 0}. (17). Additionally, notice that x2 (t) possesses decoupled and marginally stable controllable linear dynamics. On the other hand, if u1 (t) is held constant and zero, then x(t) dynamics becomes marginally stable linear controllable with respect to u2 (t). Consequently, the complete nonlinear system f (x, u) is controllable. To further illustrate the aforementioned controllability property, consider an initial state xi ∈ U and a desired final state xf ∈ U (see figure 5). Notice that the straight trajectory between them is unattainable since u1 actuation in x˙ 1 is canceled by x1 = 0, and x2 = 0 drives x out of U imperatively. A feasible trajectory is to let x drive away from U, and steer x back to U at appropriate reentry points by means of the fully actuated u (which is possible in arbitrary trajectories in A). x2. s0. s1 s2 s3. xd (1). xf. U. R2 xi. R1. x1. R0 Fig. 4. Illustration of the hypothesis: a sequence of controllers Ki where si fail to converge to 1. 3. ROA-GS FEASIBILITY – A CASE STUDY Consider the following nonlinear system d x1 x 1 u1 + x 2 = f (x, u) = u2 dt x2. Fig. 5. An example of a feasible trajectory starting and ending at the underactuated region U. (12). where x = (x1 , x2 ) ∈ R2 and u = (u1 , u2 ) ∈ R2 are, respectively, state variable and control input. Straightforward linearization with respect to (xi , ui ) yields (for subsequent local linear control design in section 3.2) u1,i 1 x1,i 0 Bi = (13) Ai = 0 0 0 1 where the subscript notation alludes x1,i u1,i = xi = ui x2,i u2,i. (14). Notice that (Ai , Bi ) is controllable for all (xi , ui ) ∈ R2 ×R2 since the associated Kalman controllability matrix is full rank, i.e., x1,i 0 u1,i x1,i 1 rank ([Bi Ai Bi ]) = rank = 2 (15) 0 1 0 0 3.1 Nonlinear controllability and underactuation analysis By inspection we conclude that f (x, u) is fully actuated in A ⊂ R2 , where A is given by (16) A = {(x1 , x2 ) ∈ R2 : x1 = 0}. 3.2 LQR design Consider the problem of tracking the rectilinear equilibrium trajectory xd (s) : [0, 1] → R2 given by xi x f − xi xd (s) = + s (18) 0 0. with xi > 0 and xf < 0. Accordingly, we apply RoA-GS with LQR as the chosen linear control design technique for all controllers Ki due to its simple methodology (numerous Ki might be required depending on the application) and adequate stability margins (Safonov and Athans (1977)). Each LQR design iteration yields a suboptimal linear control policy ∆u∗i (t) = −Ki ∆xi that locally minimizes the cost ∞ T J ∆xi (ti ), ∆ui = ∆xi Qi ∆xi + ∆uTi Ri ∆ui dt ti. (19) where ∆xi (ti ) ∈ R2 , Qi , Ri ∈ R2×2 , are, respectively, initial state of the Ki controller at the switching moment instant ti , positive semi-definite state penalty and positive definite actuator penalty matrices. For instance, let us consider. 11998.

(27) Proceedings of the 20th IFAC World Congress. Toulouse, France, July 9-14, 2017. and.   −1   1    |x1,i |      2 ,  −1 Qi = |x | x21,i   1,i   10   ,  01 Ri =. . if x1,i = 0. (20). which is positive-definite and thus the unique solution of (27). Substitution of (28) and (29) into (24) yields  x1,i  0  |x1,i | √  ∆ui (t) = −  (30) 2  ∆xi (t) 0 |x1,i |. (21). We omitted the (rather simple to solve) case x1,i = 0 since the RoA-GS algorithm never reaches xi = 0, as we shall prove next.. if x1,i = 0. . 10 01. The perhaps peculiar choice of Qi for x1,i = 0 is justified by the following identity   −1 T 1  |x1,i |  ∆x1,i ∆x1,i T   ∆xi Qi ∆xi = 2  ∆x2,i =  −1 ∆x2,i |x1,i | x21,i T    ∆x1,i ∆x1,i 1 −1  ∆x  2,i (22) =  ∆x2,i  −1 2 |x1,i | |x1,i | which illustrates larger allowance of ∆x2,i usage in view of distant (from the origin) operating points x1,i . In view of the chosen weights, by means of the HamiltonJacobi-Bellman equation, one can show (Anderson and Moore (1990)) that the control policy

(28). (23) ∆u∗i (t) = arg min J ∆xi (ti ), ∆ui ∆ui. is independent of ∆xi (ti ) and is given by (24) ∆u∗i (t) = −BiT Pi ∆xi (t) where Pi is the unique semi-positive definite solution of the algebraic Riccati equation (ARE) (25) Pi Ai + ATi Pi − Pi Bi BiT Pi + Qi = 0 1/2. if (Ai , Bi ) is stabilizable and (Ai , Qi ) is detectable. The solution Pi is normally computed numerically (Laub (1978); Petkov et al. (1991)), but our reachability assessment study calls for an analytical expression. Accordingly, we denote Pi by α β Pi = (26) β γ and substitute it back in (25) to obtain the following nonlinear algebraic system of equations  1 − β 2 − α2 x21,i = 0       α − 1 − βγ − αβx2 = 0 1,i (27) |x1,i |    2   −β 2 x21,i + 2β − γ 2 + =0  x21,i by additionally recalling that Ai and Bi (equation 13) over the trajectory described by (18) reduce to 01 x 0 Bi = 1,i (28) Ai = 00 0 1 We invite the reader to check that the following Pi is a solution to (27)   1 0  |x1,i |  √   (29) Pi =  2  0 |x1,i |. 3.3 Region-of-attraction computation Due to its complexity regions-of-attraction are often computed numerically. However, in the present study, analytical intervals-of-attraction in the direction of the desired trajectory are provided to prove RoA-GS infeasibility. Initially, since we are interested in ∆xi convergence, we rewrite (12) by substituting (4), (5) and (30) to obtain x1,i ∆x21,i + ∆x2,i (31) ∆x˙ 1,i = −|x1,i |∆xi − |x1,i | and √ 2 ∆x2,i (32) ∆x˙ 2,i = − |x1,i | The ∆x2,i component has decoupled linear dynamics that yields the well-known exponential decay given by √ − 2 (t−ti ) |x1,i |. ∆x2,i (t) = ∆x2,i (ti )e (33) that clearly converges to zero when t → ∞ for all xi,1 ∈ R∗ . Substitution of (33) in (31) yields √ − 2 x1,i (t−ti ) ∆x21,i + ∆x2,i (ti )e |x1,i | ∆x˙ 1,i = −|x1,i |∆xi − |x1,i | (34) Consider the following subset Π ⊂ R2 of initial conditions (35) Π = {∆xi (ti ) ∈ R2 : ∆x2,i (ti ) = 0} For ∆xi (ti ) ∈ Π, equation 34 reduces to x1,i ∆x21,i (36) ∆x˙ 1,i = −|x1,i |∆xi − |x1,i | which is nonlinear time-invariant. Its equilibrium points p1 and p2 are solutions of the quadratic equation x1,i 2 −|x1,i |pi − p =0 (37) |x1,i | i i.e., (38) p1 = 0, p2 = −xi,1 Its associated phase portrait is illustrated in figures 6 and 7 (be aware of the change of variables from ∆x1,i to x1,i ). Interestingly, the phase portrait reveals that x = 0 is not stable thus not included in any Ri for any controller Ki designed at any operation point x1,i = 0. Therefore the sequence of regions-of-attraction Ri never cross the origin as figure 8 illustrates. From another point of view, given a Ki controller centered in xi,1 > 0, we conclude that the next controller, namely Ki+1 , is scheduled at (39) xi+1,1 = (1 − µ)xi,1 such that (40) xi+N,1 = (1 − µ)N xi,1 > 0. 11999.

(29) Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017. nor gain scheduling quasi-static trajectory tracking. Future work might include sufficient conditions for RoA-GS trajectory tracking.. x˙ 1. ACKNOWLEDGEMENTS The authors support free education for everyone, everywhere, and highly praise Prof. Russ Tedrake for his beautiful course on underactuated robotics at edX.org.. x1 x1,i. 0. REFERENCES Fig. 6. Phase portrait of closed-loop f (x, −Ki ∆xi ) when ∆xi (ti ) ∈ Π+ . x˙ 1 x1 x1,i. 0. Fig. 7. Phase portrait of closed-loop f (x, −Ki ∆xi ) when ∆xi (ti ) ∈ Π− . x2 R0 R1 R2. Π. x1 xf. x2 x1. x0. Fig. 8. Sequence of RoA-GS controllers Ki with associated Ri fails to cross the origin and reach xf . and never reaches xf since xf < 0. RoA-GS generates an infinite sequence of controllers Ki with operating points xi asymptotically converging to 0 but never crossing it. Finally, consider the scenario of gain scheduling design without RoA for tracking the same trajectory with an uniform distribution per unit length ρ of LQR controllers used instead. By the foregoing development we conclude that no ρ renders tracking possible. Therefore, in general, discrete gain scheduling is inappropriate for this example. 4. CONCLUSION. Anderson, B.D.O. and Moore, J.B. (1990). Optimal control, linear quadratic methods. Dover. Burridge, R.R., Rizzi, A.A., and Koditschek, D.E. (1999). Sequential composition of dynamically dexterous robot behaviors. International Journal of Robotics Research, 18(6), 534–555. Chiang, H.D., Hirsh, M.W., and Wu, F.F. (1988). Stability regions of nonlinear autonomous dynamical systems. IEEE Transactions on Automatic Control, 33(1), 16–27. Gahinet, P. and Apkarian, P. (2011). Decentralized and fixed-structure H∞ control in MATLAB. In Proc. IEEE Conference on Decision and Control, 8205–8210. Johansen, T.A. (2000). Computation of Lyapunov functions for smooth nonlinear systems using convex optimization. Automatica, 36(11), 1617–1626. Laub, A.J. (1978). A Schur method for solving algebraic Riccati equations. Technical Report 859, Massachussets Institute of Technology, Laboratory for Information and Decision systems, LIDS. Lustosa, L.R., Defay, F., and Moschetta, J.M. (2015). Longitudinal study of a tilt-body vehicle: modeling, control and stability analysis. In Proc. Int. Conf. on Unmanned Aircraft Systems (ICUAS), 816–824. Moore, J., Cory, R., and Tedrake, R. (2014). Robust post-stall perching with a simple fixed-wing glider using LQR-trees. Bioinspiration and Biomimetics, 9(2). Parrilo, P.A. (2000). Structured semidefinite programs and semialgebraic geometry methods in roburobust and optimization. Ph.D. thesis, California Institute of Technology. Petkov, P., Christov, N., and Konstantinov, M. (1991). Computational methods for linear control systems. New York: Prentice. Safonov, M.G. and Athans, M. (1977). Gain and phase margins for multiloop LQG regulators. IEEE Transactions on Automatic Control. Tan, W. and Packard, A. (2008). Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming. IEEE Transactions on Automatic Control, 53(2), 565–571. Tedrake, R., Manchester, I.R., Tobenkin, M.M., and Roberts, J.W. (2010). LQR-trees: feedback motion planning via sums of squares verification. International Journal of Robotics Research, 29, 1038–1052. Topcu, U. and Packard, A. (2009). Local stability analysis for uncertain nonlinear systems. IEEE Transactions on Automatic Control, 54(5), 1042–1047. Topcu, U., Packard, A., Seiler, P., and Balas, G.J. (2010). Robust region-of-attraction estimation. IEEE Transactions on Automatic Control, 55(1), 137–142.. We prove by means of an analytical counterexample that global controllability does not imply successful RoA-GS 12000.

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