Estimation of Regions of Attraction for Nonlinear Systems via Coordinate-Transformed TS Models

Estimation of Re gions of Attra ction for Nonlinear Systems via Coordinate-T ransformed TS Models Artun Sel Electrical and Computer Engineering The Ohio State Univ ersity Columbus, OH 43210, USA artunsel@ieee.org Mehmet K o ruturk Electrical an d Com p uter Eng ineering V irginia T ech Blacksburg, V A 2 4061 , USA mkoruturk @vt.edu Erdi Say ar Robotics, Ar tificial I ntelligence and Real- time Systems T echnical University of Mu nich Munich, Ge r many erdi.sayar@tum .de Abstract —This paper presents a novel method for estimating larger Region of Attractions (RO As) fo r continuous-t i me non- linear systems modeled via t he T akagi-Sugeno (TS) framework. While cl assical approaches rely on a single TS representation derive d from the original n onlinear system to compute an RO A using L yapunov-based analysis, th e proposed meth od enh an ces this process thro ugh a system atic coordinate transfo rmation strategy . Specifically , we construct multiple TS models, each ob- tained fro m the original n onlinear system under a distinct li n ear coordinate transfo rmation. Each transf ormed system yields a local R O A estimate, and the ov erall R O A is taken as the union of these i ndividual estimates. Th is strategy leverag es th e variability introduced by th e transformations to r educe conservatism and ex- pand the certified stable re gion. Numerical examples demonstrate that thi s app roach consisten tly provides larger RO As compared to con ventional single-model TS-b ased techniques, highlighting its effective ness and potential f or improv ed nonlinear stabili t y analysis. Index T erms —TS models, R O A, coordinate transformation, nonlinear stability , L i near Matrix Ineq uality (LMI). I . I N T RO D U C T I O N The analy sis and con tr ol of non linear systems remains a fund amental challenge in modern control theory [1]–[3] . Especially with th e rec ent developments in autonom ous aer ial vehicles with their applications in remo te sensing [4] and cyber security issues in com munication s [5]–[1 0] and energy systems [1 1]–[1 6 ], the saf ety co nsideration s are becoming more relev ant. Addition ally , the stability conc e r ns may ap p ear in co mputation al mo dels [1 7]–[2 0] and optimization algo- rithms [21]– [27] that are especially importan t in estimation problem s related to aer ospace ind ustry [28 ]–[31 ]. Am ong various m e thodolo g ies developed to add ress this, the TS fuzzy modeling framework ha s proven to be particu larly effecti ve. I t provides a systematic way to app roximate nonlinear dynamical systems throug h a con vex co m bination of linear submodels, enabling the application of p owerful LMI tech niques for controller d esign an d stability verification. When constructed via the sector nonlin e arity a p proach , TS mode ls p reserve local equiv alence to the origina l non linear dyn amics within a specified comp act doma in , makin g them a pr actical too l fo r analyzing local stability pr operties [32 ]. A key application of TS modeling lies in th e estimation of the R O A around an eq uilibrium point, often the origin. T r aditional TS- based stability analysis typically inv o lves con- structing a single TS model of the nonline a r system and verifying local or g lobal stability condition s u sing a commo n L yapun ov fu nction. These methods, while comp utationally tractable, often yield co nservati ve R O A estimates due to their reliance on a fixed co ordina te representation and restrictive L yapun ov structur es [33 ]. T o mitigate these limitations, various enhan cements hav e been proposed , such as the use of nonqu a d ratic L yap unov function s, piece wise or par ameter-dependen t f unctions, and membersh ip -functio n-dependent relaxations. Howe ver , the p o- tential of explo iting multiple coor dinate representations of the sy stem to enrich the stab ility analysis rem ains largely unexplored [34] . In th is work, we introd uce a novel approa ch to expa n d the estimated R OA by systematically transformin g th e origina l system through a family of linear coordinate changes. Each transform ed system yields a distinct T S mo d el, and an indi- vidual R OA is comp uted via standard LMI-b ased techniq u es. The final RO A is th en obtain ed as the union of these individual estimates. This fr a mew ork leverages the structu ral variation introdu c ed by coo r dinate tran sformation s to redu ce the con - servati veness inhe r ent in single-m o del metho d s [35], [ 3 6]. W e de m onstrate th rough a re p resentative example th at o ur method p roduc es significantly larger RO A estimates c ompared to conventional a pproach es, while preserving compu ta tio nal tractability . The proposed method offers a prom ising dir e ction for extend ing the applicab ility and effectiveness of TS-based stability ana ly sis fo r no nlinear systems. The r est o f the pap er is organized as follows: Section II introdu c es th e prelimin aries. Section I I I reviews the standar d TS mod eling and R OA co m putation framework. Section IV presents the pr oposed m ethod based o n coordin ate tran sfor- mation an d R OA union. Section V co ncludes the paper and discusses f uture dir e c tions. I I . P R E L I M I N A R I E S T ab le I summarize s the mathematical notation used thro ugh- out th is pa per for quick refer ence. T ABLE I M AT H E M A T I C A L N O TA T I O N Symbol Descripti on R n n -dimensiona l Euclidean s pace R m × n Set of all m × n real matric es S n Space of n × n symmetric matrices S n + Cone of n × n symmetric posi ti ve semidefinite matrice s S n ++ Cone of n × n symmetric positi ve definite matrices X  0 Matrix X is positi ve s emidefinit e X ≻ 0 Matrix X is positi ve definite X  Y Matrix X − Y is positi ve semidefinite tr ( X ) Tra ce of matrix X (sum of diagona l elements) det ( X ) Dete rminant of matrix X diag ( X ) diagona l of matrix X X ⊤ Tra nspose of matrix X x ⊤ Tra nspose of vec tor x A ⊗ B Kronecker product of m atrice s A and B h A, B i Inn er product of matrices, defined as tr ( A ⊤ B ) λ i ( X ) The i -th eigen v alue of matrix X v ( P ) Optimal valu e of optimizati on problem ( P ) con v ( S ) Con vex hull of set S h X i s X + X ⊤ A. Definitions for Mathematica l Op timization Related Con- cepts 1) Linear Matrix Inequality : An LMI is a c o nstraint of the form: F ( x ) = F 0 + m X i =1 x i F i  0 (1) where x = ( x 1 , x 2 , . . . , x m ) is th e variable vector, the sym- metric m atrices F 0 , F 1 , . . . , F m ∈ R n × n are g iven, an d the inequality symb ol  0 den otes p ositiv e semidefiniteness. An LMI defines a con vex constraint on x . Multiple LMIs can be expressed as a single LMI using blo c k -diago n al struc ture. LMIs are ce ntral to semidefinite pr o grammin g, where they serve as constrain ts in conv ex op tim ization problem s that can be solved efficiently using interior-point metho ds [37]. 2) Semi Definite Programming : Semidefinite Program- ming (SDP) is a p owerful op timization framework th at extends linear pro grammin g to th e con e of Positive Semidefinite (PSD) matrices. Forma lly , a n SDP can be expressed as: minimize h C, X i subject to h A i , X i ≤ b i , i = 1 , 2 , . . . , m X  0 (2) where X , C, A 1 , . . . , A m are symmetric n × n m atrices, b 1 , . . . , b m are scalars, and X  0 denotes that X is PSD. The constra in t X  0 means that all eig e n values of X are non-n egati ve [38]. SDP has significant app lications in control theory , com- binatorial o ptimization, and mach ine learnin g. It gen eralizes se veral o ptimization p roblems, includin g linear and q uadratic progr amming, while remain ing solvable in polynom ial time using inter ior-point methods. W hat m akes SDP p articularly valuable is its ability to provide tight relaxatio ns for many NP-hard pro blems, often yielding hig h-quality approxim ate solutions [39]. 3) Relaxation in Mathema tical Op timization : Relaxation is a fundam ental tech nique in mathematical op timization where a com plex prob lem is appro x imated by a simp ler one that is more tr actable to solve. Formally , given an op timization problem ( P ) : ( P ) : min x ∈F f ( x ) (3) a relaxation ( R ) o f ( P ) is an other op timization pro blem: ( R ) : min x ∈F ′ g ( x ) (4) where F ⊆ F ′ and g ( x ) ≤ f ( x ) for all x ∈ F . Th e key proper ty of a relaxation is that it p rovides a lower b o und on the op timal value o f the or ig inal prob lem. That is, if v ( P ) and v ( R ) d enote the optimal values of problems ( P ) and ( R ) respectively , then v ( R ) ≤ v ( P ) . Relaxation s are par tic- ularly useful for addr essing difficult optimization problems, especially th ose that a r e non-convex or NP-hard . By replacing hard co nstraints with easier ones or ap proxim ating non-conve x objective func tio ns with co nvex o nes, relaxa tio ns transform intractable pro blems into on es that can be efficiently solved. The q uality of a relaxation is me a sured by the gap b etween th e original prob lem’ s op timal value and th e relaxation’ s optimal value—a smaller g ap indicates a tighter relaxation [ 4 0]. 4) SDP Relaxatio n : SDP relaxation is a powerful tech- nique th at tran sforms difficult op timization prob lems, partic- ularly non-co nvex q uadratic problems, into tractable SDP s. Giv en a Quad ratically Constrained Quadratic Prog ramming (QCQP) of th e for m: min x ∈ R n x ⊤ P 0 x + q 0 ⊤ x + r 0 subject to x ⊤ P i x + q i ⊤ x + r i ≤ 0 , ∀ i ∈ { 1 , 2 , . . . , m } X  0 (5) the SDP relaxation pro ceeds by intr o ducing a matrix variable X = xx ⊤ and relaxing this n on-convex equality constraint to X  xx ⊤ . By the Sch u r comp lement condition, this is equiv alent to req uiring th at:  1 x ⊤ x X   0 (6) The r esulting SDP relax ation is: min x ∈ R n ,X ∈ S n tr ( P 0 X ) + q 0 ⊤ x + r 0 subject to tr ( P i X ) + q i ⊤ x + r i ≤ 0 , ∀ i ∈ { 1 , 2 , . . . , m }  1 x ⊤ x X   0 (7) where S n denotes the space of n × n symmetric matrices [ 41]. B. Definitions for Applica tions of the Methods 1) Dynamical System : A dyn amical sy stem is a mathe m at- ical model th at describes the tem poral evolution of a physical system’ s state accor ding to a fixed rule. Formally , it co nsists of a state space X and a fu nction f : X × R → X th at specifies how the state x ( t ) ∈ X e volves over time. For con tinuous-time systems, this ev olution is ty pically describ ed by a differential equation: ˙ x ( t ) = f ( x ( t ) , t ) (8) where ˙ x ( t ) represents the der i vati ve of the state with respect to time . Dynam ical systems can b e classified as lin ear o r nonlinear, time-inv a riant o r time-varying, and auton omous or non-au tonomo us. Th e an alysis of these systems f o cuses on characterizin g their stab ility pr operties, equilibr ium points, limit cycles, and other qualitative beh aviors [4 2 ]. Remark 1 . In this study , we primarily focus on polyn omial- type n onlinear systems that ar e time- invariant. F or analysis pr oblems, we specific a lly e xamine au tonomo u s systems that possess th ese p r operties. 2) Stability Definitions : Co nsider a non linear d ynamical system described b y: ˙ x = f ( x ) , x (0) = x 0 (9) where x ∈ R n is the state vector, f : R n → R n is a locally Lipschitz f unction, and f (0) = 0 (i.e., the origin is an equilibrium point) . W e define various notion s of stability as f o llows: Definition 1 ( Sta bility in the Sense of L yapunov ) . The equilibrium p o int x = 0 is stab le in the sense of Lyapunov if, for any ǫ > 0 , ther e exists a δ > 0 such tha t: k x (0) k < δ = ⇒ k x ( t ) k < ǫ, ∀ t ≥ 0 (10) Definition 2 ( A sy mptotic Sta bility ) . The equilibrium p oint x = 0 is a symptotically stab le if it is stable in the sense of L yapunov and there exists a δ > 0 such that: k x (0) k < δ = ⇒ lim t →∞ k x ( t ) k = 0 (11) Definition 3 ( Exponential Stability ) . The equilibrium point x = 0 is exponentially stab le if there exist p ositive c o nstants α , β , a n d δ such that: k x (0) k < δ = ⇒ k x ( t ) k ≤ α k x (0) k e − β t , ∀ t ≥ 0 (12) Definition 4 ( G lobal Stability ) . A stability pr ope rty (Lya- punov , asymptotic, o r exponen tial) is said to be globa l if it holds fo r a ny initia l state x (0) ∈ R n (i.e., δ → ∞ ). Definition 5 ( Local Stability ) . A stability p r operty is sa id to be local if it h olds only for initial states within some bound ed neighbo rhood of the equilibrium (i.e., k x (0) k < δ for some finite δ > 0 ). 3) Analysis Problem in LTI systems : T he analysis prob le m in L TI systems inv olves determ ining system proper ties such as stability , perfo rmance, and robustness giv en a comp lete system description but we mainly r efer to stability in this r eport. Consider a c o ntinuou s-time L TI system : ˙ x ( t ) = Ax ( t ) (13) where x ( t ) ∈ R n is the state vecto r and A ∈ R n × n is the system matrix. The stab ility analy sis prob lem can be formu lated a s veri- fying whethe r all eig e n values o f A have negative r eal parts. Using L y apunov theory , this is e quiv a lent to deter mining whether the r e exists a p ositiv e definite matrix P such that: −h P A i s ∈ S ++ (14) This verification can be expressed as a conve x feasibility problem : find P subject to P ∈ S ++ − h P A i s ∈ S ++ (15) This fo r mulation p resents a sign ifica n t advantage: the p roblem can be transfo rmed into a LMI constraint, making it co nvex and solvable in p o lynomia l time. The framework extends naturally to o ther analysis prob lems includin g H 2 and H ∞ perfor mance assessment, input-to-state stability , and robust stability against mo del un certainties, with each c ase express- ible as a n SDP with appr o priate LMI c o nstraints. 4) Synthesis Prob lem in LTI systems : Th e synthesis p rob- lem in L TI systems in volves designing con trollers that achieve desired clo sed-loop p roperties such as stability , perf o rmance, and robustness but we m a inly r efer to the stabilizin g co ntroller design problem in this report. Consider a controllable L TI system: ˙ x ( t ) = Ax ( t ) + B u ( t ) (16) where x ( t ) ∈ R n is the state vector, u ( t ) ∈ R m is the con trol input, A ∈ R n × n is the system matrix , an d B ∈ R n × m is the input matrix. For a state f eedback control law u ( t ) = K x ( t ) , the closed - loop system bec o mes: ˙ x ( t ) = ( A + B K ) x ( t ) (17) A typical synthesis p roblem a ims to find a stabilizing feedback gain K th at minimize s a perform ance criterion, suc h as an H 2 or H ∞ norm. T his can be form ulated as: minimize Performan ce Measure ( A + B K ) subject to ( A + B K ) is stable (18) If the re is n o per forman c e criter io n to be optimized, then the resulting problem is known as stabilizing controller d esign problem and it can be g iv en b y the fo llowing feasibility problem : find ( P, K ) subject to P ∈ S ++ −h P A c i s ∈ S ++ A c = A + B K (19) While this pro blem ap pears non- conv ex d ue to the p roduct term P B K , it can be transfor med into a conve x p roblem throug h a chan ge of variables. By d efining X = P − 1 and M = K X wher e P ∈ S ++ , the stability constrain t becom es: h AX + B M i s ∈ S ++ (20) which r esults in the fo llowing pr o blem: find ( X , M ) subject to X ∈ S ++ −h AX + B M i s ∈ S ++ (21) where the ( P , K ) te r ms are given by ( X − 1 , M X − 1 ) . Remark 2. It should be noted that since P ∈ S n ++ , it is an in vertible ma trix and K ca n be compu ted after compu ting the ( X, M ) pair . 5) L yap unov Function and Stability of No nlinear Dynam- ical Systems : Theorem 1. F or (9 ) , whe r e the origin is the e q uilibrium point, If ∃ V ( x ) : D → R co ntinuou sly differ en tiable such th at V (0) = 0 , ˙ V (0) = 0 , V ( x ) > 0 , ∀ x ∈ D \ { 0 } , ˙ V ( x ) < 0 , ∀ x ∈ D \ { 0 } , then x = 0 is asymptotically stable; if, in addition, D = R n and V ( x ) is radially unb ounde d , then x = 0 is glo bally asymptotically sta b le. I f V ( x ) satisfies the constraints in Ω ⊆ D , then x = 0 is loca lly asymptotically stable an d Ω is called R O A . 6) Con vex Optimization Based Nonlinear Dyna mical Sys- tem Analysis : Definition 6. A conve x model for auto nomous systems is a set of first-or de r Or d inary Differ ential Equ ationss (ODEs) whose right-han d side ca n b e expr e ssed as a conve x su m of v ector fields (m o dels), n amely ˙ x ( t ) = r X i =1 h i ( z ) f i ( x ) (22) wher e r ∈ N repr esents the number of mo dels in the conve x sum, z ∈ R p may d e pend on the state x , time t , d isturbances or exogenous parameters, and th e non linear function s h i ( z ) , i ∈ { 1 , 2 , . . . , r } hold the conve x sum pr o perty in a comp act set C ⊆ R p , i.e.: r X i =1 h i ( z ) = 1 , 0 ≤ h i ( z ) ≤ 1 , ∀ z ∈ C . (2 3) and the spec ia l case is given by ˙ x ( t ) = r X i =1 h i ( z ) A i x ( t ) (24) Definition 7. A conve x mo del for no nauton omous systems is a set of first-or der ODEs whose right-han d side can be e x p r essed as a conve x sum of vector fields ( models), na mely ˙ x ( t ) = r X i =1 h i ( z ) f i ( x , u ) (25) wher e r ∈ N r epr esen ts the number of mod els in the conve x sum, z ∈ R p may depend on the state x , input u , time t , disturb a nces or exogenous p arameters, and the no n linear functions h i ( z ) , i ∈ { 1 , 2 , . . . , r } hold the con vex su m pr ope rty in a co mpact set C ⊆ R p , i.e.: r X i =1 h i ( z ) = 1 , 0 ≤ h i ( z ) ≤ 1 , ∀ z ∈ C . (2 6) Definition 8. The vecto r z in a co n ve x mod el (25) is kno wn as the pr emise or schedulin g vector; it is assumed to be boun ded and continu ously differ entiable in a com p act set C ⊆ R p of the schedulin g/pr emise sp a ce. Definition 9. The nonlinea r functions h i ( z ) , i ∈ { 1 , 2 , . . . , r } in a conve x mod e l (25) are known as Memb ership Function s (MFs); they hold the c on vex sum pr ope rty in a co mpact set C ⊆ R p of the scheduling space. Definition 10. I f f i ( x , u ) a r e linear , i.e., f i ( x , u ) = A i x + B i u , (25 ) is called a TS mod el [4 3]; if, in addition , z doe s n o t include the sta te x it is called a linear po lytopic model [44]; if th e TS model comes fr o m fuzzy modeling techniques it is also referr ed to as TS fuzzy mo del. Definition 1 1 . A TS mod e l is a set o f first-or der ODEs wh o se right-han d side can b e expr essed as a co n ve x sum of linea r models (kno wn as co nsequents or vertex models) via non linear functions (known as membership func tio ns, MFs) th a t hold the conve x sum pr operty [45], i.e. ˙ x ( t ) = r X i =1 h i ( z )  A i x ( t ) + B i u ( t )  , y ( t ) = r X i =1 h i ( z )  C i x ( t ) + D i u ( t )  . (27) where r ∈ N is the nu mber of vertex m o dels, x ∈ R n is the state vector , u ∈ R m is th e inp ut vector, y ∈ R q is th e output vector , z ∈ R p is the premise vector (wh ich may d epend on the state, time, or exog enous parameters and is a ssumed to be bound ed and smoo th in a compact set C ⊆ R p ), A i , B i , C i , D i , i ∈ { 1 , 2 , . . . , r } are matrices of prop er dime n sions, and the MFs h i ( z ) , i ∈ { 1 , 2 , . . . , r } hold the conve x sum pr op erty in C , g iv en by (26). Theorem 2. The origin x = 0 o f the un for ce d TS mo del (24) with δ x ( t ) = ˙ x ( t ) , t ∈ R , is asymp totically stable if there exis ts a matrix P = P ⊤ > 0 such that th e follo win g LMIs hold: P A i + A i ⊤ P < 0 , i ∈ { 1 , 2 , . . . , r } . (28) In this case, any trajectory x ( t ) starting within th e out- ermost Lyapunov level V ( x ) = x ⊤ P x ≤ k , k > 0 , in a compact set of th e state space X guaranteein g z ∈ C ⊆ R p goes a symptotically to x = 0 . Pr oof. Since P = P ⊤ > 0 , V ( x ) = x ⊤ P x is a valid L yapun ov functio n candidate; its time deriv ative alo ng the trajectories of the u nforced mo del (24) is ˙ V ( t ) = ˙ x ⊤ P x + x ⊤ P ˙ x (29) = r X i =1 h i ( z ) A i x ! ⊤ P x + x ⊤ P r X i =1 h i ( z ) A i x ! = r X i =1 h i ( z ) x ⊤  A ⊤ i P + P A i  x . (30) where the last equality made use of the pro p erty P r i =1 h i ( z ) = 1 . Since h i ( z ) ≥ 0 , i ∈ { 1 , 2 , . . . , r } , ∀ z ∈ C , no t all simultaneou sly 0, it is clear that LMIs (28 ) imply ∀ x 6 = 0 then ˙ V < 0 , which mean s V ( x ) is a valid L yapun ov function establishing asympto tic stability of x = 0 . There exists a compact set X o f th e state space with 0 ∈ X , guar anteeing z ∈ C , which means that any trajectory x ( t ) in the largest L yapun ov level with in X goes asymptotically to 0. Remark 3. If th e conve x sum pr operty holds everywher e, i.e. if z ∈ C ir r espective of x , the r esults above are glo bal, i.e. X = R n . I I I . RO A C O M P U TA T I O N U S I N G T S M E T H O D In this section, th e RO A estimation using TS metho d is described b y mea n s of an num e rical example. Consider the dy n amical system d escribed by th e following equations ˙ x 1 = − ( x 1 ) 2 − 2( x 2 ) − 2( x 1 ) ˙ x 2 = ( x 2 ) 3 − x 2 (31) where an estimate of the R OA need s to be com puted by u sing TS-conve x mo deling. Considering th e region giv en by D =  x ∈ R 2 : x 1 ∈ [ − 1 , + 1] , x 2 ∈ [ − 0 . 5 , +0 . 5]  (32) the dynam ics c a n be rewritten as  ˙ x 1 ˙ x 2  =  − x 1 − 2 − 2 0 ( x 2 ) 2 − 1)   x 1 x 2  (33) and th is can be written as ˙ x ( t ) = 1 X i 1 =0 1 X i 2 =0 w 1 i 1 ( z ) w 2 i 2 ( z )  − z i 1 1 − 2 − 2 0 z i 2 2 − 1)   x 1 x 2  = X i ∈ B 2 w i ( z ) ( A i x ( t )) = A w x ( t ) , (34) where the term s are defined as z 1 = x 1 , z 2 = ( x 2 ) 2 . (35) and a re defin ed o n the set given b y C =  z ∈ R 2 : ∃ x ∈ D s.t. z 1 = x 1 , z 2 = ( x 2 ) 2  . (36) Explicitly , the terms are written a s z 1 = 1 X i 1 =0 w 1 i 1 ( x ) z i 1 1 , z 2 = 1 X i 2 =0 w 2 i 2 ( x ) z i 2 2 . (37) where the bo undary -terms are comp uted by min x 1 ( t ) , x 2 ( t ) z 1 ( t ) = z i 1 =0 1 , max x 1 ( t ) , x 2 ( t ) z 1 ( t ) = z i 1 =1 1 , min x 1 ( t ) , x 2 ( t ) z 2 ( t ) = z i 2 =0 2 , max x 1 ( t ) , x 2 ( t ) z 2 ( t ) = z i 2 =1 2 . (38) and th e cor respond in g MFs are given by w 1 i 1 =1 ( x ) = z 1 1 − z 1 z 1 1 − z 0 1 , w 1 i 1 =0 ( x ) = 1 − w 1 i 1 =1 ( x ) . w 2 i 2 =1 ( x ) = z 1 2 − z 2 z 1 2 − z 0 2 , w 2 i 2 =0 ( x ) = 1 − w 2 i 2 =1 ( x ) . (39) Using Theo rem 2, the prob lem can be stated by the f ollowing feasibility p roblem: find P subject to P ∈ S ++ − h P A i i s ∈ S ++ , ∀ i (40) The m atrix is com p uted to be P =  0 . 2017 − 0 . 1326 − 0 . 132 6 0 . 7656  (41) and RO A is param eterized by a term, k which is compu ted by max k subject to Ω =  x ∈ D : x ⊤ P x ≤ k  Ω ⊆ D (42) and k is comp uted to be 0 . 17 using th e metho d described in [46] an d the co rrespond ing RO A is illustrated in Fig-1. Fig. 1. R OA for the system computed with the TS method I V . ROA C O M P U TA T I O N U S I N G C O M B I N E D S TA T E T R A N S F O R M AT I O N A N D T S M E T H O D For the stability analysis meth od, TS-metho d is useful in that it o ffers a straightforward algorithm to construct a L yapun ov fu n ction to g uarantee a local asym ptotical stability . Unfortu n ately , the results depen ds on the MFs and, as stated previously , that p rocess is n ot straightf orward, since there is not a un ique set of M Fs f o r a given system. Howe ver , instead of finding the ’best’ set of MFs to max imize the RO A , we can have a relatively simple ’metho d for comp uting th e set of MFs’ for a given dyna mical system and we can ge t one R OA , and then by implementin g a ch a nge of variables ¯ x = T ( x ) resulting in ˙ ¯ x = ¯ f ( ¯ x ) , and applying the same TS-method to th is system r esults in another R O A , we can compute the boun dary o f the ellipso idal region s (for example by r epresenting them as a set of points that are defined in ¯ x - space) and transfo rming th em into the x -space would give us another r egion th at is gu aranteed to b e locally asymp totically stable. Addition ally , the aforemen tioned transforma tio n for th e change o f variables can be as simple as a linear transfo rmation where T ( x ) = T x , whe r e T ∈ R n × n is a ny in vertible matrix. F or th e p r oblem stated in Section III, this ’chan ge of variables idea’ c an be impleme nted. By cho osing a simple transform ation matrix T =  1 2 0 1  (43) which r esults in the fo llowing system ˙ ¯ x 1 = − ( ¯ x 2 1 ) + 4( ¯ x 1 )( ¯ x 2 ) − 2( ¯ x 1 ) + 2( ¯ x 3 2 ) − 4( ¯ x 2 2 ) ˙ ¯ x 2 = ( ¯ x 3 2 ) − ¯ x 2 (44) and b y ch oosing the following set of MFs : z 1 = ¯ x 1 , z 2 = ¯ x 2 , z 3 = ¯ x 2 2 . (45) By c hoosing th e following region o f inter est ¯ D = { x : ¯ x 1 ∈ [ − 0 . 55 , +0 . 55 ] , ¯ x 2 ∈ [ − 0 . 55 , +0 . 55 ] } (46) and the RO A for the ¯ x -d omain an d the region that is comp uted by map ping that region b ack to th e origina l x -domain is illustrated in Fig -2. It can be seen that the red-ellipse which represents the bound aries of the e llipsoidal region that is comp uted using th e presented ’ change of variables meth od’ ha s some are a s outside of the o riginal RO A th at is com puted in the Section III an d that suggests that iteratively using a method (by auto matizing the set of MFs for a given nonlin ear d ynamical system ) results in a larger R OA. V . C O N C L U S I O N S In this study , we exp lo red the u se of SDP relaxatio ns and LMIs as powerful tools fo r analyzin g a n d designin g control systems. As a key application, we employed the TS modeling framework to study RO A estimation prob lem. The TS a p - proach ena b led us to leverage co n vex o ptimization techn iq ues to hand le no nlinear dynamics effecti vely . Furthermore , we propo sed an extension to the standard meth ods, aimed at Fig. 2. 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