Passivity and Passivity Indices of Nonlinear Systems Under Operational Limitations using Approximations

P assivit y and P assivit y Indices of Nonlinear Systems Under Op erational Limitations using Approxima tions Hasan Zak eri and P anos J. An tsaklis The authors are with the Department of Electrical Engineering, Univ ersity of Notre Dame, Notre Dame, IN 46556 USA. AR TICLE HISTOR Y Compiled January 4, 2019 ABSTRACT In this paper, we w ill discuss ho w operational limitations affect input- output b eha viours of the system. In particular, w e will provide form ulations for p assivity and passivity indices of a nonlinear system given operational limitations on the input and state v ariables. This formulatio n is presented in t h e form of lo cal p ass ivity and indices. W e will provide optimisation based formulatio n to derive passivit y prop erties of the system through polynomial ap p ro ximations. Tw o different a pproaches are taken to app ro ximate the nonlinear dynamics of a system through p olynomial functions; namely , T aylor’ s theorem and a multiv aria te generalisation of Bernstein p olynomials . F or each approach, conditions for stability , dissipativity , and passivit y of a system, as w ell as met h ods to fin d i ts passivit y indices, are giv en. Two different metho ds are also prese nted to reduce the size of the optimisation problem in T aylor’ s theorem approac h. Examples are provided to sho w the app lica bilit y of the results. KEYWO RDS Op erational Limitations; passivity indices; approximatio n; nonlinear systems; dissipativit y 1. In tro duction Ph ysical systems us ually ha v e inh eren t or imp osed op erational limitations. Whether it is a w all that limits th e range of motion in a rob ot arm, or th e limited force that we can apply to go v ern a system, these limitations should c hange o ur analysis of t he system. In this pap er, we will pr esen t ho w designers can consider knowledge of a system’s op eration in the inp ut-output analysis of the system. In doing so, w e fo cus on lo cal dissipativit y and extend it to lo cal passivit y and lo cal passivity indices of a sys tem giv en kno wn op erational limitations. P assivit y and dissipativit y are fun d amen ta l concepts in control theory (Willems 1972a; Willems 1972b) and ha v e b een used in man y applications (Bao and Lee 2007; Brogliato et al. 2007; Sepulc hre, Jank o vi ´ c, and Kokot o vi ´ c 199 7). T raditionally , they are used to guarant ee the stability of intercon- nected systems w ith robustness un der parameter v ariations. Passivit y can b e seen as an abstraction of a system’s b ehavio ur, wh ere the increase of energy stored in the sys tem is less than or equal to the supplied energy . P assivit y and dissipativit y h a v e sho wn great pr omise in the desig n of Cyb er- Ph ysical Systems (CPS ) (P . J . Ant saklis et al. 2013). Th eir impact on CPS design comes fr om their comp ositional prop ert y in negativ e feedbac k and parallel (more generally , energy conserving) in terconn ections (Bao and Lee 2007; Hill and P . Mo ylan 1976 ; v an der Sc haft 2017). Besides, und er CONT ACT H. Zakeri. Email : hzak eri@ nd.edu mild assu mptions, passivit y implies stabilit y (in the s ense of L 2 or asymp totic stability) . A su rv ey of applications of passivit y ind ice s in design of C PS can b e foun d in (Zake ri and P . Antsaklis 2018). P assivity and dissipativit y ha ve b een treated the same wa y for linear systems and nonlinear systems; ho w ev er, nonlin ea r systems require more d eta iled stud y . Sp ecifically , w e are intereste d in passivit y/dissip ati vit y b ehavio ur of nonlinear systems under different op erational conditions and differen t inputs. This distin ction h as alw ays b een made for lo cal internal stabilit y in the Ly apunov sense (Sastry 2013; U. T op cu and A. P ac k ard 2009), when the system is not externally excited. When there is an exogenous inpu t applied to the system, few researc hers ha v e add ressed the inpu t- output b eha v iour of the system sub ject to op eratio nal constraints. In (Ufuk T op cu and Andr ew P ack ard 2009b), the authors hav e addressed L 2 − gain of nonlinear s y s tems lo cally , and extended th e results to uncertain systems, b oth with unmo deled d ynamics or those with parametric uncertain t y . The present pap er mo dels the op erational limitations as inpu t and state constraints and fo cuses on lo cal passivity and dissipativit y of nonlinear systems. Sev eral at tempts hav e b een made to d ev elop wa ys to find Lyapuno v fun ctio nals for p articular classes of systems, lik e linear systems or nonlinear systems describ ed by p olynomial fi elds (Antonis P apachristodoulou and Stephen Pra jna 2002), Ho wev er, a general metho dology is still lac king. Th is pap er addresses this gap by p ro viding wa ys to find Ly apuno v f u nctionals through appr o x im ations, with an emph asis on d iss ipativit y applications. The most common f orm of app ro ximation is linearisation, whic h gives us a very tractable mo del with many analysis and sy nthesis to ols av ailable. The relation b etw een p assivit y of a n onlinear system and p assivity of its appro ximation is studied in (M. Xia et al. 2015; Meng Xia et al. 2017), where the authors sho w th at when the linearised mo del is simultaneously strictly passiv e and strictly input passive, the nonlin ear system is p assiv e as w ell, within a neigh b ourho o d of t he equilibrium p oin t around whic h the linearisation is done. How ev er, in general, the linearisation is only v alid within a limited neigh b our hoo d, a nd the a pproximat ion error can b e high. T h e relation b et we en appro ximation error, the neigh b ourho o d of stud y , and passivit y/dissipativit y are not evident in linearisation. In (Ufuk T op cu and An d rew Pac k ard 20 09a), the r elation b et ween linearisation and optimisation b ased study of nonlinear systems is pr esented, and conditions are pr esented b ased on linearisation for the feasibilit y of the optimisation problem. Here we prop ose approximati ons through multiv ariate p olynomial fun cti ons. T he metho dology discussed in the p resen t p ap er giv es us approximat e mo dels in a we ll-defined neigh b ourho o d of an op erating p oint along with error b oun ds. Cen tral to this approximat ion is th e Stone-Weierstr ass appr oximation the or e m, w hic h states th at under certain circumstances, an y real-v alued con tin u ous function can b e appro ximated b y a p olynomial function as closely as desired. Tw o differen t metho ds to a pproximat e a n onlinear f unction hav e b een emplo yed here. The first metho dology is T a ylor’s Theorem, wh ic h gives a p olynomial app ro ximation an d b oun ds on the error f unction. Despite the simplicit y and intuitiv en ess of the app ro ximation, find ing error b ounds in this metho d requires com- plicated calculations, and results in large optimisation pr oblems for r eal -w orld applications. Th e second appr o ximation metho d is thr ough Multiv ariate Bernstein P olynomials with more straight - forw ard calculations that lead to a more tractable optimisation p roblem. Sev eral results are give n to test b oth lo cal stability and lo cal dissipativit y of a n onlinear system through sum-of-squares optimisation and p olynomial app r o ximations of th e system. Lo cal QSR-dissip ati vit y of the system, lo cal p assivit y , and lo cal p assivit y indices are also derived from the d issipativit y results. Both these metho ds require mild assump tions on the system and are g enerally app lica ble to b road classes o f systems. The firs t app roac h requires a differentia bilit y condition, whic h is satisfied b y a ma jorit y of practical systems. The second app roac h only r equires the Lips c hitz condition to derive app ro x- imation b ound s. This is not at all a limiting factor since the Lipsc hitz condition is essen tial in uniqueness and existence of solution (Kh alil 2002). The orga nisation of this pap er is as fo llo ws: Section 2 presen ts in tro ductory ma terials on dis- sipativit y an d p assivit y of dynamical systems. Section 3 motiv ates the lo cal passivit y analysis of 2 nonlinear systems under op erational constraint s through an example and in tr odu ces defin itio ns for lo cal dissipativit y , passivit y , and p assivit y ind ice s. S ec tion 4 presents t w o differen t app r o ximations for a n onlinear system and metho ds f or studying lo cal dissipativit y and passivity of the system through eac h appr oximati on metho d. S p ecifica lly , the first part of s ec tion 4 co ve rs T a ylor’s theo- rem approac h . Th eorem 2 giv es conditions to chec k d iss ipativit y of a nonlin ear system with r espect to a giv en sup ply rate f u nction. Ho wev er, the computational complexit y of the optimisation can b e quite high when the order of appro ximation or th e order of the system’s dynamics increase. Theorem 3 r educes the size of the optimisation pr oblem b y approximati ng th e error terms by ellip- soids pr oviding optimisation constrain ts for sp ecific admissible control and state sp ac e. Corollary 4 form ulates s imila r results for lo cal s tability of a nonlinear system. The second part of section 4 presents a generalisation of Berstein p olynomials for m ultiv ariate functions follo wed b y results o n the analysis of a nonlinear system through its app r o ximation b y Bernstein p olynomials. Sp ecifically , C oroll ary 6 presents conditions for lo cal s ta bilit y of a nonlinear system, while Theorem 5 presents a metho d to c heck dissipativit y of a system with resp ect to a giv en supply r ate th rough Bernstein’s appro ximation metho d. T he rest of se ction 4 p resen ts cond itions for QS R − dissipativit y and passivit y and metho ds to find passivit y indices of a s y s tem through eac h app roac h . Section 5 giv es examples to demonstrate the applicabilit y of the results. Add itio nal mathematical details of th e Stone-W eierstrass theorem, Bernstein Polynomia ls, and generalised S − pro cedure is in the app endix. Finally , concluding remarks are giv en in section 6 . 2. Preliminaries 2.1. Passivity and Dissip ativity Consider a con tin u ous-time dy n amica l system H : u → y , where u ∈ U ⊆ R m denotes th e inp ut and y ∈ Y ⊆ R p denotes th e corresp onding output. Consider a rea l-v alued fun cti on w ( u ( t ) , y ( t )) (often referred as w ( t ) or w ( u , y ) when clear from conte n t) asso cia ted with H , called supply r ate function. W e assume that w ( t ) satisfies t 1 Z t 0 | w ( t ) | d t < ∞ , (1) for every t 0 and t 1 . No w consid er a con tinuous-time system d escrib ed b y ˙ x = f ( x , u ) y = h ( x , u ) , (2) where f ( · , · ) and h ( · , · ) are Lip sc hitz mapp in gs of prop er dimensions, and a ssume the o rigin is an equilibrium p oin t of the system; i.e., f (0 , 0) = 0 and h (0 , 0) = 0 . Definition 1. The system describ ed by (2) is called dissip ative with r esp e ct to supply r ate f unction w ( u ( t ) , y ( t )) , if there exists a n onnegati v e function V ( x ) , called the stor age fu nction, suc h that V (0) = 0 and for all x 0 ∈ X ⊆ R n , all t 1 ≥ t 0 , and all u ∈ R m , w e hav e V ( x ( t 1 )) − V ( x ( t 0 )) ≤ t 1 Z t 0 w ( u ( t ) , y ( t )) d t. (3) 3 where x ( t 0 ) = x 0 and x ( t 1 ) is the state at t 1 resulting from in itia l condition x 0 and inpu t function u ( · ) . The inequalit y (3) is called dissip ation ine quality and expresses the fact that the energy “stored” in the system at an y time t is n ot more than the initially s to red energy plus the total energy supp lied to the system d u ring this time. If th e dissipation inequalit y holds strictly , then th e system (2) is called strictly dissipativ e with r esp ect to sup ply r ate function w ( t ) . If V ( x ) in Definition 1 is d ifferen tiable, then (3) is equiv alen t to ˙ V ( x ) : = ∂ V ∂ x · f ( x , u ) ≤ w ( u, y ) . (4) According to the d efi nition of supply rate, w ( t ) can tak e an y form as long as it is lo call y in tegrable, h o w ev er, w e are particularly in terested in the case wh en w ( t ) is q u adratic in u and y . More f orm all y , a d ynamical system is called QS R -dissip ative if its su pply rate is give n by w ( u , y ) = u ⊺ R u + 2 y ⊺ S u + y ⊺ Q y , (5) where Q = Q ⊺ , S and R = R ⊺ are matrices of app ropriate dimensions. On e reason for considering suc h quadratic supply rate is that by selecting Q, S and R, we can obtain v arious n oti ons of passivit y and L 2 stabilit y . F or instance, if a system is dissipativ e with supp ly rate giv en b y (5) where R = γ 2 I , S = 0 and Q = − I , th en the syste m is L 2 stable with finite gain γ > 0 (Haddad and Chellab oina 2008). Definition 2 (P assivit y (Hi ll and P . Mo y lan 1976 ; Willems 1 972b)) . System (2) is called passive if it is d issipativ e with resp ect to the supp ly r ate fu n ctio n w ( u , y ) = u ⊺ y . The relation b etw een different n otio ns of passivit y as well as their relation to Lya punov stabilit y and L 2 stabilit y has b een extensively studied (see (Kottenstette et al. 2014) and the references therein). 2.2. Passivity Ind ic es The p assivit y index framework generalizes passivit y to systems that ma y not b e passiv e; In other w ord s, it captur es the leve l of passivity in a system. If on e of the systems in a negativ e feedbac k in terconn ection h as “shortage of passivit y ,” it is p ossible th at “excess of p assivit y in the other system can assure the passivit y or stabilit y of th e in terconn ection. More information on the comp ositional prop erties of passivit y through p assivit y indices can be found in (B ao and Lee 2007 ) and (Khalil 2002, p. 245). Definition 3 (Inpu t F eed-forw ard Passivit y Ind ex) . Th e system (2) is called input fe e d-forwar d p assive (IFP) if it is dissipativ e with resp ect to supply r ate f unction w ( u , y ) = u ⊺ y − ν u ⊺ u for some ν ∈ R , denoted as IFP( ν ). In put feed-forwa rd passivit y (IFP) index f or sy s tem (2) is the largest ν for wh ich the s y s tem is IFP . Definition 4 (Output F eedbac k P assivit y) . The system (2) is called output fe e db ack p assive (OFP) if it is dissipativ e with resp ect to supply rate function w ( u , y ) = u ⊺ y − ρ y ⊺ y for some ρ ∈ R , denoted as OFP( ρ ). Ou tput feedbac k p assivit y (OFP) in dex for system (2) is the la rgest ρ fo r whic h the system is O FP . 4 3. P assivity Under Op erational Limitations Unlik e linear systems, imp ortant p rop er ties of nonlin ear sys tems, lik e stabilit y , are typical ly studied in a neigh b ourho o d of an equilibr ium p oin t or other stationary sets and lo cal analysis d oes not necessarily imply global stability . Lo cal s ta bilit y and region of con verge nce h a v e b een s tudied b efore using differen t tec hniques (Henrion and Kord a 2014; U. T op cu and A. P ac k ard 2009; U. T op cu, A.K. P ack ard, et al. 2 010); ho w ev er, dissipativit y and passivit y o f nonlinear systems under constrain ts still r equire more in-depth study . T o further expand this p oin t, we start with an example (Z ak er i and P . J. Antsaklis 2016). Consider a non lin ea r system go verned by the follo w ing d ynamics. ˙ x = − x + x 3 + ( − x + 1) u y = x − x 2 + ( 1 2 x 2 + 1) u (6) It is p ro v ed in (Zak eri an d P . J. Ant saklis 2016) that this system is passiv e f or X =  x | x 2 − 1 ≤ 0  (7) with a qu artic s torage f u nction V ( x ) = − 0 . 4581 x 4 + 1 . 416 x 2 . (8) Ho wev er , a closer lo ok at the system’s dynamics sho w s w h y it can not b e glo bally passive. This system has a stable equ ilibr ium p oin t at x = 0 . It also h as t w o un s ta ble equilibrium p oints at x = 1 and x = − 1 . The linearization of the system around x = − 1 is ˙ x = 2 x + u, y = 3 x + u, whic h is observ able but not Lyapuno v stable. Therefore, the n onlinear system (6) cannot b e globally passiv e (Haddad and Chellab oina 2008, Corollary 5.6). F urtherm ore, the passivit y in dices also dep end on the o p erating r egi on of the system. An example of this depend ence is given in (Zak er i and P . J . An tsaklis 2016), where the system is pro ved to h a v e an output p assivit y ind ex of 0 . 35 for X = { x | k x k 2 ≤ 2 . 47 } , (9) but the index decreases as the state space r adius increases, and at some p oint b ecomes negativ e and renders the system non-passive . Figure 1 plots the p r o v ab le O FP index of the system for differen t v alues of r ; wh er e X = { x | k x k 2 ≤ r } . A simpler example c an b e foun d in (S epulc hre, Ja nko vi´ c, and Kok oto vi ´ c 1997, Ch ap. 2). Defining dissipativit y prop erties for nonlinear systems with resp ect to constrain ts requ ires care- ful consideration of the admissible con trol and how w e restrict the state space (operational limi- tations in this case are modeled as constraints ov er the in put and state spaces). There a re a few attempts in the literature to add ress this pr oblem u sing different approac hes. In (Na v arro-L´ op ez and F ossas-Colet 2004), the authors defi n ed lo cal p assivit y in a neighbour hoo d of x = 0 , u = 0 with no further restriction. O n the other hand, in (Nijmeijer et al. 1992), lo cal passivit y is defin ed through a dissipation inequalit y holding for all x 0 ∈ B 0 and for all con trol u suc h that Φ( t, x 0 , u ) ∈ B 0 for t ≥ 0 , wh ere Φ( t, x 0 , u ) is the f ull system resp onse. In other w ord s, lo cal passivit y is defined in a ball around the origin for th e initial condition and for all inp uts that do n ot drive the states “a wa y” f rom th e origin. While this assump tio n is useful, we are lo oking for a more explicit for- m u lati on of the adm iss ible in put space as well. In (Bourles and C oll edani 1995), lo cal passivit y is defined b y pu tting constrain ts on th e magnitude of the inpu t signal and its deriv ativ e by using Sob olev spaces. This defin itio n is based on suitable norm s and inner pr odu cts defin ed o v er th e 5 space. In (Hemanshu Ro y Po ta and Peter J. Mo ylan 1990; H. R. Pota and P . J . Mo ylan 1993), lo cal dissipativit y is d efined in terms of local internal stability regions and sm all gain in puts. Ho wev er, w e are lo oking for an approac h that can b e naturally extended to passivit y in dices and has the same useful imp lica tions as passivit y in the global sense. Here W e discuss lo cal passivit y indices, and we introd u ce app roac h es to determine th ese ind ices u sing p olynomial app ro ximations. T o the b est of our kno w ledge, lo cal p assivit y indices for nonlinear sys tems were considered in (Z akeri and P . J. Antsa klis 2016) firs t. Definition 5 (Lo cal Dissipativit y) . A giv en system of th e form (2) is called locally dissipativ e if (3) holds for ev ery u ( t ) ∈ U ⊂ R m and x ∈ X ⊂ R n , such that for ev ery input signal u ( t ) ∈ U , the resulting s ta te tra ject ories alwa ys remain in X . I t is assumed that X cont ains th e origin. Definition 6. A system is lo c al ly p assive if it is lo cally dissipativ e with resp ect to the sup ply rate function w ( u , y ) = u ⊺ y f or every u ( t ) ∈ U ⊂ R m and x ∈ X ⊂ R n , such that f or every inp ut signal u ( t ) ∈ U , the state tra jectories will alwa ys remain in X . Definition 7. Th e local output feedb ack passivit y (OFP) ind ex is the largest gain th at can b e placed in p ositiv e feedbac k such that the inte rconnected system is p assiv e and for ev ery u ( t ) ∈ U ⊂ R m , the state remains in X , i.e., x ∈ X ⊂ R n for all times, where X and U satisfy th e same assumptions as in Definition 2. Th is n otio n is equiv alen t to the f ollo wing dissipativ e in equalit y holding f or the largest ρ , and for ev ery u ∈ U and x ∈ X (Zak eri and P . J . Antsakli s 2016) Z T 0 u ⊺ y d t ≥ V ( x ( T )) − V ( x (0)) + ρ Z T 0 y ⊺ y d t. (10) Definition 8. The local input feedforw ard passivit y (IFP) index is the largest g ain that ca n b e put in a negativ e parallel interconnectio n with a system such that the interco nnected system is passiv e and for ev ery u ( t ) ∈ U ⊂ R m , the state remains in X , i.e., x ∈ X ⊂ R n for all times, where X an d U satisfy the same assum ptions as in Definition 2. Th is n oti on is equ iv alent to the follo w ing dissipativ e in equalit y holding for the largest ν , and for ev er y u ∈ U and x ∈ X Z T 0 u ⊺ y d t ≥ V ( x ( T )) − V ( x (0)) + ν Z T 0 u ⊺ u d t. (11) A p ositiv e index in d icat es that the system has a p ositiv e feedforwa rd path for all x ∈ X and that the zero d ynamics are lo cally asymptotically stable. Otherw ise, th e index will b e negativ e. Lo cal p assivit y and lo cal dissipativit y as defin ed in this section can offer many p racti cal adv an- tages. In most control applications, the aim is to k eep the system w orking around an equilibrium, and giv en the practical limitations, global analysis is n ot alw a ys meaningful. F or example, a p endu - lum, when it is up righ t, has very d ifferen t b eha viours than wh en it is hanging, ev en though b oth are equilibria of the s y s tem. T his definition of lo cal p assivity and diss ipativit y addresses these k in ds of op erational conditions an d actuator limitations. The same adv an tages th at passivity and d issipativ- it y ha v e provided in th e design and an alysis of systems hold for lo cal passivity and dissipativit y as w ell. F or example, if b ound s on the signals are met, we will ha ve the same comp ositional pr operties for lo cal passivit y as wel l; And this is not a limiting requirement, as most often the feedbac k lo op is arranged to k eep the signals within a desired region. Th is is a contrasting view to, for example, the notion of Equ ilibrium indep endent p assivity, w here th e dissipation inequalit y needs to hold against ev ery p ossible equilibrium p oin t (Hines, Arcak, and A. K. P ac k ard 2011). Equilibrium ind ep en den t passivit y is a generalisat ion of passivit y to the case s where the e xact lo cat ion of the equilibrium p oin t is un kno wn, mostly due to in terconn ection, un ce rtain t y , and v ariation in parameters. On the 6 other hand, lo cal passivit y enables us to ha v e a more precise kno wledge of the system within its op erational conditions. 4. P olynomial Appro ximations Here, we will discuss metho ds to study certain b eha viour s of a sy s tem through its appro ximations. W e will present t w o different m etho ds of appr o x im ation along with related optimisation problems. First, recall a well- kno wn theorem in appr oximati on th eory . Theorem 1 (W eierstrass Approximat ion Theorem (Ap ostol 1974)) . Supp ose f ( · ) is a r e al-value d and c ontinuous function define d on the c omp act r e al interval [ a, b ] . Then for every ε > 0 , ther e exi sts a p olynomial p ( x ) (which might dep end on ε ) such that for al l x ∈ [ a, b ] , we have | f ( x ) − p ( x ) | < ε, or e qui v alent ly, the supr emum no rm k f − p k < ε. This theorem w as then generalise d (by Ma rshall H. Stone) in tw o regards. First, it c onsiders an a rbitrary compact Ha usdorff space X (here w e tak e neigh b orho o ds in R n ) instead of the real in terv al [ a, b ]. Second, it inv estigates a m ore general subalgebra (multiv ariate p olynomials in R n in this case), rather than the algebra of p olynomial fu nctions. Th is theorem is included in the app endix, but we will d iscuss direct results later on. 4.1. App r o ach Base d on T aylor’s The or em A direct result of the Stone-W eierstrass theorem is T aylor’s the or em, w hic h giv es a metho d of finding a p olynomial app ro ximation of a fun cti on and determining b ound s on approxi mation error. The multiv ariate case of T a ylor’s theorem is r eported in the App endix. T o c h ec k lo cal dissipativit y of the system u sing T a ylor’s appro x im ation, the dissipation inequal- it y (3) needs to b e rewritten by substituting f ( x , u ) w ith its T a ylor app ro ximation (1), and solv ed for every v alue of x and u in X and U . Th e r emainder term is of course non-p olynomial, and the exact v alue is not k n o wn. Ho w ev er, it can b e b ound ed by (3 ), so (3 ) holds f or ev ery v alue of R in those b oun d s. This is an infinite dimen s ional optimization problem, since x , u , and R tak e infinite v alues. One w a y to deal with this p roblem is to b ound R inside a p olytop e, by sa ying r ≤ R ≤ r , and rewrite the in equ ali t y f or ev ery vertex of this p olytop e. A similar approac h is taken in (Chesi 2009), for a simp ler case wh ere nonlinearit y is only a function of one of the s ta te v ariables and app ears affinely in the dynamics. T his is not an efficien t wa y to hand le the uncertaint y in R , since w e n eed to solv e the optimisation for all 2 n 2 k v ertices of the p olytop e at the same time. On th e other hand, g iv en th e general s tructure of X and U , the same approac h migh t not apply to tak e these b ounds into accoun t. Even when there is sparsity or other desir ab le prop erties in the prob lem, this is still a large problem to solve. T o h andle th is pr oblem one could use the generalised S -Pro cedure to redu ce the size of the program. These c onditions should hold for a neigh b ourho o d around the origin, and this fact should r eflect in the formulation as w ell. Th e follo wing theorems addr ess these issues, b ut first, we will state the assu mptions needed in the th eo rems. Assumption 1. The input to system (2) is cont ained in U , i.e. u ∈ U , f or all t ≥ 0 and for ev ery u ∈ U , the resu lting tra jecto ries of the system sta y in X foreve r, i.e . x ∈ X , where X and U are defined app ropriately . Theorem 2. Consider the system define d in e qu ation (2) that ho lds Assumption 1. Also assume 7 that f ( · , · ) and h ( · , · ) satisfy the assumptions for T aylor’s The or em. Define sets X and U as X = { x | x ( t ) ∈ R n | g i ( x ( t )) ≤ 0 , i = 1 , . . . , I X , ∀ t ≥ 0 } (12) U =  u | u ( t ) ∈ R m | g ′ j ( u ( t )) ≤ 0 , j = 1 , . . . , I U , ∀ t ≥ 0  . (13) This system is dissip ative with r esp e c t to the p olynomial supply r ate function w ( u , y ) , if ther e exists a p olynomial function V ( x ) , c al le d a stor age function, that is the solution to the fol lowing optimization pr o gr am V ( x ) + I X X i =1 s 1; i ( x ) g i ( x ) ≥ 0 − n X i =1 ∂ V ( x ) ∂ x i   X | α 1 | + | α 2 |≤ k − 1 D α 1 x f i ( x , u ) D α 2 u f i ( x , u ) α 1 ! α 2 !     x =0 u =0 x α 1 u α 2 + X | β 1 | + | β 2 | = k r i ; β 1 ,β 2 x β 1 u β 2   + w ( u , ˆ y ) − n X i =1 X | β 1 | + | β 2 | = k ( s 2; i,β 1 ,β 2 ( r i ; β 1 ,β 2 − r i ; β 1 ,β 2 ) + s 3; i,β 1 ,β 2 ( r i ; β 1 ,β 2 + r i ; β 1 ,β 2 )) − p X i =1 X | δ 1 | + | δ 2 | = k  s 4; i,δ 1 ,δ 2 ( t i ; δ 1 ,δ 2 − t i ; δ 1 ,δ 2 ) + s 5; i,δ 1 ,δ 2 ( t i ; δ 1 ,δ 2 + t i ; δ 1 ,δ 2 )  + I X X i =1 s 6; i ( x , u ) g i ( x ) + I U X j =1 s 7; j ( x , u ) g ′ j ( u ) ≥ 0 (14) for some no nne g ative p olynomials s 1; i to s 7; i , wher e ˆ y = [ ˆ y 1 , . . . , ˆ y p ] ⊺ , and ˆ y j = X | γ 1 | + | γ 2 |≤ k − 1 D γ 1 x h j ( x , u ) D γ 2 u h j ( x , u ) γ 1 ! γ 2 ! x γ 1 u γ 2 + X | δ 1 | + | δ 2 | = k t j ; δ 1 ,δ 2 x δ 1 u δ 2 . (15) Her e, r i ; β 1 ,β 2 and t i ; δ 1 ,δ 2 ar e upp er b ounds for the r emainder terms of T aylor’s appr oximation of f ( · , · ) and h ( · , · ) , r esp e ctively, w hich c an b e c ompute d by (3) . Pr o of. Theorem 10 can b e app lied to app r o ximate f ( x , u ) and h ( x , u ) in nonlin ear system (2) as k -th ord er p olynomials as follo ws d x i d t = X | α 1 | + | α 2 |≤ k − 1 D α 1 x f i ( x , u ) D α 2 u f i ( x , u ) α 1 ! α 2 ! x α 1 u α 2 + X | β 1 | + | β 2 | = k R i ; β 1 ,β 2 ( x , u ) x β 1 u β 2 , y j = X | γ 1 | + | γ 2 |≤ k − 1 D γ 1 x h j ( x , u ) D γ 2 u h j ( x , u ) γ 1 ! γ 2 ! x γ 1 u γ 2 + X | δ 1 | + | δ 2 | = k T j ; δ 1 ,δ 2 ( x , u ) x δ 1 u δ 2 . (16) 8 W e rewr ite (3) and (4) and substitute f ( · , · ) and h ( · , · ) with their T a ylor’s expansions (16). Since the remainders are not necessarily p olynomial and their exact form are not known, we replace R i ; β 1 ,β 2 ( x , u ) and T i ; δ 1 ,δ 2 ( x , u ) with algebraic v ariables r i ; β 1 ,β 2 and t i ; δ 1 ,δ 2 , whose b oun ds can b e written as − r i ; β 1 ,β 2 ≤ r i ; β 1 ,β 2 ≤ r i ; β 1 ,β 2 , − t j ; δ 1 ,δ 2 ≤ t j ; δ 1 ,δ 2 ≤ t j ; δ 1 ,δ 2 . (17) T aking error b oun ds (17) and sets X and U defin ed in (12) and (13) and emplo ying the generalised S -Pro cedure to incorp orate th em with the diss ip ati on in equalit y prov es the theorem. Ev en though based on T a ylor’s theorem, the approximat ion can b e as close as desired, there is alw ays the prob lem of increasing the complexit y as th e size increases. More precisely , we will need 4 n 3 k 2 nonnegativ e p olynomials as generalised S − pro cedure m ultipliers for error b ound s. If eac h of these m ultipliers is of degree κ, th en the approximati on will im p ose a total of approximat ely κ !( n + m ) n 3 k 2 unknown v ariables to the optimisation problem. This increase in the size will b e- come a problem ev en in the most straigh tforward examples; therefore it is necessary to deriv e a more tractable solution. The follo wing theorem presents m ore tractable r esult by sur rounding the appro ximation er r ors in an ellipsoid. Theorem 3. Consider the system define d in e qu ation (2) that ho lds Assumption 1. Also assume that f ( · , · ) and h ( · , · ) satisfy the assumptions of T aylor’s The or em, and sets X and U ar e define d as (12) and (13) , r e sp e ctively. Then this system is lo c al ly dissip ative with r esp e ct to the p olynomial supply r ate function w ( u , y ) , if ther e e xists a p olynomial V ( x ) c al le d stor age function that is solution to the fol low ing fe asibility pr o gr am V ( x ) + I X X i =1 s 1; i ( x ) g i ( x ) ≥ 0 − n X i =1 ∂ V ( x ) ∂ x i   X | α 1 | + | α 2 |≤ k − 1 D α 1 x f i ( x , u ) D α 2 u f i ( x , u ) α 1 ! α 2 !     x =0 u =0 x α 1 u α 2 + X | β 1 | + | β 2 | = k r i ; β 1 ,β 2 x β 1 u β 2   + w ( u , ˆ y ) − n X i =1 s 2; i ( x , u )   r i − X | β 1 | + | β 2 | = k r 2 i ; β 1 ,β 2   − p X i =1 s 3; i ( x , u )   t i − X | δ 1 | + | δ 2 | = l t 2 i ; δ 1 ,δ 2   + I X X i =1 s 4; i ( x , u ) g i ( x ) + I U X j =1 s 5; j ( x , u ) g ′ j ( u ) ≥ 0 (18) for some nonne gative p olynomials s 1; i to s 5; i , wher e ˆ y = [ ˆ y 1 , . . . , ˆ y p ] ⊺ , ˆ y j is define d as in (15) , r i and t i ar e d efine d as r i = X | β 1 | + | β 2 | = k r 2 i ; β 1 ,β 2 t i = X | δ 1 | + | δ 2 | = k t 2 i ; δ 1 ,δ 2 . (19) 9 Pr o of. Conditions in (18) ensures, through generalised S -Pro cedure, that X | β 1 | + | β 2 | = k r 2 i ; β 1 ,β 2 ≤ X | β 1 | + | β 2 | = k r 2 i ; β 1 ,β 2 = r i and X | δ 1 | + | δ 2 | = l t 2 i ; δ 1 ,δ 2 ≤ X | δ 1 | + | δ 2 | = k t 2 i ; δ 1 ,δ 2 = t i whic h imp lies that the dissipation inequalit y holds for an y v alue of r 2 i ; β 1 ,β 2 b et wee n − r i ; β 1 ,β 2 and r i ; β 1 ,β 2 , and any v alue of t 2 i ; δ 1 ,δ 2 b et wee n − t i ; δ 1 ,δ 2 and t i ; δ 1 ,δ 2 . The ab o ve program has only 2 n + 2 I X + 2 I U m u ltipliers, w here 2 n of th ese m ultipliers are for error b ound s . This w ill yield to 2 κ !( n 2 + mn ) unkn own v ariables in optimisation if eac h m u ltiplier is of d eg ree κ. This is a muc h smaller num b er compared to th e former case. Remark 1. If the order of appr o ximation in either Theorem 2 or Theorem 3 is 1, i.e. k = 2 , and the approximati on error is n eg ligible in the region of study , then the p olynomial approxima tion will b e equiv alen t to linearization. Indeed, this is where optimisation and linearization based tec h- niques coincide. Inte rested r eaders can r efer to (Ufuk T op cu and An drew Pac k ard 2009a) for m ore information on linearizatio n based analysis versus optimisation based analysis of n onlinear systems. Stabilit y can also b e studied through dissipativit y resu lts here. Corollary 4. The nonline ar system describ e d by the fol lowing set of or dinary differ ential e quations ˙ x = f ( x ) (20) has a lo c al stable e quilibrium p oint at origin for x ∈ X , if ther e exist a p olynomial V ( x ) and nonne gative p olynomial s s 1; i , s 2; i,β , s 3; i,β and s 4; i for | β | = k satisfying the fol lowing c onditions V ( x ) − φ 1 ( x ) + I X X i =1 s 1; i ( x ) g i ( x ) ≥ 0 − n X i =1 ∂ V ( x ) ∂ x i   X | α |≤ k − 1 D α f i ( x ) α !     x =0 x α + X | β | = k r i ; β x β   − n X i =1 X | β | = k ( s 2; i,β ( r iβ − r i ; β ) + s 3; i,β ( r i ; β + r i ; β )) + I X X i =1 s 4; i ( x , u ) g i ( x ) − φ 2 ( x ) ≥ 0 (21) wher e ϕ 1 and ϕ 2 ar e a rbitr ary p ositive definite p olynomials. 4.2. App r o ach Base d on Bernstein Polynomials There is a second app roac h to Stone-W eierstrass theorem u s ing Bernstein p olynomials used here to reduce the computation cost. Details of Bernstein p olynomials alo ng with con v ergence pro of and 10 error m argin can b e foun d in the App endix. The next theorem p ro vides a n umerical to ol to test diss ipativit y of a nonlinear system through Bernstein p olynomials appr oximati ons. As men tioned b efore, QS R − dissipativit y , passivit y , and passivit y indices can b e d eriv ed from this theorem as w ell. Refer to Remark 6, Remark 7, an d Theorem 8 f or details. Theorem 5. The system define d in (2 ) is lo c al ly dissip ative with r esp e ct to the supply r ate function w ( u , y ) over X and U define d as X =  x ∈ R n | | x i | ≤ 1 2 , i = 1 , . . . , n  (22) U =  u ∈ R m | | u j | ≤ 1 2 , j = 1 , . . . , m  , (23) if ther e e xist a p olynomial f u nction V ( x ) that is the solution to the fol low ing fe asibility pr o gr am V ( x ) − φ 1 ( x ) + n X i =1  s 1 ,i ( x i − 1 2 ) − s 2 ,i ( x i + 1 2 )  ≥ 0 , n X i =1  − ∂ V ( x ) ∂ x i ( b i ( x , u ) + ε i ) + w ( u , b ′ ( x , u ) + ε ′ ) + s 3 ,i ( ε i − ε i ) − s 4 ,i ( ε i + ε i ) + s 5 ,i ( x i − 1 2 ) − s 6 ,i ( x i + 1 2 ) + s 7 ,i ( u i − 1 2 ) − s 8 ,i ( u i + 1 2 ) + s 9 ,i ( ε ′ i − ε ′ i ) − s 10 ,i ( ε ′ i + ε ′ i )  ≥ 0 (24) wher e b i ( x ) = B µ i 1 ,...,µ i n ,µ i n +1 ,...,µ i n + m ( f )( x 1 , . . . , x n , u 1 , . . . , u m ) = X 0 ≤ k j ≤ µ i j 1 ≤ j ≤ n + m f i  k 1 µ i 1 − 1 2 , . . . , k n µ i n − 1 2 , k n +1 µ i n +1 − 1 2 , . . . , k n + m µ i n + m − 1 2  × n Y j =1  µ i j k j  ( x j + 1 2 ) k j ( 1 2 − x j ) µ i j − k j  × m Y j =1  µ i j + n k j + n  ( u j + 1 2 ) k j + n ( 1 2 − u j ) µ i j + n − k j + n  (25) 11 and b ′ ( x , u ) = [ b ′ 1 , . . . , b ′ p ] ⊺ , wher e b ′ i ( x ) = B η i 1 ,...,η i n ,η i n +1 ,...,η i n + m ( h )( x 1 , . . . , x n , u 1 , . . . , u m ) = X 0 ≤ k j ≤ η i j 1 ≤ j ≤ n + m h i  k 1 η i 1 − 1 2 , . . . , k n η i n − 1 2 , k n +1 η i n +1 − 1 2 , . . . , k n + m η i n + m − 1 2  × n Y j =1  η i j k j  ( x j + 1 2 ) k j ( 1 2 − x j ) η i j − k j  × m Y j =1  η i j + n k j + n  ( u j + 1 2 ) k j + n ( 1 2 − u j ) η i j + n − k j + n  . (26) Remark 2. Th e ab o v e theorem only imp oses 6 n +2 m + 2 p multiplie rs, wh ic h is a great improv ement o ver T a ylor’s approac h . Th e dra wbac k here is that the latter is limited to X and U defin ed in (22) and (23), th er efore a scaling of v ariables is necessary if the region of stu dy is differen t. The follo wing theorem p resen ts a stabilit y test for a nonlinear system through a Bernstein appro ximation. Corollary 6. The system describ e d by (20) i s lo c al ly stable, if ther e exist a p olynomial V ( x ) and nonne gative p olynomials s j,i for 1 ≤ i ≤ n and 1 ≤ j ≤ 6 that ar e solution to the fol low ing fe asibility pr o gr am. V ( x ) − φ 1 ( x ) + n X i =1  s 1 ,i ( x i − 1 2 ) − s 2 ,i ( x i + 1 2 )  ≥ 0 n X i =1  − ∂ V ( x ) ∂ x i ( b i ( x ) + ε i ) + s 3 ,i ( ε i − ε i ) − s 4 ,i ( ε i + ε i ) + s 5 ,i ( x i − 1 2 ) − s 6 ,i ( x i + 1 2 )  − φ 2 ( x ) ≥ 0 (27) wher e b i ( x ) = B m i 1 ,...,m i n ( x 1 , . . . , x n ) = X 0 ≤ k j ≤ m i j 1 ≤ j ≤ n f i  k 1 m i 1 − 1 2 , . . . , k n m i n − 1 2  n Y j =1  m i j k j  ( x j + 1 2 ) k j ( 1 2 − x j ) m i j − k j  (28) is the Bernstein appr oximation o f func tion f i ( x ) in x ∈ [ − 1 2 , 1 2 ] n , and ε i ar e b ounds on appr oxima- tion err or which c an b e determine d thr ough (11) . Remark 3. The ab o ve theorem giv es a lo cal result for x ∈ [ − 1 2 , 1 2 ] n . If a differen t region is meant to b e studied, a scaling of state v ariables is n ece ssary in adv ance. Remark 4. In a ll of the theorems i n this section, the su pply rate is a p olynomial fu n ctio n. This assumption is not limiting, and sev eral cont rol problems h a v e a formulatio n as diss ip ati on inequalit y form with a p olynomial supply r ate function (some are present ed later on in th is section, other examples are listed in (Eb enbauer an d Allg¨ ow er 2006)). Ho wev er, if a non-p olynomia l function is 12 desired, a similar approximat ion should b e p erformed f or the supply rate fun cti on as w ell. S uc h an appro ximation can b e carried out similarly and will n ot b e r epeated here. Remark 5. T ake n ot e that the conditions on th e theorems pro vided in this section are in the form of p olynomial non n ega tivit y . This is a difficult problem to solve, ev en f or simp le cases, but the non- negativit y conditions can b e relaxed in to p olynomial optimisation. The most p opular wa y to relax the conditions is the us e of sum of squares (SOS) pr og ramming, whic h con v erts the p olynomial nonnegativit y problem into a semidefin ite optimisation program (A. Papac h ristodoulou and S. Pra- jna 2005; Lasserr e 2001). No vel appr oa c h es r ec en tly introd uced in (Ahmadi and Ma jumdar 2017) relax the cond itions in to linear programming and second-order cone programming, whic h are more efficien t to solv e. The examples in secti on 5 are solv ed using SOSTOOLS (A. Pa pac h r istodoulou and S. Pr a jna 2005). 4.3. Passivity and Passivity Indic es As m entioned in section 2, passivit y is a sp ecial case of d iss ipativit y , so we can study passivit y and passivit y ind ices of a system using either one of the approac hes discussed earlier in this section. Here, for completeness, we state the results f or QS R − dissipativit y , p assivit y , and passivit y indices. Remark 6. T h e n onlinear system defined in (2) is lo cally Q S R − d issipativ e, if it is lo cally dissipa- tiv e with r esp ec t to sup ply rate fu nction w ( u , y ) = y ⊺ Q y + 2 y ⊺ S u + u ⊺ R u (29) where Q, S , and R are constan t matrices of appropr iat e dimension and Q and R are s y m metric. This can b e c h ec ked using an y of th e T heorems 2, 3, and 5. Remark 7. Th e n onlinear system defined in (2) is lo cally passive , if it is lo cal ly diss ip ati v e w ith resp ect to supp ly rate fun ctio n w ( u , y ) = u ⊺ y . (30) Lo cal passivit y of the system can b e c hec k ed u sing Theorems 2,3, and 5 . This system is calle d lo c al ly Input F e e d- f or war d Output F e e db ack Passive (IF-O FP), if it is locally dissipativ e with resp ect t o the w ell-defined supply rate: w ( u , y ) = u ⊺ y − ρ y ⊺ y − ν u ⊺ u (31) for some ν , ρ ∈ R . The follo wing t w o th eo rems present w a ys to find passivit y in dices of a system and can b e easily deriv ed fr om previous theorems and definitions. Theorem 7. The nonline ar system (2) has lo c al output fe e db ack p assivity (OFP) index of ρ, if c onditions in The or em 3 hold for the lar ge st ρ, wher e w ( u , y ) i s g iven as w ( u , y ) = u ⊺ y − ρ y ⊺ y . (32) ν is lo c al input fe e dforwar d p assivity (IFP) for the system if it is the big g est numb er satisfying 13 c ondition in The or em 3 with w ( u , y ) define d as w ( u , y ) = u ⊺ y − ν u ⊺ u . (33) Her e, lo c al m e ans for x and u b elonging to X and U define d in (12) and (13) . Theorem 8. The no nline ar system (2) has lo c al OFP (IFP) index of ρ ( ν ) for X and U define d in (22) and (23) , if ρ ( ν ) is the lar gest value satisfying c onditions in The or e m 5, with w ( u , y ) define d in (32) (or (33) , r esp e ctive ly). 5. Examples Examples are p ro vided her e to demonstrate how to employ the giv en tec hn iques to approxi mate a nonlinear sy s tem and to ve rify stabilit y and passivit y . Example 1 d emonstrate the u se of T a ylor’s appro ximation theorem an d determining th e stabilit y of a dynamic system thr ou gh C oroll ary 4. Example 2 studies passivit y of a nonlinear system u sing T a ylor’s app ro ximation th eo rem as in Theorem 2. Example 3 u s es Bernstein polynomials to appro ximate the d ynamics of a simple pen- dulum and demonstrates the us e of C oroll ary 6 as we ll. Example 4 sho w s the use of m ultiv ariable Bernstein p olynomials and Theorem 5. Example 1 (Stabilit y) . Consider the system as ˙ x 1 = x 2 , ˙ x 2 = − 2 x 2 − x 1 cos( x 1 + x 2 ); (34) This s y s tem is nonlinear and n on-p olynomial . It is n ot trivial to fi n d a Lya punov functional to c heck stability or d issipativit y of the system. Em plo ying Lya punov’s indirect metho d will also not giv e us ev ery detail ab out the system, including h ow close to the equilibrium we n eed to sta y to remain stable, or wh at kind of inp uts can keep the system dissip ative . Assume x =  x 1 x 2  ⊺ and p ( x ) = x 1 cos( x 1 + x 2 ) . Using Theorem 10 and (16) we can rewrite p ( x ) as a 6th order app r o ximation p lus remainder as follo ws. p ( x ) = i + j ≤ 6 X i =0 ,j =0 ∂ i f ( x ) i ! ∂ x i 1 · ∂ j f ( x ) j ! ∂ x j 2 x i 1 x j 2 + 7 X i =0 R i ( x ) x i 1 x (7 − i ) 2 = x 5 1 / 24 + ( x 4 1 x 2 ) / 6 + ( x 3 1 x 2 2 ) / 4 − x 3 1 / 2 + ( x 2 1 x 3 2 ) / 6 − x 2 1 x 2 + ( x 1 x 4 2 ) / 24 − ( x 1 x 2 2 ) / 2 + x 1 + 7 X i =0 R i ( x ) x i 1 x (7 − i ) 2 . (35) Ho wev er , th e functions R i are not p olynomial, so w e b oun d them based on (3 ) as | R 0 | ≤ 2 . 0 × 10 − 4 | R 1 | ≤ 0 . 0028 | R 2 | ≤ 0 . 0125 | R 3 | ≤ 0 . 0279 | R 4 | ≤ 0 . 0349 | R 5 | ≤ 0 . 0252 | R 6 | ≤ 0 . 0097 | R 7 | ≤ 0 . 0016 (36) for | x 1 | ≤ 1 , | x 2 | ≤ 1 . Applying Corollary 4 to ab o ve app ro ximation will pro v e that the origin is a stable equ ilibrium p oin t f or the system for | x 1 | ≤ 1 , | x 2 | ≤ 1 . Stabilit y is p ro v ed b y a q u artic 14 Ly apu no v functional V 1 ( x ) = − 39 . 73 x 4 1 + 1204 . 0 x 3 1 x 2 + 99 . 79 x 2 1 x 2 2 − 106 . 1 x 2 1 + 748 . 7 x 1 x 3 2 + 0 . 0002435 x 4 2 (37) Note that th e fu n ctio n V 1 ( x ) is not p ositive (semi)defin ite, b ut it is nonn egativ e for | x 1 | ≤ 1 , | x 2 | ≤ 1 . Example 2 (P assivity) . No w consider the system ˙ x 1 = x 2 , ˙ x 2 = − 2 x 2 − x 1 cos( x 1 + x 2 ) + u ; y = x 2 (38) By appro ximating this system using Theorem 10, we can pr o ve that the s ystem is passive with the follo wing storage function V ( x ) = − 23 . 63 x 4 1 + 674 . 4 x 3 1 x 2 + 58 . 66 x 2 1 x 2 2 − 62 . 39 x 2 1 + 422 . 4 x 1 x 3 2 − 4 . 08 × 10 − 4 x 4 2 (39) Example 3 (Simple P en d ulum) . The equ ations of motion for a simple p endu lum are giv en as ˙ θ = ω , ˙ ω = − sin θ − ω . (40) Here, w e will use the ap p roac h based on Bernstein P olynomilas to study this sys tem. Assuming b ounds on states as | θ | ≤ 0 . 5 , | ω | ≤ 0 . 5 and change of v ariables as x 1 = θ + 1 2 , x 2 = ω + 1 2 (41) result in the f ollo wing d ynamical equation ˙ x 1 = x 2 − 1 2 , ˙ x 2 = − sin( x 1 − 1 2 ) − x 2 + 1 2 . (42) A 6th-order app roaximat ion of th is system based on Bernstein approac h can b e derived as ˙ x 1 = x 2 − 1 2 , ˙ x 2 =8 . 9 × 10 − 16 x 6 1 − 7 . 6 × 10 − 4 x 5 1 + 1 . 9 × 10 − 3 x 4 1 (43) + 0 . 089 x 3 1 − 0 . 14 x 2 1 − 0 . 91 x 1 + 0 . 48 + ε − x 2 + 1 2 . where | ε | ≤ 0 . 04 is the appro ximation error. Assuming u = 0 , Corollary 6 prov es that the system is lo cally stable based on the f oll o win g Lyapuno v f unction: V = − 1 . 49 ω 6 + 2 . 45 ω 5 θ + 13 . 62 ω 4 θ 2 + 37 . 74 ω 3 θ 3 − 3 . 67 ω 2 θ 4 + 6 . 13 ω θ 5 − 0 . 90 θ 6 − 46 . 15 ω 5 − 29 . 77 ω 4 θ − 58 . 22 ω 3 θ 2 − 54 . 43 ω 2 θ 3 − 21 . 75 ω θ 4 − 34 . 48 θ 5 + 29 . 35 ω 4 + 1 . 80 ω 3 θ + 58 . 86 ω 2 θ 2 − 20 . 33 ω θ 3 + 42 . 83 θ 4 − 0 . 046 ω 3 − 0 . 01 ω 2 θ − 0 . 058 ω θ 2 − 0 . 044 θ 3 + 1 . 09 × 10 − 4 ω 2 − 9 . 15 × 10 − 5 ω θ + 2 . 04 6 × 10 − 4 θ 2 The next example demonstr ate ho w to emplo y the approac h b ased on Bernstein p olynomials on a multiv ariate nonlinearit y . 15 Example 4. Consid er the system in (38). T h is system can b e approxi mated as a 4th order p oly- nomial as ˙ x 1 = x 2 , ˙ x 2 = − 2 x 2 − ( − 9 . 5 × 10 − 4 x 4 1 x 3 2 + 0 . 015 x 4 1 x 2 − 7 . 1 × 10 − 4 x 1 3 x 2 4 + 0 . 067 x 3 1 x 2 2 − 0 . 18 x 3 1 + 0 . 044 x 2 1 x 3 2 − 0 . 7 x 2 1 x 2 + 3 . 6 × 10 − 3 x 1 x 4 2 − 0 . 34 x 1 x 2 2 + 0 . 89 x 1 + 3 . 7 × 10 − 3 x 3 2 − 0 . 059 x 2 + ε ) where ε is the app ro ximation er r or and is b ounded by − 0 . 04 ≤ ε ≤ 0 . 04 . Coroll ary 6 pr o v es that the system is lo cally stable for u = 0 based on the follo wing 4th ord er Ly apunov f unctional V ( x ) = 0 . 0658 02 x 4 1 − 0 . 094308 x 3 1 x 2 − 0 . 036597 x 2 1 x 2 2 + 0 . 0096327 x 1 x 3 2 + 0 . 0002283 x 4 2 − 1 . 3876 x 3 1 + 0 . 037105 x 2 1 x 2 − 1 . 4013 x 1 x 2 2 − 0 . 036844 x 3 2 + 2 . 0697 x 2 1 + 0 . 33552 x 1 x 2 + 1 . 5356 x 2 2 , for − 0 . 5 ≤ x 1 , x 2 ≤ 0 . 5 . It can b e sho wn that this s ystem is also lo cally passive , usin g a 6th-order Ly apu no v function for | x 1 | ≤ 0 . 5 , | x 2 | ≤ 0 . 5 and | u | ≤ 0 . 5. The Lyapuno v function can b e found using Theorem 5 and Remark 7, how ev er, it is n ot listed here for the sak e of brevit y . 6. Conclusions In this pap er, w e prop osed an optimisation-based approac h to study certain energy-related b e- ha viour s of a nonlinear system through p olynomial approximat ions. The b eha viours of interest included stabilit y , dissipativit y , and passivit y , c h arac trized b y passivit y indices. A motiv ating ex- ample was give n to s h o w that dissipativit y and passivit y of a system sh ould b e stud ied lo cally . Therefore, the f ocus here wa s on local prop erties of the system in w ell-defined admissible con- trol an d state spaces. The metho dologies facilitate the systematic searc h for Ly apuno v fu nctionals through p olynomial app ro ximations. Two different appr oa c hes appr o ximate the system’s dyn amics with p olynomial fu nctio ns. The fir st app r oac h was through the well-kno w n T a ylor’s theorem. Th is approac h resulted in large optimisation p rograms, so w e show ed ho w we could redu ce the size of the optimisation p roblem b y using a generalised S -pro ce dure and by b oun ding the appro x im ation errors in an ellipsoid. T h e second app roac h w as thr ough a m ultiv ariate generalisation of Bernstein p olyno- mials. Examples were giv en to demons tr at e th e effectiv eness and applicabilit y of eac h ap p roac h . W e sho wed that the approac h based on T aylo r’s theorem pr o vides a more in tuitiv e appr o ximation and is easier to deriv e for different regions; how ev er, it may lead to larger optimisation p rograms, and there is a trade-off b et ween accuracy and computational complexit y . 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(1) Her e, the multi-index v e c tors α ∈ R n ar e the de gr e es of the monomials c omprising the whole ap- pr oximation and ther efor e, if α = ( α 1 , . . . , α n ) , then x α = x α 1 1 · x α 2 2 · · · x α n n . Also | α | = P n i =1 α i , the derivative symb ol i n (1) is define d as D α f ( x ) = ∂ | α | f ( x ) ∂ x α 1 1 · · · ∂ x α n n , (2) and α ! = α 1 ! α 2 ! · · · α n ! . If the function f : R n → R is k + 1 times con tinuously different iable in th e closed b all B , then we 18 can derive the remaind er in terms of ( k + 1)-th ord er p artial deriv ativ es of f in this n eig hborho o d: f ( x ) = X | α |≤ k D α f ( a ) α ! ( x − a ) α + X | β | = k +1 R β ( x )( x − a ) β , R β ( x ) = | β | β ! Z 1 0 (1 − t ) | β |− 1 D β f  a + t ( x − a )  dt. Here, based on th e contin uit y of ( k + 1)-th order p artial deriv ativ es in th e compact s et B , w e can obtain the un iform estimates | R β ( x ) | ≤ 1 β ! max | α | = | β | max y ∈ B | D α f ( y ) | , x ∈ B . (3) A Bernstein p olynomial is a linear combinatio n of Bernstein basis p olynomials. F or the u n i- v ariate case, the m + 1 Bernstein b asis p olynomials of d eg ree m are d efined as follo ws (Loren tz 1986) b ν,m ( x ) =  m ν  x ν (1 − x ) m − ν , ν = 0 , . . . , m. (4 ) The multiv ariate case can b e d efined similarly . Definition 9 (Multiv ariate Bern stein Polynomials (F eng and K oz ak 1992)) . Let m 1 , . . . , m n ∈ N and f b e a fu nctio n of n v ariables. T he p olynomials B m 1 ,...,m n ( f )( x 1 , . . . , x n ) := X 0 ≤ k j ≤ m j 1 ≤ j ≤ n f  k 1 m 1 , . . . , k n m n  n Y j =1  m j k j  x k j j (1 − x j ) m j − k j  (5) are called the m ultiv ariate Bernstein p olynomials of f . W e note that B m 1 ,...,m n ( f )( · ) is a linear op erator. The Bernstein p olynomials of degree m are a basis for the vect or space of p olynomials of degree m or lo w er. A Bernstein p olynomial is a linear combinatio n of Bernstein b asis p olynomials B m ( x ) = m X ν =0 β m b ν,m ( x ) . (6) It is also called a p olynomial in Berns tein form of degree m. Theorem 11. Consider a c ontinuous fu nc tion f on the interval [0 , 1] and the Bernstein p olynomial B m ( f )( x ) = m X ν =0 f  ν m  b ν,m ( x ) . (7) It c an b e shown that lim m →∞ B m ( f )( x ) = f ( x ) . (8) 19 The limit hold s uniformly on the interval [0 , 1] . This statement is str onger than p ointwise c on- ver g e nc e (wher e the limit holds for e ach value of x sep ar ately). Sp e cific al ly, uniform c onver genc e signifies that lim m →∞ sup { | f ( x ) − B m ( f )( x ) | : 0 ≤ x ≤ 1 } = 0 . (9) Theorem 12 (Uniform Con vergence ) . L et f : [0 , 1] n → R b e a c ontinuous function. Then the multivariate Bernstein p olynomials B m 1 ,...,m n ( f )( · ) c onver ge uniformly to f for m 1 , . . . , m n → ∞ . In other wor ds, The set of al l p olynomials is dense in C ([0 , 1] n ) . By assuming more knowledge ab out the fun ctio n, sp ecifically a Lipschitz co ndition, an err or b ound can b e obtained. Theorem 13 (Error Bound for Lip sc hitz C ondition) . If f : [0 , 1] n → R is a c ontinuous function satisfying the Lipschitz c ondition k f ( x ) − f ( y ) k 2 < L k x − y k 2 (10) on [0 , 1] n , then the ine quality k B m 1 ,...,m n ( f )( x ) − f ( x ) k 2 < L 2  n X j =1 1 m j  1 2 (11) holds. The f oll o win g asymp tot ic formula giv es us information ab out the rate of con v ergence. Theorem 1 4 (Asymptotic F ormula) . L e t f : [0 , 1] n → R b e a C 2 function and x ∈ [0 , 1] n , then lim m →∞ m ( B m,...,m ( f )( x ) − f ( x )) = n X j =1 x j (1 − x j ) 2 ∂ 2 f ( x ) ∂ x 2 j ≤ 1 8 n X j =1 ∂ 2 f ( x ) ∂ x 2 j . (12) The asymptotic formula states that the rate of conv ergence dep ends only on the partial deriv a- tiv es ∂ 2 f ( x ) /∂ x 2 j . This is notewo rth y , since it is often the case that the s moother a f unction is and the more is kno wn ab out its higher deriv ativ es, the more p r operties can b e prov en , but in th is case only the second order deriv ativ es play a role. The follo win g theorem p la ys a n imp ortan t role in set in clusion results of p olynomial n onnega- tivit y . It is a sim p lified, and more tractable version of a well-kno wn th eo rem called Positivstel len- satz (Parrilo 2003). Theorem 15 (Generalized S -Pro cedure (See (Zake ri and Ozgoli 2014; Zak eri and Ozgoli 2011) and the references th er ein)) . Given p olynomials { p i } m i =0 ⊂ R n , if ther e e xi sts { s i } m i =1 ⊂ Σ n such that p 0 − m X i =1 s i p i ∈ Σ n (13) then ∩ { x ∈ R n | p i ( x ) ≥ 0 } ⊆ { x ∈ R n | p 0 ( x ) ≥ 0 } . (14) Or e quiv alently, the fol lowing set is e mpty { x ∈ R n | p 1 ( x ) ≥ 0 , . . . , p m ( x ) ≥ 0 , − p 0 ( x ) > 0 } (15) 20 0 2 4 6 8 10 12 14 16 18 20 0 0 . 2 0 . 4 r ρ ρ v ersus r Figure 1. OFP index ρ v ersus upper b ound r on state norm (Zakeri and P . J. An tsaklis 2016) 21