A New Stability Result for the Feedback Interconnection of Negative Imaginary Systems with a Pole at the Origin

A New Stability Result f or the Feedback Inter connection of Negativ e Imaginary Systems with a P ole at the Origin Mohamed A. Mabrok, Abhijit G. Kallapur , Ian R. Peterse n, and Alexander Lanzon Abstract — This paper is concerned with stability conditions fo r the positive feedback in terconnection of negative imaginary systems. A generalization of th e negati ve imaginary lemma is derive d, which remains true even if the transfer function has poles on th e imaginary axis includin g the origin. A suf ficient condition for the internal stability of a f eedback interconnection fo r NI systems includ ing a p ole at the origin is given and an illustrative example i s presented to sup port the r esult. I . I N T RO D U C T I O N Structural modes in m achines and robots, gro und and aerospace vehicles, and precision instrumentation , such as atomic force microscope s and optical systems, can limit the ability o f co ntrol systems to achieve the de sired perfo rmance [1]. This problem is s implified to some e xtent by using force actuators co mbined with collo cated measuremen ts of velocity , po sition, or acceleration. The use of fo rce ac tuators com bined with velocity m ea- surements has been studied using the positiv e real (PR) theory for linear time in variant (L TI) systems; e.g. , see [2], [3]. PR systems, in the single-inpu t single-outp ut (SISO) case, can be defined as systems where th e real part of the tran sfer function is nonnegative. M any systems that dissipate energy f all under th e category of PR system s. For instance, they can arise in electric circu its with linear passi ve compon ents and mag netic couplings. In spite of its success, a drawback of the PR theory is the requ irement f or th e relative degree of the u nderlyin g system transfer fun ction to be eith er zero or one [3] . Hence, the c ontrol of flexible structu res with force actuato rs co mbined with position measurements, cannot use the theory of PR systems. Lanzon and Petersen introduce a new class of systems in [4], [5] called negative im aginary (NI) systems, which has fewer restrictions on the relati ve degree of the system tr ansfer function than in the PR case. In the SISO case, such systems are d efined by co nsidering the prope rties of the imagin ary part of the transfer f unction G ( j ω ) = D + C ( j ωI − A ) − 1 B , and requiring th e condition j ( G ( j ω ) − G ( j ω ) ∗ ) ≥ 0 for all ω ∈ (0 , ∞ ) . This work was supported by the Australian Research Council Mohamed Mabrok, Abhijit Kallapur and Ian Petersen are with the Sc hool of Engineeri ng and Information T ec hnology , Uni versit y of New South W a les at the Australia n Defence Force Academy Canberra A CT 2600, Australia abdal lamath@gmail.c om, abhijit.kalla pur@gmail.com, i.r.petersen@ gmail.com Alex ander Lanzon is with the Control Systems Centre, School of Electrical and Electroni c Engineering, Univ ersity of Manchest er , Manchest er M13 9PL, United Kingdom Alexander.Lan zon@manchester .ac.uk In ge neral, NI systems ar e stable systems ha ving a phase lag between 0 and − π f or all ω > 0 . Th at is, their Nyq uist plot lies below the real axis wh en the frequ ency varies in the open interval (0 , ∞ ) (for strictly n egati ve-imaginar y systems, the Ny quist plo t shou ld not touch the real axis except at zero frequen cy or at infinity ). This is similar to PR systems wh ere the Nyquist plot is constrained to lie in the right half of the complex pla ne [ 2], [3]. Howev er, in contra st to PR systems, transfer function s for NI systems can have relative degree more than u nity . NI systems can b e transform ed in to PR systems an d vice versa u nder some technical assumptions. Ho wever , this equiv alence is not co mplete. For instance, such a tra nsfor- mation applied to a strictly negative imagin ary (SNI) system always leads to a non-strict PR system. Hence, the passi vity theorem [2], [3] canno t captu re the stability o f the closed- loop interconnection o f an N I and an SNI system. I n addition, any controller design appro ach based on strictly PR sy nthesis cannot be used for the con trol of an NI system irrespectiv e of whether it is strict or non- strict. Also, transformatio ns of NI systems to bou nded- real systems for application of the small-gain th eorem su ffers from the exact sam e dif ficulty o f giving a no n-strict boun ded real system despite the original system bein g SNI; see [6] fo r details. Many practical systems can be consider as NI systems. F or example, when considering the tran sfer function from a fo rce actuator to a correspo nding collocated p osition sen sor (fo r instance, p iezoelectric sensor) in a lightly d amped structure [1], [4], [ 5], [7]–[9]. Also, stability r esults for interconn ecting systems w ith an NI frequ ency respo nse have b een ap plied to d ecentralized control o f large vehicle platoo ns in [10] . Here, the author s discuss the a vailability of v ariou s designs to enhance the robust stability of the system with respect to small variations in neighb or-coupling gains. NI systems theory h as been exten ded by Xiong et. al. in [11]– [13] by allowing for simple p oles on the imagin ary axis o f the c omplex plane excep t at the orig in. Fu rther- more, NI controller synthe sis has also be en discussed in [4], [5]. In addition, it has b een shown in [4], [5] that a necessary and sufficient co ndition fo r the internal stability of a po siti ve-feedback interco nnection o f an NI system w ith transfer function matrix M ( s ) an d an SNI system with transfer function m atrix N ( s ) is giv en by the DC gain condition λ max ( M (0) N (0)) < 1 . Here, the no tation λ max ( · ) denotes the maximum eigen value of a matrix with only real eigenv alues. A generalization o f th e NI lemma in [12], [13 ] to includ e a simple p ole at the orig in was pr esented in [1 4]. In [14 ], stability analysis f or a sp acial class of g eneralized NI systems with the inclusion of an in tegrator conn ected in parallel with an NI system was discussed . The assump tion in [14 ] restricts the app lication o f the pro posed stability result to NI systems which can be decomposed into the parallel c onnection of an NI system and a n integrator . In this pape r , we extend the results in [ 1], [4 ], [5 ], [11 ]– [14] f or NI systems to a llow for the existence o f a p ole at the origin with a mo re general structure than allowed in the result of [14]. This extension allows us to stabilize any NI system with a pole at th e origin withou t any parallel deco mposition assumption. Also, stabilizing NI systems with a po le at the origin can be u sed for controller design with integral action. This paper is further organ ized as fo llows: Section I I introdu ces the conc ept of PR and NI systems and presents a relation ship between them. The m ain results of this paper are p resented in Section I II. Section IV p rovides a numer ical example and th e p aper is conclu ded with a sum mary and remarks on f uture work in Section V. I I . P R E L I M I N A R I E S In this section, we introduce the definitio ns of PR a nd NI systems. W e also pr esent a lem ma describin g the tran sforma- tion betwee n PR and NI systems, and some technical r esults which will b e u sed in d eriving th e main r esults of th e paper . The definition of PR systems has be en motiv ated by the study of linea r elec tric circ uits com posed of resistors, capacitors, and induc tors. For a detailed discussion of PR systems, see [2 ], [3] and refe rences therein. Definition 1: A squ are transfer fu nction m atrix F ( s ) is positive rea l if: 1) F ( s ) has no p ole in R e [ s ] > 0 . 2) F ( j ω ) + F ( j ω ) ∗ ≥ 0 for all positi ve real j ω such that j ω is not a pole o f F ( j ω ) . 3) If j ω 0 , finite o r in finite, is a pole of F ( j ω ) , it is a simple p ole and the co rrespon ding residual matrix K 0 = lim s − → j ω 0 ( s − j ω 0 ) F ( s ) is positive semidefinite Hermitian. T o establish the main results o f this paper, we co nsider a generalized d efinition fo r NI systems which allows for a simple pole at th e origin as follows: Definition 2: A squ are transfer functio n matrix G ( s ) is NI if the following condition s are satisfied: 1) G ( s ) h as no po le in Re [ s ] > 0 . 2) For a ll ω ≥ 0 such that j ω is not a pole of G ( s ) , j ( G ( j ω ) − G ( j ω ) ∗ ) ≥ 0 . 3) if s = j ω 0 is a pole of G ( s ) th en it is a simple pole. Furthermore if ω 0 > 0 , the re sidual m atrix K 0 = lim s − → j ω 0 ( s − j ω 0 ) j G ( s ) is positive semidefinite Hermitian. Definition 3: A squ are transfer fu nction m atrix G ( s ) is SNI if th e following conditions are satisfied: 1) G ( s ) h as no po le in Re [ s ] ≥ 0 . 2) For all ω > 0 , j ( G ( j ω ) − G ( j ω ) ∗ ) > 0 . Due to ad vances in the theo ry of PR systems and th e compleme ntary definitions of PR a nd NI systems, it is useful to establish a lemma which considers the r elationship between these notions to further d ev elop th e th eory of NI systems. In order to do so, we con sider the possibility o f having a simp le po le at th e orig in, and rela x the condition det( A ) 6 = 0 considere d in [5], [11 ], [1 5]. T his leads to a modification o f the relationship between PR a nd NI systems as follows: Lemma 1: (see also [14]) Given a real ratio nal pro per transfer function matrix G ( s ) with state space realizatio n " A B C D # and the transfer function m atrix ˜ G ( s ) = G ( s ) − D , th e transfer function matrix G ( s ) is NI if an d on ly if the transfer fun ction matrix F ( s ) = s ˜ G ( s ) is PR. Here, we assume that a ny pole zero cancellation which occurs in s ˜ G ( s ) has been c arried out to ob tain F ( s ) . Pr oof: (Necessity) It is straightforward to show that if ˜ G ( s ) is NI then G ( s ) is NI a nd vice-versa. Sup pose that j  ˜ G ( j ω ) − ˜ G ( j ω ) ∗  ≥ 0 , for all ω > 0 such that j ω is not a pole of G ( s ) . Then g i ven any su ch ω > 0 , F ( j ω ) + F ( j ω ) ∗ = j ω  ˜ G ( j ω ) − ˜ G ( j ω ) ∗  ≥ 0 , and ( F ( j ω ) + F ( j ω ) ∗ ) ≥ 0 . This means that F ( − j ω ) + F ( − j ω ) ∗ ≥ 0 f or all ω > 0 which im plies that F ( j ω ) + F ( j ω ) ∗ ≥ 0 for all ω < 0 such th at j ω is no t a p ole of G ( s ) . Hence, ( F ( j ω ) + F ( j ω ) ∗ ) ≥ 0 for all ω ∈ ( −∞ , ∞ ) such that j ω is no t a pole of ˜ G ( j ω ) . Now , con sider the case wher e j ω 0 is a pole of ˜ G ( s ) and ω 0 = 0 . Since ˜ G ( s ) has on ly a simple p ole at the origin, F ( s ) = s ˜ G ( s ) will have no p ole at the o rigin because of th e pole z ero c ancellation. T his implies that F (0) is finite. Since F ( j ω ) + F ( j ω ) ∗ ≥ 0 f or all ω > 0 and F ( j ω ) is continuo us, this implies tha t F (0) + F (0) ∗ ≥ 0 . Also, if j ω 0 is a p ole of ˜ G ( s ) and ω 0 > 0 , then ˜ G ( s ) can be factored as 1 s 2 + ω 2 0 R ( s ) , which accor ding to the defin ition fo r NI systems im plies that the r esidual m atrix K 0 = 1 2 ω 0 R ( j ω 0 ) is po siti ve semidefinite Hermitian. This implies that R ( j ω 0 ) = R ( j ω 0 ) ∗ ≥ 0 . Now , the residu al matrix of F ( s ) at j ω 0 with ω 0 > 0 is given by , lim s − → j ω 0 ( s − j ω 0 ) F ( s ) = lim s − → j ω 0 ( s − j ω 0 ) s ˜ G ( s ) , = lim s − → j ω 0 ( s − j ω 0 ) s 1 s 2 + ω 2 0 R ( s ) , = 1 2 R ( j ω 0 ) which is p ositiv e semidefinite Hermitian. Hence, F ( s ) is positive rea l. (Sufficiency) Supp ose that F ( s ) is positive real. Then , F ( j ω ) + F ( j ω ) ∗ ≥ 0 for all ω ∈ ( −∞ , ∞ ) such th at j ω is not a pole of F ( s ) . Th is im plies j ω  ˜ G ( j ω ) − ˜ G ( j ω ) ∗  ≥ 0 for all ω ≥ 0 such that j ω is not a p ole of G ( s ) . Then ˜ G ( j ω ) − ˜ G ( j ω ) ∗ ≥ 0 fo r all such ω ∈ [0 , ∞ ) . I n addition, if j ω 0 is a p ole of F ( s ) , then it follows from the definition of PR systems that the r esidual matrix lim s − → j ω 0 ( s − j ω 0 ) F ( s ) is positive semid efinite Hermitian. Also, lim s − → j ω 0 ( s − j ω 0 ) F ( s ) = lim s − → j ω 0 ( s − j ω 0 ) s ˜ G ( s ) , = ω 0 lim s − → jω 0 ( s − j ω 0 ) j ˜ G ( s ) . Then using Definition 2, we can con clude th at ˜ G ( s ) is NI . Remark 1: Note that a po le zero cancellatio n at the origin in F ( s ) = s ˜ G ( s ) will not affect the use o f the PR lemm a when ap plied to F ( s ) since the min imality con dition is relaxed in the gener alized version of the PR lemma [1 6], [17]. Now , we present a generalized NI lemma, which allows for a p ole at the orig in. Consider the following L TI system, ˙ x ( t ) = Ax ( t ) + B u ( t ) , (1) y ( t ) = C x ( t ) + D u ( t ) , (2) where, A ∈ R n × n , B ∈ R n × m , C ∈ R m × n , and D ∈ R m × m . Lemma 2: (see also [1 4]) Le t " A B C D # be a minim al realization of the transfe r fu nction matr ix G ( s ) ∈ R m × m for the system in (1)-(2). Th en, G ( s ) is NI if and only if ther e exist m atrices P = P T ≥ 0 , W ∈ R m × m , and L ∈ R m × n such that th e following LMI is satisfied: " P A + A T P P B − A T C T B T P − C A − ( C B + B T C T ) # = " − L T L − L T W − W T L − W T W # ≤ 0 . (3) Pr oof: Suppose that G ( s ) is NI, which implies fr om Lemma 1 that F ( s ) = s ˜ G ( s ) with state sp ace r ealization " A B C A C B # is PR. It fo llows fro m Corollary 2 and Corollary 3 in [17] that there exists a matrix P = P T ≥ 0 , such that th e LMI in (3) is satisfied. On the othe r hand, supp ose that L MI in (3) is satisfied, then F ( s ) is PR via Corollary 1 and Cor ollary 3 in [17 ], which implies f rom Lemma 1 that G ( s ) is NI. In studying th e in ternal stability of an interconnectio n of NI and SNI systems, we shall use the following SNI lemm a: Lemma 3: [5], [1 1], [15] Supp ose that the p roper transfe r function matrix G ( s ) = C ( sI − A ) − 1 B + D with a minimal realization " A B C D # is SNI, the n the fo llowing conditions are satisfied: 1) det( A ) 6 = 0 , D = D T . 2) There exists a square matrix P = P T > 0 , W ∈ R m × m and L ∈ R m × n such that the f ollowing LMI is satisfied: " P A + A T P P B − A T C T B T P − C A − ( C B + B T C T ) # = " − L T L − L T W − W T L − W T W # . (4) Also, co nsider the following lemma, which will be u sed to der i ve the main results of this paper in Section II I, Lemma 4: [5] Given A ∈ C n × n with j ( A − A ∗ ) ≥ 0 and B ∈ C n × n with j ( B − B ∗ ) > 0 , then det( I − AB ) 6 = 0 . I I I . M A I N R E S U LT S The key result of this paper is a ge neralization of the result in [14 ], which gi ves stability con ditions for an interconnec - tion between an N I system ( which may contain a simple pole a t the o rigin) and an SNI system. Th e g eneralization is stated in The orem 1. Now , suppo se the transfer function matrix G 1 ( s ) with a minimal realization " A 1 B 1 C 1 D 1 # is NI, and G 2 ( s ) with a minimal realizatio n " A 2 B 2 C 2 D 2 # is SNI. According to Lem ma 2 and Lemm a 3, we have, P 1 A 1 + A T 1 P 1 = − L T 1 L 1 , P 2 A 2 + A T 2 P 2 = − L T 2 L 2 , P 1 B 1 − A T 1 C T 1 = − L T 1 W 1 , P 2 B 2 − A T 2 C T 2 = − L T 2 W 2 , C 1 B 1 + B T 1 C T 1 = W T 1 W 1 , C 2 B 2 + B T 2 C T 2 = W T 2 W 2 , (5) where P 1 ≥ 0 and P 2 > 0 . The inter nal stab ility of the closed-loo p positiv e-feed back interconn ection of G 1 ( s ) and G 2 ( s ) can b e guar anteed by co nsidering the stab ility of the transfer fun ction matrix, ( I − G 1 ( s ) G 2 ( s )) − 1 = ˘ D + ˘ C ( sI − ˘ A ) − 1 ˘ B , where, ˘ A = " A 1 B 1 C 2 0 A 2 # + " B 1 D 2 B 2 # ( I − D 1 D 2 ) − 1  C 1 D 1 C 2  ˘ B = " B 1 D 2 B 2 # ( I − D 1 D 2 ) − 1 , ˘ C = ( I − D 1 D 2 ) − 1  C 1 D 1 C 2  , ˘ D = ( I − D 1 D 2 ) − 1 . (6) Now , con sider the fo llowing resu lt, wh ich is the main result of th is paper: Theor em 1: Supp ose that G 1 ( s ) is strictly proper and NI and G 2 ( s ) is SNI. Then the closed -loop positive feed back interconn ection between G 1 ( s ) and G 2 ( s ) is internally stable if G 2 (0) < 0 and the matrix A 1 + B 1 G 2 (0) C 1 is not singular . Pr oof: T o prove this the orem, we prove th at the matrix ˘ A in ( 6) is Hurwitz; i.e., all of its poles lie in th e left-half of the co mplex p lane. Let T = " P 1 − C T 1 D 2 C 1 − C T 1 C 2 − C T 2 C 1 P 2 # be a candid ate L yapunov m atrix. Since G 2 (0) < 0 , P 1 ≥ 0 , we claim that P 1 − C T 1 G 2 (0) C 1 > 0 . ( 7) In order to prove this claim, consid er M = P 1 − C T 1 G 2 (0) C 1 ≥ 0 an d N ( M ) = { x : M x = 0 } , where N ( · ) den otes the null space. Also, g i ven any x ∈ N we have P 1 x = 0 and C 1 x = 0 . Now , consider the equations P 1 A 1 + A T 1 P 1 = − L T 1 L 1 , (8) B T 1 P 1 − C 1 A 1 = − W T 1 L 1 (9) outlined in (5). Now p re-multip lying a nd po st-multiplying (8) by x T and x respectively , we get, L 1 x = 0 . (10) Also, po st-multiplying (8) by x r esults in P 1 A 1 x = 0 . (11) Subsequen tly , post-multip lying (9 ) by x , gi ves C 1 A 1 x = 0 . (12) Now , let y = A 1 x , which fro m (11) and (1 2) giv es P 1 y = 0 , C 1 y = 0 (13) which implies y ∈ N ( M ) . Thus, we ha ve established that A 1 N ( M ) ⊂ N ( M ) and N ( M ) ⊂ N ( C 1 ) (14) which leads to the fact that N ( M ) is a su bset of th e unobser vable subspace of ( A 1 , C 1 ) ; e.g., see Chap ter 1 8 of [18]. It now follows fr om the minimality of ( A 1 , B 1 , C 1 , D 1 ) that N ( M ) = { 0 } . Hence, M = P 1 − C T 1 G 2 (0) C 1 > 0 . Th is completes the pr oof of the claim . Now , using this cla im, we have P 2 > 0 an d P 1 − C T 1 ( D 2 + G 2 (0) − D 2 ) C 1 > 0 , ⇒ P 2 > 0 an d P 1 − C T 1 D 2 C 1 − C T 1 C 2 P − 1 2 C T 2 C 1 > 0 , ⇒ " P 1 − C T 1 D 2 C 1 − C T 1 C 2 − C T 2 C 1 P 2 # > 0 . That is, T > 0 . Now , the corre sponding L yap unov inequality is given by , T ˘ A + ˘ A T T = " P 1 − C T 1 D 2 C 1 − C T 1 C 2 − C T 2 C 1 P 2 # × " A 1 + B 1 D 2 C 1 B 1 C 2 B 2 C 1 A 2 # + " A 1 + B 1 D 2 C 1 B 1 C 2 B 2 C 1 A 2 # T × " P 1 − C T 1 D 2 C 1 − C T 1 C 2 − C T 2 C 1 P 2 # , = − "  C T 1 D 2 W T 1 + L T 1  C T 1 W T 2 C T 2 W T 1  L T 2  # × " ( W 1 D 2 C 1 + L 1 ) W 1 C 2 W 2 C 1 ( L 2 ) # ≤ 0 . This imp lies that ˘ A h as all its poles in the closed left half of the complex plane . W e now show that det( ˘ A ) 6 = 0 . Indeed , using the assumption ( A 1 + B 1 G 2 (0) C 1 ) , we ob tain det( ˘ A ) = det( A 2 ) det(( A 1 + B 1 D 2 C 1 − B 1 C 2 ( A 2 ) − 1 B 2 C 1 ) = det( A 2 ) det( A 1 + B 1 G 2 (0) C 1 ) = det( A 2 ) det( A 1 + B 1 G 2 (0) C 1 ) 6 = 0 since ( A 1 + B 1 G 2 (0) C 1 ) is non singular an d det( A 2 ) 6 = 0 . Also, using Lemma 4 and the fact that G 1 ( s ) is NI and G 2 ( s ) is SNI, we conclud e th at det( I − G 1 ( j ω ) G 2 ( j ω )) 6 = 0 . This implies that ˘ A has no e igen values on the imaginary axis f or ω > 0 . Hence, the matrix ˘ A is Hu rwitz. This completes the proof of the theorem. I V . I L L U S T R A T I V E E X A M P L E T o illustrate the main result of th is pape r , consid er th e SNI tr ansfer functio n G 2 ( s ) = 1 s +3 − 1 , which satisfies G 2 (0) = − 2 3 < 0 and the strictly proper NI transfer fu nction G 1 ( s ) = 1 s ( s +1) , wh ich h as a pole at the origin . Thus, the assumptions in Theorem 1 are satisfied an d we can conclu de that the closed-loo p system is stable. Also, the poles of the closed-loo p tran sfer fu nction correspo nding to G 2 ( s ) and G 1 ( s ) are the roots of the polyn omial (1 − G 1 ( s ) G 2 ( s )) = s 3 + 4 s 2 + 4 s + 2 which are {− 2 . 8 4 , − 0 . 58 ± 0 . 61 i } . This verifies that the closed-loop transfer function is ind eed asymptotically stable. V . 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