Continuous observer design for nonlinear systems with sampled and delayed output measurements

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1 Preprints of th9th Worl Congress The International Feeration of Automatic Control Continuous observer esign for nonlinear systems with sample an elaye output measurements Daoyuan Zhang Yanjun Shen Xiaohua Xia College of Science, China Three Gorges University, Yichang, Hubei, 4432, China, College of Electrical Engineering an New Energy, China Three Gorges University, Yichang, Hubei, 4432, China, Department of Electrical, Electronic an Computer Engineering, University of Pretoria, Pretoria 2, South Africa Abstract: We esign observers for nonlinear systems with sample an elaye output measurements The observers are of continuous an hybri in nature Base on an auxiliary integral technique, the exponential stability of the estimation errors is achieve, an the sampling perio an the maximum elay are also given Finally, numerical examples are provie to illustrate the esign methos 1 INTRODUCTION In this paper, we consier the following system ẋ 1 = x 2 + f 1 (x 1 ), ẋ 2 = x 3 + f 2 (x 1, x 2 ), (1) ẋ n 1 = x n + f n 1 (x 1, x 2,, x n 1 ), ẋ n = f n (x 1, x 2,, x n ) + u, y = x 1, where the state x R n, the input u R We assume that the output y is sample at instants an is available for the observer at instants + τ k, where { } enotes a strictly increasing sequence such that lim t =, an τ k represents the transmission elay The sampling interval T = +1 is a positive constant The transmission elays τ k are unknown, but have an upper boun τ, that is, max{τ k } τ for all k =, 1,, We also make the assumption: τ T, that is, the measures sample at instants are available for the observer before the next measures sample at instants +1 In aition, f i ( ) (i = 1,, n) satisfy the following globally Lipschitz conition f i (x 1, x 2,, x i ) f i (ˆx 1, ˆx 2,, ˆx i ) l( x 1 ˆx 1 + x 2 ˆx x i ˆx i ), (2) where l is a positive constant There are some results on observer esign for (1) without sample an time elaye measurements, for example, (Deaza [1991], Gauthier et al [1992], Gauthier an upka [1994], Praly [23], Anrieu et al [29], Shen an Xia This work was supporte by the National Science Founation of China (No , , , ), the Grant National Science Founation of Hubei Province (213CFA5), the National Science Founation of Hubei Province (211CDB187), the Scientific Innovation Team Project of Hubei Provincial Department of Eucation (T2113) [28], Shen an Huang [29], Li et al [213a,b]) In recent years, the observer esign for continuous nonlinear systems with sample an elaye measurements have attracte more an more attention For linear systems, observers can be esigne base on the iscrete time moel of the continuous systems However, the esign metho cannot be extene to nonlinear systems because it is ifficult to obtain the exact iscrete time moel In such a case, two main approaches are propose to eal with this problem The first one is to esign a iscrete observer by introucing a consistent approximation of the exact iscretize moel (Arcak an Nešić [24]) The secon one is base on a mixe continuous an iscrete esign For instance, a high gain exponential observer is presente in (Deza an Busvelle [1992]) In (Raff et al [28]), the observers for Lipschitz nonlinear continuous time systems with nonuniformly sample measurements are esigne base on LMI an sample ata control techniques In (Ahme-Ali an Lamnabhi-Lagarrigue [212]), some conitions on the maximum allowable transmission interval are given to guarantee an exponential convergence of the observation error for networke control systems The authors propose a sample-ata nonlinear observer esign by using a continuous-time esign couple with an inter-sample output preictor (arafyllis an ravaris [29]) In (Nari et al [213]), the problem of observer esign was investigate for uniformly observable systems with sample output measurements There are some results on observer esign for nonlinear uniformly observable systems with time-varying elaye output measurement (Van Assche et al [211], Ahme-Ali et al [213a,b]) For example, in (Ahme-Ali et al [213b]), the authors aresse two classes of global exponential observers of continuous systems with sample an elaye output measurements There are two parts: a preiction part an a correction part Sufficient conitions on the maximum allowable elay an the maximum allowable sampling perio are also given to ensure exponential stability of the observation error Copyright 214 IFAC 269

2 In this paper, we consier high gain observer esign for the system (1) with sample an elaye measurements The observer is continuous an hybri By using an auxiliary integral technique, sufficient conitions are presente to ensure the global exponential stability of the observation error Compare with the existing results, the systems we consiere are more general, an the esign metho is simpler This paper is organize as follows In Section 2, we present our main results: the continuous observer is esigne for nonlinear systems with sample an elaye output measurements In Section 3, two examples are use to illustrate the valiity of the propose esign metho Finally, the paper is conclue in Section 4 2 CONTINUOUS OBSERVER DESIGN FOR THE SYSTEM (1) We will esign a continuous observer for the system (1), an present the upper boun of the maximum allowable sample perio an the elay, so that the observation errors are globally exponentially convergent The following lemma is useful for our main results Lemma 1 (Liu et al [26] ) For any positive efinite matrix M R n n, scalar γ >, vector function ω : [, γ] R n such that the integrations concerne are well efine, the following inequality hols: γ w(s)s M γ w(s)s γ γ w(s) Mw(s)s For the system (1), we esign the following observer, ˆx 1 = ˆx 2 + La 1 ( ) + f 1 (ˆx 1 ), ˆx 2 = ˆx 3 + L 2 a 2 ( ) + f 2 (ˆx 1, ˆx 2 ), ˆx n 1 = ˆx n + L n 1 a n 1 ( ) +f n 1 (ˆx 1, ˆx 2,, ˆx n 1 ), ˆx n = L n a n ( ) + f n (ˆx 1, ˆx 2,, ˆx n ) +u, t [ + τ k, + T + τ k+1 ), k, where L 1 is a constant, ( ) = x 1 ( ) ˆx 1 ( ), a i > (i = 1,, n) are given such that there exists a symmetric positive efinite matrix P such that a 1 1 where A = a n 1 1 a n (3) A P + P A I, (4) Note that ( ) is a constant on [ + τ k, + T + τ k+1 ) for k an f i ( ) (i = 1,, n) are continuous an satisfy the conition (2), then, lim t tk +T +τ ˆx i = k+1 lim t tk +T +τ + ˆx i (i = 1,, n) Therefore, ˆx i (i = k+1 1,, n) are continuous on [t, ) Remark 1 Note that the state estimate is escribe in continuous time while the evolution process x 1 ( ) ˆx 1 ( ) is only upate at time instants +τ k, that is, the current evolution process is use until the new evolution process is available Therefore, the ynamics of observer (3) is of continuous an of hybri nature Remark 2 Even though τ k an τ k+1 are unknown, ( ) is upate automatically in the observer whenever sample an elaye measurement arrives In (Ahme-Ali et al [213b]), the sample an elaye measurement is available at instant + τ k, however, ( ) is upate at instant + τ, that is, there exists time elay τ τ k Remark 3 In this paper, the sampling perio T an the maximum allowable elay τ epen on the Observer (3) From (1)-(3), for k, we obtain the observation error: ė 1 = La 1 ( ) + f 1, ė 2 = e 3 L 2 a 2 ( ) + f 2, ė n 1 = e n L n 1 a n 1 ( ) + f (5) n 1, ė n = L n a n ( ) + f n, t [ + τ k, + T + τ k+1 ), where e i = x i ˆx i, f i = f i (x 1, x 2,, x i ) f i (ˆx 1, ˆx 2,, ˆx i ), 1 i n Consier the following change of coorinates ε i = e i, i = 1, 2,, n, (6) Li 1+b where b is a positive real number Then, we have ε 1 = Lε 2 La 1 ε 1 ( ) + f 1 L b, ε 2 = Lε 3 La 2 ε 1 ( ) + f 2 L 1+b, ε n 1 = Lε n La n 1 ε 1 ( ) + f (7) n 1 L n 2+b, f n ε n = La n ε 1 ( ) + L n 1+b, t [ + τ k, + T + τ k+1 ) Further, (7) can be rewritten as follows: ε 1 = Lε 2 La 1 ε 1 + La 1 (ε 1 ε 1 ( )) + f 1 L b, ε 2 = Lε 3 La 2 ε 1 + La 2 (ε 1 ε 1 ( )) + f 2 L 1+b, (8) ε n 1 = Lε n La n 1 ε 1 + La n 1 (ε 1 ε 1 ( )) + f n 1 L n 2+b, f n ε n = La n ε 1 + La n (ε 1 ε 1 ( )) + L n 1+b, t [ + τ k, + T + τ k+1 ) Now, we will give our main results Theorem 1 Consier the system (1) with the conition (2) The output y is assume to be sample at instants an is available for the observer at instants 27

3 + τ k Let T = κ1 L, τ = κ2 3 L (κ 3 1 > κ 2 > are two positive constants) If, a i > (i = 1,, n) are selecte such that the conition (4) hols, an L 1 is given such that T + τ satisfies { L 2λ1 ˆρ 1 T + τ min, L 12nL 2 ā 1 p 2 + ˆρ, ˆρλ } 2 12nL 3 ā 2 1 p, τ < T, (9) 2 then, the state observation error system (5) is globally exponentially stable, where p 2 = λ max (P P ), ā 1 = max{a 2 i }, λ 1 = λ max (P ), λ 2 = λ min (P ), ˆρ = L 1 Proof: Consier the positive efinite function V 1 = ε P ε, (1) where ε = [ε 1,, ε n ], then the erivative of V 1 along the system (8) is given as follows: t V 1 (8) = ε P ε + ε P ε n n Lε ε + 2l ε i P ij i=1 j=1 Lb ε 1 + L 1+b ε L j 1+b ε j +2Lε P L j 1+b a 1 (ε 1 ε 1 ( )) a 2 (ε 1 ε 1 ( )) a n (ε 1 ε 1 ( )) ( 3 4 L + 2n p 1l)ε ε + 4Lnā 1 p 2 (ε 1 ε 1 ( )) 2, t [ + τ k, + T + τ k+1 ), (11) where P ij is the element of P at the ith line an jth column, p 1 = max{ P ij } Note that there exists L 1 > 1 such that for L > L 1 Then, L > 8n p 1 l, t V 1 (8) 1 2 Lε ε + 4nLā 1 p 2 (ε 1 ε 1 ( )) 2, t [ + τ k, + T + τ k+1 ) (12) By Lemma 1, we have ε 1 ε 1 ( ) 2 = ε 1 (s) 2 s L(t ) 3(t )L 2 [ε 2 (s) 2 + ε 1 (s)s 2 (t ) [ε 2 (s) a 1 ε 1 ( ) + f 1 L b+1 ]2 s l 2 L 2(b+1) ε 1(s) 2 + ā 1 ε 1 ( ) 2 ]s, Then, there exists L 2 > 1 such that when L > L 2, we have L b+1 > l If follows from (12) an (13) that t V 1 (8) 1 2 Lε ε + 12nL 3 ā 2 1 p 2 (t ) 2 ε 1 ( ) 2 +12nL 3 ā 1 p 2 (t ) t [ + τ k, + T + τ k+1 ) [ε 1 (s) 2 + ε 2 (s) 2 ]s, Note that when t [ + τ k, + T + τ k+1 ), we have t τ k T + τ k+1 τ k, that is, t T τ k+1 < Then, t V 1 (8) 1 2 Lε ε + 12nL 3 ā 2 1 p 2 (t ) 2 ε 1 ( ) 2 +12nL 3 ā 1 p 2 (t ) t T τ k+1 [ε 1 (s) 2 + ε 2 (s) 2 ]s, t [ + τ k, + T + τ k+1 ) (14) Construct the following auxiliary integral function V 2 = ρ [ε 1 (s) 2 + ε 2 (s) ε n (s) 2 ]sρ, t [, ), (15) where k = min{k : T + τ < } Then, t V 2 = (T + τ)[ε ε ε n 2 ] [ε 1 (s) 2 + ε 2 (s) ε n (s) 2 ]s (16) Now, we consier the following Lyapunov-rasovskii function V = V 1 + LV 2 (17) Calculate the erivative of V efine in (17) along the system (8), we have t V (8) ( (T + τ))lε ε + 12nL 3 ā 2 1 p 2 (T + τ) 2 ε 1 ( ) 2 + (12L 2 nā 1 p 2 (T + τ) 1)L Note that +ε 2 (s) ε n (s) 2 ]s, t [ + τ k, + T + τ k+1 ), k k V 2 (T + τ) [ε 1 (s) 2 + ε 2 (s) 2 + [ε 1 (s) 2 (18) t [ + τ k, + T + τ k+1 ) (13) +ε n (s) 2 ]s, t [ + τ k, + T + τ k+1 ) (19) 271

4 Thus, (18) an (19) imply t V (8) ( 1 2 (T + τ))lε ε 1 12L2 nā 1 p 2 (T + τ) LV nL 3 ā 2 1 p 2 (T + τ) 2 (T + τ) ε 1 ( ) L2 nā 1 p 2 (T + τ) LV 2 (T + τ) 1 (1 (T + τ))lv nL3 ā 2 1 p 2 (T + τ) 2 λ 2 V 1 ( ), t [ + τ k, + T + τ k+1 ), k k (2) Since T + τ satisfies (9), then, we have t V (8) ˆρV + (T + τ)ˆρv ( ), t [ + τ k, + T + τ k+1 ), k k From the above ifferential inequality, we obtain V e ˆρ(t τ k ) V ( + τ k ) + (T + τ)v ( ) (T + τ)e ˆρ(t τ k ) V ( ), t [ + τ k, + T + τ k+1 ), k k (21) Let t = + T + τ k+1 an t = + T, respectively, we have V ( + T + τ k+1 ) e ˆρ(T +τ k+1 τ k ) V ( + τ k ) an, Thus, +(T + τ)v ( ) (T + τ)e ˆρ(T +τ k+1 τ k ) V ( ), V ( + T ) e ˆρ(T τ k) V ( + τ k ) + (T + τ)v ( ) (T + τ)e ˆρ(T τ k) V ( ) V ( + T + τ k+1 ) + ρv ( + T ) e ˆρ(T τ) (1 + ρ)v ( + τ k ) + 2(T + τ)v ( ), (22) where < ρ < 1 is a positive constant an will be etermine later Note that T = κ1 L, τ = κ2 3 L (κ 3 1 > κ 2 ), an ˆρ = L 1, then, there exists L 3 > 1 such that when L > L 3, we have Since an then, 2(κ 1 + κ 2 ) < ( L 1)(κ 1 κ 2 ) ˆρ(κ 1 κ 2 ) = ( L 1)(κ 1 κ 2 ), eˆρ(κ1 κ2)/l3 > 1 + ˆρ(κ 1 κ 2 )/L 3, 2(κ 1 + κ 2 ) L 3 < ˆρ(κ 1 κ 2 ) L 3 < eˆρ(κ1 κ2)/l 3 1 Therefore, there exist L 4 > 1, < ρ < 1 such that 2(κ 1 + κ 2 ) L 3 < ρ < eˆρ(κ1 κ2)/l 3 1, for L > L 4 Thus, e ˆρ(T τ) (1 + ρ) < 1, 2(T + τ) < ρ (23) Therefore, we can select L > max{l 1, L 2, L 3, L 4 } such that η = max{e ˆρ(T τ) 2(T + τ) (1 + ρ), ρ } < 1 Therefore, it follows from (22) that V (+1 + τ k+1 ) + ρv ( + T ) = V ( + T + τ k+1 ) + ρv ( + T ) η[v ( + τ k ) + ρv ( )], k k (24) Applying iteratively (24), for k k, we have V ( + τ k ) + ρv ( ) η k k [V ( + τ k ) + ρv ( )](25) It follows from (21) an (25) that V V ( + τ k ) + ρv ( ) η k k [V ( + τ k ) + ρv ( )], t [ + τ k, + T + τ k+1 ), k k For any t > +τ k, there exists k k such that t [ + τ k, +T +τ k+1 ) Note that t τ k T 1 k t τ k T Then, V η t τ k T 1 [V ( + τ k ) + ρv ( )] = η t T η τ k T 1 [V ( + τ k ) + ρv ( )], t [ + τ k, + T + τ k+1 ), k k (26) That is, the system (8) is globally exponentially stable Remark 4 The value of η has an effect on the convergent time The state observation error has a faster convergent spee with a smaller value of η A large value of L can be selecte to make η small However, it may yiel a small sampling perio T In practical application, we can select a proper L (on T an τ) to achieve the esire effect Remark 5 The change of coorinates (6) iffers by the positive real number b from the stanar change of coorinates use in high-gain observers where b = If the lipschitz constant l is very large, we can choose a proper constant b such that L is not very large Then, the parameter b gives more flexibility on selection of the high gain L 3 EXAMPLE AND SIMULATION In this section, we use two examples to show the effectiveness of our high gain observer esign for nonlinear systems with sample an elaye measurements Exampl An acaemic bioreactor is reactor in which microorganisms grow by eating a substrate Let x 1 an x 2 enote the concentrations of microorganisms an substrate, respectively Then, the following stanar equations for the bioreactor (Contois [1959]) can be obtaine: ẋ 1 = a 1x 1x 2 a 2 x 1 + x 2 1, ux ẋ 2 = a 3a 1 x 1x 2 a 2 x 1 + x 2 (27) 2 + ua ux 4, y = x 1, 272

5 where the control u is in the interval Θ u = [u min, u max ] (, 1) an we choose a 1 = a 2 = a 3 = 1, a 4 = 1 In (Gauthier et al [1992]), it is observe that the set Θ x = (x 1, x 2) R 2 : x 1 γ 1, x 2 γ 2, x 1 + x 2 1 is forwar invariant, where γ 1 = (1 u max γ 2 /u max ) an u min = γ 2 /(1 γ 2 ) This means that the bioreactor state solutions are boune an actually remain in a known compact set After the state transformation x 1 = x 1, x 2 = x 1x 2/(x 1 + x 2), we have ẋ 1 = x 2 ux 1, ẋ 2 = x3 2 + (x 1 x 2 ) 2 (1u x 2 ) x 2 1 x 2 u We get the following equation for the observer: ˆx 1 = ˆx 2 uˆx 1 + 3L(y( ) ˆx 1 ( )), ˆx 2 = ˆx3 2 + (ˆx 1 ˆx 2 ) 2 (1u ˆx 2 ) ˆx 2 1 ˆx 2 u + 2L 2 (y( ) ˆx 1 ( )), t [ + τ k, + T + τ k+1 ), k In the following simulation, we apply the input u = 8, for t 1, u = 2, for 1 < t, x = [7, 3], ˆx = [2, 5], L = 1, the sampling perio T = 1s an the elay τ k satisfying τ k 5s The trajectories of the error states are shown in Fig 1 In practice, the measure is often isturbe by a white noise Fig 2 also shows the trajectories of the errors with white noise Fig 1 Trajectories of the errors an Fig 2 Trajectories of the errors an with white noise Exampl A single-link robot arm system can be moele by (Isiori [1995]) or (Marino an Tomei [1995]) ż 1 = z 2, ż 2 = N z 3 F 2 z 2 z 1 mg cos(z 1 ), ż 3 = z 4, ż 4 = 1 u + F 1 z 4, y = z 1, N z 1 N z 3 (28) where,,, N, m, g, are known parameters, F 1 an F 2 are viscous friction coefficients that are not precisely known Suppose F 1 an F 2 are boune by an unknown constant C > Consier the change of coorinates x 1 = z 1, x 2 = z 2, x 3 = N z 3 mg, x 4 = N z 4 an the pre-feeback υ = N ( 1 u mg ), which transforms (28) into ẋ 1 = x 2, ẋ 2 = x 3 F 2 x 2 x 1 mg (cos(x 1 ) 1), ẋ 3 = x 4, ẋ 4 = υ + 2 N 2 x 1 N x 3 F 1 x 4, y = x 1 Construct the following observer: ˆx 1 = ˆx 2 + 4L(y( ) ˆx 1 ( )), ˆx 2 = ˆx 3 F 2 ˆx 2 ˆx 1 mg (cos(ˆx 1 ) 1) + 6L 2 (y( ) ˆx 1 ( )), ˆx 3 = ˆx 4 + 4L 3 (y( ) ˆx 1 ( )), ˆx 4 = υ + 2 N 2 ˆx 1 N ˆx 3 F 1 ˆx 4 +L 4 (y( ) ˆx 1 ( )), t [ + τ k, + T + τ k+1 ), k In the following simulation, we apply the system parameters: / = 5, mg/ = 4, 2 /( N 2 ) = 2, /( N) = 3 The initial conitions of the whole system are (x 1 (), x 2 (), x 3 (), x 4 (), ) = ( 5, 1, 4, 2) an (ˆx 1 (), ˆx 2 (), ˆx 3 (), ˆx 4 (), ) = (5, 3, 1, 4), a = (4, 6, 4, 1), an L = 1 The sampling perio T an the elay τ k are chosen as T = 1s an τ k 5s The simulation results is shown in Fig 3 In practice, the measure may be isturbe by a white noise The trajectories of the errors with white noise are also shown in Fig 4 273

6 e 3 e Fig 3 Trajectories of the errors e i (1 i 4) e 3 e Fig 4 Trajectories of the errors e i (1 i 4) with white noise 4 CONCLUSION In this paper, we esigne observers for nonlinear systems with sample an elaye output measurements The observers were continuous an hybri in nature Base on an auxiliary integral technique, the exponential stability of the estimation errors was achieve, an the sampling perio an the maximum elay were also given Finally, numerical examples were provie to illustrate the esign methos REFERENCES T Ahme-Ali, an F Lamnabhi-Lagarrigue, High gain observer esign for some networke control systems, IEEE Trans Automat Contr, vol 57, no 4, pp 995-1, Apr 212 T Ahme-Ali, I arafyllis, an F Lamnabhi-Lagarrigue, Global exponential sample-ata observers for nonlinear systems with elaye measurements, Syst Control Lett, vol 62, pp , 213 T Ahme-Ali, V Van Assche, J Massieu an P Dorleans, Continuous-iscrete observer for state affine systems with sample an elaye measurements, IEEE Trans Automat Contr, vol 58, pp , Apr 213 V Anrieu, L Praly, A Astolfi, High gain observers with upate high-gain an homogeneous correction terms, Automatica, vol 45, no 2, pp , 29 M Arcak an D Nešić, A framework for nonlinear sample-ata observer esign via approximate iscretetime moels an emulation, Automatica, vol 4, pp , 24 D Contois, inetics of bacterial growth relationship between population ensity an specific growth rate of continuous cultures, J Genet Macrobio, vol 21, pp 4-5, 1959 F Deza, Contribution to the synthesis of exponential observers, PH D Dissertation, INSA Rouen, France, 1991 F Deza, E Busvelle, J Gauthier, an D Rakotopora, High gain estimation for nonlinear systems, Syst Contr Lett, vol 18, pp , 1992 J Gauthier, H Hammouri, an S Othman, A simple observer for nonlinear systems applications to bioreactors, IEEE Trans Automat Contr, vol 37, no 6, pp , Jun992 J Gauthier, an I A upka, Observability an observers for nonlinear systems, SIAM J Contr an Optimiz, vol 32, no 4, pp , 1994 Isiori, Nonlinear Control Systems New York: Springer, 1995 I arafyllis, an C ravaris, From continuous-time esign to sample-ata esign of observers, IEEE Trans Automat Contr, vol 54, no 9, pp , Sep 29 Y Li, X Xia, an Y Shen, A high-gain-base global finitetime nonlinear observer, Int J Contr, vol 86, no 5, pp , 213 Y Li, Y Shen, an X Xia, Global finite-time observers for a class of nonlinear systems, ybernetika, vol 49, no 2, pp , 213 Y Liu, Z Wang an X Liu, On global exponential stability of generalize stochastic netural networks with mixe time-elays, Neurocomputing, vol 7, pp , 26 R Marino, an P Tomei, Nonlinear Control Design Uk: Prentice Hall International, 1995 H Nari, H Hammouri, an R Mota, Observer esign for uniformly observable systems with sample measurements, IEEE Trans Automat Contr, vol 58, no 3, pp , Mar 213 L Praly, Asymptotic stabilization via output feeback for lower triangular systems with output epenent incremental rate, IEEE Trans Automat Contr, vol 48, pp , 23 T Raff, M ögel, an F Allgöwer, Observer with sample-an-hol upating for lipschitz nonlinear systems with nonuniformly sample measurements, in ACC 8 (American Control Conference), Seattle, Washington, USA, 28 Y Shen, an X Xia, Semi-global finite-time observers for nonlinear systems, Automatica, vol 44, no 12, pp , 28 Y Shen, an Y Huang, Uniformly observable an glaobally lipschitzian nonlinear systems amit global finite-time observers, IEEE Trans Automat Contr, vol 54, no 11, pp , Nov 29 V Van Assche, T Ahme-Ali, C Ham, an F Lamnabhi- Lagarrigue, High gain observer esign for nonlinear systems with time varying elaye measurements in 18th IFAC Worl Congress, Milan, Italy,