Haq, Aminul

Kucukdemiral, Ibrahim Beklan

English

In this paper a H∞type observer is proposed for linear time delay systems with delay in states. The stability of the observer is proved by Lyapunov approach. The novelty of the study is to include the state derivatives in the design.As a result, better delay margin and reliability is obtained. Two numerical examples have been illustrated to Show the validity and effectiveness of this prescribed approach and a comparison table shows the achievement of better delay margin in comparison with corresponding Luenberger type observer

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Observer design with better delay margin for linear time-delay systemsHaq, Aminul; Kucukdemiral, Ibrahim BeklanPublication date:2016Document VersionPublisher's PDF, also known as Version of recordLink to publication in ResearchOnlineCitation for published version (Harvard):Haq, A & Kucukdemiral, IB 2016, 'Observer design with better delay margin for linear time-delay systems',ELECTRICA, vol. 16, no. 2, pp. 2065-2071.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.Take down policyIf you believe that this document breaches copyright please view our takedown policy at https://edshare.gcu.ac.uk/id/eprint/5179 for detailsof how to contact us.Download date: 12. Sep. 2020 Received on: 28.02.2016 Accepted on: 28.04.2016          Aminul HAQ and Ibrahim Beklan KUCUKDEMIRAL / IU-JEEE Vol. 16(2), (2016), 2065-2071                 Observer Design With Better Delay Margin for Linear Time-Delay Systems   Aminul HAQ1 and Ibrahim Beklan KUCUKDEMIRAL2   1 Department of Control and Automation Engineering, Yildiz Technical University, Istanbul, Turkey  2 Department of Control and Automation Engineering, Yildiz Technical University, Istanbul, Turkey ocean_blue04@yahoo.com, beklan@yildiz.edu.tr  Abstract: In this paper a  type observer is proposed for linear time delay systems with delay in states. The stability of the observer is proved by Lyapunov approach. The novelty of the study is to include the state derivatives in the design.As a result, better delay margin and relaibility is obtained.Two numerical examples have been illustreted to Show the validity and effectiveness of this prescribed approach and a comparison table shows the achievement  of better delay margin in comparison with corresponding Luenberger type observer.  Keywords:  observer, Lyapunov, LMI, Time-delay.  1. Introduction  Time-delay system (TDS) is a system having delays in its states, inputs or outputs and occurs in many natural and engineering events. Time-delay is commonly encountered in chemical processes, biological systems, hydraulic systems etc and usually a very common source of instability. TDS actually belongs to the class of functional differential equation (FDE), which has infinite dimensions. making it more complex. Consideration of delay terms in system analysis[14] and designs is necessary for engineers to make models to behave like more to real process.   observer design is one of the fruitful research area and  has an inmate connection with fundamental system concepts. Last few decades different methods  such as Riccati Equation approach  [2,3,4], Lyapunov approach [1,6] are applied for observer design. Observer itself has different classification such as delay independent [5], delay dependent [6], delay free [8,9], positive state bounding [13]. Due to advances  in  computational capability, Linear Matrix Inequality (LMI) [15] is greatly used to analysis the stability of  TDS. It is well known that   filtering problem is dual to the  control one for linear systems without uncertainty.  Controller (observer) design procedure has been proposed and developed in [7, 10, 11, 12], which could be adopted for observer design too because of duality. The main motivation for the study stems from the fact that if PD (Proportional differential) controller is better than only ”proportional” controller then, why not thinking of Proportional-differential type of observer design and developing it in LMI structure. The proposed state estimation scheme is based on several concepts. This observer is the result of integration of following 3 ideas to be named Lyapunov-Krasovskii Theory, Luenberger Obser-ver, Linear Matrix Inequality.  Fig 1. Block diagram of proposed observer  To design an observer for TDSS we use simple Luenberger approach, but we introduced here two feedback line instead of one. The first feedback line contains a proportional gain matrix (L1) and second feedback line has a gain matrix (L2, given) followed by a differentiator block. So here we are considering not only the difference between real states and estimator states or error signals but also the rate of change of error signals. Taking into consideration both error and rate of change of error data would make the observer more reliable than simple Luenberger type one.  2. Problem Formulation  Consider the following linear time-delay system,  (t)=Ax(t)+Adx(t-h)+Bu(t)+Nw(t)                                                           y=Cx(t) Aminul HAQ and Ibrahim Beklan KUCUKDEMIRAL / IU-JEEE Vol. 16(2), (2016), 2065-2071     2066  x(t+ ) =  ( )                                                (1)  x Rn     : The State vector w(t) Rq      : The exogenous disturbance input which                      belongs to  L2[0, ).    y(t) Rp      : The output vector. A, Ad , B , N , C . The above matrices are constant and known system matrices. h  0  :a positive scalar denoting the time delay.   ( )   : a continuously differentiable function on [- ]             representing the  initial condition.                       3. Main Result  Let us formulate an observer dynamics as follows, (t)=F (t)+G (t-h)+Hu(t)+Mw(t)+L1(y(t)- (t))+L2( (t)- (t))                       (t)=C (t)                                                                                     (2)                                                                                   (t) Rn           :The estimator state vector L1,L2  :The  constant observer gain matrix                                         to be selected appropriately. (t) Rp              :The estimated  output vector.  F, G , H , M , C .  Theorem: Observer in form of (2) can be constructed if there exists matrices P=PT>0, R1= >0, R2= >0 and X  for a given noise attenuation level , satisfying the following LMI,                                                                                < 0                     (3)  where  =(ATPZ-1 -CTXTZ-1+AdTPZ-1+ Z-TPA- Z-TXC+ Z-TPAd +CTC3.1 Proof  Subtracting equation (2) from equation (1) we get,  (t) - (t)   = Ax(t)+Adx(t-h)+Bu(t)+Nw(t) - F (t)-G (t-h)                       -Hu(t)-Mw(t)-L1(y(t)- (t))-L2( (t)- (t))   (t)           =Ax(t)+Adx(t-h)+Bu(t)+Nw(t) - F (t)-G (t-h)                     -Hu(t)-Mw(t)-L1(y(t)- (t))-L2( (t)- (t))+Fx(t)                     +Gx(t-h)-Fx(t)-Gx(t-h)  (t)    =(A-F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)+(N-M)w(t)                   +F(x(t) (t))+G(x(t-h)- (t-h))-L1(Cx(t)-C (t))                    -L2(C (t)-C (t))  (t)          =(A-F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)+(N-M)w(t)                    +Fe(t)+Ge(t-h)-L1C(x(t)- (t)) -L2C( (t)- (t))  (t)+L2C (t) = (A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)                          +(N-M)w(t)+Fe(t)+Ge(t-h)-L1Ce(t)   (I+ L2C) (t)  =  (A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)                          +(N-M)w(t)+(F-L1C)e(t)+Ge(t-h)   (t)           =( I+ L2C)-1[(A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)                     +(N-M)w(t)+(F-L1C)e(t)+Ge(t-h)]  (t)           = Z[(A -F)x(t)+(Ad–G)x(t-h)+(B-H)u(t)                      +(N-M)w(t)+(F-L1C)e(t)+Ge(t-h)]     where ,  Z=( I+ L2C)-1         Here, we will choose L2 arbitrarily and calculate the gain L1 accordingly.Obviously, e(t)→0 as t→∞ if the following conditions are satisfied: (1)           The system is stable and observable. (2)           ( I+ L2C) is invertible. (3)           A=F,          Ad=G,            B=H,                N=M,                  Then the error dynamics reduces to,                                                (t)              =( I+ L2C)-1[(F-L1C)e(t)+Ge(t-h)]               (t)              =   Z[(F-L1C)e(t)+Ge(t-h)] (4)  We will utilize following the Leibniz rule   Lemma 1:   A(t-h) = A(t)-     We will also use the following lemma in our proof   Lemma 2:  -2UTV  UTRU+VT V  Then we have the error dynamics as follows, (t)              =   Z[(F-L1C)e(t)+Ge(t-h)]   Using Leibniz rule given in Lemma 1, we can write,  e(t-h) = e(t)-                 = e(t)-    (t)     =Z(F-L1C)e(t)              +ZG{e(t)- }     Aminul HAQ and Ibrahim Beklan KUCUKDEMIRAL / IU-JEEE Vol. 16(2), (2016), 2065-2071     2067  The error dynamics (4) is now transformed into the following equation.                                                                                                                    (t)    = Z[(F-L1C)+G]e(t)                                                                                                                                          (5) (t) 0 as t  means error in (5) tends to ‘0’ as time evolves.  Delay-Dependent Approach: Consider the following  Lyapunov–Krasovskii functional  V(e,t)= e(t)T Z-TPZ-1 e(t)                  +                   +   (e,t) = (t)T Z-TPZ-1e(t)+ e(t)TZ-TPZ-1 (t)                  +h e(t)T  e(t)                  -                   + h e(t)TGTR2Ge(t)                  -             =  e(t)T[(F-L1C)+G]TZT Z-TPZ-1e(t)              + e(t)T Z-TPZ-1Z[(F-L1C)+G] e(t)              -2e(t)TZ-TPZ-1ZGZ                        *               + h e(t)T(F-L1C)TR1(F-L1C)e(t)              -               + h e(t)TGTR2Ge(t)-               e(t)T[(F-L1C)+G]TPZ-1e(t)              + e(t)T Z-TP[(F- L1C)+G]e(t)               + d               +               + d               +              + h e(t)T(F-L1C)TR1(F-L1C)e(t) + h e(t)TGTR2Ge(t)                               e(t)T[FTPZ-1-CTL1TPZ-1+GTPZ-1+Z-TPF- Z-TPL1C            +Z-TPG]e(t)+he(t)TZ-TPGZR1-1ZTGTPZ-1e(t)       +h e(t)TZ-TPGZR2-1ZTGTPZ-1e(t)       +he(t)T(F-L1C)TR1(F-L1C)e(t)+ h e(t)TGTR2Ge(t)   Now applying Schur complement we get,  e(t)T                                                                                         e(t)                                                                                                              Here,  𝛺= (FTPZ-1 -CTL1TPZ-1+GTPZ-1+ Z-TPF- ZTPL1C+Z-TPG)   If above matrix is less than 0,then  (e,t)  is negative so e(t) 0 as t .                                                                                                                                                               < 0     (6)      Pre and post multiplying (6) by diag {I,I,I,I,P} and replacing –hR2-1 by h(R2-2I) as we know –R2-1<(R2-2I), we get                                                                                             <0  We can replace –hPR1-1P by h(R1-2P) as,   h(R1-P) (R1-P) >0 hR1-hP-hP+ hPR1-1P>0  –hPR1-1P<h(R1-2P)                                                                                                                    < 0 Aminul HAQ and Ibrahim Beklan KUCUKDEMIRAL / IU-JEEE Vol. 16(2), (2016), 2065-2071     2068  Now let PL1=X and defining the matrix (right hand side of the equation ) as ,   = < 0  Here =(FTPZ-1 -CTXTZ-1+GTPZ-1+ Z-TPF- Z-TXC+ Z-TPG)  For  observer,it has to satisfy the following equation, +z(t)Tz(t)- 2w(t)Tw(t)]dt<0                                                                                                                  (7) If      +z(t)Tz(t)- 2w(t)Tw(t)<0       then (7) will be true also. e(t)T  e(t)+ e(t)TCTCe(t))- 2w(t)Tw(t)<0         [here, z=y(t)- (t)= Cx(t)- C (t)= Ce(t)]                                    (8) if 𝛇(t)=[e(t) ; w(t)] and applying Schur complement to (8) ,  𝛇 (t)T    𝛇 (t)< 0  where  =(FTPZ-1 -CTXTZ-1+GTPZ-1+ Z-TPF- Z-TXC+ Z-TPG +CTC)             < 0  According to necessary condition, replacing F=A and G=Ad we get the following final LMI             < 0  where  =(ATPZ-1 -CTXTZ-1+AdTPZ-1+ Z-TPA- Z-TXC+ Z-TPAd +CTC) Solving the LMI for P and X we can get L1=P-1X.    Aminul HAQ and Ibrahim Beklan KUCUKDEMIRAL / IU-JEEE Vol. 16(2), (2016), 2065-2071     2069  4. Numerical Example  In this section, we will demonstrate the theory developed in this paper by means of simple examples. Here to solve problem we have used Matlab software,Yalmip Optimization Toolbox and Sedumi solver.  Consider the linear continuous time-delay system (9) and (10) with parameters given by   Example (1): A=         Ad=             (9)                   C= [  1      0  ]      L2=[0.5    0.4]T (chosen)  (10)   Where 0 < h  is an unknown positive scalar. The purpose is to design observer using equation (3) according to the block diagram. The transfer function from exogenous disturbances to error state outputs meets the prescribed  norm upper bound constraint   0.8 Here, we take the value =0.3  Solving the LMI, we get   P=     R1=     R2=               X=                             L1 =  Here in the example plant initial state is [5;-2] and estimator initial state is [0;0].                  Fig 2. Trajectories of state (t) and (t)    Fig 3.  Trajectories of state (t) and (t)  Example (2):  A=                 Ad=                   (11)  C= [     0          1   ]               L2=[0.5     0.4]T (chosen)          (12)    Where  0 < h  is an unknown positive scalar.  The transfer function from exogenous disturbances to error state outputs meets the prescribed  norm upper bound constraint   0.8  Here, we chose =0.3.Solving the LMI, we can get the values as follows,   P=          R1=     R2=            X=     L1 =   Here in the example plant initial state is [4;-3] and estimator initial state is [0;0].          Fig 4. Trajectories of state (t) and (t)  Aminul HAQ and Ibrahim Beklan KUCUKDEMIRAL / IU-JEEE Vol. 16(2), (2016), 2065-2071     2070          Fig 5. Trajectories of state (t) and (t) From the simulation result shown on graphs,we can see that the trajectories of plant states and observer states converge within few seconds,which is pretty good performance by the observer designed using the method developed in this paper.  5. Conclusions The advantges of such type observer is better estimation of actuall plant states as both state values and rate of change of state values have been taken into consideration in the observer equation. One of the principle goal for time-dealy systems community is designing observer or controller to achieve longer time delay without interrupting stability.Using the methodology developed in this paper would increase the delay margin.It would be clear in the following comparison table, here same examples are simulated in both Luenberger type observer and proposed observer and obtained delay margin is compared.  Table 1. Comparison table of time delays  Plant Luenberger type Observer Proposed Observer Example 1 0.71 sec 0.77 sec Example 2 0.28 sec 0.32 sec  It is obvious from the table that when we utilize Luenberger type observer with a system, the system can have states with 0.71 seconds maximum delay (example 1). But utilizing proposed observer, the system can have states with 0.77 seconds maximum delay(example 1). So it means for a system with, lets say 0.75 seconds delay in any of it’s state, the Luenberger type observer will not work correctly while the proposed observer will still track down the unknown data. In case of example 2, Luenberger type observer can be used with system having 0.28 seconds maximum delay while proposed observer offer 0.32 seconds delay.  In this paper an observer design procedure for systems with delays in states has been studied. An appropriate gain matrix for observer is calculated while the gain matrix for differentiator block has been predetermined. Necessary and sufficient conditions have also been derived. Numerical examples provided here described the effectiveness of this method.  6. References  [1]  Oliver Sename, “New trends in design of observers for time-delay  systems”, Kybernetika, Vol 37 no. 4(2001) 427-458. [2] Zidong Wang, Biao Huang, H. Unbehauaen,”Robust observer design of linear time-delay systems with       parametric uncertainty”,Systems & Control Letters 42 (2001)  [3] A.Fattouh, O.Sename, J.M. Dion,”  observer design for time-delay systems”,in proc 37th IEEE Confer. on Decision & Control (Tampa,Florida,USA),(1998) 4545-4546. [4] M. Boutayeb, “Obsevers design for linear time-delay systems”,  Systems & Control Letters 44 (2001) 103-109. [5] Yan-Ming, Fu, Guang-Ren Duan, Shen-Min Song, “Design of Unknown Input Observer for Linear Time-delay Systems”, International Journal of Control, Automation and Systems,vol.2 no.4 (2004) 530-535. [6] Md. Aminul Haq, Ibrahim Beklan Kucukdemiral, “  observer design for linear time-delay systems”, Int. Conference on Electrical and Electronics Engineering, ELECO 2015. [7] Fatma Yildiz Tascikaraoglu, Levent Ucun, Ibrahim B Kucukdemiral,”Receding horizon   control of time delay systems”,Transaction of the institute of measurement and Control (2014) 1-10. [8] M.N. Alpaslan Parklakci,”Robust delay-free observer-based controller design for uncertain neutral time delay”, Systems, Systems Science (2006).  [9] M.N. Alpaslan Parklakci,”Robust delay-free observer-based controller design for uncertain neutral time delay”, Systems, Systems Science (2006). [10] Mai Viet Thuan, Vu Ngoc Phat, Hieu Trinh,”Observer-based controller design of time-delay systems with an interval time-varying delay”, Int. J. Appl. Math. Comput. Sci., 2012, Vol. 22, No. 4, 921–927. [11] Cao, Y.-Y., Sun, Y.-X., & Lam, J. ,”Delay dependent robust control for uncertain systems with time varying delays”, IEE Proceedings: Control Theory and Applications, (1998).  143, 338–344. [12] Vladimir B. Kolmanovskii , “On the Liapunov-Krasovskii functionals for stability analysis of linear delay systems”, International Journal of Control,(1999) 72:4, 374-384. [13] Ping Li,James Lam,”Positive state-bounding observer for positive interval continuous-time systems with time delay”, Int. J. Robust. Nonlinear Control 2012; 22:1244–1257. [14] Jean-Pierre Richard, “Time-delay systems: an overview of some recent advances and open problems”, Automatica 39 (2003) 1667-1694.  Aminul HAQ and Ibrahim Beklan KUCUKDEMIRAL / IU-JEEE Vol. 16(2), (2016), 2065-2071     2071  [15] Sheng yuan Xu & James Lam, “A survey of linear matrix inequality techniques in stability analysis of delay systems”, International Journal of Systems Science, (2008), 39:12,1095-1113.  Note:  Md. Aminul Haq was born in 1986 in chittagong, Bangladesh. He finished his B.Sc. in Electrical and Electronics Engineering  in 2009 and M.Sc. in Control and Automation Engineering in 2016.  Ibrahim Beklan Kucukdemiral received his B.Sc., M.Sc. and Ph.D. degrees from Yildiz Technical University,Department of Electrical Engineering in 1997, 1999, 2002 respectively.He is currently professor of Control and Automation Engineering in Yildiz Technical University.He is a member of several IFAC Technical committee and served as editorial member of different international control congress.His field of research interest is robust,optimal and  adaptive control systems, particular focuses are LMI based analysis and synthesis for uncertain TDS, LPV, Convex optimization etc. 

Observer design with better delay margin for linear time-delay systems

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Observer design with better delay margin for linear time-delay systems

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