Observer design based on linearization via input–output injection of time-delay systems

10th IFAC Symposium on Nonlinear Control Systems 10th IFAC Symposium on Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA 10th IFAC Symposium on Control Systems 10th IFAC Symposium on Nonlinear Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA Available August 23-25, 2016. Monterey, California, USA August 23-25, 2016. Monterey, California, USA online at www.sciencedirect.com

ScienceDirect IFAC-PapersOnLine 49-18 (2016) 672–677

Observer design based on Observer design based on Observer design based on input–output injection input–output injection input–output injection systems systems systems

linearization via linearization via linearization via of time-delay of time-delay of time-delay

C. Califano ∗∗ L.A. M´ arquez–Mart´ınez ∗∗ ∗∗ ∗∗ ∗∗∗ ∗ C. Califano L.A. M´ a rquez–Mart´ınez ∗∗ L.A. M´ C. Califano a Moog C.C.H. Califano L.A. M´ arquez–Mart´ rquez–Mart´ınez ınez ∗ ∗∗∗ C.H. Moog ∗∗∗ ∗∗∗ C.H. Moog C.H. Moog ∗ Centro de Investigaci´ on Cient´ıfica y de Educaci´ on Superior de ∗ ∗ Centro de Investigaci´ o n Cient´ ıfica yy M´ de Educaci´ o n Superior de ∗ Centro de Investigaci´ o n Cient´ ıfica de Educaci´ o Ensenada, Baja California, exico, (e-mail: Centro de Investigaci´ o n Cient´ ıfica y de Educaci´ on n Superior Superior de de Ensenada, Baja California, M´ e xico, (e-mail: Ensenada, Baja California, M´ eexico, (e-mail: {lmarquez,edugar}@cicese.edu.mx). Ensenada, Baja California, M´ xico, (e-mail: {lmarquez,edugar}@cicese.edu.mx). ∗∗ {lmarquez,edugar}@cicese.edu.mx). a di Roma La Sapienza, Roma, Italy, (e-mail: {lmarquez,edugar}@cicese.edu.mx). ∗∗ Universit` ∗∗ a di Roma La Sapienza, Roma, Italy, (e-mail: ∗∗ Universit` Universit` a Roma [email protected]) Universit` a di di Roma La La Sapienza, Sapienza, Roma, Roma, Italy, Italy, (e-mail: (e-mail: [email protected]) ∗∗∗ [email protected]) IRCCyN, UMR CNRS 6597 , Nantes, France, (e-mail: [email protected]) [email protected]) ∗∗∗ ∗∗∗ IRCCyN, UMR CNRS 6597, Nantes, France, (e-mail: [email protected]) ∗∗∗ IRCCyN, UMR CNRS 6597, Nantes, France, (e-mail: [email protected]) IRCCyN, UMR CNRS 6597, Nantes, France, (e-mail: [email protected]) Abstract: An observer design is derived for a nonlinear time–delay system from its equivalence Abstract: An observer design is derived for a nonlinear time–delay its equivalence Abstract: An observer is for nonlinear system from its equivalence to the observer form up to nonlinear injection.system These from results obtained Abstract: An canonical observer design design is derived derived for aainput–output nonlinear time–delay time–delay system from itsare equivalence to the observer canonical form up to nonlinear input–output injection. These results are obtained to the observer canonical form up to nonlinear input–output injection. These results are obtained by using algebraic tools. form It is up shown that it isinput–output not necessaryinjection. to guarantee equivalence through to the observer canonical to nonlinear These results are obtained by using algebraic tools. It is shown that it is not necessary to guarantee equivalence through by using algebraic tools. It is shown that it is not necessary to guarantee equivalence through bicausal change of coordinates. A practical case study is worked out for an electromechanical by using algebraic tools. It is shown that it is not necessary to guarantee equivalence through bicausal change of coordinates. A practical case study is worked out for an electromechanical bicausal change of coordinates. A practical case study is worked out for an electromechanical system. Simulation results are included to show the feasibility of the presented observers. bicausal change of coordinates. A practical case study is worked out for an electromechanical system. Simulation results are included to show the feasibility of observers. system. Simulation results are to the of the the presented presented observers. system. Simulation resultsFederation are included included to show show Control) the feasibility feasibility presented © 2016, IFAC (International of Automatic Hosting of bythe Elsevier Ltd. Allobservers. rights reserved. Keywords: Nonlinear time-delay systems, observers, bicausal change of coordinates. Keywords: Nonlinear time-delay systems, observers, bicausal change of coordinates. Keywords: Keywords: Nonlinear Nonlinear time-delay time-delay systems, systems, observers, observers, bicausal bicausal change change of of coordinates. coordinates. 1. INTRODUCTION As an application of the results considered in this paper, 1. INTRODUCTION INTRODUCTION As observer an application application of the the results results considered in this this paper, 1. As an of considered in paper, an is designed a quarter-car model subject to 1. INTRODUCTION As an application of thefor results considered in this paper, an observer is designed for a quarter-car model subject to an observer is designed for a quarter-car model subject to delays in the measurement (see Hu and Wang (2002)). an observer ismeasurement designed for a(see quarter-car model(2002)). subject to delays in the Hu and Wang delays in the measurement (see and Wang (2002)). delays in of the measurement (see Hu Hu Wang (2002)). rest this work is organized as and follows. The algebraic A state observer is an important component in many The The rest of this work is organized as follows. The algebraic A state observer is an important component in many The rest of this work is organized as follows. The algebraic setting is recalled in §2, while the notion of invertible A state observer is an important component in many control systems. This concept has been extensivelyinstudied The restisofrecalled this workinis§2, organized as follows. The algebraic A state observer is an important component many setting while the notion of invertible control systems. This concept has been been extensively extensively studied is recalled in §2, while the of invertible change of coordinates is presented innotion §3. These are capcontrol systems. concept has for the linear caseThis since its introduction in Kalman studied (1963). setting setting is recalled in §2, while the notion of invertible control systems. This concept has been extensively studied change of coordinates is presented inwhether §3. These These are capcapfor the the linear case since since itsthe introduction in Kalman (1963). change of coordinates is presented in §3. are ital for the procedure to determine a system is for linear case its introduction in Kalman (1963). For the nonlinear case, problem is more involved, change of coordinates is presented in §3. These are capfor the linear case since itsthe introduction in Kalman (1963). ital for the procedure to determine whether a system is For the nonlinear case, problem is more involved, ital for the procedure to determine whether a system is equivalent to a linear form via input-output injection, For the nonlinear case, the problem is more involved, and several techniques have been employed to tackle it ital for the procedure to determine whether a system is For the nonlinear case, the problem is more involved, equivalent to a linear form via input-output injection, andfairly several techniques have been employed to tackle tackleand it equivalent to aa linear via input-output which is presented in §4. form Simulation results of theinjection, observer and several techniques have employed to it (a complete review canbeen be found in Nijmeijer equivalent to linear form via input-output injection, and several techniques have been employed to tackle it which is is presented presented in §4. Simulation Simulation results of the the observer (a fairly fairly complete review can be be found founddelays in Nijmeijer Nijmeijer and which §4. results of for the in quarter-car model are presented in §5. (a complete review can in and Van Der Schaft (1990)). However, are present, which is presented in §4. Simulation results of the observer observer (a fairly complete review can be when founddelays in Nijmeijer and developped developped for the quarter-car model are presented in §5. §5. Van Der Schaft (1990)). However, when are present, developped for the quarter-car model are presented in Finally, some concluding remarks are presented in §6. Van Der Schaft (1990)). However, when delays are present, these techniques cannot be directly employed. This has developped for the quarter-car model are presented in §5. Van Der Schaft (1990)). However, when delays areThis present, Finally, some some concluding concluding remarks remarks are are presented presented in in §6. §6. these techniques cannot be directly employed. has Finally, these techniques cannot directly employed. This has motivated the development new definitions observthese techniques cannot be be of directly employed.of has Finally, some concluding remarks are presented in §6. motivated the development of new definitions definitions ofThis observmotivated development of new observability, andthe new classes of observers (see e.g.,of Germani 2. DEFINITIONS AND ALGEBRAIC SETTING motivated the development of new definitions of observability, and new new classesand of observers observers (seeAnguelova e.g., Germani Germani 2. DEFINITIONS DEFINITIONS AND ALGEBRAIC ALGEBRAIC SETTING ability, and classes of (see e.g., 2. et al. (2001), Germani Pepe (2005), and ability, and new classes of observers (see e.g., Germani 2. DEFINITIONS AND AND ALGEBRAIC SETTING SETTING et al. (2001), Germani and Pepe (2005), Anguelova and et al. (2001), Germani and Pepe (2005), Anguelova and Wennberg (2010), Bejarano and Zheng (2014), up to Vafei In the present paper, the class of nonlinear single inet al. (2001), Germani and Pepe (2005), Anguelova and Wennberg (2010), Bejarano Bejarano and Zheng (2014), (2003) up to to Vafei In the the present paper, the class class of time-delay nonlinear systems single ininWennberg (2010), Zheng (2014), up and Yazdanpanah (2016), asand well as Richard and In present the nonlinear single put single outputpaper, dynamical causalof is Wennberg (2010), Bejarano and Zheng (2014), (2003) up to Vafei Vafei In the present paper, the class of time-delay nonlinear systems single inand Yazdanpanah (2016), as well as Richard and put single output dynamical causal is and Yazdanpanah (2016), as well as Richard (2003) and the references therein). put single output dynamical causal time-delay systems is considered which are affected by constant commensurable and Yazdanpanah (2016), as well as Richard (2003) and put single output dynamical causal time-delay systems is the references references therein). therein). considered which are affected by constant commensurable the considered which are affected by constant commensurable Suchwhich a class ofaffected systems by canconstant be represented, without the references therein). considered are commensurable The development of observers for nonlinear time-delay delays. delays. Such aa class class ofthe systems can be be represented, represented, without delays. Such of systems can without loss of generality, by equations The development of observers for nonlinear time-delay delays. Such a class of systems can be represented, without The development of observers for nonlinear time-delay systems have followed, and practical applications have loss of generality, by the equations The development of observers for nonlinear time-delay loss of generality, by the equations systems have followed, followed, and practical applications have loss of generality, by the equations systems have and practical applications have been reported (see, e.g., Chen et al. (2016); He and Liu systems have followed, and applications have been reported reported (see, e.g.,(2015); Chenpractical et al. al. (2016); (2016); He and Liu x(t) ˙ = F (x(t), x(t − D), . . . , x(t − sD)) + been (see, e.g., Chen et Liu (2016a); Hansen et al. Khosravian et He al. and (2016)). been reported (see, e.g., Chen et al. (2016); He and Liu x(t) =F F (x(t), (x(t), x(t − − D), D), .. .. .. ,, x(t x(t − − sD)) sD)) + +  (2016a); Hansen et al. (2015); Khosravian et al. (2016)). s x(t) ˙˙˙ = x(t (2016a); Hansen (2015); Khosravian et al. (2016)). Although this is et anal. active research area (see Fall et al. x(t) = F (x(t), x(t(x(t), − D), . .− . , D), x(t − sD)) + sD))u(t − jD)  (2016a); Hansen et al. (2015); Khosravian et al. (2016)). s G x(t . . . , x(t i s Although this isLiu an (2016b); active research research area (see Fall et etand al. sj=0 Gi (x(t), x(t − D), . . . , x(t − Although an active area Fall al. (2015); Hethis andis Murguia et (see al. (2015) − sD))u(t sD))u(t − jD) jD) Although this is an active research area (see Fall et al. G (x(t), x(t − D), . . . , x(t i (x(t), x(t − D), . . . , x(t − j=0 − sD))u(t − − jD) (2015); He Hetherein), and Liu Liumost (2016b); Murguia et al. (2015) and i j=0 G (2015); and (2016b); (2015) references worksMurguia consider et theal. case when and the y(t) = H(x(t), (1) j=0 x(t − D), . . . , x(t − sD)). (2015); He and Liu (2016b); Murguia et al. (2015) and references therein), most works consider the case case when when the the y(t) = H(x(t), x(t − D), .. .. .. ,, x(t − sD)). (1) references therein), most consider the delay appears only in theworks output measurement. y(t) = H(x(t), x(t − D), x(t − sD)). (1) references therein), most works consider the case when the y(t) = H(x(t), x(t − D), . . . , x(t − sD)). (1) where D is a constant delay, which after rescaling can delay appears only in the output measurement. delay appears only in the output measurement. where D is a constant delay, which after rescaling can n delay appears only in the output measurement. where D is a constant delay, which after rescaling can In this paper, based on the results in Garc´ıa-Ram´ırez et al. be , y(t) ∈ R, set equal to 1, that is D = 1. x(t) ∈ R D is atoconstant delay, which after rescaling can n n , y(t) ∈ R, In this this paper, paper, based on on the theobserver results in in Garc´ ıa-Ram´ ırez et et al. al. where be set equal 1, that is D = 1. x(t) ∈ R n , y(t)of∈ the In based results Garc´ ıa-Ram´ ırez R, be set equal to 1, that is D = 1. x(t) ∈ R (2016), a Luenberger-like is derived for nonlinear u(t) ∈ R denote respectively the current values , y(t)of∈ the R, In this paper, based on theobserver results in Garc´ ıa-Ram´ ırez et al. be set∈ equal to 1, respectively that is D =the 1. x(t) ∈ Rvalues u(t) R denote denote current (2016), Luenberger-like is derived for by nonlinear (2016), aa Luenberger-like observer derived for nonlinear R the current values the time-delay systems by means of a is linearization input- u(t) state,∈ output and respectively input variables. The state initialof conu(t) ∈ R denote respectively the current values of the (2016), a Luenberger-like observer is derived for nonlinear state, output output and0] input input variables. Theto state state initial concontime-delay systems by means means of aa linearization linearization by inputtime-delay systems by of state, and The initial output injection. Under appropriate conditions,by theinputesti- dition χ : [−sD, → Rnnvariables. is assumed be smooth. The state, output and0] input variables. The state initial contime-delay systems by means of a linearization by n is assumed to be smooth. The dition χ ::notation [−sD, → R output can injection. Under appropriate conditions, theinputestin isfrom output injection. Under appropriate conditions, the estidition χ [−sD, 0] → R assumed to be smooth. The mates then be used to stabilize the given system as following is taken Califano et al. (2011): K χ :notation [−sD, 0]is→taken R isfrom assumed to be smooth. The output injection. appropriate conditions, the estifollowing Califano et al. (2011): K mates can can then beUnder used to stabilize stabilize the given system system as dition mates then be used to the given as following notation is taken from Califano et al. (2011): K shown in the didactic Example 1. More general feedback denotes the field of meromorphic functions of the symbols notation ismeromorphic taken from Califano etofal. (2011): K mates can then be used to stabilize the general given system as following denotes the field of functions the symbols shown in the didactic Example 1. More feedback (k) + symbols shown in the didactic Example 1. More general feedback denotes the field of meromorphic functions of the laws using predictors can also 1. beMore used general as discussed in denotes i ∈ Z, k ∈ofZthe }; d is the {x(t − i), u(tfield − i),of. .meromorphic . , u(k) (t − i), functions the symbols shown in the didactic Example feedback + (k) (t − i), i ∈ Z, k ∈ Z+ }; d is the {x(t − i), u(t − i), . . . , u laws using predictors can also be used as discussed in laws using can used as in (t ii ∈ ∈ is the − − ,, u Germani et predictors al. (2002); Ahmed-Ali al. (2013); Battilotti differential elements Kd = (t − − i), i), ∈ Z, Z, k k from ∈Z Z+ }; }; dto isE the {x(t − i), i), u(t u(toperator − i), i), .. .. ..that u(k)maps laws using can also also be beet as discussed discussed in {x(t differential operator that maps elements from K to E = Germani et predictors al. (2002); (2002); Ahmed-Ali etused al. (2013); (2013); Battilotti (k) Germani et al. Ahmed-Ali et al. Battilotti differential operator that maps elements from K to E (2015); Khosravian et al. (2015). span {dx(t−i), du(t−i), . . . , du (t−i), i ∈ Z, k ∈ N}; δ is K operator that. .maps elements from K to E = = Germani et al. (2002); Ahmed-Ali et al. (2013); Battilotti differential (k) (k) (t−i), i ∈ Z, k ∈ N}; δ is span {dx(t−i), du(t−i), . , du (2015); Khosravian et al. (2015). K (k) (t−i), i ∈ Z, k ∈ N}; δ is (2015); Khosravian et al. (2015). span {dx(t−i), du(t−i), . . . , du K spanK {dx(t−i), du(t−i), . . . , du (t−i), i ∈ Z, k ∈ N}; δ is (2015); Khosravian et al. (2015). E. E. E. E.

Garc´ıa–Ram´ırez ∗∗ ∗ Garc´ ıa–Ram´ırez Garc´ Garc´ıa–Ram´ ıa–Ram´ırez ırez ∗

Copyright © 2016, 2016 IFAC 684Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 684 Copyright © 2016 IFAC 684 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 684Control. 10.1016/j.ifacol.2016.10.243

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the time-shift operator defined as follows: if a(·), f (·) ∈ K, then δ(a(t)df (t)) = a(t − 1)δdf (t) = a(t − 1)df (t − 1). Using the time-shift operator δ as indeterminate, the noncommutative Euclidean (left) ring of polynomials with coefficients over K is denoted as K(δ]; R[δ] is the ring of polynomials in δ with coefficients in R. M is defined as the left-module over the ring K(δ]: M = spanK(δ] {dξ | ξ ∈ K}. Denoting by deg(·) the polynomial degree in δ of its argument, the elements of K(δ] may be written as rα i α (t)δ , with αi ∈ K, and rα = deg(α(δ]). α(δ] = i=0 i Addition and multiplication on this ring are defined by max{rα ,rβ } α(δ] + β(δ] = (αi (t) + βi (t))δ i , and α(δ]β(δ] = i=0  rα  rβ i+j . j=0 αi (t)βj (t − i)δ i=0 Let us define for p, s ≥ 0, by (x[p,s] ) = (x(t + p), . . . , x(t − s)); (z[p,s] ), and (u[p,s] ), are defined similarly. We will use x[s] for x[0,s] . Define u ¯ = (u(t), u(t), ˙ . . . , u(n−1) (t))T , and y ¯ is defined in a similar way. For simplicity y(t), x(t), and u(t) will stand for y[0] , x[0] and u[0] . A matrix M (x[p,s] , δ) ∈ Kn×n (δ] is called unimodular if it has a polynomial inverse. It is called polymodular if there exists a polynomial matrix M  (x[p,s] , δ) such that M (x[p,s] , δ)M  (x[p,s] , δ) = diag{δ k1 , . . . δ kn }, for some ki ∈ Z+ . Consider also f¯(x[l] ) |x[l] (−j) := f¯(x(t − j), x(t − j − 1), . . . , x(t − j − l)). Then, it is possible to rewrite equation s (1) as x˙ [0] = F (x[s] ) + j=0 Gj (x[s] )u[0] (−j) (2) y[0] = H(x[s] ). The corresponding differential-form representation is then dx˙ [0] = f (x[s] , u[s] , δ)dx[0] + g(x[s] , δ)du[0] where   s s   ∂F (x[s] ) ∂Gj (x[s] )  δi f (x[s] , u[s] , δ)=  + u[0] (−j) ∂x (−i) ∂x (−i) [0] [0] i=0 j=0 s g(x[s] , δ) = Gj (x[s] )δ j j=0

s

i=0

Definition 1. System (2) is said to be weakly-observable if the matrix O has full rank around x[0] .

Definition 2. System (2) is said to be strongly observable if the matrix O is unimodular around x[0] . 3. INVERTIBLE CHANGE OF COORDINATES When dealing with time-delay systems a classical assumption is to consider bicausal change of coordinates, that is change of coordinates which are causal and admit a causal inverse. In fact, such a class of change of coordinates guarantees that causality is preserved under change of coordinates. However, when dealing with observer design for time-delay systems, the causality property can be lost still guaranteeing that a feasible solution exists for the problem. As it will be shown later on in the observer design, the requirement of bicausality of the change of coordinates can be weakened to invertibility. Definition 3. Given the system defined by (2), z[0] = φ(x[p,s] ) is an invertible change of coordinates if there ¯ [p ,s ] ) ∈ K, p, s, p , s ∈ exists a differentiable function φ(z ¯ N, such that φ(z[p ,s ] ) |z[0] =φ(x[p,s] ) = x[0] . To the invertible change of coordinates z[0] = φ(x[p,s] ) we can associate a list of integers ri = max{l ∈ Z | ∂φi (x[p,s] ) ≡ 0}. Its differential representation can be ∂x[0] (l) written as diag(δ r1 , . . . , δ rn )dz[0] = N (x[0,¯s] , δ)dx[0] where diag(·) stands for a diagonal matrix. For the inverse transformation, the corresponding indices are defined by ¯ ∂ φ(z   ) ki = max{l ∈ Z | ∂z[0][p(l,s )] ≡ 0}. The differential   representation is dz[0] (k1 )  .. ˜ (z[p ,s ] , δ)  dx[0] = N   . dz[0] (kn )

dy[0] = h(x[s] , δ)dx[0] ,

h(x[s] , δ) =

673

∂H(x[s] ) i δ dx[0] . ∂x[0] (−i)

Ω = spanK(δ] {ωi (x, δ), i = 1, . . . , p} is the module spanned over K(δ] by the row vectors ω1 (x, δ), ω2 (x, δ), . . . , ωp (x, δ) ∈ Kn (δ].

Let us consider the extended Lie derivative for nonlinear time-delay systems, given in the framework of Xia et al. (2002), and expressed in Zheng et al. (2011) as s  ∂H(x[s] ) i LF H(x[s] ) = δ F (x[s] ), (3) ∂x[0] (−i) i=0

Accordingly, consider the l-th extended Lie derivative LlF H(x[s] ) := LF (Ll−1 F H(x[s] )). The observability matrix is then given by    dy  dH(x) dy˙   dLF H(x[p] )    = O(x[s] , δ)dx[0] =  .. .  . (4)     ..  .

H(x[p] ) dLn−1 dy(n−1) F The notions of weak and strong observability are borrowed from (Califano et al. (2013)). 685

Consequently   dz[0] (k1 ) diag(δ r1 , . . . , δ rn )dz[0] =  .. ˜ (z[p ,s ] , δ) N (x[0,¯s] , δ)|x=ψ(z[p ,s ] ) N   . dz[0] (kn ) It follows that diag(δ r1 +k1 , . . . , δ rn +kn ) = ˜ N (x[0,p] ¯ , δ)|x=ψ(z[p ,s ] ) N (z[p ,s ] , δ) that is, the differential representation of the change of coordinates, is characterized by a polymodular matrix. If p = j = 0 the transformation is a bicausal change of coordinates and the associated differential representation is characterized by a unimodular matrix. Remark 1. While the property of invertibility of the change of coordinates may be sufficient to solve the observer design problem, it should be noted that the properties of weak and strong observablity of the given system are invariant only under bicausal transformation. 4. EQUIVALENCE TO A LINEAR FORM VIA INPUT-OUTPUT INJECTION In this section we recall some results given in Garc´ıaRam´ırez et al. (2016) concerning the equivalence of system

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

(2) to the canonical observer form up to input–output injection s z˙ [0] = i=0 Ai z[0] (−i) + ϕ(y[s] , u[s] ), s (5) y[0] = j=0 Cj z[0] (−j).

In (5), z[0] ∈ Rn , u[0] ∈ R, y[0] ∈ R, Ai ∈ Rn×n for i = 0, . . . , s, and Cj ∈ R1×n for j = 0, . . . , s. (A0 , C0 ) is assumed as an observable pair. As a consequence they are assumed to be in the Brunovsky form, since this can always be achieved through a linear change of coordinates. When Ai = Ci = 0 ∀i ≥ 1, one gets the equivalence to the stronger structure z˙1,[0] z˙n−1,[0] z˙n,[0] y[0]

= .. . = = =

z2,[0] + ϕ1 (y[s] , u[s] ) zn,[0] + ϕn−1 (y[s] , u[s] ) , ϕn (y[s] , u[s] ) z1,[0]

(6)

A first property that must be pointed out, is that while a dynamical system with time delays is in general represented by an input–output equation of the form (n)

(n−1)

ψ(y[s] , y[s]

(n−1)

, . . . , y[s] , u[s]

, . . . , u[s] ) = 0,

(7)

the existence of a solution to Problems 1 and 2 is based on the fundamental property that the given system admits an input–output equation of retarded type as highlighted by next result. Lemma 1. (Garc´ıa-Ram´ırez et al., 2016) Problem 1 and Problem 2 are solvable, only if the given system admits an input-output equation of retarded type, and of the form (n)

y[0] =

n

i=1

Φi (y[s] , u[s] )(i−1) .



 ..   .   dy˙ [0]     dy  0 = λ(x[s] , u[s] , δ)  (n−1)   du[0]     ..   .  du ˙ [0] du[0]

¯ [s] , u[s] , δ) = λ(x[s] , u[s] , δ)A(x

(9) 

dx[0] d¯ u[0]



¯ [s] , u[s] , δ) is Since the dimension of the columns of A(x 2n + 1, and the system is claimed to be observable, ¯ [s] , u[s] , δ)) = 2n, so that there is one solution rank(A(x in the left kernel. This fact is used in (Garc´ıa-Ram´ırez et al., 2016) for the proof of the following Lemma: Lemma 2. (Garc´ıa-Ram´ırez et al., 2016) Assume that the system is given in its state-space representation, and let ¯ [s] , u[s] , δ) ∈ K(2n+1)×2n (δ] be A(x ¯ [s] , u ¯ [s] , δ) = A(x

where the linear part, up to input and output injection, is not affected by delays. Accordingly one can state the following problems: Problem 1. Given the observable time-delay system (2) find, if possible, an invertible change of coordinates z[0] = φ(x[p,s] ), such that (2) is transformed into (5). Problem 2. Given an observable time-delay system described by equation (2) find, if possible, an invertible change of coordinates z[0] = φ(x[p,s] ), such that (2) is transformed into (6).

(n)

dy[0]



ˆ [s] ,¯ ˆ [s] ,¯ u[s] ,δ) B(x u[s] ,δ) A(x 0 I



(10)

where setting u ¯ = (u, u, ˙ · · · , u(n−1) ), ˆ [s] , u A(x ¯ [s] , δ) =

s  ∂(H (n) , H (n−1) , . . . , H) i=0

ˆ [s] , u ¯ [s] , δ) = B(x

∂x[0] (−i)

s  ∂(H (n) , H (n−1) , . . . , H) i=0

∂¯ u[0] (−i)

δi , (11) i

δ.

Then the given system admits an input-output equation of retarded type, if and only if the left-annihilator of ¯ [s] , u ¯ [s] , δ) is generated by a normalized the matrix A(x ¯ [s] , δ) (Definition 4 above). covector λ(x[s] , u The normalized vector plays a key role in understanding if Problems 1 and 2 are solvable. As a matter of fact, starting from the normalized vector, an Algorithm is proposed hereafter which allows to compute a candidate state transformation. The necessary and sufficient conditions for the solvability of Problem 1 are given in Theorem 3, while Corollary 4, states the conditions to solve Problem 2. Algorithm 1. ¯ , δ) := [1, χ0n−1 , · · · , χ00 , µ0n−1 , · · · , µ00 ] be a norLet λ(x, u malized covector satisfying Lemma 2. n−1  (i) (i) 0 ¯ , δ)dy[0] − n−1 ¯ , δ)du[0] , Set Ψ1 := − i=0 χ0i (x, u i=0 µi (x, u

and set dh0 (x) := dH(x). (8)

Functions Φi (y[s] , u[s] ) can be computed using the linearization algorithm presented in M´ arquez-Mart´ınez et al. (2002). Now, let us recall the concept of normalized covector introduced in Garc´ıa-Ram´ırez et al. (2016). Definition 4. Let λ(x, u, δ) = [λ1 , · · · , λn ] ∈ Kn¯ (δ]. λ is called a normalized covector if λi = 0, for i ∈ [1, j − 1] and λj = 1. Given the system in the state space form (2), consider the set of equations 686

STEP 1. ¯ , δ)dy[0] − µ0n−1 (x, u ¯ , δ)du[0] Set ω1 := −χ0n−1 (x, u Check: dω1 = 0? NO: Stop, YES: Compute Φ1 (x, u, δ) such that ω1 = dΦ1 (x, u, δ), ˙ − dΦ1 (x, u, δ), and and set dh1 (x) := dH(x) Ψ2 := − with

n−2  i=0

(i)

¯ , δ)dy[0] − χ1i (x, u

n−2  i=0

(i)

¯ , δ)du[0] , µ1i (x, u

IFAC NOLCOS 2016 August 23-25, 2016. Monterey, California,E.USA García–Ramírez et al. / IFAC-PapersOnLine 49-18 (2016) 672–677

¯ , δ) = χ1i (x, u



 n−1 0 ¯ , δ))(n−1−i) ¯ , δ) − χi (x, u (χ0n−1 (x, u n−1−i

¯ , δ) = µ1i (x, u

¯ , δ) µ0i (x, u



 n−1 ¯ , δ))(n−1−i) − (µ0n−1 (x, u n−1−i

STEP k. ¯ , δ)dy[0] − µk−1 Set ωk := −χk−1 ¯ , δ)du[0] n−k (x, u n−k (x, u Check: dωk = 0? NO: Stop, YES: Compute Φk (x, u, δ) such that ωk = dΦk (x, u, δ). k−1 ¯ , δ)(j) , and Set dhk (x) := dH(x)(k) − j=0 dΦk−j (x, u n−k−1 

Ψk+1 := −

with

i=0

(i)

n−k−1 

¯ , δ)dy[0] − χki (x, u

i=0



k−1 ¯ , δ) = χi (x, u χki (x, u ¯ , δ) −

(i)

¯ , δ)du[0] , µki (x, u

 n−k ¯ , δ))(n−k−i) (χk−1 n−k (x, u n−k−i

¯ , δ) = µk−1 ¯ , δ) − µki (x, u (x, u i  n−k  k−1 (n−k−i) ¯ n−k−i (µn−k (x, u, δ))

(12) 

Note that, if ωi in Algorithm 1 is an exact differential for all i = 1, · · · k, then i) ωi = dΦi (y[0] , · · · , y[0] (−s), u[0] , · · · , u[0] (−s)) i−1 (n−l) (n) ¯) ii) Ψi = y[0] − l=1 Φl (¯ y, u

and, by construction, ¯ , δ)dy[0] − µi−1 ¯ , δ)du[0] ωi = −χi−1 n−i (x, u n−i (x, u Accordingly the following result can be stated. Theorem 3. Problem 1 is solvable if and only if

675

      x1,[0] (-2)δ 3 + x1,[0] (-3)δ 2 0 d¨ y[0] δ2 dy˙ [0] =−x1,[0] (-3)δ 2 −x1,[0] (-2)δ 3 δ 2 dx[0] +  0 du[0] dy[0] 0 δ2 0 so that the normalized covector   λ(x, u ¯ , δ) = 1, 0, −x1,[0] (−2)δ − x1,[0] (−3), 0, 0, −δ 2 From Algorithm 1, Ψ1 = (x1,[0] (−2)δ + x1,[0] (−3))dy[0] + du[0] (−2) and we get that dh0 = dx1,[0] (−2) and ω1 = 0 = dΦ1 which is exact.   We thus compute dh1 = d x2,[0] (-2) − x1,[0] (-2)x1,[0] (-3) , and using equation (12), we get that χ10 = −x1,[0] (−2)δ − x1,[0] (−3), µ10 = −1, χ11 = µ11 = 0.

Accordingly Ψ2 = (x1,[0] (−2)δ + x1,[0] (−3))dy[0] + δ 2 du[0] 2 and  [0] + δ du[0] = dΦ2 =  ω2 = (x1,[0] (−2)δ + x1,[0] (−3))dy d x1,[0] (−2)x  1,[0] (−3) + u[0] (−2) = d y[0] y[0] (−1) +u[0] (−2) .     δ2 0 dh0 Thus = dx[0] dh1 −x1,[0] (−2)δ 3 − x1,[0] (−3)δ 2 δ 2 which is characterized by a polymodular matrix. In fact we have that can be written as  it   1 0 δ2 0 . −x1,[0] δ − x1,[0] (−1) 1 0 δ2 The change of coordinates is thus z1,[0] = x1,[0] (−2), z2,[0] = x2,[0] (−2) − x1,[0] (−2)x1,[0] (−3) (15) In the new coordinates the system reads z˙1,[0] = z2,[0] z˙2,[0] = z1,[0] z1,[0] (−1) + u[0] (−2)

i) The system admits an input–output equation of retarded type ii) The one-forms ωi defined by Algorithm 1 are exact for all i = 1, . . . , n. iii) There exist a polymodular matrix T (x[p,j] , δ), and a full–rank matrix Q(δ) ∈ R[δ] such that, from Algorithm 1, Q(δ)T (x[p,j] , δ)dx[0] (p) = P (x[s] , δ)dx[0] = (dhT0 , . . . , dhTn−1 )T . Corollary 4. Problem 2 is solvable if and only if i) and ii) from Theorem 3 are fulfilled, and iiib) There exists a polymodular matrix T (x[p,j] , δ), that fulfills T (x[p,j] , δ)dx[0] = (dh0 , . . . , dhn−1 )T from the Algorithm 1. Example 1. Consider the following system x˙ 1,[0] = x2,[0] − x1,[0] x1,[0] (−1), x˙ 2,[0] = x1,[0] x1,[0] (−1) + x2,[0] x1,[0] (−1) − x1,[0] x21,[0] (−1) +x1,[0] x2,[0] (−1) − x1,[0] x1,[0] (−2)x1,[0] (−1) + u[0] , y[0] = x1,[0] (−2) (13) which is neither in form (5) nor (6). Now, computing the two first derivatives of the output we have y˙ [0] = x2,[0] (−2) − x1,[0] (−2)x1,[0] (−3), (14) y¨[0] = x1,[0] (−2)x1,[0] (−3) + u[0] We thus have that 687

(16) y[0] = z1,[0] To design an observer, let us consider ζ˙1,[0] = ζ2,[0] + k1 (y[0] − ζ1,[0] ) ζ˙2,[0] = y[0] y[0] (−1) + k2 (y[0] − ζ1,[0] ) + u[0] (−2) with k1 and k2 positive. In the coordinates ξ1,[0] = ζ1,[0] , ξ2,[0] = ζ2,[0] + ζ1,[0] ζ1,[0] (−1), which are defined from the inverse map of (15), the observer reads ξ˙1,[0] = ξ2,[0] − ξ1,[0] ξ1,[0] (−1) + k1 (y[0] − ξ1,[0] ) ξ˙2,[0] = y[0] y[0] (−1) + k2 (y[0] − ξ1,[0] ) + u[0] (−2)

+(ξ2,[0] − ξ1,[0] ξ1,[0] (−1) + k1 (y[0] − ξ1,[0] ))ξ1,[0] (−1) +ξ1,[0] (ξ2,[0] (−1) − ξ1,[0] (−1)ξ1,[0] (−2)

+k1 (y[0] (−1) − ξ1,[0] (−1)))

(17)

ξ1,[0] is used to estimate x1,[0] (−2), while ξ2,[0] is used to estimate x2,[0] (−2). Set (18) u[0] = −1.25y[0] − 2(ξ2,[0] − y[0] y[0] (−1)) which stabilizes (13). Remark 2. In Figure 1 the estimation error of x0 (−2D) obtained through the observer (17) is shown. Furthermore the trajectories of system (13) are displayed both in openloop for u = 0 (unstable system) and in closed-loop with the stabilizing feedback (18) which uses the measured output and the estimates of the states computed in (17).

IFAC NOLCOS 2016 676 García–Ramírez et al. / IFAC-PapersOnLine 49-18 (2016) 672–677 August 23-25, 2016. Monterey, California,E.USA

error

0 −20

error for x1,[0](−2D) error for x2,[0](−2D)

−40 0

1

2 time t

3

4

10 x2,[0]

5 0 −5 −3

−2

−1

0

1

2

x1,[0]

Fig. 1. On the top: Observation error on x[0] (−2D) for system (13), with D=0.2sec. At the bottom: Openloop (in solid line) and closed-loop (dashed line) trajectories, for system (13), with D=0.2sec. 5. NUMERICAL RESULTS FOR A QUARTER CAR MODEL OF SUSPENSION WITH DELAYED INPUT In this section we present numerical results observer design for a quarter car model of suspension with delayed input, presented in Hu and Wang (2002). With the notation adopted in the paper, this model is represented by the equations ¨[0] = −cs [x˙ [0] − y˙ [0] ]−ks [x[0] −y[0] ]−u[0] (−D) mb x mt y¨[0] = cs [x˙ [0] − y˙ [0] ] + ks [x[0] − y[0] ] (19) −f (y[0] − z[0] ) + u[0] (−D)

with delayed feedback u[0] (−D) = u ˜x[0] (−D) + ν˜x˙ [0] (−D), and f (y[0] − z[0] ) = kt (x1,[0] + sin(2πt)). In the following the values mb = 290kg, mt = 59kg, ks = 16, 812N/m, kt = 190, 000N/m, cs = 800N s/m, and v = 500N s/m are used for numerical computations. Such values are inside of the region of stability defined in Hu and Wang (2002). System (19) can be written as x˙ 1,[0] = x2,[0] , x˙ 2,[0] = a1 (x3,[0] − x1,[0] ) + a2 (x4,[0] − x2,[0] ) + a3 (u[s] (−D) − f (x1,[0] − z[0] )),

x˙ 3,[0] = x4,[0] ,

(20)

x˙ 4,[0] =−c(a1 (x3,[0] −x1,[0] )+a2 (x4,[0] −x2,[0] )+a3 u[0] (−D)) y[0] = x1,[0] (−D).

To find if system (20) is linearizable by input-output equation, we compute the first four derivatives of y(t) y[0] = x1,[0] (−D), y˙ [0] = x2,[0] (−D), y¨[0] = a1 x3,[0] (−D) + a2 x4,[0] (−D) − a1 y[0] + a3 [u[0] (−2D) − f[0] (−D)] − a2 y˙ [0] , (3) y[0] = a1 x4,[0] (−D) − ca2 a3 f[0] (−D) − a1 y˙ [0] + a3 [u˙ [0] (−2D) − f˙(−D)] − a2 (c + 1)¨ y[0] .

From y (4) (t), we get that the non linear part on the new coordinates is given by the functions ϕ1 (y[s] , u[s] ) = a2 (c+ 1)y[0] , ϕ2 (y[s] , u[s] ) = a3 [u[0] (−2D) − f[0] (−D)] − (1 +

688

c)a1 y[0] , ϕ3 (y[s] , u[s] ) = ca2 a3 f[0] (−D), and ϕ4 (y[s] , u[s] ) = ca1 a3 f[0] (−D). Using Corollary 4 we can determinate that system (20) is linearizable by input-output injection with a change of coordinates given by the next set of equations z1,[0] = x1,[0] (−D), z2,[0] = a2 (c + 1)x1,[0] (−D) + x2,[0] (−D), z3,[0] = ca1 x1,[0] (−D) + ca2 x2,[0] (−D) + a1 x3,[0] (−D)+ a2 x4,[0] (−D), z4,[0] = ca1 x2,[0] (−D) + a1 x4,[0] (−D). (21) The inverse transformation of (21) given by x1,[0] (−D) = z1,[0] , x2,[0] (−D) = z2,[0] − a(c + 1)z1,[0] , (22) x3,[0] (−D) = −cz1,[0] + z3,[0] /a1,[0] − a2 z4,[0] /a21 , x4,[0] (−D) = a2 c(c + 1)z1,[0] − cz2,[0] + z4,[0] /a1 . This implies that the observer below will estimate the state variable with a delay D. The change of coordinates (21) takes system (20) into the equivalent form z˙1,[0] = z2,[0] − a2 (c + 1)y[0] , z˙2,[0] = z3,[0] + a3 [u[0] (−2D) − f[0] (−D)] − (1 + c)a1 y[0] , z˙3,[0] = z4,[0] − ca2 a3 f[0] (−D), z˙4,[0] = −ca1 a3 f[0] (−D), y[0] = z1,[0] , and we can set the next observer dynamics as ξ˙1,[0] = ξ2,[0] + k1 ξ1,[0] − k1 y[0] − a2 (c + 1)y[0] , ξ˙2,[0] = ξ3,[0] + k2 ξ1,[0] − k2 y[0] +a3 [u[0] (−2D) − f[0] (−D)] − (1 + c)a1 y[0] , ξ˙3,[0] = ξ4,[0] + k3 ξ1,[0] − k3 y[0] − ca2 a3 f[0] (−D), ξ˙4,[0] = k4 ξ1,[0] − k4 y[0] − ca1 a3 f[0] (−D), with coefficients defined as k1 = −261.32, k2 = −3.41431× 104 , k3 = −2.6132 × 106 ,and k4 = −1 × 108 according with the Butterworth filter design procedure. In Figure 2, simulations concerning the delayed state variables x(−D) of system (20) and their corresponding estimates are carried out with a delay D = 1sec. 6. CONCLUSION Though the structural theory of nonlinear time-delay systems is still at the dawn of its history, it was shown that effective design of observers can be processed. This is mandatory for a state feedback implementation. The observer that is derived yields an error dynamics which is the solution of a linear time-delay system. More precisely, the results on the observer linearization up to input and output injection problem for time–delay systems are used to design an observer in the time–delay context. The results, which do not need to deal with bicausal change of coordinates, are obtained by using tools within a non commutative algebraic setting. They are used to deal with a practical case study consisting of an automotive system. Simulation results are included to show the feasibility of the presented observers. Future works should consider the case of unknown and time-varying delays. REFERENCES Ahmed-Ali, T., Karafyllis, I., and Lamnabhi-Lagarrigue, F. (2013). Global exponential sampled-data observers

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3

x1,[0]

2

observed x1,[0](−D)

x1,[0]

1 0 −1 −2 −3 40

x2,[0]

20 0 −20

x2,[0] observed x2,[0](−D)

−40 300

x3,[0]

200

100 x3,[0] observed x3,[0](−D)

x4,[0]

0 400

x4,[0]

300

observed x4,[0](−D)

200 100 0 −100 0

1

2

3

4

time t

5

6

7

8

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