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Cascade High Gain Observers Based State and Unknown Input Reconstruction
Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Automatic Control Proceedings of the 20th9-14, World Congress Toulouse, France, July 2017 The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 1445–1450 Cascade High Gain Observers Based State Cascade High Gain Observers Based State Cascade High Gain Observers Based State and Unknown Input Reconstruction Cascade High Gain Observers Based State and Unknown Input Reconstruction and Unknown Input Reconstruction and Unknown Input Reconstruction ∗ ∗∗ S. Hadj Sa¨ıd ∗∗ M. Farza ∗∗ ∗∗ F. M’Sahli ∗ T. Menard ∗∗
S. Hadj Sa¨ıd ∗∗ M. Farza ∗∗ F. M’Sahli ∗∗ T. Menard ∗∗ S. Hadj Sa¨ıd ∗ M. Farza ∗∗ F. M’Sahli ∗ T. Menard ∗∗ ∗∗ S. Hadj Sa¨ ıd M. Farza F. M’Sahli T. Menard ∗∗ Department of Electrical Engineering, Department of Electrical Engineering, Monastir Monastir Engineering Engineering School, School, Ibn Eljazzar, 5019 Monastir, DepartmentRoad of Electrical Engineering, MonastirTunisia, Engineering School, Road Ibn Eljazzar, 5019 Monastir, Tunisia, Department of Electrical Engineering, Monastir Engineering School, (e-mail: [email protected], [email protected]). Road Ibn Eljazzar, 5019 Monastir, Tunisia, [email protected]). ∗∗ (e-mail: [email protected], Road Ibn Eljazzar, 5019 Monastir, Tunisia, 6072 6 (e-mail: UMR [email protected], [email protected]). ∗∗ GREYC, GREYC, 6072 CNRS, CNRS, Caen Caen University, University, 6 Bd Bd Marechal Marechal Juin, Juin, ∗∗ (e-mail: UMR [email protected], [email protected]). ∗∗ GREYC, 14050 Caen Cedex, France, UMR 6072 CNRS, Caen University, 6 Bd Marechal Juin, 14050 Caen Cedex, France, ∗∗ GREYC, UMR 6072 CNRS, Caen University, 6 Bd Marechal Juin, (e-mail: [email protected]). 14050 Caen Cedex, France, (e-mail: [email protected], [email protected], [email protected]). 14050 Caen Cedex, France, (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). Abstract: Abstract: This This paper paper presents presents a a systematic systematic routine routine for for jointly jointly reconstruct reconstruct the the state state variables variables and the unknown inputs for a large class of nonlinear MIMO systems. After an appropriate Abstract: This paper presents a systematic routine for jointly reconstruct the state variables and the unknown inputs for a large class of nonlinear MIMO systems. After an appropriate Abstract: This paper presents aofsystematic routine for MIMO jointly are reconstruct the state variables change state gain designed in a and theof inputs fora aset class ofhigh nonlinear systems. After an appropriate change ofunknown state coordinates, coordinates, setlarge of cascade cascade high gain observers observers are designed in such such a way way that that and the unknown inputs foraa aset large class of nonlinear MIMO systems. After an appropriate each of them provides an estimation of only one component of the unknown input vector, except change of state coordinates, of cascade high gain observers are designed in such a way that each of them provides an estimation of only one component of the unknown input vector, except change state coordinates, a set of cascade high gain observers are designed in such a way that the one gives a of whole Such design achieves each of of them provides an estimation of only one component of thevariables. unknown input vector, except the last last one which which gives a final final adjustment adjustment of the the whole state state variables. Such design achieves each of them provides an estimation of only one component of the unknown input vector, except aa boundary the error can be small properly the last one of which gives estimation a final adjustment of the state variables. designspecifying achieves boundary of the state state estimation error which which canwhole be arbitrarily arbitrarily small by bySuch properly specifying the last one which gives estimation a final adjustment ofexample the statethe variables. design achieves the sole synthesis An illustrative effectiveness of the proposed a boundary of the parameter. state error which canwhole beverifies arbitrarily small bySuch properly specifying the sole synthesis parameter. An illustrative example verifies the effectiveness of the proposed a boundary of the parameter. state estimation error which can beverifies arbitrarily small by properly specifying scheme. the sole synthesis An illustrative example the effectiveness of the proposed scheme. the sole synthesis parameter. An illustrative example verifies the effectiveness of the proposed scheme. © 2017, IFAC (International Federation ofEstimation, Automatic Control) Hosting by Elsevier Ltd. All rights reserved. scheme. Keywords: Keywords: State State Estimation, Estimation, Robust Robust Estimation, Nonlinear Nonlinear Systems, Systems, Cascade Cascade Observers, Observers, Unknown Estimation. Keywords: State Estimation, Unknown Inputs Inputs Estimation. Robust Estimation, Nonlinear Systems, Cascade Observers, Keywords: State Estimation, Unknown Inputs Estimation. Robust Estimation, Nonlinear Systems, Cascade Observers, Unknown Inputs Estimation. for 1. for multiple multiple model model presentation presentation is is proposed proposed in in Chadli Chadli et et 1. INTRODUCTION INTRODUCTION al. (2009). For state affine systems, a necessary and suffor multiple model presentation is proposed in Chadli et 1. INTRODUCTION al. (2009). For state affine systems, a necessary and suffor multiple model presentation is proposed in Chadli et 1. INTRODUCTION ficient condition is for the of al. (2009). For state affine a necessary and sufficient condition is given given forsystems, the existence existence of an an unknown unknown Since (2009). For state affine systems, a &necessary and sufinput observer (UIO) in (Hammouri Tmar, 2010). In Since four four decades decades ago, ago, the the study study of of dynamical dynamical systems systems al. ficient condition is given for the existence of an unknown input observer (UIO) in for (Hammouri & Tmar, 2010). In in presence of unknown inputs has ficient condition is given thefor existence of anobservable unknown Since decades the study of (UIs) dynamical systems input Mohamed et al. (2012), UIO a canonical in the thefour presence of ago, unknown inputs (UIs) has strongly strongly observer (UIO) inan (Hammouri & Tmar, 2010). In Since four decades ago, the study of dynamical systems Mohamed et al. (2012), an UIO for a canonical observable motivated the activities in theory input observer in with (Hammouri & Tmar, 2010). In in the presence of unknown inputs (UIs)control has strongly form system uncertainties is motivated the research research activities in both both control theory Mohamed et al. (UIO) (2012), UIOparameters for a canonical observable in the presence of unknown inputs (UIs) has strongly form of of nonlinear nonlinear systeman with parameters uncertainties is and diagnostic The unknown input identification motivated the analysis. research activities in both control theory form Mohamed et LMI al. (2012), an UIO forrecent a canonical observable developed in terms. In a more work Bejarano et and diagnostic analysis. The unknown input identification of nonlinear system with parameters uncertainties is motivated the research activities in both control theory developed in LMI terms. In a more recent work Bejarano et techniques contribute not only to solve the control issue form of nonlinear system with parameters uncertainties is and diagnostic analysis. The unknown input identification al. an observation nonlinear techniques contribute not only to solve the control issue developed terms. Inscheme a moreof recent work differentialBejarano et and diagnostic analysis. The unknown input al. (2015), (2015), in anLMI observation scheme of nonlinear differentialfor systems with unmeasurable or weakly developed in LMI terms. Inscheme a more recent work Veluvolu Bejarano et techniques contribute not only todisturbances solve the identification control issue al. algebraic systems subject to UI is designed. et for systems with unmeasurable disturbances or weakly (2015), an observation of nonlinear differentialtechniques contribute not only to solve the control issue al. algebraic systems subject scheme to UI is designed. Veluvolu et modeled but also, to generate a disturbance-decoupled (2015), an observation of nonlinear differentialfor systems with unmeasurable disturbances or weakly al. (2014) extend the standard high gain observer (HGO) modeled but also, to generate a disturbance-decoupled algebraic systems subject to UI is designed. Veluvolu et for systems with unmeasurable disturbances or the weakly al. (2014) extend the standard high gain observer (HGO) residual signals for detecting the plant faults. On one modeled but also, to generate a disturbance-decoupled systems subject to UIhigh isfollows designed. Veluvolu et by sliding mode based term the residual signals for detecting the aplant faults. On the one algebraic al. standard gain observer (HGO) modeled but also, to which generate disturbance-decoupled by a a(2014) slidingextend mode the based term that that follows the disturbance disturbance side, the earlier works was dedicated the linear al. (2014) extend the standard high gain observer (HGO) residual signals for detecting the plant faults.to On the one by vector. Other consistent results on UIO synthesis using side, the earlier works which was dedicated to the linear a sliding mode based term that follows the disturbance residual for detecting the faults.toOn one by vector. Other consistent results on UIO the synthesis using systems have focused on alleviating the input a sliding mode basedmodels term that disturbance side, thesignals earlier works which wasplant dedicated thethe linear Fuzzy and/or neutrals are provided in Haoussi & systems have focused on alleviating the unknown unknown input vector. Other consistent results onfollows UIO synthesis using side, the earlier works which was dedicated to the linear Fuzzy and/or neutrals models are provided in Haoussi & effects in the quality of state reconstruction. The main vector. Otherand consistent results UIO synthesis using systems on state alleviating the unknown Tissir (2009) Du et al. (2015). effects inhave the focused quality of reconstruction. The input main Fuzzy and/or neutrals models areonprovided in Haoussi & systems have focused on alleviating the unknown input Tissir (2009) and Du et al. (2015). idea consists in decoupling the influence of the UI and effects in the quality of state reconstruction. The main Fuzzy and/or neutrals models are provided in Haoussi & In way to idea consists in decoupling the influence of the UImain and Tissir (2009) andand Duthanks et al. (2015). effects in the quality of state The In parallel parallel way and thanks to the the technological technological progress progress guarantees that the state estimation error converge the Tissir (2009) andand Duthanks et al. (2015). idea consists in decoupling thereconstruction. influence of the UIto and in terms of computing time for real-time implementaguarantees that the state estimation error converge to the In parallel way to the technological progress idea consists in decoupling the influence of the UI and in terms of computing time for real-time implementaorigin least possess a bound Kudva et al. In parallel and thanks to the real-time technological progress guarantees theit estimation converge tion, cascade observers increasingly investigated last origin or or at atthat least itstate possess a little little error bound Kudva to et the al. in terms ofway computing time for implementaguarantees theit state estimation error converge the tion, cascade observers were were increasingly investigated last (1980), & Muller (1992). These developments origin orHou atthat least possess a little bound Kudva to ethave al. tion, in terms of computing timein for real-time implementadecade. Bevly demonstrates Bevly & Parkinson (2007) (1980), Hou & Muller (1992). These developments have cascade observers were increasingly investigated last origin or at least it possess a little bound Kudva et al. decade. Bevly demonstrates in Bevly & Parkinson (2007) led to a generalization of the UI decoupling concept and (1980), Hou & Muller (1992). These developments have tion, cascade observers were increasingly investigated last that cascaded estimation provides better led to a Hou generalization of the UI decoupling concept have and decade. demonstrates in technique Bevly & Parkinson (1980), & Muller (1992). These developments that the theBevly cascaded estimation technique provides (2007) better interesting methods have emerged such as Kell & Sauter decade. Bevly demonstrates in technique Bevly & conventional Parkinson (2007) led to a generalization of the UI decoupling concept and that estimation of the vehicle states over a estiinteresting methods have emerged such as Kell & Sauter the cascaded estimation provides better led to a generalization of the UI decoupling concept and estimation of the vehicle states over a conventional esti(2013) and Ichal & Mammar (2015). side, that thescheme. cascaded estimation technique providessuggest better interesting methods have emerged suchOn as the Kellother & Sauter mation In Ng et al. (2012), the authors (2013) and Ichal & Mammar (2015). On the other side, estimation of the vehicle states over a conventional estiinteresting methods have emerged suchOn as the Kell &UIs Sauter mation scheme. In Ng et al. (2012), the authors suggest some authors Xiong & Saif (2003) considered the as a (2013) and Ichal & Mammar (2015). other side, of theInvehicle states over afault conventional estia sliding mode for reconstruction some authors Xiong & Saif (2003) considered the UIs as a estimation mation scheme. Ng etobserver al. (2012), suggest (2013) andstate Ichal && Mammar (2015). On their thethe other a cascaded cascaded sliding mode observer for the faultauthors reconstruction part of the vector upon assuming that variations some authors Xiong Saif (2003) considered UIs side, as a mation scheme. Inthe Ng faults etobserver al. and (2012), the authors suggest when considering disturbances as UIs part of the state vector upon assuming that their variations a cascaded sliding mode for fault reconstruction some Xiong & upon Saif (2003) considered thevariations UIs time as a awhen considering the faults and for disturbances as UIs of of are relatively slow w.r.t the state dynamics cascaded sliding mode observer fault of reconstruction part ofauthors the state vector assuming that (constant, their the system. Throughout the two stages HGO, both are relatively slow w.r.t the state dynamics (constant, time when considering the faults and disturbances as UIs of part of the state vector upon assuming that their variations the system. Throughout the two stages of HGO, both polynomial, The corresponding observers are then when considering the faults and as UIs of are relativelyetc). slow w.r.t state dynamics (constant, state variables and parameters are reconstructed polynomial, etc). The the corresponding observers are time then the Throughout two disturbances stages HGO, both are relativelyin slow w.r.t the state dynamics (constant, time statesystem. variables and critical criticalthe parameters areof reconstructed constructed such aa way to estimate both the state polynomial, etc). The corresponding observers arevector then the system. Throughout the two stages ofreconstructed HGO, both in induction motor application in Hadj Said et al. (2011). constructed in such way to estimate both the state vector state variables and critical parameters are polynomial, etc). The corresponding observers arevector then state in induction motor application in Hadj Said et al. (2011). and The common approach for this variables and critical parameters areobservable reconstructed constructed suchmost a way to estimate both the Output injection observers for a class casand the the UIs. UIs.in The most common approach for state this second second in induction motor application in Hadj of Said et al. (2011). constructed in such a way to estimate both the state vector Output injection observers for a class of observable cascategory consists in transforming the UI in induction motor application in Hadj affine Said etsystems al. (2011). and the UIs. The most common approach foridentification this second Output cade systems which include the state are category consists in transforming the UI identification injection observers for a class of observable casand the UIs. The most common approach for this second cade systems which include the state affine systems are problem into a constrained optimization that can Output injection observers for a state class(2014). of observable cascategory consists in transforming the problem UI identification discussed in Sahnoun & Hammouri In Yang et problem into a constrained optimization problem that can cade systems which include the affine systems are category consists in transforming thematrix UI identification discussed in Sahnoun & Hammouri (2014). In Yang et be easily solved by adopting the linear inequalities problem into a constrained optimization problem that can cade systems which include the state(2014). affine systems are al. (2015), adaptive together with a be easily into solved by adoptingoptimization the linear matrix inequalities discussed inan Sahnoun &SMO Hammouri Inhigh Yanggain et problem a constrained problem that can al. (2015), an adaptive SMO together with a high gain (LMIs) formalism. The issues of both state estimation and discussed in Sahnoun & Hammouri (2014). In Yang et be easily solved by adopting the linear matrix inequalities SMO are in estimate respectively the (LMIs) formalism. The issues of both state estimation and al. (2015), an adaptive SMOto a high gain be easilyformalism. solved by adopting linear matrix inequalities SMO are developed developed in order order totogether estimatewith respectively the unknown input reconstruction the nonlinear case still al. (2015), an adaptive SMOnetworks together with a derivative high gain (LMIs) The issuesthe of in both state estimation and states of uncertain complex and the unknown input reconstruction in the nonlinear case still SMO are developed in order to estimate respectively the (LMIs) formalism. The issues of both state estimation and states of uncertain complex networks and the derivative provide an open research field. Recent academic works SMO are developed inwhereas order tothe estimate respectively the unknown the nonlinear caseworks still states of output vector, reconstructed by provide aninput openreconstruction research field.in Recent academic uncertain complex the derivative unknown the some nonlinear caseworks still of the the of output vector, whereasnetworks the UI UI is isand reconstructed by are investigated in the synthesis for sub-classes of provide aninput openreconstruction research field.inRecent academic states of uncertain complex networks and the derivative an algebraic method. An interconnected adaptive observer are investigated in the synthesis for some sub-classes of of the output vector, whereas the UI is reconstructed by provide ansystems. open inresearch field. Recent academic works an the algebraic method. An interconnected adaptive observer nonlinear A sliding mode observer (SMO) design of output vector, whereas the UI is reconstructed by are investigated the synthesis for some sub-classes of nonlinear systems. Athe sliding mode observer (SMO) design algebraic method. An interconnected adaptive observer are investigated in A synthesis some (SMO) sub-classes of an an algebraic method. An interconnected adaptive observer nonlinear systems. sliding mode for observer design nonlinear systems. A sliding mode observer (SMO) design ∗ ∗ ∗ ∗ ∗
Copyright © 2017 IFAC 1481 Copyright 1481Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) © 2017 IFAC 1481 Copyright Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 1481Control. 10.1016/j.ifacol.2017.08.288
Proceedings of the 20th IFAC World Congress 1446 S. Hadj Saïd et al. / IFAC PapersOnLine 50-1 (2017) 1445–1450 Toulouse, France, July 9-14, 2017
for flying capacitor multilevel converters is highlighted in Morales et al. (2013). In such case, the observer for the whole system which is modeled by a set of cascade interconnected subsystem, is approached by designing a separate observer for each individual subsystem. This paper builds on the work in Farza et al. (2011) where a HGO is designed for a non-triangular class of nonlinear systems. Under the existence of the two state coordinates which transform the original system into the observable canonical form, a set of cascade sub-observers is systematically planned for achieving simultaneous estimation of the non-measured states and the UIs without using any prior information about the waveform of the latter. This paper is organized as follows. In the next section, the class of nonlinear systems which is the basis of the observer design is introduced. Sections 2 and 3 detailed respectively the two state transformations and the cascade observer synthesis. Section 4 is devoted to a simulation example that highlight the performance of the proposed observation scheme. 2. PROBLEM STATEMENT AND PRELIMINARIES Let us consider the MIMO class of continuous nonlinear systems that can be represented by: x(t) ˙ = f (u(t), x(t)) + g(u(t), x(t), v(t)) + ε˜(t) Σ0 (1) y(t) = x1 (t)
with 1 :
1 f (u, x[2] ) x1 .. .. . . µ −1 µ1 −1 [µ1 ] 1 (u, x ) x f µ1 f µ1 (u, x[µ1 +1] ) x . . . . x = . , f (.) = . f µr −1 (u, x[µr ] ) xµr −1 µr µr [µr +1] x f (u, x ) . . . . .. s−1 s−1 x (u, x) f s s x f (u, x) 1 g (u, x1 ) .. . µ1 −1 [µ1 −1] g (u, x ) µ g 1 (u, x[µ1 ] , v 1 ) . .. and g(.) = g µr −1 (u, x[µr −1] , v [r−1] ) g µr (u, x[µr ] , v) . .. s−1 [s−1] , v) g (u, x s g (u, x, v) wherethe state x ∈ Rn with xk ∈ Rnk for k = 1, · · · , s s and ˜ = k=1 nk = n with n1 ≥ n2 ≥ · · · ≥ ns . ε T T T sT 0n1 , · · · , 0ns−1 , ε˜ is an unknown function.u ⊂ U subset of absolutely continuous functions with bounded derivatives from R+ into U a compact subset of Ru and 1
for the convenient of description, we omit the time variable t in the remainder of the paper and we note x[i] = x1 , x2 , · · · , xi and v [i] = v 1 , v 2 , · · · , v i
y ∈ Rn1 denote respectively the known input and the output of the system. v ∈ Rm denotes the unknown input r vector with v i ∈ Rmi for i = 1, · · · , r and i=1 mi = m with m1 ≥ m2 ≥ · · · ≥ mr . The r characteristic indices µj associated to the m UIs are defined as follows: Definition 1: The characteristic index (or the relative degree) µj associated to the parameter v j for system (1) is the smallest integer j such that i ∂g (u, x, v) ≡ 0, ∀i < µj ∂v j (2) µj ∂g (u, x, v) = 0 ∂v j with 1 ≤ µ1 ≤ µ2 · · · ≤ µr ≤ s. Remark 1: • When g(x, u, v) = 0 (in the absence of UIs) the system (1) is the same as the one studied in Farza et al. (2005). • In the case where µ1 = 1, all first µ1 − 1 equations of the model (1) can be omitted. The studied class of nonlinear MIMO systems is large enough to include a variety of real plant, ranging from chemical and biological processes to electromechanical actuator and robot platform. The main objective consists in the synthesis of an observer to simultaneously estimate the vector of UIs v(t) and the non measured states x(t) without assuming any model for the UIs. The synthesis of the observer requires the adoption of some hypothesis which will be stated in due courses. Consider at the first the following assumptions : Assumption 1 Let for: 1 ≤ k ≤ s − 1, the application : Rnk+1 → Rnk (u, x[k+1] ) → f k (u, x[k+1] )
(3)
is injective for all (u, x[k+1] ). Moreover, we suppose that ∃ α1 , β1 > 0 such as: ∀k ∈ {1, ..., s − 1} , ∀x ∈ Rn , ∀u ∈ U (α1 )2 Ink+1 ≤
∂f k (u, x) ∂xk+1
T
∂f k (u, x) ≤ (β1 )2 Ink+1 (4) ∂xk+1
with Ink+1 is the matrix identity with (nk+1 ) × (nk+1 ) dimension Assumption 2 The application:
(u, x vj
R
nµ
[µj +1]
)
j
+1 +mj
→R
nµ
j
→ ϕ¯µj (u, x[µj +1] , v j )
(5)
is injective for all (u, xµj , v j ) with ϕ¯k (u, x, v) = f k (u, x) + g k (u, x, v) One also assumes that ∃ α2 , β2 > 0 such as for x ∈ Rn , ∀u ∈ U , ∀v ∈ Rm : (α2 )2 Inµj +1 +mj ≤ (F˜ µj (.))T F˜ µj (.) ≤ (β2 )2 Inµj +1 +mj (6) ∂ϕ ¯µj ∂ϕ ¯µj (u, x, v) (u, x, v) with F˜ µj (x, u, v) = ∂x µj +1 ∂v j
The inequality given by assumption 2 implies that the rank of matrix F˜ µj (x, u, v) is equal to nµj +1 + mj . Yet, this matrix is rectangular includes nµj lines and nµj +1 + mj columns, so:
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Proceedings of the 20th IFAC World Congress S. Hadj Saïd et al. / IFAC PapersOnLine 50-1 (2017) 1445–1450 Toulouse, France, July 9-14, 2017
nµj +1 + mj ≤ nµj
(7)
Thereafter, one can deduces that: m=
r j=1
mj ≤
r j=1
(nµj − nµj +1 )
(8)
Remark 2: • In the absence of UIs, the given assumptions are the same as these adopted in Farza et al. (2005), whereas the other hypotheses that involves the UI are simply issued from the rank condition. • The number of UIs must be strictly lower than the number of the measured output, except when µr = s. In this last case the number of UIs can be equal to the number of the measured outputs. 3. COORDINATE CHANGE Now, one shall introduce a set of coordinate transformations in order to put the system (1) under the canonical form which corresponds to the high gain observer synthesis. Precisely, an augmented system which is composed with r + 1 blocks such that there output’s are equal of these of the original system (1) is planned. The associated variables of each individual j th block are: x1 , ..., xµj +1 and v j , whereas the variables which are associated to the last one are simple copies of the whole system state’s. So, the considered immersion can be interpreted as an injective map ψ(.) which is defined as follows: r nµ +1 i nk
+m+n
i=1 k=1 ψ : Rn+m → R z1 ψ1 (x, v) x → z = ... = ... v r+1 (x, v) ψ z r+1 i k z = x f or k = 1, · · · , µ i = 1, · · · , r i k i µ; +1 zµi +1,1 x i i : zµi +1 = = vi zµi i +1,2 r+1 k zk = x , k = 1, · · · , s
(9)
f 1 (u, z i[2] ) 1 .. f (u, z r+1 [2] ) . r+1 . F i = f µi (u, z i = .. ; F ; [µi +1] ) s r+1 f µi +1 (u, z i f (u, z ) [µi +2] ) 0 1 g (u, z1i ) 1 .. g (u, z1r+1 ) . r+1 .. = . Gi = g µi (u, z i ; G , [µi +1] ) s r+1 g µi +1 (u, z i g (u, z ) ) 0
[µi +1]
0 .. . εi = 0 0
µi +1
and εr+1 = ε˜.
εi Let now the second injective map Φ such that: r nµ +1 r i Φ : R
with i ζ1 ζ2i i ζ3 . ..
nk
i=1
+m+n
n1 [(µi +1)+s]
→ Ri=1 ζ 1 = Φ1 .. .
k=1
z1 .. ζ= . → z r+1 ζ r+1 = Φr+1
(11)
ζji and ζjr+1 for i = 1, · · · , r and j = 1, · · · , µi + 1 : = z1i = f 1 (u, z i[2] ) ∂f 1 (u, z i[2] ) 2 = f (u, z i[3] ) ∂z2i
µ −2 k i ∂f (u, z i[k+1] ) i f µi −1 (u, z i[µ1 ] ) ζµi = i ∂z k=1 µ −1 k k+1 i i ∂f (u, z [k+1] ) i ζµi +1 = i ∂zk+1 k=1
µi i i × f µi (u, z i , z i ) + g (u, z , z ) [µ1 ] µ1 +1,1 [µ1 ] µ1 +1,2 r+1 r+1 ζ = z 1 1 ζ2r+1 = f 1 (u, z r+1 [2] ) 1 ∂f (u, z r+1 [2] ) 2 ζ3r+1 = f (u, z r+1 r+1 [3] ) ∂z2 .. . s−2 k ∂f (u, z r+1 ) [k+1] r+1 f s−1 (u, z r+1 ζs = [s] ) r+1 ∂z k+1 k=1 Performing simple manipulation yields: i ζ˙ = Ai ζ i + ϕi (u, u, ˙ ζ [i] ) + ε¯i Σ2 i yi = ζ1
Consequently, in the z coordinates, the (r + 1) subsystems are presented as follows: i z˙ = F i (u, z i ) + Gi (u, z i ) + εi Σ1 (10) yi = z1i with:
1447
where:
0 I(µi )n1 0 In1 (s−1) Ai = ; Ar+1 = ; 0 0 0 0 0 ϕi2 (u, u, ˙ ζ i[2] , ζ [i−1] ) ˙ ζ [i] ) = ϕi (u, u, ; ϕi (u, u, i [i−1] ˙ ζ [µ ] , ζ ) µi i ϕiµi +1 (u, u, ˙ ζ, ζ [i−1] ) 0 ϕr+1 ˙ ζ) 2 (u, u, ϕr+1 (u, u, ˙ ζ) = .. .
(u, u, ˙ ζ) ϕr+1 s
T r+1 T (u,z) r+1 ε¯i = 0T(µi )n1 , i ; ε¯r+1 = ∂Φ∂zr+1 ε µ −1 k i µ i i ∂f (u,z[k+1] ) ∂g i (u,ζ ) i ε and i = ∂v i ∂z i
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k=1
k+1
(12)
Proceedings of the 20th IFAC World Congress 1448 S. Hadj Saïd et al. / IFAC PapersOnLine 50-1 (2017) 1445–1450 Toulouse, France, July 9-14, 2017
4. OBSERVER SYNTHESIS WITH r SET OF UNKNOWN INPUTS By referring to the framework given in Farza et al. (2011), one can adopt the HGO designed for the class of MIMO nonlinear systems with non-triangular canonical form. Let at this stage the following assumption : Assumption 3 The nonlinear functions ϕi (u, ζ) are globally Lipchitz with respect to ζ uniformly in u. A candidate HGO that corresponds to the form (12) can be given as follows: [i] ˙i i i ˙ ζˆ ) ζˆ = Aj ζˆ + ϕ (u, u, −1 T ˆi i −θδi ∆−1 i (θ)Si Ci (ζ1 − ζ1 ) for i = 1 : r ˆ Σ 2 ˙ r+1 r+1 r+1 ˆ ζˆ = Ar+1 ζˆ + ϕ (u, u, ˙ ζ) −1 r+1 δr+1 −1 T ˆ −θ ∆r+1 (θ)Sr+1 Cr+1 (ζ1 − ζˆ1r+1 ) (13) with: 1 1 Cµi +1 In1 Cs In1 Cµ2 +1 In1 C 2 In i s 1 −1 T Cr+1 = . Si−1 CiT = . , Sr+1 .. .. s µi +1 I C Cµi +1 In1 s n1 ∆i (θ) = diag In1 , θ1δi In1 , · · · , θµ1i δi In1 ; n! ∆r+1 (θ) = diag In1 , θ1 In1 , · · · , θ1s In1 and Cin = (n−i)!i! The δi which indicates the power of θ are defined as the same as given in Farza et al. (2011). According to the theorem 3.1 established in the above reference, the estimation error which remains in a ball with a radius depends only on θ, can made as small as desired when the latter parameter is chosen large enough. However, due to the sensitivity of the HGO to the output noise, a compromise between fast convergence and filtering noise should be achieved. Now, when returning to the z coordinate one obtains: i zˆ˙ = F i (u, zˆ) + Gi (u, zˆ) + −1 T − (Λi (u, zˆ)) θδi ∆−1 z1i − z1i ) i (θ)Si Ci (ˆ ˆ Σ 1 r+1 r+1 r+1 zˆ˙ =F (u, zˆ) + G (u, zˆ) + −1 δr+1 −1 T ∆r+1 (θ)Sr+1 Cr+1 (ˆ z1r+1 − z1r+1 ) −Λr+1 (.)θ (14) where: + Λi (u, zˆ) = diag In1 , Λ+ ˆi ), · · · , Λ+ ˆi ) i,2 (u, z i,µi +1 (u,z + Λ+ ˆ) = diag In1 , Λ+ r+1 (u, z r+1,2 (.), · · · , Λr+1,s (.) such as: k−1 ∂f j i Λ (u, z ˆ ) = (u, zˆ) f or k = 2, · · · , µi i,k i ∂zj+1 j=1 µ i −1 ∂f j ¯ µi F (u, zˆ) Λi,µi +1 (u, zˆi ) = i ∂zj+1 j=1 k−1 ∂f j r+1 (u, zˆ) f or k = 2, · · · , s Λ (u, z ˆ ) = r+1,k ∂z i
with:
F¯ µi (u, zˆ) =
j=1
j+1
µj ∂f µj (u, x, v) ∂f ∂v j ∂xµj +1
Fig. 1. UIO block diagram 1 x ˆ˙ = f 1 (u, x ˆ [2] ) + g 1 (u, x ˆ1 ) − Cµ1i +1 θδi x ˜i . .. [i] ˆ [µi +1] ) + g ρr (u, x ˆ [µi ] , vˆ ) x ˆ˙ µi = f µi (u, x i i ˆ x −Cµµii+1 θµi δi Λ+ Σ i,µi (u, x )˜ 0(i) µ +1 µ +1 µi +1 i i ˙ x ˆ (.) + g (.) f = 0 vˆ˙ i µi +1 (µi +1)δi + −Cµi +1 θ xi Λi,µi +1 (u, xi , vˆ[i] )˜ f or i = 1, .., r; 1 x ˆ˙ = f 1 (u, x ˆ [2] ) + g 1 (u, x ˆ1 ) − Cs1 θ˜ xi .. Σ . ˆ 0(r+1) ˆ) + g s (u, x ˆ, vˆ) x ˆ˙ s = f s (u, x s s + −C θ Λ xi (u, x, vˆ[i] )˜ s
r+1,s
with x ˜i = x ˆi − xi
Proposition: Assume that system (1) satisfies assumptions (A1)-(A2) and that there exists a diffeomorphism (11) that transforms the augmented system (10) (in z coordinates) to the canonical form (12) (in ζ coordinates) with assumption (A3) holds, then ∀ M > 0; ∃ θ0 > 0; ∀ θ > θ0 ; ∃ λθ > 0; ∃ µθ > 00 ; ∃ αθ > 0, the system (15)
k an observer
constitutes
for (1) with:
x ˆk (t) − xk (t) ≤ λθ e−µθ t x ˆ (0) − xk (0) + αθ β. for every admissible control u s.t u∞ ≤ M , where βk is the upper bound of εk . Moreover, λθ is a polynomial in θ, lim µθ = +∞ and lim αθ = 0. (See the Appendix for θ→∞
θ→∞
a sketch of proof.) The block diagram of the whole system of the observation is depicted in figure (1). It is composed of (r + 1) subsystems in cascade form. The first one provides an estimation of v 1 , the second gives an estimation of v 2 , and so on until the rth subsystem which estimates the v r component. The ultimate block of the observer that exploit the previous unknown input estimates, achieves a reconstruction of all the state variables. Note that each of the ’r’ first subobservers include a copy of µi + 1 first blocks (i = 1...r) of the original system augmented by the corresponding unknown input component.
(u, x, v)
5. EXAMPLE
Finally, in the original coordinate x, the (r + 1) cascade sub-systems of observation have the following forms:
(15)
Let consider the following nonlinear MIMO system:
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x˙ 1 = x1 − x2 + c(t)x4 − s(t)x5 x˙ 2 = −.2x1 x2 + x3 + s(t)x4 + c(t)x5 x˙ 3 = x34 + x55 + s(x3 ) + v1 + v13 x˙ = x x − (x + .3)(v + v 3 ) − v 4 3 6 3 2 1 2 x˙ 5 = (x3 − a)x6 + x3 v2 − x5 s(v1 ) x˙ 6 = x37 + x7 − 3v1 s2 (v2 ) − x5 + v1 v2 x˙ = −x6 + 3x5 c(v1 ) + 3x5 s(v2 ) + ε7 7 T y = [ x1 x2 x3 ]
(16)
with s(.) = sin(.), c(.) = cos(.) and ε7 = −0.5v1 v2 . Our aim is to conjointly estimate the unmeasured state variables x4 , x5 , x6 and x7 together with the two UIs v1 and v2 . We show that the whole state vector can be arranged into four sub-state vectors: x1 = [x1 x2 x3 ]T ; x2 = [x4 x5 ]T ; x3 = x6 and x4 = x7 . So, we have: µ1 = 1, µ2 = 2, s = 4 with n1 = nµ1 = 3, n2 = nµ2 = 2, n3 = ns−1 = 1, n4 = ns = 1, m = r = 2 ≤ nµ1 − ns = 2. Consequently, such state distribution gives rise to the design of three cascade observation blocks. The first and the second blocks provide respectively an estimation of the UIs v 1 an v 2 , whereas the last subsystem gives a reconstruction of the whole system’s state x. The differential equations of the three cascade modules are given as follows:
x ˆ˙ 1 x ˆ1 − x ˆ2 + c(t)ˆ x4 − s(t)ˆ x5 x x1 x ˆ2 + x ˆ3 + s(t)ˆ x4 + c(t)ˆ x5 ˆ˙ 2 = −.2ˆ 3 5 3 x ˆ4 + x ˆ5 + s(ˆ x3 ) + vˆ1 + vˆ1 x ˆ˙ 3 δ1 1 −2θ x ˜ ˙ 3 x ˆ4 x ˆ − (ˆ x + .3)(ˆ v + v ˆ ) − v ˆ x ˆ 3 6 3 2 1 ˆ 2 Σ01 ˙ x3 + .3)ˆ x6 + x ˆ3 vˆ2 − x ˆ5 s(ˆ v1 ) x ˆ5 = (ˆ ˙ 0 vˆ1 −1 c(t) −s(t) 0 x 0 −θ2δ1 s(t) c(t) ˜1 2 4 2 3ˆ x4 5ˆ x5 1 + 3ˆ v1 (17)
x ˆ˙ 1 x ˆ1 − x ˆ2 + c(t)ˆ x4 − s(t)ˆ x5 x x1 x ˆ2 + x ˆ3 + s(t)ˆ x4 + c(t)ˆ x5 ˆ˙ 2 = −.2ˆ 3 5 3 x ˆ4 + x ˆ5 + s(ˆ x3 ) + vˆ1 + vˆ1 x ˆ˙ 3 δ2 1 −3θ x ˜ ˙ x ˆ4 x ˆ3 x ˆ6 − (ˆ x3 + .3)(ˆ v2 + vˆ23 ) − vˆ1 ˆ Σ02 = (ˆ x3 + .3)ˆ x6 + x ˆ3 vˆ2 − x ˆ5 s(ˆ v1 ) x ˆ˙ 5 2δ2 + 1 ˜ −3θ 3 Λ22 x 2 ˙6 x ˆ x ˆ + x ˆ ˆ5 + v1 v2 7 7 − 3v1 s (v2 ) − x = 0 vˆ˙ 2 −θ3δ2 Λ+ ˜1 23 x (18)
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ˆ˙ 1 x ˆ1 − x ˆ2 + c(t)ˆ x4 − s(t)ˆ x5 x x x1 x ˆ2 + x ˆ3 + s(t)ˆ x4 + c(t)ˆ x5 ˆ˙ 2 = −.2ˆ 3 5 3 ˙ x ˆ4 + x ˆ5 + s(ˆ x3 ) + vˆ1 + vˆ1 x ˆ3 1 −4θ˜ x x ˆ˙ 4 ˆ6 − (ˆ x3 + .3)(ˆ v2 + vˆ23 ) − vˆ1 x ˆ3 x = x (ˆ x3 + .3)ˆ x6 + x ˆ3 vˆ2 − x ˆ5 s(ˆ v1 ) ˆ˙ 5 2 + 1 −6θ Λ x ˜ 22 ˆ 03 Σ 3 2 ˙6 = x x ˆ ˆ + x ˆ ˆ5 + v1 v2 7 7 − 3v1 s (v2 ) − x + x ˆ3 3 −4θ Λ22 x ˜1 x ˆ3 − a x ˆ˙ 7 = (−ˆ x6 + 3ˆ x5 c(ˆ v1 ) + 3ˆ x5 s(ˆ v2 )) + 4 θ x ˆ3 − 2 x ˜1 Λ22 x ˆ3 + .3 3ˆ x7 + 1 (19) c(t) −s(t) x ˆ3 b where: Λ22 = s(t) c(t) ; Λ23 = Λ22 x ˆ3 + .3 x ˆ3 2 4 3ˆ x4 5ˆ x5 T
˜2 x ˜3 ] . ˜1 x with b = −(ˆ x3 + .3)(1 + 3ˆ v22 ) and x ˜1 = [ x
In order to evaluate the performances issued from the application of the proposed observation algorithm, we have numerically simulated the previous system and its corresponding observer with the choice of the synthesis parameter θ as θ = 20. The initial conditions are given as ˆ 01 : x the follows: x(0) = [1 .5 1 .3 1 − .2 0.9]T ; Σ ˆ(0) = T ˆ [1 .5 1 −.25 .4 0] ; Σ02 : x ˆ(0) = [1 .5 1 −.25 .4 −.3 −0.5]T ; ˆ 03 : x Σ ˆ(0) = [1 .5 1 − .25 .4 − .3 0]T ; The simulation results are illustrated in fig. (2). On the one hand, we can perceive, after some fluctuations in the transient, that the states x4 , x5 , x6 and x7 are perfectly reconstructed with a quasi-null steady state error. On the other hand the waveform signals of the UIs v1 et v2 are restituted about only 0.5s. Furthermore, the correction gains of all sub-observers which require the calibration of the sole synthesis parameter θ can be explicitly computed; and thereafter they lead to reasonable computations for a real-time implementation.
6. CONCLUSION The modularity of this purpose is the design of an observation scheme that allows to jointly estimate the state variables and the UIs for a large class of nonlinear MIMO systems. As solution a set of cascade sub-observers is developed for sequentially achieving the reconstruction of UI’s components without any prior information about its waveforms. It is shown that the bounded limit of the state estimation error can be reduced as desired by a calibration of the sole synthesis parameter θ. Of fundamental interest, the systematic method planed here seems more general to include a wider class of nonlinear systems and it exhibits quite state estimation in presence of UIs which can loop up external disturbance or non-modeled dynamics. Moreover, it offers promising technique for other recent research fields such as ’fault detect and diagnostic’ (FDD) and ’Fault tolerant control’ (FTC). Nonetheless, the dependency of the studied model in the UI remains restrictive enough. Our further works should be focused on this point.
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(a)
x4
5 0 −5
0
0.5
1
1.5
2 (b)
2.5
3
3.5
4
0
0.5
1
1.5
2 (c)
2.5
3
3.5
4
0
0.5
1
1.5
2 (g)
2.5
3
3.5
x5
4 2 0 5 x6
0 −5 −10
x7
4 x7 x7e
5 0 −5
0
0.5
1
1.5
2 (e)
2.5
3
3.5
0
0.5
1
1.5
2 (f)
2.5
3
3.5
2 time(s)
2.5
1
4 v1 v1e
0 −1
4 v2 v2e
0.5 0 −0.5
0
0.5
1
1.5
3
3.5
4
Fig. 2. Conjoint estimation results of state and UIs REFERENCES [1] A. Bobtsov, A. Pyrkin, S. Aranovskiy (2011) Cancellation of Unknown Harmonic Disturbance for Nonlinear System with Input Delay. IFAC WC, Milan, 1516-1521. [2] Bejarano, F. J.,Perruquetti, W., & Zheng, G. (2015) Observation of nonlinear differential-algebraic systems with UIs. IEEE Trans. Aut. Cont., 60(7), 1957-1962. [3] Bevly, D.M., & Parkinson, B. (2007) Cascaded Kalman filters for accurate estimation of multiple biases, deadreckoning navigation and full state feedback control of ground vehicles. IEEE Tr. Con. Sys. Tech., 15, 199-208. [4] Chadli, M., Akhenak, A., Ragot, J., & Maquin, D. (2009) State and unknown input estimation for discrete time multiple model. J. Franklin Institute, 346, 593-610. [5] Du, X., Zhao, H., & Chang, X.H. (2015) UIO design for fuzzy systems with uncertainties. Applied Mathematics and Computation, 266(1), 108-118. [6] Farza, M., M’Saad, M., & Sekher, M. (2005) A set of observers for a class of nonlinear systems. in: Proceeding of16th IFAC World Congress, 4-8. [7] Farza, M., M’Saad, M., Triki, M., & Maatoug, T. (2011) High gain observer for a class of non-triangular systems. Systems & Control Letters, 60(1), 27-35. [8] Hadj Said, S., Mimouni, M.F., M’Sahli, F., & Farza, M. (2011) HGO based on-line rotor and stator resistances estimation for IMs. Sim. Mod. Prac. Th., 19, 1518-1529. [9] Hammouri, H., & Tmar, Z. (2010) Unknown input observer for state affine systems: A necessary and sufficient condition. Automatica, 46, 271-278. [10] Haoussi, F., & Tissir H. (2009) Robust Hinf Controller Design for Uncertain Neutral Systems via Dynamic Observer Based Output Feedback. I.J.A.C. 6, 164-170. [11] Hou, M. & Muller, P. C. (1992) Design of observers for linear systems with UI. IEEE T.A.Contr., 37, 871-875.
[12] Ichalal, D., & Mammar, S. (2015) On UIOs for LPV systems. IEEE Trans. Ind. Elect., 12, 5870-5880. [13] Keller, J., & Sauter, D. (2013) Kalman filter for discrete-time stochastic linear systems subject to intermittent UI. IEEE Tr. Automat. Cont, 58, 1882-1887. [14] Kudva, J., Viswanadham, N., & Ramakrishna, A. (1980) Observers for linear systems with unknown inputs. IEEE Trans. Automat. Control, 25, 113-115. [15] Mohamed, K., Chadli, M., & Chaabane, M. (2012) Unknown inputs observer for a class of nonlinear uncertain systems: An LMI approach. Internat. J. of Automation and Computing, 9(3), 331-336. [16] Morales, J.L., Jimenez, M.T., & Escalante, M.F. (2013) An interconnected adaptive observer for flying capacitor multilevel converters, Elec. P.S.R., 100, 7-14. [17] Ng, K.Y., Tan, C.P., & Oetomo, D. (2012) Disturbance decoupled fault reconstruction using cascaded sliding mode observers. Automatica, 48, 794-799. [18] Nicosia S., Tornambe A., (1989) High-gain observers in the state and parameter estimation of robots havingelastic joints, Syst. Cont. Lett., 13(4):331-337. [19] Sahnoun, M., & Hammouri, H. (2014) Nonlinear observer based on observable cascade form. in: Proceeding of 13th European control conference. [20] Veluvolu, K. C., Defoort, M., & Soh, Y. (2014) High-gain observer with sliding mode for nonlinear state estimation and fault reconstruction. J. Franklin Institute 351(4) 1995-2008. [21] Xiong, Y., & Saif, M. (2003) Unknown disturbance inputs estimation based on a state functional observer design. Automatica, 39, 1389-1398. [22] Yang, J., Zhu, F., Yu, K., & Bu, X. (2015) Observerbased state estimation and UI reconstruction for nonlinear complex dynamical systems. Communications in Nonl. Sc. and Num. Sim., 20(3), 927-939. APPENDIX : SKETCH OF CONVERGENCE PROOF Since the injective map Φ ◦ ψ brought system (1) under the form (12), the observer (13) can be written in the original coordinates under form (15). Indeed, let r nµ +1 i zˆ ∈ R i=1 r
nk
k=1
n1[(µ1 +1)+s]
+m+n
, then there exists a unique
ˆ Differentiating ζˆ ∈ Ri=1 such that Φ(ˆ z ) = ζ. each term of the above equality with respect to time gives: i + ˙ z) zˆ˙ i = ∂Φ∂z(ˆ ζˆi i + z) −1 T i ˜1 = F i (ˆ z ) + Gi (ˆ z ) − θδi ∂Φ(ˆ ∆−1 i (θ)Si Ci z ∂z Proceeding as in Farza et al. (2005) and according to the particular structure of the Jacobian transformation ∂Φ(z) ∂z which is lower triangular, only diagonal terms are taken into account in the compute of observer’s gain. Consequently, we can obtain the observer (14) with Λi (u, zˆ) k are the diagonals of ∂Φ∂z(ˆz) . Moreover, By referring to the th. 3.1 (Farza et al. (2011)), when canonical nontriangular form is used, the observation error remains in a 1 ball with radius proportional to θ1−η . The injective map ˆ Φ(ˆ z ) = ζ allows to preserve this convergence property and the estimation error can be made as small as desired in the z coordinates by an appropriate choice of the design parameter θ (see the demarche given in Farza et al. (2004)).
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IFAC PapersOnLine 50-1 (2017) 1445–1450 Cascade High Gain Observers Based State Cascade High Gain Observers Based State Cascade High Gain Observers Based State and Unknown Input Reconstruction Cascade High Gain Observers Based State and Unknown Input Reconstruction and Unknown Input Reconstruction and Unknown Input Reconstruction ∗ ∗∗ S. Hadj Sa¨ıd ∗∗ M. Farza ∗∗ ∗∗ F. M’Sahli ∗ T. Menard ∗∗
S. Hadj Sa¨ıd ∗∗ M. Farza ∗∗ F. M’Sahli ∗∗ T. Menard ∗∗ S. Hadj Sa¨ıd ∗ M. Farza ∗∗ F. M’Sahli ∗ T. Menard ∗∗ ∗∗ S. Hadj Sa¨ ıd M. Farza F. M’Sahli T. Menard ∗∗ Department of Electrical Engineering, Department of Electrical Engineering, Monastir Monastir Engineering Engineering School, School, Ibn Eljazzar, 5019 Monastir, DepartmentRoad of Electrical Engineering, MonastirTunisia, Engineering School, Road Ibn Eljazzar, 5019 Monastir, Tunisia, Department of Electrical Engineering, Monastir Engineering School, (e-mail: [email protected], [email protected]). Road Ibn Eljazzar, 5019 Monastir, Tunisia, [email protected]). ∗∗ (e-mail: [email protected], Road Ibn Eljazzar, 5019 Monastir, Tunisia, 6072 6 (e-mail: UMR [email protected], [email protected]). ∗∗ GREYC, GREYC, 6072 CNRS, CNRS, Caen Caen University, University, 6 Bd Bd Marechal Marechal Juin, Juin, ∗∗ (e-mail: UMR [email protected], [email protected]). ∗∗ GREYC, 14050 Caen Cedex, France, UMR 6072 CNRS, Caen University, 6 Bd Marechal Juin, 14050 Caen Cedex, France, ∗∗ GREYC, UMR 6072 CNRS, Caen University, 6 Bd Marechal Juin, (e-mail: [email protected]). 14050 Caen Cedex, France, (e-mail: [email protected], [email protected], [email protected]). 14050 Caen Cedex, France, (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). Abstract: Abstract: This This paper paper presents presents a a systematic systematic routine routine for for jointly jointly reconstruct reconstruct the the state state variables variables and the unknown inputs for a large class of nonlinear MIMO systems. After an appropriate Abstract: This paper presents a systematic routine for jointly reconstruct the state variables and the unknown inputs for a large class of nonlinear MIMO systems. After an appropriate Abstract: This paper presents aofsystematic routine for MIMO jointly are reconstruct the state variables change state gain designed in a and theof inputs fora aset class ofhigh nonlinear systems. After an appropriate change ofunknown state coordinates, coordinates, setlarge of cascade cascade high gain observers observers are designed in such such a way way that that and the unknown inputs foraa aset large class of nonlinear MIMO systems. After an appropriate each of them provides an estimation of only one component of the unknown input vector, except change of state coordinates, of cascade high gain observers are designed in such a way that each of them provides an estimation of only one component of the unknown input vector, except change state coordinates, a set of cascade high gain observers are designed in such a way that the one gives a of whole Such design achieves each of of them provides an estimation of only one component of thevariables. unknown input vector, except the last last one which which gives a final final adjustment adjustment of the the whole state state variables. Such design achieves each of them provides an estimation of only one component of the unknown input vector, except aa boundary the error can be small properly the last one of which gives estimation a final adjustment of the state variables. designspecifying achieves boundary of the state state estimation error which which canwhole be arbitrarily arbitrarily small by bySuch properly specifying the last one which gives estimation a final adjustment ofexample the statethe variables. design achieves the sole synthesis An illustrative effectiveness of the proposed a boundary of the parameter. state error which canwhole beverifies arbitrarily small bySuch properly specifying the sole synthesis parameter. An illustrative example verifies the effectiveness of the proposed a boundary of the parameter. state estimation error which can beverifies arbitrarily small by properly specifying scheme. the sole synthesis An illustrative example the effectiveness of the proposed scheme. the sole synthesis parameter. An illustrative example verifies the effectiveness of the proposed scheme. © 2017, IFAC (International Federation ofEstimation, Automatic Control) Hosting by Elsevier Ltd. All rights reserved. scheme. Keywords: Keywords: State State Estimation, Estimation, Robust Robust Estimation, Nonlinear Nonlinear Systems, Systems, Cascade Cascade Observers, Observers, Unknown Estimation. Keywords: State Estimation, Unknown Inputs Inputs Estimation. Robust Estimation, Nonlinear Systems, Cascade Observers, Keywords: State Estimation, Unknown Inputs Estimation. Robust Estimation, Nonlinear Systems, Cascade Observers, Unknown Inputs Estimation. for 1. for multiple multiple model model presentation presentation is is proposed proposed in in Chadli Chadli et et 1. INTRODUCTION INTRODUCTION al. (2009). For state affine systems, a necessary and suffor multiple model presentation is proposed in Chadli et 1. INTRODUCTION al. (2009). For state affine systems, a necessary and suffor multiple model presentation is proposed in Chadli et 1. INTRODUCTION ficient condition is for the of al. (2009). For state affine a necessary and sufficient condition is given given forsystems, the existence existence of an an unknown unknown Since (2009). For state affine systems, a &necessary and sufinput observer (UIO) in (Hammouri Tmar, 2010). In Since four four decades decades ago, ago, the the study study of of dynamical dynamical systems systems al. ficient condition is given for the existence of an unknown input observer (UIO) in for (Hammouri & Tmar, 2010). In in presence of unknown inputs has ficient condition is given thefor existence of anobservable unknown Since decades the study of (UIs) dynamical systems input Mohamed et al. (2012), UIO a canonical in the thefour presence of ago, unknown inputs (UIs) has strongly strongly observer (UIO) inan (Hammouri & Tmar, 2010). In Since four decades ago, the study of dynamical systems Mohamed et al. (2012), an UIO for a canonical observable motivated the activities in theory input observer in with (Hammouri & Tmar, 2010). In in the presence of unknown inputs (UIs)control has strongly form system uncertainties is motivated the research research activities in both both control theory Mohamed et al. (UIO) (2012), UIOparameters for a canonical observable in the presence of unknown inputs (UIs) has strongly form of of nonlinear nonlinear systeman with parameters uncertainties is and diagnostic The unknown input identification motivated the analysis. research activities in both control theory form Mohamed et LMI al. (2012), an UIO forrecent a canonical observable developed in terms. In a more work Bejarano et and diagnostic analysis. The unknown input identification of nonlinear system with parameters uncertainties is motivated the research activities in both control theory developed in LMI terms. In a more recent work Bejarano et techniques contribute not only to solve the control issue form of nonlinear system with parameters uncertainties is and diagnostic analysis. The unknown input identification al. an observation nonlinear techniques contribute not only to solve the control issue developed terms. Inscheme a moreof recent work differentialBejarano et and diagnostic analysis. The unknown input al. (2015), (2015), in anLMI observation scheme of nonlinear differentialfor systems with unmeasurable or weakly developed in LMI terms. Inscheme a more recent work Veluvolu Bejarano et techniques contribute not only todisturbances solve the identification control issue al. algebraic systems subject to UI is designed. et for systems with unmeasurable disturbances or weakly (2015), an observation of nonlinear differentialtechniques contribute not only to solve the control issue al. algebraic systems subject scheme to UI is designed. Veluvolu et modeled but also, to generate a disturbance-decoupled (2015), an observation of nonlinear differentialfor systems with unmeasurable disturbances or weakly al. (2014) extend the standard high gain observer (HGO) modeled but also, to generate a disturbance-decoupled algebraic systems subject to UI is designed. Veluvolu et for systems with unmeasurable disturbances or the weakly al. (2014) extend the standard high gain observer (HGO) residual signals for detecting the plant faults. On one modeled but also, to generate a disturbance-decoupled systems subject to UIhigh isfollows designed. Veluvolu et by sliding mode based term the residual signals for detecting the aplant faults. On the one algebraic al. standard gain observer (HGO) modeled but also, to which generate disturbance-decoupled by a a(2014) slidingextend mode the based term that that follows the disturbance disturbance side, the earlier works was dedicated the linear al. (2014) extend the standard high gain observer (HGO) residual signals for detecting the plant faults.to On the one by vector. Other consistent results on UIO synthesis using side, the earlier works which was dedicated to the linear a sliding mode based term that follows the disturbance residual for detecting the faults.toOn one by vector. Other consistent results on UIO the synthesis using systems have focused on alleviating the input a sliding mode basedmodels term that disturbance side, thesignals earlier works which wasplant dedicated thethe linear Fuzzy and/or neutrals are provided in Haoussi & systems have focused on alleviating the unknown unknown input vector. Other consistent results onfollows UIO synthesis using side, the earlier works which was dedicated to the linear Fuzzy and/or neutrals models are provided in Haoussi & effects in the quality of state reconstruction. The main vector. Otherand consistent results UIO synthesis using systems on state alleviating the unknown Tissir (2009) Du et al. (2015). effects inhave the focused quality of reconstruction. The input main Fuzzy and/or neutrals models areonprovided in Haoussi & systems have focused on alleviating the unknown input Tissir (2009) and Du et al. (2015). idea consists in decoupling the influence of the UI and effects in the quality of state reconstruction. The main Fuzzy and/or neutrals models are provided in Haoussi & In way to idea consists in decoupling the influence of the UImain and Tissir (2009) andand Duthanks et al. (2015). effects in the quality of state The In parallel parallel way and thanks to the the technological technological progress progress guarantees that the state estimation error converge the Tissir (2009) andand Duthanks et al. (2015). idea consists in decoupling thereconstruction. influence of the UIto and in terms of computing time for real-time implementaguarantees that the state estimation error converge to the In parallel way to the technological progress idea consists in decoupling the influence of the UI and in terms of computing time for real-time implementaorigin least possess a bound Kudva et al. In parallel and thanks to the real-time technological progress guarantees theit estimation converge tion, cascade observers increasingly investigated last origin or or at atthat least itstate possess a little little error bound Kudva to et the al. in terms ofway computing time for implementaguarantees theit state estimation error converge the tion, cascade observers were were increasingly investigated last (1980), & Muller (1992). These developments origin orHou atthat least possess a little bound Kudva to ethave al. tion, in terms of computing timein for real-time implementadecade. Bevly demonstrates Bevly & Parkinson (2007) (1980), Hou & Muller (1992). These developments have cascade observers were increasingly investigated last origin or at least it possess a little bound Kudva et al. decade. Bevly demonstrates in Bevly & Parkinson (2007) led to a generalization of the UI decoupling concept and (1980), Hou & Muller (1992). These developments have tion, cascade observers were increasingly investigated last that cascaded estimation provides better led to a Hou generalization of the UI decoupling concept have and decade. demonstrates in technique Bevly & Parkinson (1980), & Muller (1992). These developments that the theBevly cascaded estimation technique provides (2007) better interesting methods have emerged such as Kell & Sauter decade. Bevly demonstrates in technique Bevly & conventional Parkinson (2007) led to a generalization of the UI decoupling concept and that estimation of the vehicle states over a estiinteresting methods have emerged such as Kell & Sauter the cascaded estimation provides better led to a generalization of the UI decoupling concept and estimation of the vehicle states over a conventional esti(2013) and Ichal & Mammar (2015). side, that thescheme. cascaded estimation technique providessuggest better interesting methods have emerged suchOn as the Kellother & Sauter mation In Ng et al. (2012), the authors (2013) and Ichal & Mammar (2015). On the other side, estimation of the vehicle states over a conventional estiinteresting methods have emerged suchOn as the Kell &UIs Sauter mation scheme. In Ng et al. (2012), the authors suggest some authors Xiong & Saif (2003) considered the as a (2013) and Ichal & Mammar (2015). other side, of theInvehicle states over afault conventional estia sliding mode for reconstruction some authors Xiong & Saif (2003) considered the UIs as a estimation mation scheme. Ng etobserver al. (2012), suggest (2013) andstate Ichal && Mammar (2015). On their thethe other a cascaded cascaded sliding mode observer for the faultauthors reconstruction part of the vector upon assuming that variations some authors Xiong Saif (2003) considered UIs side, as a mation scheme. Inthe Ng faults etobserver al. and (2012), the authors suggest when considering disturbances as UIs part of the state vector upon assuming that their variations a cascaded sliding mode for fault reconstruction some Xiong & upon Saif (2003) considered thevariations UIs time as a awhen considering the faults and for disturbances as UIs of of are relatively slow w.r.t the state dynamics cascaded sliding mode observer fault of reconstruction part ofauthors the state vector assuming that (constant, their the system. Throughout the two stages HGO, both are relatively slow w.r.t the state dynamics (constant, time when considering the faults and disturbances as UIs of part of the state vector upon assuming that their variations the system. Throughout the two stages of HGO, both polynomial, The corresponding observers are then when considering the faults and as UIs of are relativelyetc). slow w.r.t state dynamics (constant, state variables and parameters are reconstructed polynomial, etc). The the corresponding observers are time then the Throughout two disturbances stages HGO, both are relativelyin slow w.r.t the state dynamics (constant, time statesystem. variables and critical criticalthe parameters areof reconstructed constructed such aa way to estimate both the state polynomial, etc). The corresponding observers arevector then the system. Throughout the two stages ofreconstructed HGO, both in induction motor application in Hadj Said et al. (2011). constructed in such way to estimate both the state vector state variables and critical parameters are polynomial, etc). The corresponding observers arevector then state in induction motor application in Hadj Said et al. (2011). and The common approach for this variables and critical parameters areobservable reconstructed constructed suchmost a way to estimate both the Output injection observers for a class casand the the UIs. UIs.in The most common approach for state this second second in induction motor application in Hadj of Said et al. (2011). constructed in such a way to estimate both the state vector Output injection observers for a class of observable cascategory consists in transforming the UI in induction motor application in Hadj affine Said etsystems al. (2011). and the UIs. The most common approach foridentification this second Output cade systems which include the state are category consists in transforming the UI identification injection observers for a class of observable casand the UIs. The most common approach for this second cade systems which include the state affine systems are problem into a constrained optimization that can Output injection observers for a state class(2014). of observable cascategory consists in transforming the problem UI identification discussed in Sahnoun & Hammouri In Yang et problem into a constrained optimization problem that can cade systems which include the affine systems are category consists in transforming thematrix UI identification discussed in Sahnoun & Hammouri (2014). In Yang et be easily solved by adopting the linear inequalities problem into a constrained optimization problem that can cade systems which include the state(2014). affine systems are al. (2015), adaptive together with a be easily into solved by adoptingoptimization the linear matrix inequalities discussed inan Sahnoun &SMO Hammouri Inhigh Yanggain et problem a constrained problem that can al. (2015), an adaptive SMO together with a high gain (LMIs) formalism. The issues of both state estimation and discussed in Sahnoun & Hammouri (2014). In Yang et be easily solved by adopting the linear matrix inequalities SMO are in estimate respectively the (LMIs) formalism. The issues of both state estimation and al. (2015), an adaptive SMOto a high gain be easilyformalism. solved by adopting linear matrix inequalities SMO are developed developed in order order totogether estimatewith respectively the unknown input reconstruction the nonlinear case still al. (2015), an adaptive SMOnetworks together with a derivative high gain (LMIs) The issuesthe of in both state estimation and states of uncertain complex and the unknown input reconstruction in the nonlinear case still SMO are developed in order to estimate respectively the (LMIs) formalism. The issues of both state estimation and states of uncertain complex networks and the derivative provide an open research field. Recent academic works SMO are developed inwhereas order tothe estimate respectively the unknown the nonlinear caseworks still states of output vector, reconstructed by provide aninput openreconstruction research field.in Recent academic uncertain complex the derivative unknown the some nonlinear caseworks still of the the of output vector, whereasnetworks the UI UI is isand reconstructed by are investigated in the synthesis for sub-classes of provide aninput openreconstruction research field.inRecent academic states of uncertain complex networks and the derivative an algebraic method. An interconnected adaptive observer are investigated in the synthesis for some sub-classes of of the output vector, whereas the UI is reconstructed by provide ansystems. open inresearch field. Recent academic works an the algebraic method. An interconnected adaptive observer nonlinear A sliding mode observer (SMO) design of output vector, whereas the UI is reconstructed by are investigated the synthesis for some sub-classes of nonlinear systems. Athe sliding mode observer (SMO) design algebraic method. An interconnected adaptive observer are investigated in A synthesis some (SMO) sub-classes of an an algebraic method. An interconnected adaptive observer nonlinear systems. sliding mode for observer design nonlinear systems. A sliding mode observer (SMO) design ∗ ∗ ∗ ∗ ∗
Copyright © 2017 IFAC 1481 Copyright 1481Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) © 2017 IFAC 1481 Copyright Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 1481Control. 10.1016/j.ifacol.2017.08.288
Proceedings of the 20th IFAC World Congress 1446 S. Hadj Saïd et al. / IFAC PapersOnLine 50-1 (2017) 1445–1450 Toulouse, France, July 9-14, 2017
for flying capacitor multilevel converters is highlighted in Morales et al. (2013). In such case, the observer for the whole system which is modeled by a set of cascade interconnected subsystem, is approached by designing a separate observer for each individual subsystem. This paper builds on the work in Farza et al. (2011) where a HGO is designed for a non-triangular class of nonlinear systems. Under the existence of the two state coordinates which transform the original system into the observable canonical form, a set of cascade sub-observers is systematically planned for achieving simultaneous estimation of the non-measured states and the UIs without using any prior information about the waveform of the latter. This paper is organized as follows. In the next section, the class of nonlinear systems which is the basis of the observer design is introduced. Sections 2 and 3 detailed respectively the two state transformations and the cascade observer synthesis. Section 4 is devoted to a simulation example that highlight the performance of the proposed observation scheme. 2. PROBLEM STATEMENT AND PRELIMINARIES Let us consider the MIMO class of continuous nonlinear systems that can be represented by: x(t) ˙ = f (u(t), x(t)) + g(u(t), x(t), v(t)) + ε˜(t) Σ0 (1) y(t) = x1 (t)
with 1 :
1 f (u, x[2] ) x1 .. .. . . µ −1 µ1 −1 [µ1 ] 1 (u, x ) x f µ1 f µ1 (u, x[µ1 +1] ) x . . . . x = . , f (.) = . f µr −1 (u, x[µr ] ) xµr −1 µr µr [µr +1] x f (u, x ) . . . . .. s−1 s−1 x (u, x) f s s x f (u, x) 1 g (u, x1 ) .. . µ1 −1 [µ1 −1] g (u, x ) µ g 1 (u, x[µ1 ] , v 1 ) . .. and g(.) = g µr −1 (u, x[µr −1] , v [r−1] ) g µr (u, x[µr ] , v) . .. s−1 [s−1] , v) g (u, x s g (u, x, v) wherethe state x ∈ Rn with xk ∈ Rnk for k = 1, · · · , s s and ˜ = k=1 nk = n with n1 ≥ n2 ≥ · · · ≥ ns . ε T T T sT 0n1 , · · · , 0ns−1 , ε˜ is an unknown function.u ⊂ U subset of absolutely continuous functions with bounded derivatives from R+ into U a compact subset of Ru and 1
for the convenient of description, we omit the time variable t in the remainder of the paper and we note x[i] = x1 , x2 , · · · , xi and v [i] = v 1 , v 2 , · · · , v i
y ∈ Rn1 denote respectively the known input and the output of the system. v ∈ Rm denotes the unknown input r vector with v i ∈ Rmi for i = 1, · · · , r and i=1 mi = m with m1 ≥ m2 ≥ · · · ≥ mr . The r characteristic indices µj associated to the m UIs are defined as follows: Definition 1: The characteristic index (or the relative degree) µj associated to the parameter v j for system (1) is the smallest integer j such that i ∂g (u, x, v) ≡ 0, ∀i < µj ∂v j (2) µj ∂g (u, x, v) = 0 ∂v j with 1 ≤ µ1 ≤ µ2 · · · ≤ µr ≤ s. Remark 1: • When g(x, u, v) = 0 (in the absence of UIs) the system (1) is the same as the one studied in Farza et al. (2005). • In the case where µ1 = 1, all first µ1 − 1 equations of the model (1) can be omitted. The studied class of nonlinear MIMO systems is large enough to include a variety of real plant, ranging from chemical and biological processes to electromechanical actuator and robot platform. The main objective consists in the synthesis of an observer to simultaneously estimate the vector of UIs v(t) and the non measured states x(t) without assuming any model for the UIs. The synthesis of the observer requires the adoption of some hypothesis which will be stated in due courses. Consider at the first the following assumptions : Assumption 1 Let for: 1 ≤ k ≤ s − 1, the application : Rnk+1 → Rnk (u, x[k+1] ) → f k (u, x[k+1] )
(3)
is injective for all (u, x[k+1] ). Moreover, we suppose that ∃ α1 , β1 > 0 such as: ∀k ∈ {1, ..., s − 1} , ∀x ∈ Rn , ∀u ∈ U (α1 )2 Ink+1 ≤
∂f k (u, x) ∂xk+1
T
∂f k (u, x) ≤ (β1 )2 Ink+1 (4) ∂xk+1
with Ink+1 is the matrix identity with (nk+1 ) × (nk+1 ) dimension Assumption 2 The application:
(u, x vj
R
nµ
[µj +1]
)
j
+1 +mj
→R
nµ
j
→ ϕ¯µj (u, x[µj +1] , v j )
(5)
is injective for all (u, xµj , v j ) with ϕ¯k (u, x, v) = f k (u, x) + g k (u, x, v) One also assumes that ∃ α2 , β2 > 0 such as for x ∈ Rn , ∀u ∈ U , ∀v ∈ Rm : (α2 )2 Inµj +1 +mj ≤ (F˜ µj (.))T F˜ µj (.) ≤ (β2 )2 Inµj +1 +mj (6) ∂ϕ ¯µj ∂ϕ ¯µj (u, x, v) (u, x, v) with F˜ µj (x, u, v) = ∂x µj +1 ∂v j
The inequality given by assumption 2 implies that the rank of matrix F˜ µj (x, u, v) is equal to nµj +1 + mj . Yet, this matrix is rectangular includes nµj lines and nµj +1 + mj columns, so:
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nµj +1 + mj ≤ nµj
(7)
Thereafter, one can deduces that: m=
r j=1
mj ≤
r j=1
(nµj − nµj +1 )
(8)
Remark 2: • In the absence of UIs, the given assumptions are the same as these adopted in Farza et al. (2005), whereas the other hypotheses that involves the UI are simply issued from the rank condition. • The number of UIs must be strictly lower than the number of the measured output, except when µr = s. In this last case the number of UIs can be equal to the number of the measured outputs. 3. COORDINATE CHANGE Now, one shall introduce a set of coordinate transformations in order to put the system (1) under the canonical form which corresponds to the high gain observer synthesis. Precisely, an augmented system which is composed with r + 1 blocks such that there output’s are equal of these of the original system (1) is planned. The associated variables of each individual j th block are: x1 , ..., xµj +1 and v j , whereas the variables which are associated to the last one are simple copies of the whole system state’s. So, the considered immersion can be interpreted as an injective map ψ(.) which is defined as follows: r nµ +1 i nk
+m+n
i=1 k=1 ψ : Rn+m → R z1 ψ1 (x, v) x → z = ... = ... v r+1 (x, v) ψ z r+1 i k z = x f or k = 1, · · · , µ i = 1, · · · , r i k i µ; +1 zµi +1,1 x i i : zµi +1 = = vi zµi i +1,2 r+1 k zk = x , k = 1, · · · , s
(9)
f 1 (u, z i[2] ) 1 .. f (u, z r+1 [2] ) . r+1 . F i = f µi (u, z i = .. ; F ; [µi +1] ) s r+1 f µi +1 (u, z i f (u, z ) [µi +2] ) 0 1 g (u, z1i ) 1 .. g (u, z1r+1 ) . r+1 .. = . Gi = g µi (u, z i ; G , [µi +1] ) s r+1 g µi +1 (u, z i g (u, z ) ) 0
[µi +1]
0 .. . εi = 0 0
µi +1
and εr+1 = ε˜.
εi Let now the second injective map Φ such that: r nµ +1 r i Φ : R
with i ζ1 ζ2i i ζ3 . ..
nk
i=1
+m+n
n1 [(µi +1)+s]
→ Ri=1 ζ 1 = Φ1 .. .
k=1
z1 .. ζ= . → z r+1 ζ r+1 = Φr+1
(11)
ζji and ζjr+1 for i = 1, · · · , r and j = 1, · · · , µi + 1 : = z1i = f 1 (u, z i[2] ) ∂f 1 (u, z i[2] ) 2 = f (u, z i[3] ) ∂z2i
µ −2 k i ∂f (u, z i[k+1] ) i f µi −1 (u, z i[µ1 ] ) ζµi = i ∂z k=1 µ −1 k k+1 i i ∂f (u, z [k+1] ) i ζµi +1 = i ∂zk+1 k=1
µi i i × f µi (u, z i , z i ) + g (u, z , z ) [µ1 ] µ1 +1,1 [µ1 ] µ1 +1,2 r+1 r+1 ζ = z 1 1 ζ2r+1 = f 1 (u, z r+1 [2] ) 1 ∂f (u, z r+1 [2] ) 2 ζ3r+1 = f (u, z r+1 r+1 [3] ) ∂z2 .. . s−2 k ∂f (u, z r+1 ) [k+1] r+1 f s−1 (u, z r+1 ζs = [s] ) r+1 ∂z k+1 k=1 Performing simple manipulation yields: i ζ˙ = Ai ζ i + ϕi (u, u, ˙ ζ [i] ) + ε¯i Σ2 i yi = ζ1
Consequently, in the z coordinates, the (r + 1) subsystems are presented as follows: i z˙ = F i (u, z i ) + Gi (u, z i ) + εi Σ1 (10) yi = z1i with:
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where:
0 I(µi )n1 0 In1 (s−1) Ai = ; Ar+1 = ; 0 0 0 0 0 ϕi2 (u, u, ˙ ζ i[2] , ζ [i−1] ) ˙ ζ [i] ) = ϕi (u, u, ; ϕi (u, u, i [i−1] ˙ ζ [µ ] , ζ ) µi i ϕiµi +1 (u, u, ˙ ζ, ζ [i−1] ) 0 ϕr+1 ˙ ζ) 2 (u, u, ϕr+1 (u, u, ˙ ζ) = .. .
(u, u, ˙ ζ) ϕr+1 s
T r+1 T (u,z) r+1 ε¯i = 0T(µi )n1 , i ; ε¯r+1 = ∂Φ∂zr+1 ε µ −1 k i µ i i ∂f (u,z[k+1] ) ∂g i (u,ζ ) i ε and i = ∂v i ∂z i
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k=1
k+1
(12)
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4. OBSERVER SYNTHESIS WITH r SET OF UNKNOWN INPUTS By referring to the framework given in Farza et al. (2011), one can adopt the HGO designed for the class of MIMO nonlinear systems with non-triangular canonical form. Let at this stage the following assumption : Assumption 3 The nonlinear functions ϕi (u, ζ) are globally Lipchitz with respect to ζ uniformly in u. A candidate HGO that corresponds to the form (12) can be given as follows: [i] ˙i i i ˙ ζˆ ) ζˆ = Aj ζˆ + ϕ (u, u, −1 T ˆi i −θδi ∆−1 i (θ)Si Ci (ζ1 − ζ1 ) for i = 1 : r ˆ Σ 2 ˙ r+1 r+1 r+1 ˆ ζˆ = Ar+1 ζˆ + ϕ (u, u, ˙ ζ) −1 r+1 δr+1 −1 T ˆ −θ ∆r+1 (θ)Sr+1 Cr+1 (ζ1 − ζˆ1r+1 ) (13) with: 1 1 Cµi +1 In1 Cs In1 Cµ2 +1 In1 C 2 In i s 1 −1 T Cr+1 = . Si−1 CiT = . , Sr+1 .. .. s µi +1 I C Cµi +1 In1 s n1 ∆i (θ) = diag In1 , θ1δi In1 , · · · , θµ1i δi In1 ; n! ∆r+1 (θ) = diag In1 , θ1 In1 , · · · , θ1s In1 and Cin = (n−i)!i! The δi which indicates the power of θ are defined as the same as given in Farza et al. (2011). According to the theorem 3.1 established in the above reference, the estimation error which remains in a ball with a radius depends only on θ, can made as small as desired when the latter parameter is chosen large enough. However, due to the sensitivity of the HGO to the output noise, a compromise between fast convergence and filtering noise should be achieved. Now, when returning to the z coordinate one obtains: i zˆ˙ = F i (u, zˆ) + Gi (u, zˆ) + −1 T − (Λi (u, zˆ)) θδi ∆−1 z1i − z1i ) i (θ)Si Ci (ˆ ˆ Σ 1 r+1 r+1 r+1 zˆ˙ =F (u, zˆ) + G (u, zˆ) + −1 δr+1 −1 T ∆r+1 (θ)Sr+1 Cr+1 (ˆ z1r+1 − z1r+1 ) −Λr+1 (.)θ (14) where: + Λi (u, zˆ) = diag In1 , Λ+ ˆi ), · · · , Λ+ ˆi ) i,2 (u, z i,µi +1 (u,z + Λ+ ˆ) = diag In1 , Λ+ r+1 (u, z r+1,2 (.), · · · , Λr+1,s (.) such as: k−1 ∂f j i Λ (u, z ˆ ) = (u, zˆ) f or k = 2, · · · , µi i,k i ∂zj+1 j=1 µ i −1 ∂f j ¯ µi F (u, zˆ) Λi,µi +1 (u, zˆi ) = i ∂zj+1 j=1 k−1 ∂f j r+1 (u, zˆ) f or k = 2, · · · , s Λ (u, z ˆ ) = r+1,k ∂z i
with:
F¯ µi (u, zˆ) =
j=1
j+1
µj ∂f µj (u, x, v) ∂f ∂v j ∂xµj +1
Fig. 1. UIO block diagram 1 x ˆ˙ = f 1 (u, x ˆ [2] ) + g 1 (u, x ˆ1 ) − Cµ1i +1 θδi x ˜i . .. [i] ˆ [µi +1] ) + g ρr (u, x ˆ [µi ] , vˆ ) x ˆ˙ µi = f µi (u, x i i ˆ x −Cµµii+1 θµi δi Λ+ Σ i,µi (u, x )˜ 0(i) µ +1 µ +1 µi +1 i i ˙ x ˆ (.) + g (.) f = 0 vˆ˙ i µi +1 (µi +1)δi + −Cµi +1 θ xi Λi,µi +1 (u, xi , vˆ[i] )˜ f or i = 1, .., r; 1 x ˆ˙ = f 1 (u, x ˆ [2] ) + g 1 (u, x ˆ1 ) − Cs1 θ˜ xi .. Σ . ˆ 0(r+1) ˆ) + g s (u, x ˆ, vˆ) x ˆ˙ s = f s (u, x s s + −C θ Λ xi (u, x, vˆ[i] )˜ s
r+1,s
with x ˜i = x ˆi − xi
Proposition: Assume that system (1) satisfies assumptions (A1)-(A2) and that there exists a diffeomorphism (11) that transforms the augmented system (10) (in z coordinates) to the canonical form (12) (in ζ coordinates) with assumption (A3) holds, then ∀ M > 0; ∃ θ0 > 0; ∀ θ > θ0 ; ∃ λθ > 0; ∃ µθ > 00 ; ∃ αθ > 0, the system (15)
k an observer
constitutes
for (1) with:
x ˆk (t) − xk (t) ≤ λθ e−µθ t x ˆ (0) − xk (0) + αθ β. for every admissible control u s.t u∞ ≤ M , where βk is the upper bound of εk . Moreover, λθ is a polynomial in θ, lim µθ = +∞ and lim αθ = 0. (See the Appendix for θ→∞
θ→∞
a sketch of proof.) The block diagram of the whole system of the observation is depicted in figure (1). It is composed of (r + 1) subsystems in cascade form. The first one provides an estimation of v 1 , the second gives an estimation of v 2 , and so on until the rth subsystem which estimates the v r component. The ultimate block of the observer that exploit the previous unknown input estimates, achieves a reconstruction of all the state variables. Note that each of the ’r’ first subobservers include a copy of µi + 1 first blocks (i = 1...r) of the original system augmented by the corresponding unknown input component.
(u, x, v)
5. EXAMPLE
Finally, in the original coordinate x, the (r + 1) cascade sub-systems of observation have the following forms:
(15)
Let consider the following nonlinear MIMO system:
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x˙ 1 = x1 − x2 + c(t)x4 − s(t)x5 x˙ 2 = −.2x1 x2 + x3 + s(t)x4 + c(t)x5 x˙ 3 = x34 + x55 + s(x3 ) + v1 + v13 x˙ = x x − (x + .3)(v + v 3 ) − v 4 3 6 3 2 1 2 x˙ 5 = (x3 − a)x6 + x3 v2 − x5 s(v1 ) x˙ 6 = x37 + x7 − 3v1 s2 (v2 ) − x5 + v1 v2 x˙ = −x6 + 3x5 c(v1 ) + 3x5 s(v2 ) + ε7 7 T y = [ x1 x2 x3 ]
(16)
with s(.) = sin(.), c(.) = cos(.) and ε7 = −0.5v1 v2 . Our aim is to conjointly estimate the unmeasured state variables x4 , x5 , x6 and x7 together with the two UIs v1 and v2 . We show that the whole state vector can be arranged into four sub-state vectors: x1 = [x1 x2 x3 ]T ; x2 = [x4 x5 ]T ; x3 = x6 and x4 = x7 . So, we have: µ1 = 1, µ2 = 2, s = 4 with n1 = nµ1 = 3, n2 = nµ2 = 2, n3 = ns−1 = 1, n4 = ns = 1, m = r = 2 ≤ nµ1 − ns = 2. Consequently, such state distribution gives rise to the design of three cascade observation blocks. The first and the second blocks provide respectively an estimation of the UIs v 1 an v 2 , whereas the last subsystem gives a reconstruction of the whole system’s state x. The differential equations of the three cascade modules are given as follows:
x ˆ˙ 1 x ˆ1 − x ˆ2 + c(t)ˆ x4 − s(t)ˆ x5 x x1 x ˆ2 + x ˆ3 + s(t)ˆ x4 + c(t)ˆ x5 ˆ˙ 2 = −.2ˆ 3 5 3 x ˆ4 + x ˆ5 + s(ˆ x3 ) + vˆ1 + vˆ1 x ˆ˙ 3 δ1 1 −2θ x ˜ ˙ 3 x ˆ4 x ˆ − (ˆ x + .3)(ˆ v + v ˆ ) − v ˆ x ˆ 3 6 3 2 1 ˆ 2 Σ01 ˙ x3 + .3)ˆ x6 + x ˆ3 vˆ2 − x ˆ5 s(ˆ v1 ) x ˆ5 = (ˆ ˙ 0 vˆ1 −1 c(t) −s(t) 0 x 0 −θ2δ1 s(t) c(t) ˜1 2 4 2 3ˆ x4 5ˆ x5 1 + 3ˆ v1 (17)
x ˆ˙ 1 x ˆ1 − x ˆ2 + c(t)ˆ x4 − s(t)ˆ x5 x x1 x ˆ2 + x ˆ3 + s(t)ˆ x4 + c(t)ˆ x5 ˆ˙ 2 = −.2ˆ 3 5 3 x ˆ4 + x ˆ5 + s(ˆ x3 ) + vˆ1 + vˆ1 x ˆ˙ 3 δ2 1 −3θ x ˜ ˙ x ˆ4 x ˆ3 x ˆ6 − (ˆ x3 + .3)(ˆ v2 + vˆ23 ) − vˆ1 ˆ Σ02 = (ˆ x3 + .3)ˆ x6 + x ˆ3 vˆ2 − x ˆ5 s(ˆ v1 ) x ˆ˙ 5 2δ2 + 1 ˜ −3θ 3 Λ22 x 2 ˙6 x ˆ x ˆ + x ˆ ˆ5 + v1 v2 7 7 − 3v1 s (v2 ) − x = 0 vˆ˙ 2 −θ3δ2 Λ+ ˜1 23 x (18)
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ˆ˙ 1 x ˆ1 − x ˆ2 + c(t)ˆ x4 − s(t)ˆ x5 x x x1 x ˆ2 + x ˆ3 + s(t)ˆ x4 + c(t)ˆ x5 ˆ˙ 2 = −.2ˆ 3 5 3 ˙ x ˆ4 + x ˆ5 + s(ˆ x3 ) + vˆ1 + vˆ1 x ˆ3 1 −4θ˜ x x ˆ˙ 4 ˆ6 − (ˆ x3 + .3)(ˆ v2 + vˆ23 ) − vˆ1 x ˆ3 x = x (ˆ x3 + .3)ˆ x6 + x ˆ3 vˆ2 − x ˆ5 s(ˆ v1 ) ˆ˙ 5 2 + 1 −6θ Λ x ˜ 22 ˆ 03 Σ 3 2 ˙6 = x x ˆ ˆ + x ˆ ˆ5 + v1 v2 7 7 − 3v1 s (v2 ) − x + x ˆ3 3 −4θ Λ22 x ˜1 x ˆ3 − a x ˆ˙ 7 = (−ˆ x6 + 3ˆ x5 c(ˆ v1 ) + 3ˆ x5 s(ˆ v2 )) + 4 θ x ˆ3 − 2 x ˜1 Λ22 x ˆ3 + .3 3ˆ x7 + 1 (19) c(t) −s(t) x ˆ3 b where: Λ22 = s(t) c(t) ; Λ23 = Λ22 x ˆ3 + .3 x ˆ3 2 4 3ˆ x4 5ˆ x5 T
˜2 x ˜3 ] . ˜1 x with b = −(ˆ x3 + .3)(1 + 3ˆ v22 ) and x ˜1 = [ x
In order to evaluate the performances issued from the application of the proposed observation algorithm, we have numerically simulated the previous system and its corresponding observer with the choice of the synthesis parameter θ as θ = 20. The initial conditions are given as ˆ 01 : x the follows: x(0) = [1 .5 1 .3 1 − .2 0.9]T ; Σ ˆ(0) = T ˆ [1 .5 1 −.25 .4 0] ; Σ02 : x ˆ(0) = [1 .5 1 −.25 .4 −.3 −0.5]T ; ˆ 03 : x Σ ˆ(0) = [1 .5 1 − .25 .4 − .3 0]T ; The simulation results are illustrated in fig. (2). On the one hand, we can perceive, after some fluctuations in the transient, that the states x4 , x5 , x6 and x7 are perfectly reconstructed with a quasi-null steady state error. On the other hand the waveform signals of the UIs v1 et v2 are restituted about only 0.5s. Furthermore, the correction gains of all sub-observers which require the calibration of the sole synthesis parameter θ can be explicitly computed; and thereafter they lead to reasonable computations for a real-time implementation.
6. CONCLUSION The modularity of this purpose is the design of an observation scheme that allows to jointly estimate the state variables and the UIs for a large class of nonlinear MIMO systems. As solution a set of cascade sub-observers is developed for sequentially achieving the reconstruction of UI’s components without any prior information about its waveforms. It is shown that the bounded limit of the state estimation error can be reduced as desired by a calibration of the sole synthesis parameter θ. Of fundamental interest, the systematic method planed here seems more general to include a wider class of nonlinear systems and it exhibits quite state estimation in presence of UIs which can loop up external disturbance or non-modeled dynamics. Moreover, it offers promising technique for other recent research fields such as ’fault detect and diagnostic’ (FDD) and ’Fault tolerant control’ (FTC). Nonetheless, the dependency of the studied model in the UI remains restrictive enough. Our further works should be focused on this point.
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Proceedings of the 20th IFAC World Congress 1450 S. Hadj Saïd et al. / IFAC PapersOnLine 50-1 (2017) 1445–1450 Toulouse, France, July 9-14, 2017
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Fig. 2. Conjoint estimation results of state and UIs REFERENCES [1] A. Bobtsov, A. Pyrkin, S. Aranovskiy (2011) Cancellation of Unknown Harmonic Disturbance for Nonlinear System with Input Delay. IFAC WC, Milan, 1516-1521. [2] Bejarano, F. J.,Perruquetti, W., & Zheng, G. (2015) Observation of nonlinear differential-algebraic systems with UIs. IEEE Trans. Aut. Cont., 60(7), 1957-1962. [3] Bevly, D.M., & Parkinson, B. (2007) Cascaded Kalman filters for accurate estimation of multiple biases, deadreckoning navigation and full state feedback control of ground vehicles. IEEE Tr. Con. Sys. Tech., 15, 199-208. [4] Chadli, M., Akhenak, A., Ragot, J., & Maquin, D. (2009) State and unknown input estimation for discrete time multiple model. J. Franklin Institute, 346, 593-610. [5] Du, X., Zhao, H., & Chang, X.H. (2015) UIO design for fuzzy systems with uncertainties. Applied Mathematics and Computation, 266(1), 108-118. [6] Farza, M., M’Saad, M., & Sekher, M. (2005) A set of observers for a class of nonlinear systems. in: Proceeding of16th IFAC World Congress, 4-8. [7] Farza, M., M’Saad, M., Triki, M., & Maatoug, T. (2011) High gain observer for a class of non-triangular systems. Systems & Control Letters, 60(1), 27-35. [8] Hadj Said, S., Mimouni, M.F., M’Sahli, F., & Farza, M. (2011) HGO based on-line rotor and stator resistances estimation for IMs. Sim. Mod. Prac. Th., 19, 1518-1529. [9] Hammouri, H., & Tmar, Z. (2010) Unknown input observer for state affine systems: A necessary and sufficient condition. Automatica, 46, 271-278. [10] Haoussi, F., & Tissir H. (2009) Robust Hinf Controller Design for Uncertain Neutral Systems via Dynamic Observer Based Output Feedback. I.J.A.C. 6, 164-170. [11] Hou, M. & Muller, P. C. (1992) Design of observers for linear systems with UI. IEEE T.A.Contr., 37, 871-875.
[12] Ichalal, D., & Mammar, S. (2015) On UIOs for LPV systems. IEEE Trans. Ind. Elect., 12, 5870-5880. [13] Keller, J., & Sauter, D. (2013) Kalman filter for discrete-time stochastic linear systems subject to intermittent UI. IEEE Tr. Automat. Cont, 58, 1882-1887. [14] Kudva, J., Viswanadham, N., & Ramakrishna, A. (1980) Observers for linear systems with unknown inputs. IEEE Trans. Automat. Control, 25, 113-115. [15] Mohamed, K., Chadli, M., & Chaabane, M. (2012) Unknown inputs observer for a class of nonlinear uncertain systems: An LMI approach. Internat. J. of Automation and Computing, 9(3), 331-336. [16] Morales, J.L., Jimenez, M.T., & Escalante, M.F. (2013) An interconnected adaptive observer for flying capacitor multilevel converters, Elec. P.S.R., 100, 7-14. [17] Ng, K.Y., Tan, C.P., & Oetomo, D. (2012) Disturbance decoupled fault reconstruction using cascaded sliding mode observers. Automatica, 48, 794-799. [18] Nicosia S., Tornambe A., (1989) High-gain observers in the state and parameter estimation of robots havingelastic joints, Syst. Cont. Lett., 13(4):331-337. [19] Sahnoun, M., & Hammouri, H. (2014) Nonlinear observer based on observable cascade form. in: Proceeding of 13th European control conference. [20] Veluvolu, K. C., Defoort, M., & Soh, Y. (2014) High-gain observer with sliding mode for nonlinear state estimation and fault reconstruction. J. Franklin Institute 351(4) 1995-2008. [21] Xiong, Y., & Saif, M. (2003) Unknown disturbance inputs estimation based on a state functional observer design. Automatica, 39, 1389-1398. [22] Yang, J., Zhu, F., Yu, K., & Bu, X. (2015) Observerbased state estimation and UI reconstruction for nonlinear complex dynamical systems. Communications in Nonl. Sc. and Num. Sim., 20(3), 927-939. APPENDIX : SKETCH OF CONVERGENCE PROOF Since the injective map Φ ◦ ψ brought system (1) under the form (12), the observer (13) can be written in the original coordinates under form (15). Indeed, let r nµ +1 i zˆ ∈ R i=1 r
nk
k=1
n1[(µ1 +1)+s]
+m+n
, then there exists a unique
ˆ Differentiating ζˆ ∈ Ri=1 such that Φ(ˆ z ) = ζ. each term of the above equality with respect to time gives: i + ˙ z) zˆ˙ i = ∂Φ∂z(ˆ ζˆi i + z) −1 T i ˜1 = F i (ˆ z ) + Gi (ˆ z ) − θδi ∂Φ(ˆ ∆−1 i (θ)Si Ci z ∂z Proceeding as in Farza et al. (2005) and according to the particular structure of the Jacobian transformation ∂Φ(z) ∂z which is lower triangular, only diagonal terms are taken into account in the compute of observer’s gain. Consequently, we can obtain the observer (14) with Λi (u, zˆ) k are the diagonals of ∂Φ∂z(ˆz) . Moreover, By referring to the th. 3.1 (Farza et al. (2011)), when canonical nontriangular form is used, the observation error remains in a 1 ball with radius proportional to θ1−η . The injective map ˆ Φ(ˆ z ) = ζ allows to preserve this convergence property and the estimation error can be made as small as desired in the z coordinates by an appropriate choice of the design parameter θ (see the demarche given in Farza et al. (2004)).
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