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A switched LPV observer for actuator fault estimation
Proceedings of the 1st IFAC Workshop on Proceedings of the 1st IFAC Workshop on Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems Available online at www.sciencedirect.com Linear Parameter Varying Systems Proceedings of the 1st IFAC Workshop on Linear Parameter VaryingFrance Systems Oct. 7-9, 2015. Grenoble, Oct. 7-9, 2015. Grenoble, Linear Parameter VaryingFrance Systems Oct. 7-9, 2015. Grenoble, France Oct. 7-9, 2015. Grenoble, France
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IFAC-PapersOnLine 48-26 (2015) 194–199
A switched LPV observer for actuator fault A switched LPV observer for actuator fault A switched LPV observer for actuator A switched LPVestimation observer for actuator fault fault estimation estimation estimation M.Q. Nguyen, O. Sename, L. Dugard
M.Q. Nguyen, O. Sename, L. Dugard M.Q. M.Q. Nguyen, Nguyen, O. O. Sename, Sename, L. L. Dugard Dugard Univ. Grenoble Grenoble Alpes, Alpes, GIPSA-lab, GIPSA-lab, Control Control Systems Systems Dpt., Dpt., F-38000 F-38000 Grenoble, Grenoble, Univ. Univ. Grenoble Alpes, GIPSA-lab, Control Systems Dpt., F-38000 France Univ. Grenoble Alpes, GIPSA-lab, France Control Systems Dpt., F-38000 Grenoble, Grenoble, France CNRS, GIPSA-lab, Control Systems Dpt., F-38000 F-38000 Grenoble, Grenoble, France. France. France CNRS, GIPSA-lab, Control Systems Dpt., CNRS, Systems F-38000 E-mail:Control {manh-quan.nguyen, olivier.sename, CNRS, GIPSA-lab, GIPSA-lab, Control Systems Dpt., Dpt., olivier.sename, F-38000 Grenoble, Grenoble, France. France. E-mail: {manh-quan.nguyen, E-mail: {manh-quan.nguyen, olivier.sename, luc.dugard}@gipsa-lab.grenoble-inp.fr E-mail: {manh-quan.nguyen, olivier.sename, luc.dugard}@gipsa-lab.grenoble-inp.fr luc.dugard}@gipsa-lab.grenoble-inp.fr luc.dugard}@gipsa-lab.grenoble-inp.fr Abstract: In this paper, an actuator fault estimation problem is tackled, based on the LPV approach. Abstract: In this paper, an actuator fault estimation problem is tackled, based on the LPV approach. Abstract: In this paper, an actuator fault estimation problem is tackled, based on the LPV approach. First, the actuator fault is modeled in the form of a multiplicative fault by using a constant coefficient Abstract: In this paper, anmodeled actuatorinfault estimation problem is tackled, based on the LPV coefficient approach. First, the actuator fault is the form of a multiplicative fault by using a constant First, the actuator fault is modeled in the form of a multiplicative fault by using a constant coefficient representing the loss of efficiency α. Therefore, of aa time varying First, the actuator fault is modeledof inthe theactuator’s form of apower multiplicative faultthe by estimation using a constant coefficient representing the loss of efficiency of the actuator’s power α. Therefore, the estimation of time varying representing the loss of efficiency actuator’s power α. the estimation of aa time varying faulty actuator into the estimation aa constant coefficient α. Then, the faulty representing thecan lossbe of transformed efficiency of of the the actuator’s power of α. Therefore, Therefore, the estimation of time varying faulty actuator can be transformed into the estimation of constant coefficient α. Then, the faulty faulty actuator can be transformed into the estimation of a constant coefficient α. Then, the faulty system is rewritten in the form of a switched LPV system. The coefficient α and the system states are faulty actuator can in bethe transformed into the estimation of The a constant coefficient α. system Then, the faulty system is rewritten form of a switched LPV system. coefficient α and the states are system is rewritten in the form of a switched LPV system. The coefficient α and the system states are estimated using an extended switched observer. The stability and performance of the observer is ensured system is rewritten in the form of a switched LPV system. The coefficient α of andthethe system isstates are estimated using an extended switched observer. The stability and performance observer ensured estimated using an extended switched observer. The stability and performance of the observer is ensured considering aa switched time-dependent Lyapunov The observer gains are derived on estimated using an extended switched observer. The function. stability and performance of the observer isbased ensured considering switched time-dependent Lyapunov function. The observer gains are derived based on considering aa switched time-dependent Lyapunov function. The gains are based on LMI solutions for polytopic switched systems. Some simulation results are presented that show considering switched time-dependent Lyapunov function. The observer observer gains are derived derived based the on LMI solutions for polytopic switched systems. Some simulation results are presented that show the LMI solutions for polytopic switched systems. Some simulation results are presented that show the effectiveness of this approach. LMI solutionsof for polytopic switched systems. Some simulation results are presented that show the effectiveness this approach. effectiveness of this effectiveness this approach. approach. © 2015, IFACof(International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Multiplicative Multiplicative fault, fault, fault fault estimation, estimation, switched switched extended extended observer, observer, LPV LPV system system Keywords: Multiplicative fault, fault estimation, switched extended observer, LPV Keywords: Multiplicative fault, fault estimation, switched extended observer, LPV system system 1. INTRODUCTION time-varying fault is considered power 1. INTRODUCTION time-varying fault is considered in in the the form form of of actuator actuator power 1. time-varying fault is in of power loss. Then, constant ”fault” is to the 1. INTRODUCTION INTRODUCTION time-varying is considered considered in the the form form of actuator actuator power loss. Then, aa fault constant ”fault” coefficient coefficient is used used to model model the loss. Then, a constant ”fault” coefficient is used to model the power loss. In this way, a constant coefficient should be esloss. Then, aInconstant ”fault” coefficient is usedshould to model the power loss. this way, a constant coefficient be esFault diagnosis plays an important role in the control and superFault diagnosis plays an important role in the control and superpower loss. In this way, a constant coefficient should be estimated rather than the time-varying fault signal. The paper Fault diagnosis plays an important role in the control and superpower loss. In this way, a constant coefficient should be estimated rather than the time-varying fault signal. The paper vision of the industrial systems in order to enhance the safety, Fault diagnosis plays ansystems important in to theenhance control and super- contribution vision of the industrial in role order the safety, timated rather than the time-varying fault signal. The paper are twofold: timated rather than the time-varying fault signal. The paper vision of the industrial systems in order to enhance the safety, contribution are twofold: reliability and to reduce the loss of productivity. Nowadays, vision of the industrial systems in order to enhance the safety, reliability and to reduce the loss of productivity. Nowadays, contribution are twofold: •• First, system contribution arefaulty twofold: reliability and to the loss of Nowadays, it has attracted more attention the First, the the faulty system is is modeled modeled in in the the form form of of an an LPV LPV reliability to reduce reduce theand lossmore of productivity. productivity. Nowadays, it has been beenand attracted more and more attention from from the rere• First, the faulty system is modeled in the form an system by considering the control input as aaof schedulit has been attracted more and more attention from the re• First, the faulty system is modeled in the form of an LPV LPV searchers. In fact, fault estimation is a key step in Fault Detecsystem by considering the control input as schedulit has beenIn attracted more and more attention from the researchers. fact, fault estimation is a key step in Fault Detecsystem by considering the control input as aa scheduling parameter. Then, the LPV system is rewritten to searchers. In fact, fault estimation is a key step in Fault Detecsystem by considering the control input as schedultion and Isolation (FDI). More precisely, a FDI strategy can be ing parameter. Then, the LPV system is rewritten to aa searchers. In fact, (FDI). fault estimation is a keya step Fault Detection and Isolation More precisely, FDI in strategy can be ing parameter. Then, the LPV system is rewritten switched LPV system. tion and Isolation (FDI). More precisely, a FDI strategy can be ing parameter. Then, the LPV system is rewritten to to aa model-based or and is isolate well switched LPV system. tion and Isolation (FDI). More FDI strategy can be model-based or data-based data-based and precisely, is used used to to adetect, detect, isolate as as well switched LPV system. • Second, an LPV extended switched observer is designed model-based or and is to detect, as switchedan LPV system. as estimate faults. last decades, FDI • Second, LPV extended switched observer is designed model-based or data-based data-based and the is used used detect, isolate isolate as well well as estimate the the faults. During During the last to decades, FDI modules, modules, •• to Second, an LPV extended switched observer is designed estimate both the ”faulty” coefficient α system as estimate the faults. During the last decades, FDI modules, Second, an LPV extended switched observer based on the analytical redundancy for fault estimation (e.g in to estimate both the ”faulty” coefficient α and andisthe thedesigned system as estimate the faults. During the last decades, FDI modules, based on the analytical redundancy for fault estimation (e.g in to estimate both the ”faulty” coefficient α and the system states. Two case studies are considered: the system withbased on the analytical redundancy for fault estimation (e.g in to estimate both the ”faulty” coefficient α and the system Zhang and Jiang (2008)), have received a lot of attention by states. Two case studies are considered: the system withbased on theJiang analytical redundancy for fault estimation (e.g by in Zhang and (2008)), have received a lot of attention states. Two case studies are considered: the system without and with input disturbance. Some conditions on the Zhang and Jiang (2008)), have received a lot of attention by states. Two case studies are considered: the system withmany researchers. In the literature, there exist many different out and with input disturbance. Some conditions on the Zhang and Jiang (2008)), have received a lot of attention by many researchers. In the literature, there exist many different out and with input disturbance. Some conditions on decay rate of observer are added in order to increase the many researchers. In the literature, there exist many different out and with input disturbance. Some conditions on approaches to estimate a fault occurring either on the actuator decay rate of observer are added in order to increase the many researchers. In the literature, thereeither exist on many approaches to estimate a fault occurring the different actuator decay are added convergence’s speed of estimation. approaches to aa mention fault either on decay rate rate of of observer observer are added in in order order to to increase increase the the or on the sensor. Let us the classical based convergence’s speed of estimation. approaches to estimate estimate fault occurring occurring either methods, on the the actuator actuator or on the sensor. Let us mention the classical methods, based convergence’s speed of estimation. or on the sensor. Let us mention the classical methods, based convergence’s speed of estimation. on the parity space theory (see in Gertler (1997)) to generate or on sensor. Lettheory us mention classical methods, based on thethe parity space (see inthe Gertler (1997)) to generate The paper on parity space theory in Gertler (1997)) to generate the residues approximate the or bank paper is is organized organized as as follows: follows: the the next next section section recalls recalls on the the parityand space theory (see (see Gertler generate The the residues and approximate theinfault fault or the the(1997)) bank of oftoobservers observers The paper is organized as follows: the next section recalls some preliminaries on the stability of the switched the residues and approximate the fault or the bank of observers approach (see Varrier (2013)) as well as by sliding mode obThe paper is organized as follows: the next sectionsystems. recalls some preliminaries on the stability of the switched systems. the residues andVarrier approximate of mode observers approach (see (2013))the as fault well or as the by bank sliding ob- Section some preliminaries on the stability of the switched systems. 3 presents the problem statement. Section 4 gives a full approach (see Varrier (2013)) as well as by sliding mode observers (Edwards et al. (2000)). Recently, a new approach (see some preliminaries on the stability of the switched systems. Section 3 presents the problem statement. Section 4 gives approach (see Varrier well as by sliding mode (see ob- Section 3 presents the problem statement. Section 4 gives aa full servers (Edwards et al.(2013)) (2000)).asRecently, a new approach description for an actuator multiplicative fault estimation based servers (Edwards et al. (2000)). Recently, a new approach (see in Shi Patton considered the element as Section 3 presents the problem statement.fault Section 4 givesbased a full full for an actuator multiplicative estimation servers (Edwards et al. (2000)). Recently, a new approach (see description in Shi and and Patton (2014)) (2014)) considered the fault fault element as aa state state description for an actuator multiplicative fault estimation based on the switched observer. Section 5 shows some simulation in Shi and Patton (2014)) considered the fault element as a state of the augmented system and designed an extended observer to description for an actuator multiplicative fault estimation based on the switched observer. Section 5 shows some simulation in Shi and Patton (2014)) considered the fault element as a state of the augmented system and designed an extended observer to results on switched observer. Section 55 shows some the method using an example. of system and designed an extended observer to estimate at time the state the fault system. on the the of switched observer. Section shows some simulation simulation of the proposed proposed method using an academic academic example. of the the augmented augmented system extended observer to results estimate at the the same same timeand the designed state and and an the fault of of the the system. results of the proposed method using an academic example. Finally, some conclusions are drawn in the section 6. ˙ estimate at the same time the state and the fault of the system. However, it is limited to constant faults f (t) = 0. Then, Zhang results of the proposed method using an academic Finally, some conclusions are drawn in the section 6. ˙ estimate atitthe same time the statefaults and the the system. However, is limited to constant f˙(t)fault = 0.ofThen, Zhang Finally, some conclusions are drawn in the section 6. example. However, it is limited to constant faults f (t) = 0. Then, Zhang et aa method to the timeare drawn in the section 6. However, it ispresented limited to constantallowing faults f˙(t) = 0. Then, et al. al. (2008) (2008) presented method allowing to evaluate evaluate theZhang time- Finally, some conclusions 2. PRELIMINARIES et al. (2008) presented aa method allowing to evaluate the time2. PRELIMINARIES varying fault by using fast adaptive fault estimation (FAFE) et al. (2008) presented method allowing to evaluate the time2. PRELIMINARIES varying fault by using fast adaptive fault estimation (FAFE) 2. PRELIMINARIES varying using a fast adaptive fault (FAFE) methodology based on adaptive But the varying fault fault by by using fast adaptiveobserver. fault estimation estimation (FAFE) methodology based onaan an adaptive observer. But therein, therein, the This This section section is is devoted devoted to to recall recall some some results results on on the the stability stability methodology based on an adaptive observer. But therein, the authors solved the problem with a regular Linear Time InvariThis section is devoted to recall some results on the stability analysis for the continuous time, switched system using methodology on an adaptive observer. ButTime therein, the analysis authors solvedbased the problem with a regular Linear InvariThis section is devoted to recall some results on theby stability for the continuous time, switched system by using authors solved the problem with a regular Linear Time Invariant (LTI) system without considering the disturbances. Next, analysis for the continuous time, switched system by the multiple Lyapunov function. Let us consider the following authors solved the problem with a regular Linear Time Invariant (LTI) system without considering the disturbances. Next, the analysis for Lyapunov the continuous time, system by using using multiple function. Letswitched us consider the following ant (LTI) system without considering the disturbances. Next, Rodrigues et al. (2014) proposed an adaptive polytopic observer the multiple Lyapunov function. Let us consider the following linear system: ant (LTI) system without considering the disturbances. Next, switched Rodrigues et al. (2014) proposed an adaptive polytopic observer the multiple Lyapunov function. Let us consider the following switched linear system: Rodrigues et proposed adaptive polytopic observer for time-varying fault estimation in spite of the disturbances for switched Rodrigues et al. al. (2014) (2014) proposed an an adaptive polytopic observer for time-varying fault estimation in spite of the disturbances for switched linear linear system: system: time-varying fault estimation in spite of the disturbances for xx˙˙(t) x(t), x(0) = xx0 ,, (1) aafor class of descriptor Linear Parameter Varying (LPV) systems. forclass time-varying faultLinear estimation in spite of the disturbances for (t) = =A Aσ(t) of descriptor Parameter Varying (LPV) systems. x(0) = σ(t) x(t), xx˙˙(t) = A x(t), x(0) = xx∈00 ,,{A , ..., A } , A(1) (1) aa class of descriptor Linear Parameter Varying (LPV) systems. σ(t)signal Most of these works considered the fault as an additive one or (t) = A x(t), x(0) = where σ(t) is the switching and A class of descriptor Linear Parameter Varying (LPV) systems. Most of these works considered the fault as an additive one or where σ(t) is the switching σ(t)signal and Aσ(t) ∈ 0 {A1 , ..., A M } , A(1) i∈ σ(t) 1 M i∈ Most of these works considered the fault as an additive one or n×n transformed a multiplicative fault into an additive one. ∈ {A , ..., A } , A ∈ where σ(t) is the switching signal and A σ(t) 1 M iin Most of thesea multiplicative works considered as an additive , i = 1...M. Obviously, this model imposes discontinuity n×n transformed faultthe intofault an additive one. one or R ∈ {A , ..., A } , A ∈ where σ(t) is the switching signal and A , i = 1...M. Obviously, this model imposes discontinuity R σ(t) 1 M n×n , i = 1...M. Obviously, this model imposes discontinuity iin transformed in R A since this matrix jumps instantaneously from A to A for n×n transformed aa multiplicative multiplicative fault fault into into an an additive additive one. one. σ(t) i j A since this matrix jumps instantaneously from A to A for , i = 1...M. Obviously, this model imposes discontinuity in R σ(t) i j The since σ(t)j. The main main purpose purpose of of this this paper paper is is to to propose propose aa methodology methodology iiA A since this this matrix matrix jumps jumps instantaneously instantaneously from from A Aii to to A A jj for for σ(t)j. The main purpose of this paper is to propose aa methodology i j. Now, the stability of the switched system (1) is guaranteed to estimate multiplicative faults for the actuators. An actuator The main purpose of this paper is to propose methodology to estimate multiplicative faults for the actuators. An actuator iNow, the stability of the switched system (1) is guaranteed j. to to estimate estimate multiplicative multiplicative faults faults for for the the actuators. actuators. An An actuator actuator Now, Now, the the stability stability of of the the switched switched system system (1) (1) is is guaranteed guaranteed 2405-8963 2015, IFAC IFAC (International Federation of Automatic Control) Copyright © 2015 194 Hosting by Elsevier Ltd. All rights reserved. Copyright © 2015 IFAC 194 Peer review©under responsibility of International Federation of Automatic Copyright 2015 IFAC 194Control. Copyright © 2015 IFAC 194 10.1016/j.ifacol.2015.11.136
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if there exists a family of symmetric and positive Lyapunov matrices {P1 , ...P M }, each one associated to the correspondent Ai such that the Lyapunov function Vσ(t) (x(t)) is non increasing for all t ≥ 0 (Liberzon (2003)). Indeed we need to ensure that the Lyapunov function is non increasing at the switching instants (V(x(tk ) ≤ V(x(tk− )) with tk is the switching instant). This condition is conservative and it is replaced by a weaker condition that the sequence V(x(tk )) is decreasing for tk , k = 0, ..., ∞ is switching instant (Geromel and Colaneri (2006)) i.e if tk and tk+1 are successive switching times such that σ(tk ) = i and σ(tk+1 ) = j then V j (x(tk+1 )) ≤ Vi (x(tk )). To ensure the global stability under slow switching, the concept of ”dwell time τd ” is considered imposing that the interval between 2 successive switching instants satisfies th+1 −th ≥ τd , for all switching times. However, in the design step, it is not easy to handle these conditions in terms of convex optimization (and therefore as LMIs), specially for systems with uncertainties or polytopic systems. An interesting solution has been recently proposed by Allerhand et al. (2011), using a time-dependent parametrized Lyapunov function: Vσ (t) = x(t) Pσ (t)x(t), where Pσ(t) (t) is the Lyapunov matrix and is given by: t − τ s,k Pi,k + (Pi,k+1 − Pi,k ) T/K := Pˆ i,k i f t ∈ [τ s,k , τ s,k+1 ) Pσ(t) (t) = Pi,K i f t ∈ [τ s,K , τ s+1,0 ) P i f t ∈ [0, τ1 ) i0 ,K
(2)
where i = 1, ...N with N is number of subsystems, i0 = σ(0). τ1 , τ2 , ... are the switching instants. T is the dwell time satisfying τ s+1 −τ s ≥ T , and τ s,k = τ s +k(T/K) for k = 0, ...K , τ s = τ s,0 , τ s,K = τ s +T ). Pi,k are symmetric matrices of compatible dimensions, where K is an integer that may be chosen a priori. This Lyapunov function allows to solve the conditions at switching instants easier and to deal with uncertainties or LPV systems. Throughout this paper, this Lyapunov function will be used. 3. PROBLEM FORMULATION Consider the following single input continuous time linear invariant system: x˙(t) = Ax(t) + B2 u(t) + B1 w(t)
(3)
y(t) = Cx(t) where x(t) ∈ Rn , u(t) ∈ R, w(t) ∈ R p and y(t) ∈ Rm are the state, the control input, the input disturbance and the measured output vectors, respectively. Matrices A, B2 , B1 ,C are real matrices of appropriate dimensions. Assume that the pair (A,C) is observable. Now, assume that the system (3) is in the faulty actuator
195
As mentioned previously, the objective of this work is to estimate the actuator fault. To do this, the main problem consists in estimating the coefficient α, which will be tackled, based on an extended switched observer and presented in the sequel. Remark : It can be seen that if the control input u(t) = 0, then the fault information α in (5) becomes unobservable. The solution in the synthesis step becomes therefore infeasible and the coefficient α cannot be estimated . Thus, in this work, it is assumed that in the synthesis step, the control input u(t) is kept always from zero. To account for the change the sign of u(t), the system will be rewritten as a switched LPV system in the next section. It is worth noting that the corresponding additive fault f (t) = (1 − α)u(t) is a time variant signal and depends on the value of the control input u(t), which is more complex to handle and estimate. 4. ACTUATOR FAULT ESTIMATION BASED SWITCHED OBSERVER The actuator fault estimation in this work is done by estimating a effciency coefficient α of the actuator, and is given step by step in the sequel. Let us consider 2 cases studies: system without input disturbance and system with input disturbance. 4.1 System without input disturbance In this section, it is assumed that the input disturbance w(t) is null or vanishes. Then, from the system (5), one obtains a system with the fault actuator but without input disturbance as follows: x˙(t) = Ax(t) + B2 αu(t)
(6)
y(t) = Cx(t) where α has to be estimated. Since the control input u(t) is known, in this work, one can model the system (6) as an LPV system by choosing u(t) as a time-varying parameter. Moreover, it is assumed that the control input u(t) is bounded, i.e: 0 <| u(t) |≤ umax , umax > 0 (7) Let us rewrite u(t) = |u(t)|sign(u(t)), and denote ρ(t) = |u(t)| as a time varying parameter. Then, ρ(t) i f u(t)) > 0 u(t) = ρ(t)sign(u(t)) = (8) −ρ(t) i f u(t) < 0 Note that from (7), ρ(t) is also bounded: 0 < ρ(t) ≤ umax . Moreover, the bounds of ρ shoud be chosen to be able to apply the polytopic approach in the latter. Here, one chooses ρ(t) ∈ [ρ ρ] ¯ where ρ = 0.01, ρ¯ = umax . Then (6) becomes: x˙(t) = Ax(t) + B2 αρ(t)sign(u(t)) = Ax(t) + B(ρ)αsign(u(t))(9)
Fig. 1. System with actuator fault situation, e.g loss of actuator power (Fig.1). Let us model this failure by a multiplicative fault. In fact, denoting u¯ is the output of faulty actuator, then: u¯ (t) = αu(t) (4) where α stands for the faulty coefficient i.e if α = 1, the actuator is healthy , and when α = 0.8, the actuator loses 20% of its efficiency. Then, the system (6) becomes: x˙(t) = Ax(t) + B2 αu(t) + B1 w(t)
y(t) = Cx(t) where B(ρ) = B2 ρ(t). As seen before, α is constant, then α˙ = 0. Hence the LPV system (9) can be augmented and rewritten in the following form: x˙ A B(ρ)σ x (10) = α α˙ 0 0 Ae (ρ)
(5)
y(t) = Cx(t)
Ce
where σ = sign(u(t)) 195
x y = [C 0] α
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Furthermore, it can be rewritten in a switched LPV system form (Lu and Wu (2004)) as follows: x x˙ (11) = Ae,σ (ρ) α α˙ x y = [C 0] α Ce
where Ae,σ (ρ) is switched between two modes Ae,1 (ρ), Ae,2 (ρ) according to the switched signal σ(t) = sign(u(t)) as follows: A B(ρ) i f u(t) > 0 0 0 Ae,1 (ρ) Ae,σ (ρ) = (12) A −B(ρ) i f u(t) < 0 0 0 Ae,2 (ρ)
It is worth noting that the previous definition of switched system allows to handle control input signals that crosses the 0 value, which cannot be done when choosing u(t) as scheduling parameter. Now, the following LPV extended switched observer is used to estimate the extended state of the switched system (11): xˆ x˙ˆ (ρ) + Kσ (t)(y − yˆ ) (13) = A e,σ αˆ α˙ˆ xˆ yˆ = Ce αˆ From (11) and (13), the estimation error e(t) satisfies: e e˙x = (Ae,σ (ρ) − Kσ (t)Ce ) x e˙ = e˙ α eα
(14)
The first main result about stability of the estimation error is given below. Theorem 1. Consider the switched LPV system (11) and the extended switched observer (13), if there exists a collection of matrices Pi,k > 0, Yi,k , k = 0, ...K, i = 1, ...M (M = 2: number of subsytems), of appropriate dimensions and K is prescribed integer, such that for all i = 1, ..M and j = 1, ...N, (N = 2: number of the vertices of the polytope), the following LMIs hold: (Pi,k+1 − Pi,k ) + Pi,h A( j) e,σ − Yi,h Ce < 0 + A( j) e,σ Pi,h − Ce Yi,h T/K for k = 0, ...K − 1, h = k, k + 1,
+ Pi,K A( j) e,σ + Pi,K A( j) e,σ − Yi,K Ce < 0 A( j) e,σ Pi,K − Ce Yi,K
then
Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M.
−1 Pˆ Yˆ i,k i,k −1 −1 P Kσ(t) (t) = Pσ(t) (t) Yσ(t) (t) = i,K Yi,K P−1 Yi ,K i0 ,K 0
i f t ∈ [τ s,k , τ s,k+1 ) i f t ∈ [τ s,K , τ s+1,0 ) i f t ∈ [0, τ1 )
(15)
(16) (17)
if if if
t ∈ [τ s,k , τ s,k+1 )
t ∈ [τ s,K , τ s+1,0 ) t ∈ [0, τ1 )
(Ae,σ (ρ) − Kσ (t)Ce ) Pσ (t) + Pσ (t)(Ae,σ (ρ) − Kσ (t)Ce ) + P˙ σ (t) < 0
(20)
⇔ Ae,σ (ρ)
(21)
Pσ (t) − Ce Yσ (t) + Pσ (t)Ae,σ (ρ) − Yσ (t)Ce + P˙ σ (t) < 0
From the formula of Pσ (t) in (2), we ensure (21)⇔ Ae,σ (ρ) Pi,h − Ce Yi,h + Pi,h Ae,σ (ρ) − Yi,h Ce +
(Pi,k+1 − Pi,k ) <0 T/K
holds for h = k, k + 1, i = 1, 2, k = 0, ...K − 1. and Ae,σ (ρ) Pi,K − Ce Yi,K + Pi,K Ae,σ (ρ) + Pi,K Ae,σ (ρ) − Yi,K Ce < 0
(19)
Proof: Define the Lyapunov function candidate: Vσ (t) = e(t) Pσ (t)e(t) where Pσ (t) = Pσ (t) > 0. Denoting Yσ(t) (t) = Pσ(t) (t)Kσ(t) (t). 196
(23)
The equation (22) guarantees that the Lyapunov function Vσ (t) decreases during the time interval t ∈ [τ s,0 , τ s,K ). The LMIs (23) ensures that Vσ (t) decreases after the dwell time and before the next switching time, i.e t ∈ [τ s,K , τ s+1,0 ). From the definition of Pσ(t) (t), consider that at the switching instant τk , the system switches from the mode i to the mode l, and V(τ−k ) = x(τk ) Pi,K x(τk ), V(τk ) = x(τk ) Pl,0 x(τk ). To guarantee the non-increasing of the Lyapunov function at the switching instants, we must ensure that V(τk ) ≤ V(τ−k ) which corresponds to the inequality (17), i.e: Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M. In our case, we have 2 subsystems, M = 2, i.e P1,1 − P2,0 ≥ 0 and P2,1 − P1,0 ≥ 0. Now, in order to resolve the LMIs in (22), (23), that are parameter dependent, the polytopic approach for LPV systems is considred where the 2-vertices polytope is given by Ωρ = ρ ρ¯ . The LMIs (22), (23) is therefore solved at the vertices of the polytope (i.e ρ, ρ). ¯ Then, one obtains the LMIs in (15), (16). Finally, Kσ (t) = P−1 σ Yσ (t) is the gain of the switched extended observer (13). The following theorem extends the previous one imposing a pole placement constraint for the observer design, imposing a prefixed decay rate for the exponential stability of the estimation error. Theorem 2. Consider the switched LPV system (11) and the switched extended observer (13), if there exists a collection of matrices Pi,k > 0, Yi,k , k = 0, ...K, i = 1, ...M (M = 2: number of subsytems), of appropriate dimensions, K is prescribed integer, and a positive scalar β such that for all i = 1, ..M and j = 1, ...N, (N = 2: number of the vertices of the polytope), the following LMIs hold: (Pi,k+1 − Pi,k ) + A( j) e,σ Pi,h − Ce Yi,h + Pi,h A( j) e,σ − Yi,h Ce + 2βPi,h < 0 T/K
(18)
(22)
hold for i = 1, 2.
for k = 0, ...K − 1, h = k, k + 1,
is the gain of the extended observer (13) and the error estimation asymptotically converges to zero for a given dwell time T . where t − τ s,k := Yˆ i,k Yi,k + (Yi,k+1 − Yi,k ) T/K Yσ(t) (t) = Yi,K Y i0 ,K
Then the estimation error in (14) is asymptotically stable if: V˙ σ < 0 ⇔ e˙ Pσ e + e Pσ e˙ + e P˙ σ e < 0 ⇔
A( j) e,σ Pi,K − Ce Yi,K + Pi,K A( j) e,σ + Pi,K A( j) e,σ − Yi,K Ce + 2βPi,K < 0
Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M.
(24)
(25) (26)
then Kσ(t) (t) = Pσ(t) (t)−1 Yσ(t) (t) is the gain of the extended observer (13) and this observer converges with a decay rate of β, i.e the estimation error e(t) asymptotically exponentially converges to zero for a dwell time of T . Proof: The idea of the proof is similar to the ”Theorem 1” and is omitted here. We need only to note that in order to increase the convergence’s speed of the estimation, one introduces another term of ’decay rate’, i.e, if there exists a positive scalar β such that V˙ σ (t) ≤ −2βVσ (t), then the estimation error converges to zero with a decay rate of β.
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4.2 System with input disturbance In this section, the fault estimation in the presence of input disturbance is considered, i.e, the system (5) is taken into account: x˙(t) = Ax(t) + B2 αu(t) + B1 w(t)
(27)
y(t) = Cx(t) Similarly, one has u(t) = |u(t)|sign(u(t)), and denote ρ(t) = |u(t)| as a time varying parameter and ρ(t) is also bounded: ρ ≤ ρ(t) ≤ ρ, ¯ where ρ = 0.01, ρ¯ = umax . Then, (27) becomes: x˙(t) = Ax(t) + B(ρ)αsign(u(t)) + B1 w(t)
(28)
y(t) = Cx(t) The system (28) is augemented as follows: x B x˙ + 1 w = Ae,σ (ρ) α 0 α˙ x y = [C 0] α
(29)
Ce
where Ae,σ (ρ) is switched between two modes Ae,1 (ρ), Ae,2 (ρ) according to the signal σ(t) = sign(u(t)), as in (12). Then, the following switched extended observer is proposed to estimate the system’s state and the coefficient α : x˙ˆ xˆ = Ae,σ (ρ) (30) + Kσ (t)(y − yˆ ) αˆ αˆ˙ xˆ yˆ = Ce αˆ
From (29) and (30), the estimation error e(t) satisfies: e e˙ (31) e˙ = x = (Ae,σ (ρ) − Kσ (t)Ce ) x + B1e w e˙ α eα B where B1e = 1 and Kσ (t) is the observer gain which has to 0 be determined. The extended observer (30) and the estimation error (31) are affected by the disturbance effect w(t). The observer design problem in this case involves the calculation of the observer gain Kσ (t) such that: • when w(t) ≡ 0, the observer is asymptotically stable • when w(t) 0, the following L2 -induced gain performance criterion is satisfied : min γ s.t e 2 < γ w 2
(32)
where . 2 stands for L2 norm. The following theorem solves the above problem. Theorem 3. Consider the switched system (29) and the switched extended observer (30), if there exists a collection of matrices Pi,k > 0, Yi,k , k = 0, ...K, i = 1, ...M (M = 2: number of subsytems), of appropriate dimensions and K is prescribed integer, such that for all i = 1, ..M and j = 1, ...N, (N = 2: number of the vertices of the polytope), the following LMIs hold: (P i,k+1 − Pi,k ) + A( j) P − C Y + P A( j) − Y C P B e,σ i,h i,h e i,h 1e I e,σ i,h e i,h T/K <0 ∗ −γ2 I 0 ∗ ∗ −I (33)
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for k = 0, ...K − 1, h = k, k + 1, ( j) A e,σ Pi,K − Ce Yi,K + Pi,K A( j) e,σ + Pi,K A( j) e,σ − Yi,K Ce Pi,K B1e I ∗ −γ2 I 0 < 0 ∗ ∗ −I (34) Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M. (35)
then
−1 Pˆ i,k Yˆ i,k −1 P Kσ(t) (t) = Pσ(t) (t) Yσ(t) (t) = i,K Yi,K P−1 Yi ,K i0 ,K 0 −1
i f t ∈ [τ s,k , τ s,k+1 ) i f t ∈ [τ s,K , τ s+1,0 ) i f t ∈ [0, τ1 )
(36)
is the gain of the extended observer (30) and the error estimation asymptotically converges to zero for a dwell time of T . where t − τ s,k := Yˆ i,k Yi,k + (Yi,k+1 − Yi,k ) T/K Yσ(t) (t) = Yi,K Y i0 ,K
if if if
t ∈ [τ s,k , τ s,k+1 )
t ∈ [τ s,K , τ s+1,0 ) t ∈ [0, τ1 )
(37)
Proof: From the Bounded Real Lemma, the condition (32) is satisfied if the following condition holds: ˙ + e e − γ2 w w < 0 V(t) (38) ⇔
(Ae,σ (ρ) − Kσ (t)Ce ) Pσ (t) + Pσ (t)(Ae,σ (ρ) − Kσ (t)Ce ) + P˙ σ (t) Pσ (t)B1e I 2 ∗ −γ I 0 < 0 (39) ∗ ∗ −I
From the formula of Pσ (t) in (2), (39) is satisfied if
A (ρ) P − C Y + P A (ρ) − Y C + (Pi,k+1 − Pi,k ) P B i,h i,h e,σ i,h e i,h 1e I e i,h e,σ T/K <0 ∗ −γ2 I 0 ∗ ∗ −I (40) holds for h = k, k + 1, i = 1, 2, k = 0, ...K − 1. and + Pi,K Ae,σ (ρ) + Pi,K Ae,σ (ρ) − Yi,K Ce Pi,K B1e I Ae,σ (ρ) Pi,K − Ce Yi,K 2 ∗ −γ I 0 < 0 ∗ ∗ −I (41) hold for i = 1, 2.
The equation (40) guarantees that the Lyapunov function Vσ (t) decreases and (38) holds during the time intervals i.e t ∈ [τ s,0 , τ s,K ). The LMIs (41) ensure that Vσ (t) decreases and that (38) holds after the dwell time and before the next switching instant, i.e t ∈ [τ s,K , τ s+1,0 ). From the definition of Pσ (t), consider that at instant τk , the system switches from the mode i to the mode l, to guarantee the non-increasing of the Lyapunov function at the switching instants, we must ensure: Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M. (42) In our case, we have 2 subsystems, M = 2, i.e P1,1 − P2,0 ≥ 0 and P2,1 − P1,0 ≥ 0. Now, in order to resolve the LMIs in (40), (41), we apply the polytopic solution for LPV system where the polytope is given by Ωρ = ρ ρ¯ and obtain the LMIs in (33), (34). The next result extends the previosu one imposing a prefixed decay rate on the convergence of the estimation error. Theorem 4. Consider the switched system (29) and the switched observer (30), if there exists a collection of matrices Pi,k > 0, Yi,k , k = 0, ...K, i = 1, ...M (M = 2: number of subsytems), of appropriate dimensions, K is prescribed integer, and a positive scalar β such that for all i = 1, ..M and j = 1, ...N, (N = 2: number of the vertices of the polytope), the following LMIs hold:
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(P i,k+1 − Pi,k ) + A( j) P − C Y + P A( j) − Y C + 2βP P B e,σ i,h i,h e i,h i,h 1e I e,σ i,h e i,h T/K <0 2 ∗ −γ I 0 ∗ ∗ −I (43) for k = 0, ...K − 1, h = k, k + 1, ( j) A e,σ Pi,K − Ce Yi,K + Pi,K A( j) e,σ + Pi,K A( j) e,σ − Yi,K Ce + 2βPi,K Pi,K B1e I ∗ −γ2 I 0 < 0 ∗ ∗ −I (44) Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M. (45)
then Kσ(t) (t) = Pσ(t) (t)−1 Yσ(t) (t) is the gain of the extended observer (30) and the error estimation asymptotically converges to zero for a dwell time of T . Proof: The proof is similar to the last cases and is omitted here for the simplification. Finally, in order to design the switched observer (30), one has to solve the following optimization problem: min γ2
Pi,k ,Yi,k
subject to (43), (44), (45) and Pi,k > 0
(46)
By solving this optimization problem, one can derive Pσ (t), Yσ (t) and the observer gain is calculated by Kσ (t) = Pσ (t)−1 Yσ (t). 5. NUMERICAL EXEMPLE The following numerical example illustrates the effectiveness of the proposed observer. Let consider a LTI system of the form (3) with the following matrices: −3 −1 1 A= , B2 = 1 0 0 0.1 1 −1 C= , B1 = 1 0 1
The system is subject to the control input u(t) = 1 + 2 sin(t) (see Fig. 2) and the initial value x(0) = 0 0 . Thus, one has : −1 ≤ u(t) ≤ 3 and 0 < |u(t)| ≤ 3.
Then, the time varying parameter ρ(t) = |u(t)| is assumed to be bounded by: 0.01 ≤ ρ ≤ 3 and Ae,σ (ρ) is switched between two modes Ae,1 (ρ), Ae,2 (ρ) according to the switched signal σ(t) = sign(u(t)), and is given by (12), and at each mode, one has 2 matrices corresponding to 2 vertices of the polytope Ωρ = ρ ρ¯ as follows: −3 −2 0.01 −3 −2 3 1 0 0 , 1 0 0 Ae,1 (ρ) = 0 0 0 0 0 0 −3 −2 −0.01 −3 −2 −3 , 1 0 0 1 0 0 (ρ) = A e,2 0 0 0 0 0 0
The figure Fig. 2 shows also the scheduling parmeter and the switched signal. Finamly, as seen in figure 3 the efficiency coefficient of the actuator is such that, for t ∈ [0 − 5]s, α = 1, the actuator is healthy, then, for t > 5s, α = 0.5, the actuator loses 50 % of its efficiency. Now, the different results are given for the two case studies (without and with input disturbance) with the observer design of section 3.
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Fig. 6. Estimation of α without decay rate β = 0 (left) and with β = 1 (right)- (case 2) estimation. The estimation is based on an extended switched observer and the simulation results show the effectiveness of the proposed approach. In the future work, this approach can be applied on a real system such as the suspension system. Therein, we want to estimate the fault on the damper malfunction (e.g. oil leakage of the damper,...) where some preliminary results are presented in Nguyen et al. (2015). Moreover, some fault tolerant control strategies can be developped, based on the proposed fault diagnosis. REFERENCES Allerhand, L., Shaked, U., et al. (2011). Robust stability and stabilization of linear switched systems with dwell time. Automatic Control, IEEE Transactions on, 56(2), 381–386. Edwards, C., Spurgeon, S.K., and Patton, R.J. (2000). Sliding mode observers for fault detection and isolation. Automatica, 36(4), 541–553. 199
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Fig. 7. State estimation without decay rate β = 0 (left) and with β = 1 (right)- (case 2) Geromel, J.C. and Colaneri, P. (2006). Stability and stabilization of continuous-time switched linear systems. SIAM Journal on Control and Optimization, 45(5), 1915–1930. Gertler, J. (1997). Fault detection and isolation using parity relations. Control engineering practice, 5(5), 653–661. Liberzon, D. (2003). Switching in systems and control. Springer Science & Business Media. Lu, B. and Wu, F. (2004). Switching lpv control designs using multiple parameter-dependent lyapunov functions. Automatica, 40(11), 1973–1980. Nguyen, M.Q., Sename, O., and Dugard, L. (2015). An LPV fault tolerant control for semi-active suspension -scheduled by fault estimation. In 9th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes SAFEPROCESS 15. Rodrigues, M., Hamdi, H., Theilliol, D., Mechmeche, C., and BenHadj Braiek, N. (2014). Actuator fault estimation based adaptive polytopic observer for a class of lpv descriptor systems. International Journal of Robust and Nonlinear Control. Shi, F. and Patton, R. (2014). Active fault tolerant control of lpv descriptor systems based on extended state observers. In American Control Conference (ACC), 2014, 4458–4463. IEEE. Varrier, S. (2013). D´etection de situations critiques et commande robuste tol´erante aux d´efauts pour l’automobile. Ph.D. thesis, Institut National Polytechnique de GrenobleINPG. Zhang, K., Jiang, B., Cocquempot, V., et al. (2008). Adaptive observer-based fast fault estimation. International Journal of Control Automation and Systems, 6(3), 320. Zhang, Y. and Jiang, J. (2008). Bibliographical review on reconfigurable fault-tolerant control systems. Annual Reviews in Control, 32(2), 229 – 252.
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A switched LPV observer for actuator fault A switched LPV observer for actuator fault A switched LPV observer for actuator A switched LPVestimation observer for actuator fault fault estimation estimation estimation M.Q. Nguyen, O. Sename, L. Dugard
M.Q. Nguyen, O. Sename, L. Dugard M.Q. M.Q. Nguyen, Nguyen, O. O. Sename, Sename, L. L. Dugard Dugard Univ. Grenoble Grenoble Alpes, Alpes, GIPSA-lab, GIPSA-lab, Control Control Systems Systems Dpt., Dpt., F-38000 F-38000 Grenoble, Grenoble, Univ. Univ. Grenoble Alpes, GIPSA-lab, Control Systems Dpt., F-38000 France Univ. Grenoble Alpes, GIPSA-lab, France Control Systems Dpt., F-38000 Grenoble, Grenoble, France CNRS, GIPSA-lab, Control Systems Dpt., F-38000 F-38000 Grenoble, Grenoble, France. France. France CNRS, GIPSA-lab, Control Systems Dpt., CNRS, Systems F-38000 E-mail:Control {manh-quan.nguyen, olivier.sename, CNRS, GIPSA-lab, GIPSA-lab, Control Systems Dpt., Dpt., olivier.sename, F-38000 Grenoble, Grenoble, France. France. E-mail: {manh-quan.nguyen, E-mail: {manh-quan.nguyen, olivier.sename, luc.dugard}@gipsa-lab.grenoble-inp.fr E-mail: {manh-quan.nguyen, olivier.sename, luc.dugard}@gipsa-lab.grenoble-inp.fr luc.dugard}@gipsa-lab.grenoble-inp.fr luc.dugard}@gipsa-lab.grenoble-inp.fr Abstract: In this paper, an actuator fault estimation problem is tackled, based on the LPV approach. Abstract: In this paper, an actuator fault estimation problem is tackled, based on the LPV approach. Abstract: In this paper, an actuator fault estimation problem is tackled, based on the LPV approach. First, the actuator fault is modeled in the form of a multiplicative fault by using a constant coefficient Abstract: In this paper, anmodeled actuatorinfault estimation problem is tackled, based on the LPV coefficient approach. First, the actuator fault is the form of a multiplicative fault by using a constant First, the actuator fault is modeled in the form of a multiplicative fault by using a constant coefficient representing the loss of efficiency α. Therefore, of aa time varying First, the actuator fault is modeledof inthe theactuator’s form of apower multiplicative faultthe by estimation using a constant coefficient representing the loss of efficiency of the actuator’s power α. Therefore, the estimation of time varying representing the loss of efficiency actuator’s power α. the estimation of aa time varying faulty actuator into the estimation aa constant coefficient α. Then, the faulty representing thecan lossbe of transformed efficiency of of the the actuator’s power of α. Therefore, Therefore, the estimation of time varying faulty actuator can be transformed into the estimation of constant coefficient α. Then, the faulty faulty actuator can be transformed into the estimation of a constant coefficient α. Then, the faulty system is rewritten in the form of a switched LPV system. The coefficient α and the system states are faulty actuator can in bethe transformed into the estimation of The a constant coefficient α. system Then, the faulty system is rewritten form of a switched LPV system. coefficient α and the states are system is rewritten in the form of a switched LPV system. The coefficient α and the system states are estimated using an extended switched observer. The stability and performance of the observer is ensured system is rewritten in the form of a switched LPV system. The coefficient α of andthethe system isstates are estimated using an extended switched observer. The stability and performance observer ensured estimated using an extended switched observer. The stability and performance of the observer is ensured considering aa switched time-dependent Lyapunov The observer gains are derived on estimated using an extended switched observer. The function. stability and performance of the observer isbased ensured considering switched time-dependent Lyapunov function. The observer gains are derived based on considering aa switched time-dependent Lyapunov function. The gains are based on LMI solutions for polytopic switched systems. Some simulation results are presented that show considering switched time-dependent Lyapunov function. The observer observer gains are derived derived based the on LMI solutions for polytopic switched systems. Some simulation results are presented that show the LMI solutions for polytopic switched systems. Some simulation results are presented that show the effectiveness of this approach. LMI solutionsof for polytopic switched systems. Some simulation results are presented that show the effectiveness this approach. effectiveness of this effectiveness this approach. approach. © 2015, IFACof(International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Multiplicative Multiplicative fault, fault, fault fault estimation, estimation, switched switched extended extended observer, observer, LPV LPV system system Keywords: Multiplicative fault, fault estimation, switched extended observer, LPV Keywords: Multiplicative fault, fault estimation, switched extended observer, LPV system system 1. INTRODUCTION time-varying fault is considered power 1. INTRODUCTION time-varying fault is considered in in the the form form of of actuator actuator power 1. time-varying fault is in of power loss. Then, constant ”fault” is to the 1. INTRODUCTION INTRODUCTION time-varying is considered considered in the the form form of actuator actuator power loss. Then, aa fault constant ”fault” coefficient coefficient is used used to model model the loss. Then, a constant ”fault” coefficient is used to model the power loss. In this way, a constant coefficient should be esloss. Then, aInconstant ”fault” coefficient is usedshould to model the power loss. this way, a constant coefficient be esFault diagnosis plays an important role in the control and superFault diagnosis plays an important role in the control and superpower loss. In this way, a constant coefficient should be estimated rather than the time-varying fault signal. The paper Fault diagnosis plays an important role in the control and superpower loss. In this way, a constant coefficient should be estimated rather than the time-varying fault signal. The paper vision of the industrial systems in order to enhance the safety, Fault diagnosis plays ansystems important in to theenhance control and super- contribution vision of the industrial in role order the safety, timated rather than the time-varying fault signal. The paper are twofold: timated rather than the time-varying fault signal. The paper vision of the industrial systems in order to enhance the safety, contribution are twofold: reliability and to reduce the loss of productivity. Nowadays, vision of the industrial systems in order to enhance the safety, reliability and to reduce the loss of productivity. Nowadays, contribution are twofold: •• First, system contribution arefaulty twofold: reliability and to the loss of Nowadays, it has attracted more attention the First, the the faulty system is is modeled modeled in in the the form form of of an an LPV LPV reliability to reduce reduce theand lossmore of productivity. productivity. Nowadays, it has been beenand attracted more and more attention from from the rere• First, the faulty system is modeled in the form an system by considering the control input as aaof schedulit has been attracted more and more attention from the re• First, the faulty system is modeled in the form of an LPV LPV searchers. In fact, fault estimation is a key step in Fault Detecsystem by considering the control input as schedulit has beenIn attracted more and more attention from the researchers. fact, fault estimation is a key step in Fault Detecsystem by considering the control input as aa scheduling parameter. Then, the LPV system is rewritten to searchers. In fact, fault estimation is a key step in Fault Detecsystem by considering the control input as schedultion and Isolation (FDI). More precisely, a FDI strategy can be ing parameter. Then, the LPV system is rewritten to aa searchers. In fact, (FDI). fault estimation is a keya step Fault Detection and Isolation More precisely, FDI in strategy can be ing parameter. Then, the LPV system is rewritten switched LPV system. tion and Isolation (FDI). More precisely, a FDI strategy can be ing parameter. Then, the LPV system is rewritten to to aa model-based or and is isolate well switched LPV system. tion and Isolation (FDI). More FDI strategy can be model-based or data-based data-based and precisely, is used used to to adetect, detect, isolate as as well switched LPV system. • Second, an LPV extended switched observer is designed model-based or and is to detect, as switchedan LPV system. as estimate faults. last decades, FDI • Second, LPV extended switched observer is designed model-based or data-based data-based and the is used used detect, isolate isolate as well well as estimate the the faults. During During the last to decades, FDI modules, modules, •• to Second, an LPV extended switched observer is designed estimate both the ”faulty” coefficient α system as estimate the faults. During the last decades, FDI modules, Second, an LPV extended switched observer based on the analytical redundancy for fault estimation (e.g in to estimate both the ”faulty” coefficient α and andisthe thedesigned system as estimate the faults. During the last decades, FDI modules, based on the analytical redundancy for fault estimation (e.g in to estimate both the ”faulty” coefficient α and the system states. Two case studies are considered: the system withbased on the analytical redundancy for fault estimation (e.g in to estimate both the ”faulty” coefficient α and the system Zhang and Jiang (2008)), have received a lot of attention by states. Two case studies are considered: the system withbased on theJiang analytical redundancy for fault estimation (e.g by in Zhang and (2008)), have received a lot of attention states. Two case studies are considered: the system without and with input disturbance. Some conditions on the Zhang and Jiang (2008)), have received a lot of attention by states. Two case studies are considered: the system withmany researchers. In the literature, there exist many different out and with input disturbance. Some conditions on the Zhang and Jiang (2008)), have received a lot of attention by many researchers. In the literature, there exist many different out and with input disturbance. Some conditions on decay rate of observer are added in order to increase the many researchers. In the literature, there exist many different out and with input disturbance. Some conditions on approaches to estimate a fault occurring either on the actuator decay rate of observer are added in order to increase the many researchers. In the literature, thereeither exist on many approaches to estimate a fault occurring the different actuator decay are added convergence’s speed of estimation. approaches to aa mention fault either on decay rate rate of of observer observer are added in in order order to to increase increase the the or on the sensor. Let us the classical based convergence’s speed of estimation. approaches to estimate estimate fault occurring occurring either methods, on the the actuator actuator or on the sensor. Let us mention the classical methods, based convergence’s speed of estimation. or on the sensor. Let us mention the classical methods, based convergence’s speed of estimation. on the parity space theory (see in Gertler (1997)) to generate or on sensor. Lettheory us mention classical methods, based on thethe parity space (see inthe Gertler (1997)) to generate The paper on parity space theory in Gertler (1997)) to generate the residues approximate the or bank paper is is organized organized as as follows: follows: the the next next section section recalls recalls on the the parityand space theory (see (see Gertler generate The the residues and approximate theinfault fault or the the(1997)) bank of oftoobservers observers The paper is organized as follows: the next section recalls some preliminaries on the stability of the switched the residues and approximate the fault or the bank of observers approach (see Varrier (2013)) as well as by sliding mode obThe paper is organized as follows: the next sectionsystems. recalls some preliminaries on the stability of the switched systems. the residues andVarrier approximate of mode observers approach (see (2013))the as fault well or as the by bank sliding ob- Section some preliminaries on the stability of the switched systems. 3 presents the problem statement. Section 4 gives a full approach (see Varrier (2013)) as well as by sliding mode observers (Edwards et al. (2000)). Recently, a new approach (see some preliminaries on the stability of the switched systems. Section 3 presents the problem statement. Section 4 gives approach (see Varrier well as by sliding mode (see ob- Section 3 presents the problem statement. Section 4 gives aa full servers (Edwards et al.(2013)) (2000)).asRecently, a new approach description for an actuator multiplicative fault estimation based servers (Edwards et al. (2000)). Recently, a new approach (see in Shi Patton considered the element as Section 3 presents the problem statement.fault Section 4 givesbased a full full for an actuator multiplicative estimation servers (Edwards et al. (2000)). Recently, a new approach (see description in Shi and and Patton (2014)) (2014)) considered the fault fault element as aa state state description for an actuator multiplicative fault estimation based on the switched observer. Section 5 shows some simulation in Shi and Patton (2014)) considered the fault element as a state of the augmented system and designed an extended observer to description for an actuator multiplicative fault estimation based on the switched observer. Section 5 shows some simulation in Shi and Patton (2014)) considered the fault element as a state of the augmented system and designed an extended observer to results on switched observer. Section 55 shows some the method using an example. of system and designed an extended observer to estimate at time the state the fault system. on the the of switched observer. Section shows some simulation simulation of the proposed proposed method using an academic academic example. of the the augmented augmented system extended observer to results estimate at the the same same timeand the designed state and and an the fault of of the the system. results of the proposed method using an academic example. Finally, some conclusions are drawn in the section 6. ˙ estimate at the same time the state and the fault of the system. However, it is limited to constant faults f (t) = 0. Then, Zhang results of the proposed method using an academic Finally, some conclusions are drawn in the section 6. ˙ estimate atitthe same time the statefaults and the the system. However, is limited to constant f˙(t)fault = 0.ofThen, Zhang Finally, some conclusions are drawn in the section 6. example. However, it is limited to constant faults f (t) = 0. Then, Zhang et aa method to the timeare drawn in the section 6. However, it ispresented limited to constantallowing faults f˙(t) = 0. Then, et al. al. (2008) (2008) presented method allowing to evaluate evaluate theZhang time- Finally, some conclusions 2. PRELIMINARIES et al. (2008) presented aa method allowing to evaluate the time2. PRELIMINARIES varying fault by using fast adaptive fault estimation (FAFE) et al. (2008) presented method allowing to evaluate the time2. PRELIMINARIES varying fault by using fast adaptive fault estimation (FAFE) 2. PRELIMINARIES varying using a fast adaptive fault (FAFE) methodology based on adaptive But the varying fault fault by by using fast adaptiveobserver. fault estimation estimation (FAFE) methodology based onaan an adaptive observer. But therein, therein, the This This section section is is devoted devoted to to recall recall some some results results on on the the stability stability methodology based on an adaptive observer. But therein, the authors solved the problem with a regular Linear Time InvariThis section is devoted to recall some results on the stability analysis for the continuous time, switched system using methodology on an adaptive observer. ButTime therein, the analysis authors solvedbased the problem with a regular Linear InvariThis section is devoted to recall some results on theby stability for the continuous time, switched system by using authors solved the problem with a regular Linear Time Invariant (LTI) system without considering the disturbances. Next, analysis for the continuous time, switched system by the multiple Lyapunov function. Let us consider the following authors solved the problem with a regular Linear Time Invariant (LTI) system without considering the disturbances. Next, the analysis for Lyapunov the continuous time, system by using using multiple function. Letswitched us consider the following ant (LTI) system without considering the disturbances. Next, Rodrigues et al. (2014) proposed an adaptive polytopic observer the multiple Lyapunov function. Let us consider the following linear system: ant (LTI) system without considering the disturbances. Next, switched Rodrigues et al. (2014) proposed an adaptive polytopic observer the multiple Lyapunov function. Let us consider the following switched linear system: Rodrigues et proposed adaptive polytopic observer for time-varying fault estimation in spite of the disturbances for switched Rodrigues et al. al. (2014) (2014) proposed an an adaptive polytopic observer for time-varying fault estimation in spite of the disturbances for switched linear linear system: system: time-varying fault estimation in spite of the disturbances for xx˙˙(t) x(t), x(0) = xx0 ,, (1) aafor class of descriptor Linear Parameter Varying (LPV) systems. forclass time-varying faultLinear estimation in spite of the disturbances for (t) = =A Aσ(t) of descriptor Parameter Varying (LPV) systems. x(0) = σ(t) x(t), xx˙˙(t) = A x(t), x(0) = xx∈00 ,,{A , ..., A } , A(1) (1) aa class of descriptor Linear Parameter Varying (LPV) systems. σ(t)signal Most of these works considered the fault as an additive one or (t) = A x(t), x(0) = where σ(t) is the switching and A class of descriptor Linear Parameter Varying (LPV) systems. Most of these works considered the fault as an additive one or where σ(t) is the switching σ(t)signal and Aσ(t) ∈ 0 {A1 , ..., A M } , A(1) i∈ σ(t) 1 M i∈ Most of these works considered the fault as an additive one or n×n transformed a multiplicative fault into an additive one. ∈ {A , ..., A } , A ∈ where σ(t) is the switching signal and A σ(t) 1 M iin Most of thesea multiplicative works considered as an additive , i = 1...M. Obviously, this model imposes discontinuity n×n transformed faultthe intofault an additive one. one or R ∈ {A , ..., A } , A ∈ where σ(t) is the switching signal and A , i = 1...M. Obviously, this model imposes discontinuity R σ(t) 1 M n×n , i = 1...M. Obviously, this model imposes discontinuity iin transformed in R A since this matrix jumps instantaneously from A to A for n×n transformed aa multiplicative multiplicative fault fault into into an an additive additive one. one. σ(t) i j A since this matrix jumps instantaneously from A to A for , i = 1...M. Obviously, this model imposes discontinuity in R σ(t) i j The since σ(t)j. The main main purpose purpose of of this this paper paper is is to to propose propose aa methodology methodology iiA A since this this matrix matrix jumps jumps instantaneously instantaneously from from A Aii to to A A jj for for σ(t)j. The main purpose of this paper is to propose aa methodology i j. Now, the stability of the switched system (1) is guaranteed to estimate multiplicative faults for the actuators. An actuator The main purpose of this paper is to propose methodology to estimate multiplicative faults for the actuators. An actuator iNow, the stability of the switched system (1) is guaranteed j. to to estimate estimate multiplicative multiplicative faults faults for for the the actuators. actuators. An An actuator actuator Now, Now, the the stability stability of of the the switched switched system system (1) (1) is is guaranteed guaranteed 2405-8963 2015, IFAC IFAC (International Federation of Automatic Control) Copyright © 2015 194 Hosting by Elsevier Ltd. All rights reserved. Copyright © 2015 IFAC 194 Peer review©under responsibility of International Federation of Automatic Copyright 2015 IFAC 194Control. Copyright © 2015 IFAC 194 10.1016/j.ifacol.2015.11.136
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if there exists a family of symmetric and positive Lyapunov matrices {P1 , ...P M }, each one associated to the correspondent Ai such that the Lyapunov function Vσ(t) (x(t)) is non increasing for all t ≥ 0 (Liberzon (2003)). Indeed we need to ensure that the Lyapunov function is non increasing at the switching instants (V(x(tk ) ≤ V(x(tk− )) with tk is the switching instant). This condition is conservative and it is replaced by a weaker condition that the sequence V(x(tk )) is decreasing for tk , k = 0, ..., ∞ is switching instant (Geromel and Colaneri (2006)) i.e if tk and tk+1 are successive switching times such that σ(tk ) = i and σ(tk+1 ) = j then V j (x(tk+1 )) ≤ Vi (x(tk )). To ensure the global stability under slow switching, the concept of ”dwell time τd ” is considered imposing that the interval between 2 successive switching instants satisfies th+1 −th ≥ τd , for all switching times. However, in the design step, it is not easy to handle these conditions in terms of convex optimization (and therefore as LMIs), specially for systems with uncertainties or polytopic systems. An interesting solution has been recently proposed by Allerhand et al. (2011), using a time-dependent parametrized Lyapunov function: Vσ (t) = x(t) Pσ (t)x(t), where Pσ(t) (t) is the Lyapunov matrix and is given by: t − τ s,k Pi,k + (Pi,k+1 − Pi,k ) T/K := Pˆ i,k i f t ∈ [τ s,k , τ s,k+1 ) Pσ(t) (t) = Pi,K i f t ∈ [τ s,K , τ s+1,0 ) P i f t ∈ [0, τ1 ) i0 ,K
(2)
where i = 1, ...N with N is number of subsystems, i0 = σ(0). τ1 , τ2 , ... are the switching instants. T is the dwell time satisfying τ s+1 −τ s ≥ T , and τ s,k = τ s +k(T/K) for k = 0, ...K , τ s = τ s,0 , τ s,K = τ s +T ). Pi,k are symmetric matrices of compatible dimensions, where K is an integer that may be chosen a priori. This Lyapunov function allows to solve the conditions at switching instants easier and to deal with uncertainties or LPV systems. Throughout this paper, this Lyapunov function will be used. 3. PROBLEM FORMULATION Consider the following single input continuous time linear invariant system: x˙(t) = Ax(t) + B2 u(t) + B1 w(t)
(3)
y(t) = Cx(t) where x(t) ∈ Rn , u(t) ∈ R, w(t) ∈ R p and y(t) ∈ Rm are the state, the control input, the input disturbance and the measured output vectors, respectively. Matrices A, B2 , B1 ,C are real matrices of appropriate dimensions. Assume that the pair (A,C) is observable. Now, assume that the system (3) is in the faulty actuator
195
As mentioned previously, the objective of this work is to estimate the actuator fault. To do this, the main problem consists in estimating the coefficient α, which will be tackled, based on an extended switched observer and presented in the sequel. Remark : It can be seen that if the control input u(t) = 0, then the fault information α in (5) becomes unobservable. The solution in the synthesis step becomes therefore infeasible and the coefficient α cannot be estimated . Thus, in this work, it is assumed that in the synthesis step, the control input u(t) is kept always from zero. To account for the change the sign of u(t), the system will be rewritten as a switched LPV system in the next section. It is worth noting that the corresponding additive fault f (t) = (1 − α)u(t) is a time variant signal and depends on the value of the control input u(t), which is more complex to handle and estimate. 4. ACTUATOR FAULT ESTIMATION BASED SWITCHED OBSERVER The actuator fault estimation in this work is done by estimating a effciency coefficient α of the actuator, and is given step by step in the sequel. Let us consider 2 cases studies: system without input disturbance and system with input disturbance. 4.1 System without input disturbance In this section, it is assumed that the input disturbance w(t) is null or vanishes. Then, from the system (5), one obtains a system with the fault actuator but without input disturbance as follows: x˙(t) = Ax(t) + B2 αu(t)
(6)
y(t) = Cx(t) where α has to be estimated. Since the control input u(t) is known, in this work, one can model the system (6) as an LPV system by choosing u(t) as a time-varying parameter. Moreover, it is assumed that the control input u(t) is bounded, i.e: 0 <| u(t) |≤ umax , umax > 0 (7) Let us rewrite u(t) = |u(t)|sign(u(t)), and denote ρ(t) = |u(t)| as a time varying parameter. Then, ρ(t) i f u(t)) > 0 u(t) = ρ(t)sign(u(t)) = (8) −ρ(t) i f u(t) < 0 Note that from (7), ρ(t) is also bounded: 0 < ρ(t) ≤ umax . Moreover, the bounds of ρ shoud be chosen to be able to apply the polytopic approach in the latter. Here, one chooses ρ(t) ∈ [ρ ρ] ¯ where ρ = 0.01, ρ¯ = umax . Then (6) becomes: x˙(t) = Ax(t) + B2 αρ(t)sign(u(t)) = Ax(t) + B(ρ)αsign(u(t))(9)
Fig. 1. System with actuator fault situation, e.g loss of actuator power (Fig.1). Let us model this failure by a multiplicative fault. In fact, denoting u¯ is the output of faulty actuator, then: u¯ (t) = αu(t) (4) where α stands for the faulty coefficient i.e if α = 1, the actuator is healthy , and when α = 0.8, the actuator loses 20% of its efficiency. Then, the system (6) becomes: x˙(t) = Ax(t) + B2 αu(t) + B1 w(t)
y(t) = Cx(t) where B(ρ) = B2 ρ(t). As seen before, α is constant, then α˙ = 0. Hence the LPV system (9) can be augmented and rewritten in the following form: x˙ A B(ρ)σ x (10) = α α˙ 0 0 Ae (ρ)
(5)
y(t) = Cx(t)
Ce
where σ = sign(u(t)) 195
x y = [C 0] α
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Furthermore, it can be rewritten in a switched LPV system form (Lu and Wu (2004)) as follows: x x˙ (11) = Ae,σ (ρ) α α˙ x y = [C 0] α Ce
where Ae,σ (ρ) is switched between two modes Ae,1 (ρ), Ae,2 (ρ) according to the switched signal σ(t) = sign(u(t)) as follows: A B(ρ) i f u(t) > 0 0 0 Ae,1 (ρ) Ae,σ (ρ) = (12) A −B(ρ) i f u(t) < 0 0 0 Ae,2 (ρ)
It is worth noting that the previous definition of switched system allows to handle control input signals that crosses the 0 value, which cannot be done when choosing u(t) as scheduling parameter. Now, the following LPV extended switched observer is used to estimate the extended state of the switched system (11): xˆ x˙ˆ (ρ) + Kσ (t)(y − yˆ ) (13) = A e,σ αˆ α˙ˆ xˆ yˆ = Ce αˆ From (11) and (13), the estimation error e(t) satisfies: e e˙x = (Ae,σ (ρ) − Kσ (t)Ce ) x e˙ = e˙ α eα
(14)
The first main result about stability of the estimation error is given below. Theorem 1. Consider the switched LPV system (11) and the extended switched observer (13), if there exists a collection of matrices Pi,k > 0, Yi,k , k = 0, ...K, i = 1, ...M (M = 2: number of subsytems), of appropriate dimensions and K is prescribed integer, such that for all i = 1, ..M and j = 1, ...N, (N = 2: number of the vertices of the polytope), the following LMIs hold: (Pi,k+1 − Pi,k ) + Pi,h A( j) e,σ − Yi,h Ce < 0 + A( j) e,σ Pi,h − Ce Yi,h T/K for k = 0, ...K − 1, h = k, k + 1,
+ Pi,K A( j) e,σ + Pi,K A( j) e,σ − Yi,K Ce < 0 A( j) e,σ Pi,K − Ce Yi,K
then
Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M.
−1 Pˆ Yˆ i,k i,k −1 −1 P Kσ(t) (t) = Pσ(t) (t) Yσ(t) (t) = i,K Yi,K P−1 Yi ,K i0 ,K 0
i f t ∈ [τ s,k , τ s,k+1 ) i f t ∈ [τ s,K , τ s+1,0 ) i f t ∈ [0, τ1 )
(15)
(16) (17)
if if if
t ∈ [τ s,k , τ s,k+1 )
t ∈ [τ s,K , τ s+1,0 ) t ∈ [0, τ1 )
(Ae,σ (ρ) − Kσ (t)Ce ) Pσ (t) + Pσ (t)(Ae,σ (ρ) − Kσ (t)Ce ) + P˙ σ (t) < 0
(20)
⇔ Ae,σ (ρ)
(21)
Pσ (t) − Ce Yσ (t) + Pσ (t)Ae,σ (ρ) − Yσ (t)Ce + P˙ σ (t) < 0
From the formula of Pσ (t) in (2), we ensure (21)⇔ Ae,σ (ρ) Pi,h − Ce Yi,h + Pi,h Ae,σ (ρ) − Yi,h Ce +
(Pi,k+1 − Pi,k ) <0 T/K
holds for h = k, k + 1, i = 1, 2, k = 0, ...K − 1. and Ae,σ (ρ) Pi,K − Ce Yi,K + Pi,K Ae,σ (ρ) + Pi,K Ae,σ (ρ) − Yi,K Ce < 0
(19)
Proof: Define the Lyapunov function candidate: Vσ (t) = e(t) Pσ (t)e(t) where Pσ (t) = Pσ (t) > 0. Denoting Yσ(t) (t) = Pσ(t) (t)Kσ(t) (t). 196
(23)
The equation (22) guarantees that the Lyapunov function Vσ (t) decreases during the time interval t ∈ [τ s,0 , τ s,K ). The LMIs (23) ensures that Vσ (t) decreases after the dwell time and before the next switching time, i.e t ∈ [τ s,K , τ s+1,0 ). From the definition of Pσ(t) (t), consider that at the switching instant τk , the system switches from the mode i to the mode l, and V(τ−k ) = x(τk ) Pi,K x(τk ), V(τk ) = x(τk ) Pl,0 x(τk ). To guarantee the non-increasing of the Lyapunov function at the switching instants, we must ensure that V(τk ) ≤ V(τ−k ) which corresponds to the inequality (17), i.e: Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M. In our case, we have 2 subsystems, M = 2, i.e P1,1 − P2,0 ≥ 0 and P2,1 − P1,0 ≥ 0. Now, in order to resolve the LMIs in (22), (23), that are parameter dependent, the polytopic approach for LPV systems is considred where the 2-vertices polytope is given by Ωρ = ρ ρ¯ . The LMIs (22), (23) is therefore solved at the vertices of the polytope (i.e ρ, ρ). ¯ Then, one obtains the LMIs in (15), (16). Finally, Kσ (t) = P−1 σ Yσ (t) is the gain of the switched extended observer (13). The following theorem extends the previous one imposing a pole placement constraint for the observer design, imposing a prefixed decay rate for the exponential stability of the estimation error. Theorem 2. Consider the switched LPV system (11) and the switched extended observer (13), if there exists a collection of matrices Pi,k > 0, Yi,k , k = 0, ...K, i = 1, ...M (M = 2: number of subsytems), of appropriate dimensions, K is prescribed integer, and a positive scalar β such that for all i = 1, ..M and j = 1, ...N, (N = 2: number of the vertices of the polytope), the following LMIs hold: (Pi,k+1 − Pi,k ) + A( j) e,σ Pi,h − Ce Yi,h + Pi,h A( j) e,σ − Yi,h Ce + 2βPi,h < 0 T/K
(18)
(22)
hold for i = 1, 2.
for k = 0, ...K − 1, h = k, k + 1,
is the gain of the extended observer (13) and the error estimation asymptotically converges to zero for a given dwell time T . where t − τ s,k := Yˆ i,k Yi,k + (Yi,k+1 − Yi,k ) T/K Yσ(t) (t) = Yi,K Y i0 ,K
Then the estimation error in (14) is asymptotically stable if: V˙ σ < 0 ⇔ e˙ Pσ e + e Pσ e˙ + e P˙ σ e < 0 ⇔
A( j) e,σ Pi,K − Ce Yi,K + Pi,K A( j) e,σ + Pi,K A( j) e,σ − Yi,K Ce + 2βPi,K < 0
Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M.
(24)
(25) (26)
then Kσ(t) (t) = Pσ(t) (t)−1 Yσ(t) (t) is the gain of the extended observer (13) and this observer converges with a decay rate of β, i.e the estimation error e(t) asymptotically exponentially converges to zero for a dwell time of T . Proof: The idea of the proof is similar to the ”Theorem 1” and is omitted here. We need only to note that in order to increase the convergence’s speed of the estimation, one introduces another term of ’decay rate’, i.e, if there exists a positive scalar β such that V˙ σ (t) ≤ −2βVσ (t), then the estimation error converges to zero with a decay rate of β.
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4.2 System with input disturbance In this section, the fault estimation in the presence of input disturbance is considered, i.e, the system (5) is taken into account: x˙(t) = Ax(t) + B2 αu(t) + B1 w(t)
(27)
y(t) = Cx(t) Similarly, one has u(t) = |u(t)|sign(u(t)), and denote ρ(t) = |u(t)| as a time varying parameter and ρ(t) is also bounded: ρ ≤ ρ(t) ≤ ρ, ¯ where ρ = 0.01, ρ¯ = umax . Then, (27) becomes: x˙(t) = Ax(t) + B(ρ)αsign(u(t)) + B1 w(t)
(28)
y(t) = Cx(t) The system (28) is augemented as follows: x B x˙ + 1 w = Ae,σ (ρ) α 0 α˙ x y = [C 0] α
(29)
Ce
where Ae,σ (ρ) is switched between two modes Ae,1 (ρ), Ae,2 (ρ) according to the signal σ(t) = sign(u(t)), as in (12). Then, the following switched extended observer is proposed to estimate the system’s state and the coefficient α : x˙ˆ xˆ = Ae,σ (ρ) (30) + Kσ (t)(y − yˆ ) αˆ αˆ˙ xˆ yˆ = Ce αˆ
From (29) and (30), the estimation error e(t) satisfies: e e˙ (31) e˙ = x = (Ae,σ (ρ) − Kσ (t)Ce ) x + B1e w e˙ α eα B where B1e = 1 and Kσ (t) is the observer gain which has to 0 be determined. The extended observer (30) and the estimation error (31) are affected by the disturbance effect w(t). The observer design problem in this case involves the calculation of the observer gain Kσ (t) such that: • when w(t) ≡ 0, the observer is asymptotically stable • when w(t) 0, the following L2 -induced gain performance criterion is satisfied : min γ s.t e 2 < γ w 2
(32)
where . 2 stands for L2 norm. The following theorem solves the above problem. Theorem 3. Consider the switched system (29) and the switched extended observer (30), if there exists a collection of matrices Pi,k > 0, Yi,k , k = 0, ...K, i = 1, ...M (M = 2: number of subsytems), of appropriate dimensions and K is prescribed integer, such that for all i = 1, ..M and j = 1, ...N, (N = 2: number of the vertices of the polytope), the following LMIs hold: (P i,k+1 − Pi,k ) + A( j) P − C Y + P A( j) − Y C P B e,σ i,h i,h e i,h 1e I e,σ i,h e i,h T/K <0 ∗ −γ2 I 0 ∗ ∗ −I (33)
197
197
for k = 0, ...K − 1, h = k, k + 1, ( j) A e,σ Pi,K − Ce Yi,K + Pi,K A( j) e,σ + Pi,K A( j) e,σ − Yi,K Ce Pi,K B1e I ∗ −γ2 I 0 < 0 ∗ ∗ −I (34) Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M. (35)
then
−1 Pˆ i,k Yˆ i,k −1 P Kσ(t) (t) = Pσ(t) (t) Yσ(t) (t) = i,K Yi,K P−1 Yi ,K i0 ,K 0 −1
i f t ∈ [τ s,k , τ s,k+1 ) i f t ∈ [τ s,K , τ s+1,0 ) i f t ∈ [0, τ1 )
(36)
is the gain of the extended observer (30) and the error estimation asymptotically converges to zero for a dwell time of T . where t − τ s,k := Yˆ i,k Yi,k + (Yi,k+1 − Yi,k ) T/K Yσ(t) (t) = Yi,K Y i0 ,K
if if if
t ∈ [τ s,k , τ s,k+1 )
t ∈ [τ s,K , τ s+1,0 ) t ∈ [0, τ1 )
(37)
Proof: From the Bounded Real Lemma, the condition (32) is satisfied if the following condition holds: ˙ + e e − γ2 w w < 0 V(t) (38) ⇔
(Ae,σ (ρ) − Kσ (t)Ce ) Pσ (t) + Pσ (t)(Ae,σ (ρ) − Kσ (t)Ce ) + P˙ σ (t) Pσ (t)B1e I 2 ∗ −γ I 0 < 0 (39) ∗ ∗ −I
From the formula of Pσ (t) in (2), (39) is satisfied if
A (ρ) P − C Y + P A (ρ) − Y C + (Pi,k+1 − Pi,k ) P B i,h i,h e,σ i,h e i,h 1e I e i,h e,σ T/K <0 ∗ −γ2 I 0 ∗ ∗ −I (40) holds for h = k, k + 1, i = 1, 2, k = 0, ...K − 1. and + Pi,K Ae,σ (ρ) + Pi,K Ae,σ (ρ) − Yi,K Ce Pi,K B1e I Ae,σ (ρ) Pi,K − Ce Yi,K 2 ∗ −γ I 0 < 0 ∗ ∗ −I (41) hold for i = 1, 2.
The equation (40) guarantees that the Lyapunov function Vσ (t) decreases and (38) holds during the time intervals i.e t ∈ [τ s,0 , τ s,K ). The LMIs (41) ensure that Vσ (t) decreases and that (38) holds after the dwell time and before the next switching instant, i.e t ∈ [τ s,K , τ s+1,0 ). From the definition of Pσ (t), consider that at instant τk , the system switches from the mode i to the mode l, to guarantee the non-increasing of the Lyapunov function at the switching instants, we must ensure: Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M. (42) In our case, we have 2 subsystems, M = 2, i.e P1,1 − P2,0 ≥ 0 and P2,1 − P1,0 ≥ 0. Now, in order to resolve the LMIs in (40), (41), we apply the polytopic solution for LPV system where the polytope is given by Ωρ = ρ ρ¯ and obtain the LMIs in (33), (34). The next result extends the previosu one imposing a prefixed decay rate on the convergence of the estimation error. Theorem 4. Consider the switched system (29) and the switched observer (30), if there exists a collection of matrices Pi,k > 0, Yi,k , k = 0, ...K, i = 1, ...M (M = 2: number of subsytems), of appropriate dimensions, K is prescribed integer, and a positive scalar β such that for all i = 1, ..M and j = 1, ...N, (N = 2: number of the vertices of the polytope), the following LMIs hold:
IFAC LPVS 2015 198 Oct. 7-9, 2015. Grenoble, France
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(P i,k+1 − Pi,k ) + A( j) P − C Y + P A( j) − Y C + 2βP P B e,σ i,h i,h e i,h i,h 1e I e,σ i,h e i,h T/K <0 2 ∗ −γ I 0 ∗ ∗ −I (43) for k = 0, ...K − 1, h = k, k + 1, ( j) A e,σ Pi,K − Ce Yi,K + Pi,K A( j) e,σ + Pi,K A( j) e,σ − Yi,K Ce + 2βPi,K Pi,K B1e I ∗ −γ2 I 0 < 0 ∗ ∗ −I (44) Pi,K − Pl,0 ≥ 0 ∀l = 1, ..i − 1, i + 1, ...M. (45)
then Kσ(t) (t) = Pσ(t) (t)−1 Yσ(t) (t) is the gain of the extended observer (30) and the error estimation asymptotically converges to zero for a dwell time of T . Proof: The proof is similar to the last cases and is omitted here for the simplification. Finally, in order to design the switched observer (30), one has to solve the following optimization problem: min γ2
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By solving this optimization problem, one can derive Pσ (t), Yσ (t) and the observer gain is calculated by Kσ (t) = Pσ (t)−1 Yσ (t). 5. NUMERICAL EXEMPLE The following numerical example illustrates the effectiveness of the proposed observer. Let consider a LTI system of the form (3) with the following matrices: −3 −1 1 A= , B2 = 1 0 0 0.1 1 −1 C= , B1 = 1 0 1
The system is subject to the control input u(t) = 1 + 2 sin(t) (see Fig. 2) and the initial value x(0) = 0 0 . Thus, one has : −1 ≤ u(t) ≤ 3 and 0 < |u(t)| ≤ 3.
Then, the time varying parameter ρ(t) = |u(t)| is assumed to be bounded by: 0.01 ≤ ρ ≤ 3 and Ae,σ (ρ) is switched between two modes Ae,1 (ρ), Ae,2 (ρ) according to the switched signal σ(t) = sign(u(t)), and is given by (12), and at each mode, one has 2 matrices corresponding to 2 vertices of the polytope Ωρ = ρ ρ¯ as follows: −3 −2 0.01 −3 −2 3 1 0 0 , 1 0 0 Ae,1 (ρ) = 0 0 0 0 0 0 −3 −2 −0.01 −3 −2 −3 , 1 0 0 1 0 0 (ρ) = A e,2 0 0 0 0 0 0
The figure Fig. 2 shows also the scheduling parmeter and the switched signal. Finamly, as seen in figure 3 the efficiency coefficient of the actuator is such that, for t ∈ [0 − 5]s, α = 1, the actuator is healthy, then, for t > 5s, α = 0.5, the actuator loses 50 % of its efficiency. Now, the different results are given for the two case studies (without and with input disturbance) with the observer design of section 3.
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IFAC LPVS 2015 Oct. 7-9, 2015. Grenoble, France
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Fig. 6. Estimation of α without decay rate β = 0 (left) and with β = 1 (right)- (case 2) estimation. The estimation is based on an extended switched observer and the simulation results show the effectiveness of the proposed approach. In the future work, this approach can be applied on a real system such as the suspension system. Therein, we want to estimate the fault on the damper malfunction (e.g. oil leakage of the damper,...) where some preliminary results are presented in Nguyen et al. (2015). Moreover, some fault tolerant control strategies can be developped, based on the proposed fault diagnosis. REFERENCES Allerhand, L., Shaked, U., et al. (2011). Robust stability and stabilization of linear switched systems with dwell time. Automatic Control, IEEE Transactions on, 56(2), 381–386. Edwards, C., Spurgeon, S.K., and Patton, R.J. (2000). Sliding mode observers for fault detection and isolation. Automatica, 36(4), 541–553. 199
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Fig. 7. State estimation without decay rate β = 0 (left) and with β = 1 (right)- (case 2) Geromel, J.C. and Colaneri, P. (2006). Stability and stabilization of continuous-time switched linear systems. SIAM Journal on Control and Optimization, 45(5), 1915–1930. Gertler, J. (1997). Fault detection and isolation using parity relations. Control engineering practice, 5(5), 653–661. Liberzon, D. (2003). Switching in systems and control. Springer Science & Business Media. Lu, B. and Wu, F. (2004). Switching lpv control designs using multiple parameter-dependent lyapunov functions. Automatica, 40(11), 1973–1980. Nguyen, M.Q., Sename, O., and Dugard, L. (2015). An LPV fault tolerant control for semi-active suspension -scheduled by fault estimation. In 9th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes SAFEPROCESS 15. Rodrigues, M., Hamdi, H., Theilliol, D., Mechmeche, C., and BenHadj Braiek, N. (2014). Actuator fault estimation based adaptive polytopic observer for a class of lpv descriptor systems. International Journal of Robust and Nonlinear Control. Shi, F. and Patton, R. (2014). Active fault tolerant control of lpv descriptor systems based on extended state observers. In American Control Conference (ACC), 2014, 4458–4463. IEEE. Varrier, S. (2013). D´etection de situations critiques et commande robuste tol´erante aux d´efauts pour l’automobile. Ph.D. thesis, Institut National Polytechnique de GrenobleINPG. Zhang, K., Jiang, B., Cocquempot, V., et al. (2008). Adaptive observer-based fast fault estimation. International Journal of Control Automation and Systems, 6(3), 320. Zhang, Y. and Jiang, J. (2008). Bibliographical review on reconfigurable fault-tolerant control systems. Annual Reviews in Control, 32(2), 229 – 252.