Virtual actuator-based FTC for LPV systems with saturating actuators and FDI delays

Transcription

1 Virtual actuator-based FTC or LPV systems with saturating actuators and FDI delays Damiano Rotondo, Jean-Christophe Ponsart, Fatiha Neari, Didier Theilliol and Vicenç Puig Abstract The main contribution o this paper consists in solving the problem o ault tolerant control FTC or linear parameter varying LPV systems subect to actuator saturation and ault detection and isolation FDI delays. The FTC is based on virtual actuators that reconigure the aulty plant to maintain the stability and to avoid the saturation o the actuators. On the other hand, a design methodology that provides the nominal output-eedback controller, which maximizes the tolerated delay between the ault occurrence and its isolation, is developed. The design process consists in inding the optimal easible solution to a inite set o linear matrix inequalities LMIs. Finally, an example is used to illustrate the theoretical results. I. INTRODUCTION Control systems are aected by saturations, which means that the control action that can be delivered by the actuators is limited in magnitude 1]. It is important to take into account the presence o saturations during the design o a control system, otherwise its real perormance could exhibit strong perormance degradation and, in some cases, instability could occur as well. For this reason, the problem o stabilizing systems with saturated inputs is extensively studied, as demonstrated by a number o works that have appeared recently in the literature, see e.g. 2] 5]. Several control techniques have been extended to cope with input saturations, such as sliding mode control 6] and adaptive control 7]. Also, a ew monographies have been published about this topic, see e.g. 8] and 9]. The presence o actuator saturations becomes even more problematic when actuator aults are considered. In the last decades, ault tolerant control FTC techniques have been investigated 10] 12] with the obective o mitigating This work has been unded by the Spanish Government MINECO through the proect CICYT ECOCIS re. DPI C2-1-R, by MINECO and FEDER through the proect CICYT HARCRICS re. DPI R, by AGAUR through the contract FI-DGR 2015 re. 2015FI_B and by the DGR o Generalitat de Catalunya SAC group Re. 2014/SGR/374. D. Rotondo also acknowledges that part o this work was carried out during the tenure o an ERCIM Alain Bensoussan Fellowship Programme and that it was partly supported by the Research Council o Norway through the Centres o Excellence unding scheme, Proect number D. Rotondo, F. Neari and V. Puig are with Advanced Control Systems Group SAC, Universitat Politècnica de Catalunya UPC, Rambla de Sant Nebridi, 11, Terrassa, Spain. D. Rotondo is also with the Centre or Autonomous Marine Operations and Systems NTNU- AMOS, Department o Engineering Cybernetics, Norwegian University o Science and Technology, Trondheim, Norway. V. Puig is also with the Institut de Robòtica i Inormàtica Industrial IRI, UPC-CSIC, Carrer de Llorens i Artigas, 4-6, Barcelona, Spain. J.-C. Ponsart and D. Theilliol are with Université de Lorraine, CRAN, UMR 7039, Campus Sciences, B.P , Vandoeuvre-les-Nancy Cedex and with CNRS, CRAN, UMR Corresponding author: Damiano Rotondo damiano.rotondo@yahoo.it the eects o aults on the closed-loop perormance and stability. These techniques usually lead to a redistribution o the control action among the healthy actuators, which can potentially become saturated. In order to avoid this potential actuator saturation, active FTC systems that include a traectory replanning algorithm have been proposed 13] 15]. Recently, an FTC strategy or unstable linear time invariant LTI systems subect to actuator saturation and ault isolation delay has been developed 16]. The proposed strategy relied on virtual actuators, a ault-hiding technique that aims at reconiguring the aulty plant instead o the controller 17] 21]. In 22], this strategy has been extended to linear parameter varying LPV systems, a standard ormalism or perorming gain scheduling o nonlinear systems 23]. However, in 22], a perect ault detection and isolation FDI has been assumed. In some cases, the time delay between the appearance o the ault and the moment in which the active strategy is activated at the ault detection or isolation time may destabilize the system. The main contribution o the present work is to include a maximization o the tolerated FDI delay within the design process, which is done by means o a linear matrix inequality LMI-based design. The paper is structured as ollows. Section II recalls some preliminary results. Section III describes the system, the aults and the nominal controller. The virtual actuator-based FTC strategy is presented in Section IV. The design o the nominal controller through maximization o the tolerated FDI delay is described in Section V. Section VI illustrates the theoretical results through an example. Finally, Section VII summarizes the main conclusions. II. PRELIMINARIES Consider the autonomous nonlinear system: ẋt = gxt 1 where x R n x is the state and g denotes a nonlinear unction. For x0 = x 0 R n x, let us denote the traectory o the system 1 as ψt,x 0. Then, the domain o attraction o the origin is: { } S := x 0 R n x : lim ψt,x 0 = 0 2 t + Let P 0, ρ > 0 and denote: EP,ρ = { x 0 R n x : x T Px ρ } 3 and let V xt = xt T Pxt be a candidate Lyapunov unction. The ellipsoid EP, ρ is said to be contractively invariant i V xt < 0 or all x EP,ρ\{0}. A contractively

2 invariant ellipsoid EP, ρ is inside the domain o attraction S 24]. Finally, throughout the paper, He{M} will be used as a shorthand notation or the matrix M + M T. III. SYSTEM, FAULT AND CONTROLLER DEFINITION Let us consider the ollowing LPV system subect to actuator saturations: ẋt = Aθtxt + Btsat ut 4 yt = Cxt 5 where x R n x is the state, u R n u is the control input, y R n y is the measured output, Aθt R n x n x is the parameter varying state matrix, whose values depend on the vector θt Θ R n θ, C R n y n x is the output matrix, and Bt R n x n u is the input matrix, deined as ollows: Bt = { B B {B 1,...,B n } t < t t t 6 B R n x n u and the corresponding LPV system obtained rom 4-5 will be reerred to as nominal input matrix and nominal system, respectively, B R n x n u and the corresponding LPV system obtained rom 4-5 will be reerred to as aulty input matrix and aulty system, respectively, and t R + is the ault occurrence time. Remark 1: The ault deinition provided by 6 is a general ormulation that may speciy both aults in a single actuator and in multiple actuators. For example, B 1 could represent the total loss o actuator 1, while B 2 could represent the total loss o actuators 1 and 2. The saturation unction sat : R n u R n u speciies the limited actuator capacity o the control input ut in 4. The saturation is assumed to be a symmetrical, decoupled, sectorbounded and static actuator nonlinearity with a constant saturation limit u MAX in the -th input, such that: sat 1 u 1. satu = sat u 7. sat nu u nu u MAX i u > u MAX sat u = u i u u MAX u MAX 8 i u < u MAX or = 1,...,n u, where u MAX = u MAX 1,...,u MAX T n u R n u is a vector with positive entries. The n matrices B 1,...,B n R n x n u are such that: rank < rank B 9 and the pairs: B h Aθ,B h B h B 10 are stabilizable, θ Θ and h = 1,...,n. The structure chosen or the controller is the LPV dynamic eedback one 25]: ẋ c t = A c θtx c t + B c θtyt 11 u c t = C c θtx c t + D c θtyt 12 where x c R n x is the controller state and u c R n u is the controller output, respectively. A strict requirement or the closed-loop system obtained by considering the system 4-5 and the controller is that EP, 1 S and EP,1 Lu,u MAX, where Lu,u MAX denotes the region o the state space in which the actuators are not saturated. In other words, the controller must be such that or any initial closed-loop state vector satisying: x0 T x c 0 T P x0 x c the control input never saturates, and the closed-loop state traectory converges to the origin. IV. VIRTUAL ACTUATOR-BASED FTC The FTC strategy used in this work relies on a reconigured control law, given by: u c t t < t I u 1 t t t I,Bt = B 1 ut = 14.. u n t t t I,Bt = B n where t I R +, t I t is the ault isolation time, that is assumed to be provided by an FDI module. The values or u 1 t,...,u n t are provided by the LPV virtual actuators, which have a structure as in 22]: ẋ v h t = Aθt + B h M h θt x v h t + B B h u c t 15 t = N h u c t M h θtx h t 16 u h where h = 1,...,n, x v h are the virtual actuators states with x v h t I = 0, M h θt R n u n x are the virtual actuators gains to be designed, and the matrices N h and B h are given by: N h = B 17 B h B h = B h v N h = B h B h B 18 Moreover, in order to obtain the ault-hiding characteristic, the output equation 5 is slightly changed ater t I, as ollows: yt = C xt + x v h t t t I, Bt = B h 19 Given a controller 11-12, satisying EP, 1 S and EP,1 Lu,u MAX, the ollowing theorem provides the conditions to design the virtual actuator guaranteeing that, i at the ault isolation time t I, the closed-loop system state is inside the ellipsoid EP,ν, with ν ]0,1], the state

3 traectory will converge to zero despite the change o the input matrix rom B to B h due to the ault. Theorem 1: Let Xva 1 S n x n x and Γ h θ R n u n x, h = 1,...,n be such that: { } ν A cl θp 1 0 He ν A h θp 1 AθXva 1 + B h Γ h 20 θ Xva 1 Γ h k θt 2 umax Γ h k θ Nh µ 0 = 1,...,n u N h 21 0 n h ũ hold θ Θ, where: Aθ + BDc θc BC A cl θ = c θ B c θc A c θ A h θ = B B h D c θc 22 B B h C c θ 23 µ = max EP,ν u c 24 n h ũ is the number o non-zero elements in N h, and k in 21 takes values corresponding to the indices o the nonzero elements in N h. Then, i the virtual actuators gains M h θ in are calculated as M h θ = Γ h θx va, EP,ν is contractively invariant or the system 4-5 with the control law 14, and EP,ν Lu,u MAX, t t I. Proo: See 22]. By relying on a polytopic representation, it is possible to transorm the ininite number o conditions provided by Theorem 1 in a inite number o conditions, as stated by the ollowing corollary. Corollary 1: Let the matrices Aθt, A c θt, B c θt, C c θt, D c θt be polytopic, i.e.: Aθt A c θt B c θt C c θt D c θt = N i=1 with coeicients µ i θt such that: N i=1 µ i θt A i A c,i B c,i C c,i D c,i 25 µ i θt = 1, µ i θt 0, i = 1,...,N, θ Θ 26 and choose the virtual actuator gain as: M h θt = N i=1 µ i θtm h i 27 I there exist Xva 1 S n x n x and Γ h i R n u n x, h = 1,...,n, i = 1,...,N such that: {} ν A cl,i P 1 O 2nx n x He ν A h,i P 1 A i Xva 1 + B h Γ h O 28 i X 1 va Γ h ik Γ h T ik 2 umax Nh µ n h ũ O = 1,...,n u 0 N h hold i = 1,...,N, where: Ai + BD A cl,i = c,i C BC c,i B c,i C A c,i A h,i = B B h D c,i C B B h C c,i µ is deined as in 24, n h ũ is the number o non-zero elements in N h, and k in 29 takes values corresponding to the indices o the non-zero elements in N h, and i the vertex virtual actuators gains in 27 are calculated as M h i = Γ h i X va, then EP,ν is contractively invariant or the system 4-5 with control law 14, and EP,ν Lu,u MAX, t t I. Proo: See 22]. V. DESIGN OF THE CONTROLLER Under the assumption o instantaneous ault isolation, i.e. t I = t, i the output-eedback controller has been designed such that EP,1 S and EP,1 Lu,u MAX and i the closed-loop state traectory has reached EP,ν, the virtual actuator-based FTC described in Section IV guarantees the state traectory convergence under ault occurrence. However, this is not the case when there is a delay in the ault isolation, i.e. t I t > 0. In act, between the occurrence o the ault, that changes the system input matrix rom B to some B h, and the ault isolation time, when the appropriate control begins to be applied, there is a time interval where the system is driven by the nominal control u c t. During this period, there is no guarantee that, i the system has reached EP,ν at time t, it will stay inside this region until t I. This act can lead to severe consequences, because i the state traectory leaves EP,ν beore t I, the system could be destabilized 26]. Hence, it is interesting to improve the u h overall system robustness against the ault isolation delay. Given xt T x c t T ] T EP,ν, let us deine, or the aulty system 4-5 with Bt = B h, h = 1,...,n, under control law ut = u c t, the critical ault isolation time t h I xt,x c t t as the time instant such that: T T ] T x c EP,ν 32 but: x t h I x t h T I +t ε t h I x c t h T ] T I +t ε / EP,ν 33 or all t ε > 0. The critical ault isolation time indicates that the guarantees o non-saturating control input and state traectory convergence to the origin given by the virtual actuator-based FTC are lost i t I > t h I xt,x c t. It is

4 worth remarking that the Lyapunov-based conditions provided in the ollowing are suicient, such that the system may exhibit state traectory convergence to zero with nonsaturating control input even i t I > t h I xt,x c t. In the ollowing, the output eedback controller will be designed in such a way that it maximizes min h=1,...,n ˆt h I xt,x c t or all xt T x c t T ] T EP,ν, where ˆt h I xt,x c t is an estimation o h t I xt,x c t. As a irst step to obtain the conditions or designing the output eedback controller 11-12, let us consider the ollowing lemma. Lemma 1: Let V xt = xt T Pxt, and let D be an LMI region 27], i.e. a subset o the complex plane such that: D = {σ C : D σ } 34 where D σ is the characteristic unction deined as: D σ = α + βσ + β T σ 35 where α S m m, β R m m, and σ denotes the complex conugate o σ. I the autonomous LPV system: ẋt = Aθtxt 36 is quadratically D stable 27], i.e. there exists P 0 such that θ Θ: α P + β PAθ + β T Aθ T P 37 then, V xt satisies, or all xt 0: 1 V xt 2 V xt D R 38 Proo: Pre-multiplying 37 by I xt T, and postmultiplying it by I xt, respectively, the ollowing is obtained or all xt 0: α xt T Pxt + β xt T PAθtxt + β T xt T Aθt T Pxt 39 Recalling that: 1 2 V xt = xt T PAθtxt = xt T Aθt T Pxt 40 and dividing 39 by V xt, this process leads to: α 1 + β 1 V xt 2 V xt + βt 1 V xt 2 V xt 41 which implies 38. Lemma 1 is used in the subsequent theorem to obtain an estimation o ˆt h I xt,x c t or all xt T x c t T ] T EP,ν. Theorem 2: Let λ h R +, h = 1,...,n, be such that θ Θ: { } 2λ h P+He P Aθ + B h D c θc B h C c θ B c θc A c θ 42 and let xt T x c t T ] T = x EP,ν. Then: xt T x c t T ] T EP,ν ] t t, ˆt h I xt,x c t 43 with: ˆt h I xt,x c t = t + 1 ν ln 2λ h x T Px 44 Proo: The aulty system 4-5, with Bt = B h, together with the output eedback controller 11-12, can be rewritten in the closed-loop autonomous orm as: ẋt = Aθt + B h D c θtc B h C c θt xt ẋ c t B c θtc A c θt x c t 45 Let us apply Corollary 1 to 45 using the region Rez < λ h, that corresponds to 35 with α = 2λ h and β = 1, such that 37 reads as: { } 2λ h P+He P Aθ + B h D c θc B h C c θ B c θc A c θ 46 Hence, i 46 holds, 38 is true or the quadratic unction: V xt,x c t = xt T x c t T P xt T x c t T T that implies: V xt,x c t V x e 2λh t t = x T Px e 2λh t t By considering the condition V xt,x c t ν, that deines EP,ν, it is straightorward to obtain 43. From 44 it can be seen that, in order to maximize min h=1,...,n ˆt h I xt,x c t, it is necessary to minimize λ = max h=1,...,n λ h. The solution to this problem is given by the ollowing theorem. Theorem 3: Let X,Y S n x n x, Fθ R n x n y, Kθ R n u n x, Lθ R n u n y, F h θ R n x n y and N h θ R n x n x, h = 1,...,n, correspond to the solution to the ollowing constrained minimization problem: subect to λ 0 and: X I 2λ I Y min λ 49 He{XAθ + FθC} 50 He{AθY + BKθ} 51 X I CL θ T I Y K θ T 2 L θc K θ u MAX 0 52 { XAθ + F h θc N h θ +He Aθ + B h LθC AθY + B h Kθ = 1,...,n u, h = 1,...,n and θ Θ. } 53

5 Then, the output eedback controller 11-12, with matrices calculated as: 1 Ac θ B c θ Z XB = C c θ D c θ O I Aθ + BLθC T 1 XAθY Fθ Y O Kθ Lθ CY I 54 Z = X Y 1 55 maximizes min h=1,...,n ˆt h I xt,x c t or all xt T x c t T ] T E P,ν, where ˆt h I xt,x c t is the estimation o t h I xt,x c t obtained as 44 with P deined as: X Z P = 56 Z Z Proo: I conditions hold, then the closed-loop system obtained by considering the system 4-5 and the controller 11-12, with matrices calculated as in 54-55, is such that EP,1 S and EP,1 Lu,u MAX, with P deined as in 56. On the other hand, the design condition 53 corresponds to the analysis condition 42 with λ = λ h. In act, by applying a congruent transormation to 42 with: I 0 Γ = 57 Y Y and λ = λ h, 53 is obtained using the ollowing change o variables: N h θ =XAθY ZA c θy + ZB c θcy 58 XB h C c θy + XB h D c θcy F h θ = ZB c θ + XB h D c θ 59 Kθ = C c θy + D c θcy 60 Lθ = D c θ 61 Since a common λ is being used, it is clear that λ = max h=1,...,n λ h, and by minimizing λ, we are maximizing min h=1,...,n ˆt h I, deined as in 44. Also in this case, by relying on a polytopic representation, it is possible to obtain conditions that can be applied or the controller design, as stated by the ollowing corollary. Corollary 2: Let the matrices Aθt, A c θt, B c θt, C c θt, D c θt be polytopic, as in 25-26, and let X,Y S n x n x, F i R n x n y, K i R n u n x, L i R n u n y, F h i R n x n y and N h i R n x n x, h = 1,...,n, i = 1,...,N, correspond to the solution o the constrained minimization problem 49: min λ 62 subect to λ 0 and: He{XA i + F i C} 63 He{A i Y + BK i } 64 X I 2λ I Y X I CLi T I Y Ki T L i C K i u MAX { +He 2 XA i + F h i C N h i A i + B h L i C AθY + B h 0 65 K i } 66 i = 1,...,N, = 1,...,n u and h = 1,...,n. Then, the controller 11-12, with matrices calculated as: 1 Ac,i B c,i Z XB = C c,i D c,i O I Ai + BL i C T 1 67 XA i Y F i Y O K i L i CY I maximizes min h=1,...,n ˆt h I xt,x c t or all xt T x c t T ] T E P,ν, where ˆt h I xt,x c t is the estimation o t h I xt,x c t obtained as 44 with P deined as in 56. Proo: It ollows rom the basic property o matrices 28] that any linear combination o negative positive deinite matrices with non-negative coeicients whose sum is positive is negative positive deinite. Hence, using the coeicients µ i θt, taking into account 26, ollow directly rom VI. EXAMPLE Let us consider the open-loop unstable LPV system subect to actuator saturations proposed in 22], i.e. 4-5, with: 2 + θt 0 Aθt = θ 1,1] B = t < t 0 1 Bt = 1 0 C = B = t t 0 0 and satu as in 7-8 with u MAX = 10, = 1,2. When the output-eedback controller designed in 22] is applied reerred in the ollowing as controller 1, i.e. when the controller is designed without maximizing the tolerated ault isolation delay, the application o Theorem 2 gives a value λ = On the other hand, i the controller is designed using Corollary 2 reerred in the ollowing as controller 2, a value λ = 0 is achieved. Notice that achieving the case λ = 0, that would correspond to ˆt I = using 44, is equivalent to the existence o a nominal controller that is robust against the considered ault. For the sake o additional comparison, let us also consider a controller designed using a preixed value o λ, i.e. λ = 1, which will be denoted as controller 3. Let us consider a simulation that lasts 20s with x0 = 1 0 T, x c 0 = 0 0 T, θt = sin5t, and t = 0.5s. Since x0 T,x c 0 T EP,1 or all controllers 1,2 and 3, the state traectories will converge towards the origin and the control input will not saturate in the time interval 0,t ], as shown in Fig. 1.

6 t t x t Proection o EP,ν on the plane x 1 controller 1 Proection o EP,ν on the plane x 1 controller 2 Proection o EP,ν on the plane x 1 controller 3 xt obtained with controller 1 xt obtained with controller 2 xt obtained with controller x 1 Fig. 1. State traectories obtained with t = 0.5s and no ault isolation during the simulation t I. Assuming that the ault is not isolated during the simulation t I, it can be seen that the state traectories obtained with controllers 1 and 3 diverge ater the ault occurrence see red and green lines in Fig. 1. On the other hand, the traectory o the closed-loop system obtained with controller 2 keeps converging despite the ault occurrence and the absence o ault isolation see blue line in Fig. 1. As a matter o act, using Theorem 2, it can be calculated that or controller 1 ˆt I = 0.525s, while various simulations have shown that t I = 0.748s. On the other hand, or controller 3, ˆt I = 0.698s and t I = 1.066s, which conirms that controller 1 is more sensitive to ault isolation delays than controller 3, whereas controller 2, which has been obtained applying the proposed methodology, is tolerant w.r.t. any ault isolation delay. The control inputs are shown in Fig. 2, where it can be seen that, due to the state traectory divergence, the control inputs obtained with controllers 1 and 3 diverge as well. On the other hand, the control inputs obtained with controller 2 are within the saturation limits both beore and ater the ault occurrence. Finally, Fig. 3 shows the state traectories obtained or t I = 3s. In this case, controller 1 is not able to recover the system s stability ater the ault is isolated. On the other hand, the traectory obtained with controller 3 converges asymptotically to the origin, although with a worse perormance than the one obtained with controller 2. VII. CONCLUSIONS AND FUTURE WORK This paper has presented a virtual actuator-based FTC strategy or LPV systems that are subect to saturation o the actuators and FDI delays. The developed design methodology allows maximizing the tolerated delay between the ault occurrence and its isolation. Under the assumption that the LPV system is polytopic, the design process consists in inding the optimal easible solution to a inite set o u 1 controller 1 u 2 controller 1 u 1 controller 2 u 2 controller 2 u 1 controller 3 u 2 controller time s] Fig. 2. Control inputs obtained with t = 0.5s and no ault isolation during the simulation t I. LMIs. The results achieved using an academic example have shown that the proposed design improves the perormances with respect to a more traditional design that does not take into account the presence o FDI delays. In particular, it has been shown that, as a particular case, the proposed design methodology provides a nominal controller that is robust against the considered aults. Future work will aim at improving the obtained results by considering: i the presence o sensor noise and process disturbances; ii parameter-varying input and output matrices; iii dierent indices, which will allow analyzing the trade-o between the tolerated ault isolation delay and the controller perormance.

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