- Email: [email protected]
Fault tolerant control of unstable LPV systems subject to actuator saturations using virtual actuators★
9th IFAC Symposium on Fault Detection, Supervision and 9th IFAC Symposium on 9th IFAC on Fault Fault Detection, Detection, Supervision Supervision and and Safety of Symposium Technical Processes 9th IFAC Symposium on Fault Detection, Supervision and Safety Processes Available online at www.sciencedirect.com Safety of of Technical Technical Processes September 2-4, 2015. Arts et Métiers ParisTech, Paris, France Safety of Technical Processes September September 2-4, 2-4, 2015. 2015. Arts Arts et et Métiers Métiers ParisTech, ParisTech, Paris, Paris, France France September 2-4, 2015. Arts et Métiers ParisTech, Paris, France
ScienceDirect
IFAC-PapersOnLine 48-21 (2015) 018–023
Fault tolerant control of unstable LPV systems Fault tolerant control of unstable LPV systems Fault tolerant control of unstable LPV systems Fault tolerant control of unstable LPV systems subject to actuator saturations using virtual subject to actuator saturations using virtual subject to actuator saturations using virtual subject to actuator saturations using virtual actuators actuators actuators actuators
∗ Jean-Christophe Ponsart ∗∗ Didier Theilliol ∗∗ Damiano Rotondo ∗∗ Jean-Christophe Ponsart ∗∗ ∗∗ ∗∗ Didier ∗∗ Damiano Rotondo Theilliol ∗ Vicenc ∗,∗∗∗ Damiano Rotondo Ponsart Theilliol ∗ Jean-Christophe ∗∗ Didier ∗∗ Fatiha Nejjari ¸ Puig Damiano Rotondo Jean-Christophe Ponsart Didier Theilliol ∗ ∗,∗∗∗ ∗ ∗,∗∗∗ Fatiha Nejjari Nejjari ∗ Vicenc Vicenc¸¸ Puig Puig ∗,∗∗∗ Fatiha Fatiha Nejjari Vicenc¸ Puig ∗ Automatic Control Department, Universitat Polit` eecnica de Catalunya ∗∗ Automatic Control Department, Universitat Polit` de Catalunya Control Department, Universitat Polit` eecnica cnica de Catalunya ∗ Automatic (UPC), Rambla de Sant Nebridi, 11, 08222 Terrassa, Spain Automatic Control Department, Universitat Polit` cnica de Catalunya (UPC), Rambla de Sant Nebridi, 11, 08222 Terrassa, Spain ∗∗ Centre (UPC), Rambla de Sant Nebridi, 11, 08222 Terrassa, Spain de Recherche en Automatique de Nancy (CRAN), Universit´ ee de (UPC), Rambla de Sant Nebridi, 11, 08222 Terrassa, Spain ∗∗ ∗∗ Centre de Recherche en Automatique de Nancy (CRAN), Universit´ de Recherche en Automatique de Nancy (CRAN), Universit´ ee de de ∗∗ Centre Lorraine, CNRS UMR 7039, 54506 Vandoeuvre-les-Nancy, France Centre de Recherche en Automatique de Nancy (CRAN), Universit´ de Lorraine, CNRS UMR 7039, 54506 Vandoeuvre-les-Nancy, France ∗∗∗ Institut Lorraine, CNRS UMR 7039, 54506 Vandoeuvre-les-Nancy, France de Robotica i Informatica Industrial (IRI), UPC-CSIC Lorraine, CNRS UMR 7039, 54506 Vandoeuvre-les-Nancy, France ∗∗∗ ∗∗∗ Institut de Robotica i Informatica Industrial (IRI), UPC-CSIC de Robotica Informatica Industrial (IRI), ∗∗∗ Institut Carrer Llorens ii iiArtigas, 4-6, 08028 Barcelona, Spain Institut de Robotica Informatica Industrial (IRI), UPC-CSIC UPC-CSIC Carrer de Llorens Artigas, 4-6, 08028 Barcelona, Spain Carrer de Llorens i Artigas, 4-6, 08028 Barcelona, Carrer de Llorens i Artigas, 4-6, 08028 Barcelona, Spain Spain
Abstract: An An active active fault fault tolerant tolerant control control (FTC) (FTC) strategy strategy for for open-loop open-loop unstable unstable linear linear parameter parameter varying varying Abstract: Abstract: An active fault tolerant control (FTC) strategy for open-loop unstable linear parameter varying (LPV) systems subject to actuator saturations and faults is proposed in this paper. At first, the nominal Abstract: An active fault tolerant control (FTC) strategy for open-loop unstable linear parameter varying (LPV) systems subject to actuator saturations and faults is proposed in this paper. At first, the nominal (LPV) systems subject to saturations proposed this paper. first, the controller is designed designed using direct approach and that faults takes is into accountin the actuator limits. Later, virtual (LPV) systems subjectusing to actuator actuator saturations and faults is proposed inthe thisactuator paper. At At first,Later, the nominal nominal controller is aaa direct approach that takes into account limits. virtual controller is designed using direct approach that takes into account the actuator limits. Later, virtual actuators are added to the control scheme, such that the faulty plant is reconfigured, and the fault controller is designed using a direct approach that takes into account the actuator limits. Later, virtual actuators are added to the control scheme, such that the faulty plant is reconfigured, and the fault actuators are added to the control scheme, such that the faulty plant is reconfigured, and the fault is hidden from the controller point of view. Conditions for designing the virtual actuator gains are actuators are added to the control scheme, such that the faulty plant is reconfigured, and the fault is hidden from the controller point of view. Conditions for designing the virtual actuator gains are is hidden from the controller point of view. Conditions for designing the virtual actuator gains are designed, aiming at guaranteeing that if at the fault isolation time the closed-loop system state is inside is hidden from the controller point of view. Conditions for designing the virtual actuator gains are designed, aiming aiming at at guaranteeing guaranteeing that that if if at at the the fault fault isolation isolation time time the the closed-loop closed-loop system system state state is is inside inside aaa designed, region defined by a value of the Lyapunov function, the state trajectory will converge to zero despite the designed, aiming at guaranteeing that if at the fault isolation time the closed-loop system state is inside a region defined by aa value of the Lyapunov function, the state trajectory will converge to zero despite the region defined by value of the Lyapunov function, the state trajectory will converge to zero despite the appearance of faults within a predefined set. The theoretical results are demonstrated using a numerical region defined by a value of the Lyapunov function, the state trajectory will converge to zero despite the appearance of faults within aa predefined set. The theoretical results are demonstrated using aa numerical appearance example. of appearance of faults faults within within a predefined predefined set. set. The The theoretical theoretical results results are are demonstrated demonstrated using using a numerical numerical example. example. example. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Linear Linear parameter parameter varying varying (LPV) (LPV) systems, systems, actuator actuator saturation, saturation, unstable unstable open open loop loop systems, systems, Keywords: Keywords: Linear parameter varying (LPV) systems, actuator saturation, unstable openactuators. loop systems, systems, linear matrix inequalities (LMIs), active fault tolerant control system (AFTCS), virtual Keywords: Linear parameter varying (LPV) systems, actuator saturation, unstable open loop linear matrix inequalities (LMIs), active fault tolerant control system (AFTCS), virtual actuators. linear matrix inequalities (LMIs), active fault tolerant control system (AFTCS), virtual actuators. linear matrix inequalities (LMIs), active fault tolerant control system (AFTCS), virtual actuators. 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION 1. In many many applications, applications, aa significant significant control control action action is is necessary necessary In In many many applications, adesired significant control action is necessary necessary in order to perform a task with a high performance. In applications, a significant control action is in perform aa desired task with aa high performance. in order order to tothe perform desired task withthe high performance. However, actuators which deliver control actions to in order to perform a desired task with a high performance. However, the which deliver the actions However, the actuators actuators which deliver the control control actions to to the controlled system are limited in magnitude (Tarbouriech However, the actuators which deliver the control actions to the controlled system are limited in magnitude (Tarbouriech the controlled system are limited in magnitude (Tarbouriech and controlled Turner, 2009). 2009). If these these limits are are not taken taken into into account, the system are limited in magnitude (Tarbouriech and Turner, limits account, Turner, 2009). If Ifofthese these limits are are not not taken into into and account, aaand strong degradation the closed-loop performance, even and Turner, 2009). If limits not taken account, of the closed-loop performance, and a strong strong degradation degradation ofFor thethis closed-loop performance, and even even instability, may occur. reason, approaches to deal with ainstability, strong degradation of the closed-loop performance, and even may occur. For this reason, approaches to deal with instability, may occur. For this reason, approaches to deal with this problem have been proposed, such as the anti-windup cominstability, may occur. For this reason, approaches to deal with this problem have been proposed, such as the anti-windup comthis problem have been proposed, such as the anti-windup compensation, where part proposed, of the controller dedicated to achieve this problem haveaabeen such asis the anti-windup compensation, where the controller is to achieve pensation, where a part part of ofand thethe controller is dedicated dedicated tothe achieve the desired performance other part deals with conpensation, where a part of the controller is dedicated to achieve the performance and the part deals the the desired desired performance and and the other other part approach, deals with withwhere the conconstraints (Mulder et al., 2001), the direct the the desired performance and the other part deals with the constraints (Mulder et al., 2001), and the direct approach, where the straints (Mulder et al., 2001), and the direct approach, where the controller is designed taking into account both desired perforstraints (Mulder et al., 2001), and the direct approach, where the controller is designed taking into account both desired perforcontroller is designed taking into account both desired performance and andissaturation saturation the account same time time (Sussmann et al., al., controller designed limits takingatinto both desired performance mance and and saturation saturation limits limits at at the the same same time time (Sussmann (Sussmann et et al., al., 1994). mance limits at the same (Sussmann et 1994). 1994). 1994). When actuator actuator faults faults are are considered, considered, mitigating mitigating the the effects effects of of When When actuator faults are considered, mitigating the effects of saturations becomes of utmost importance. In fact, in these When actuator faults of areutmost considered, mitigating the effects of saturations becomes importance. In fact, in these saturations becomes of utmost importance. In fact, in these cases, fault fault becomes tolerant control control (FTC) techniquesInare are usually apsaturations of utmost importance. fact, in these cases, tolerant (FTC) techniques usually cases, with fault the tolerant control (FTC) techniques are degradation usually apapplied aim of reducing the performance cases, fault tolerant control (FTC) techniques are usually applied with the aim of reducing the performance degradation plied with the aim of reducing the performance degradation and preserving the closed-loop stability in the event of faults plied with the aim of reducing stability the performance degradation and the in event and preserving preserving the closed-loop closed-loop stability in the thetechniques event of of faults faults (Blanke et al., al., 2006, 2006, Noura et et al., al.,stability 2009). These These usuand preserving the closed-loop in the event of faults (Blanke et Noura 2009). techniques usu(Blanke et al., 2006, Noura et al., 2009). These techniques usually redistribute, in some way, the control effort corresponding (Blanke et al., 2006, Noura et al., 2009). These techniques usually redistribute, in some way, the control effort corresponding ally redistribute, in some way, the control effort corresponding to the theredistribute, faulty actuators actuators among the healthy ones. This This ally in some way,the theremaining control effort corresponding to among healthy ones. to the the faulty faulty actuators actuators among the remaining remaining healthy ones. This This redistribution may lead to saturation of both the faulty and to faulty among the remaining healthy ones. redistribution redistribution may may lead lead to to saturation saturation of of both both the the faulty faulty and and redistribution may lead to saturation of both the faulty and This work has been funded by the Spanish Ministry of Science and Technol This work has been by Spanish Ministry of This work the has projects been funded funded by the the Spanish Ministry of Science Science and and TechnolTechnol ogy through CICYT ECOCIS (Ref. DPI2013-48243-C2-1-R) and This work the has projects been funded by the Spanish Ministry of Science and Technology through CICYT ECOCIS (Ref. DPI2013-48243-C2-1-R) and ogy through the projects CICYT ECOCIS (Ref. DPI2013-48243-C2-1-R) HARCRICS (Ref. DPI2014-58104-R), by (Ref. AGAUR through the contract and FIogy through the projects CICYT ECOCIS DPI2013-48243-C2-1-R) and HARCRICS (Ref. DPI2014-58104-R), by through the contract FIHARCRICS (Ref. DPI2014-58104-R), bytheAGAUR AGAUR through the de contract FIDGR 2014 (ref. 2014FI B1 00172) and by DGR of Generalitat Catalunya HARCRICS (Ref. DPI2014-58104-R), bytheAGAUR through the de contract FIDGR 2014 (ref. 2014FI B1 00172) and by DGR of Generalitat Catalunya DGR 2014 (ref. 2014FI B1 00172) and by the DGR of Generalitat de Catalunya (SAC 2014 group(ref. Ref.2014FI 2014/SGR/374). e-mail: [email protected] DGR B1 00172) and by the DGR of Generalitat de Catalunya (SAC group Ref. 2014/SGR/374). e-mail: [email protected] (SAC group Ref. 2014/SGR/374). e-mail: [email protected] (SAC group Ref. 2014/SGR/374). e-mail: [email protected]
the healthy actuators. It has been shown that neglecting this the healthy actuators. It has been shown that neglecting this the healthy actuators. It has been shown that neglecting this fact may lead to severe performance degradation and instability the healthy actuators. It has been shown that neglecting this fact may lead to severe performance degradation and instability fact may lead to severe performance degradation and instability (Fan et al., 2012). Even though several solutions for designing fact may lead to severe performance degradation and instability (Fan et et al., al., 2012). Even Even though though several several solutions solutions for for designing designing (Fan FTC systems subject to actuator have been proposed (Fan et al., 2012). 2012). Even though saturations several solutions for designing FTC systems subject to actuator saturations have been proposed FTC systems subject to actuator saturations have been proposed for the open-loop stable case, e.g. (Zuo et al., 2010, Mhaskar FTC systems subject to actuator saturations have been proposed for the open-loop stable case, e.g. (Zuo et al., 2010, Mhaskar for the open-loop stable case, e.g. (Zuo et al., 2010, Mhaskar et al., 2008), only a few works have considered this problem for the open-loop stable case, e.g. (Zuo et al., 2010, Mhaskar et al., 2008), only aa few works have considered this problem et al., 2008), only few works have considered this problem for open-loop unstable systems (Stoustrup and Niemann, 2004, et al., 2008), only a few works have considered this problem for open-loop open-loop unstable unstable systems systems (Stoustrup (Stoustrup and and Niemann, Niemann, 2004, for 2004, Weng et al., 2006, Qi et al., 2014). for open-loop unstable systems (Stoustrup and Niemann, 2004, Weng et al., 2006, Qi et al., 2014). Weng et al., 2006, Qi et al., 2014). Weng et al., 2006, Qi et al., 2014). The main contribution of this paper consists in designing an The main contribution of this paper consists in designing an The main contribution of this paper consists in designing an FTC strategy for open-loop unstable linear parameter varying The main contribution of this paper consists in designing an FTC strategy for open-loop unstable linear parameter varying FTC strategy for open-loop unstable linear parameter varying (LPV) systems subject to actuator saturations and faults. In FTC strategy for open-loop unstable linear parameter varying (LPV) systems subject to actuator saturations and faults. In (LPV) systems subject to actuator saturations and faults. In particular, the faults considered in this paper are undesired (LPV) systems subject to actuator saturations and faults. In particular, the the faults faults considered considered in in this this paper paper are are undesired undesired particular, changes in the input matrix that cause its loss of rank, e.g. particular, the faults considered in this paper are undesired changes in the input matrix that cause its loss of rank, e.g. changes in the input that its loss rank, total losses LPV systems are aa standard changes in of theactuators. input matrix matrix that cause cause its loss of of formalism rank, e.g. e.g. total losses of actuators. LPV systems are standard formalism total losses of actuators. LPV systems are a standard formalism that allows performing gain scheduling of nonlinear systems total losses of actuators. LPV systems are a standard formalism that allows performing gain scheduling of systems that allows performing gain scheduling of nonlinear nonlinear systems using an of invariant techniques. In that performing gaintime scheduling nonlinear systems usingallows an extension extension of linear linear time invariantof(LTI) (LTI) techniques. In using an extension of linear time invariant (LTI) techniques. In particular, this paper extends some results obtained in Rotondo using an extension of linear time invariant (LTI) techniques. In particular, this paper extends some results obtained in Rotondo particular, this paper extends some results obtained in Rotondo et al. (2015b) for the LTI case. It is worth highlighting that in particular, this paper extends some results obtained in Rotondo et al. (2015b) for the LTI case. It is worth highlighting that in et al. (2015b) for the LTI case. It is worth highlighting that in this work, it is assumed that the information about the fault is et al. (2015b) for the LTI case. It is worth highlighting that in this work, it is assumed that the information about the fault is this work, it is assumed that the information about the fault is provided by a fault detection and isolation (FDI) module. The this work, it is assumed that the information about the fault is provided by aa fault detection and isolation (FDI) module. The provided by fault detection and isolation (FDI) module. The design of this module is not considered in this work, due to provided by a fault detection and isolation (FDI) module. The design of this module is not considered in this work, due to design of this module is considered this to the fact theoretical is independent from design ofthat thisthe module is not notdevelopment considered in in this work, work, due due to the fact that the theoretical development is independent from the fact that the theoretical development is independent from strategy used to perform the FDI. The interested reader the fact that the theoretical development is independent from the strategy used to perform the FDI. The interested reader the strategy used to FDI. The reader could find useful about designing FDI strategies the strategy used information to perform perform the the FDI. The interested interested reader could find useful information about designing FDI strategies could find useful information about designing FDI strategies in Isermann (2005), Hwang et al. (2010), and the references could find useful information about designing FDI strategies in Isermann (2005), Hwang et al. (2010), and the references in Isermann therein. in Isermann (2005), (2005), Hwang Hwang et et al. al. (2010), (2010), and and the the references references therein. therein. therein. At first, aa nominal controller is designed using aa direct apAt first, controller is designed using apAt first, aa nominal nominal controller is designed using aa direct direct approach, i.e. taking into account the saturations. Then, a reconAt first, nominal controller is designed using direct approach, i.e. taking into account the saturations. Then, a reconproach, i.e. taking into account the saturations. Then, a reconfiguration block is inserted in the control scheme, such that proach, i.e. taking into account saturations. Then, a reconfiguration block block is inserted inserted in the the control scheme, scheme, such that that figuration figuration block is is inserted in in the control control scheme, such such that
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 IFAC 18 Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 18 Copyright ©under 2015 IFAC IFAC 18 Control. Peer review© of International Federation of Automatic Copyright © 2015 responsibility IFAC 18 10.1016/j.ifacol.2015.09.498
SAFEPROCESS 2015 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
the property of fault tolerance is added without affecting the other properties, e.g. stability and performance, already attained by the controller under nominal situation. This is achieved through a fault-hiding paradigm (Steffen, 2005), since the reconfiguration block hides the fault from the controller point of view. In case of actuator faults, the reconfiguration block is named virtual actuator, as initially proposed by Lunze and Steffen (2006), and further investigated by Dziekan et al. (2011), Richter et al. (2011), Seron et al. (2012), Rotondo et al. (2015a) and Blesa et al. (2014).
E (P, 1) =
x xc
∈R
2nx
:
x xc
T x ≤1 P xc
with P O is an ellipsoid, and x0 2nx ∈ R : lim ψ(t, x0 , xc0 ) = 0 S = xc0 t→∞
(6)
(7)
is the domain of attraction of the origin, with ψ(t, x0 , xc0 ) denoting the state trajectory of the closed-loop system given by (1)-(2) plus (4)-(5), obtained with x(0) = x0 and xc (0) = xc0 .
In other words, it is wished to design the controller (4)-(5) for the system (1)-(2) such that if the initial condition satisfies x(0) (8) x(0)T xc (0)T P x (0) ≤ 1 c then the control input will not saturate, and the closed-loop state trajectory ψ will converge to the origin.
The paper is structured as follows. Section 2 presents the design of the nominal controller. Section 3 describes the virtual actuator strategy. The theoretical results are illustrated using a numerical example in Section 4. Finally, Section 5 outlines the main conclusions. NOTATION Symbol Mi Mi j ∗ He {M} M ∈ Sn×n I O MO M≺O ⊗ †
19
In order to achieve this objective, the following Theorem is proposed, obtained as an extension of a similar theorem presented for the LTI case in Iwasaki and Fu (2002). Theorem 1. (LPV output feedback controller design) Let X,Y ∈ Snx ×nx , F(θ ) ∈ Rnx ×ny , K(θ ) ∈ Rnu ×nx and L(θ ) ∈ Rnu ×ny be such that He {XA(θ ) + F(θ )C} ≺ O (9) He {A(θ )Y + BK(θ )} ≺ O (10) X I CLi (θ )T I (11) Y Ki (θ )T O i = 1, . . . , nu 2 Li (θ )C Ki (θ ) αi ∀θ ∈ Θ. Then, the controller (4)-(5), with matrices calculated as −1 Z XB Ac (θ ) Bc (θ ) = ··· O I Cc (θ ) Dc (θ ) −1 T −Y O − XA(θ )Y F(θ ) − (A(θ ) + BL(θ )C) ··· CY I K(θ ) L(θ ) (12) Z = X −Y −1 (13) is such that, for the closed-loop system obtained with u(t) = uc (t), E (P, 1) ⊆ S and E (P, 1) ⊆ L (u, α) where
Description ith row of a matrix M Element i, j of a matrix M Symmetric element in a matrix Shorthand for M + M T Shorthand for a symmetric matrix M ∈ Rn×n Identity matrix of appropriate dimensions Zero matrix of appropriate dimensions Shorthand for a positive definite matrix Shorthand for a negative definite matrix Kronecker product Moore-Penrose pseudoinverse
2. DESIGN OF THE NOMINAL CONTROLLER Let us consider the following linear parameter varying (LPV) system subject to actuator saturations x(t) ˙ = A (θ (t)) x(t) + Bsat (u(t)) (1) y(t) = Cx(t) (2) n n n x u y where x ∈ R is the state, u ∈ R is the control input, y ∈ R is the measured output, A (θ (t)) ∈ Rnx ×nx is the parameter varying state matrix, whose values depend on the vector θ (t) ∈ Θ ⊂ Rnθ , B ∈ Rnx ×nu is the input matrix, C ∈ Rny ×nx is the output matrix, and sat : Rnu → Rnu is the saturation function, defined as sat1 (u1 ) .. αi (ui > αi ) . ui (|ui | ≤ αi ) sat(u) = sati (ui ) , sati (ui ) = .. −αi (ui < −αi ) . satnu (unu ) (3) where α = (α1 , . . . , αnu )T ∈ Rnu is a given vector with positive entries. For an output feedback law u(t) = g (y(t)) = g (Cx(t)), let us define L (u, α) the region of the state space in which the actuators are not saturated.
P=
X Z Z Z
(14)
Proof. The proof follows the reasoning developed by Iwasaki and Fu (2002) in the case of LTI systems, and is based on demonstrating that if (9)-(11) hold and the controller matrices are calculated as in (12), then E (P, 1) is contractively invariant, i.e. by defining the quadratic Lyapunov function V (x(t)) = x(t)T Px(t), it is obtained that V˙ (x(t)) < 0 for all x ∈ E (P, ρ)\ {0}, if the controller matrices are chosen as (12). Since E (P, 1) is contractively invariant, it is inside the domain of attraction S (Hu et al., 2002), such that the stability is guaranteed over the whole set of possible values of θ . The complete proof is omitted due to lack of space.
The problem solved hereafter is to design an LPV dynamic output feedback controller for the system (1)-(2) x˙c (t) = Ac (θ (t)) xc (t) + Bc (θ (t)) y(t) (4) uc (t) = Cc (θ (t)) xc (t) + Dc (θ (t)) y(t) (5) where xc ∈ Rnx is the controller state and uc ∈ Rnu is the controller output, such that when u(t) = uc (t), then E (P, 1) ⊆ S and E (P, 1) ⊆ L (u, α), where
The conditions provided by Theorem 1 rely on the satisfaction of infinite constraints, due to the fact that (9)-(11) should hold for all the possible values of θ . However, by considering a polytopic approach, as proposed by Apkarian et al. (1995) and Chilali and Gahinet (1996), (9)-(11) can be transformed in a finite number of linear matrix inequalities (LMIs), as shown by the following Corollary. 19
SAFEPROCESS 2015 20 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
(23) is the faulty system. It is assumed that the n f matrices
Corollary 1. (Polytopic LPV output feedback controller design) Assume that the LPV system (1) is polytopic, i.e. the matrix A (θ (t)) can be written as A (θ (t)) =
N
∑ µ j (θ (t))A( j)
(1)
(15)
and the pairs
j=1
where µ j , j = 1, . . . , N, are the coefficients of the polytopic decomposition, such that N
∑ µ j (θ (t)) = 1,
j=1
µ j (θ (t)) ≥ 0
∀θ ∈ Θ
(16)
(h) †
Bf
B
(27)
then the control input will not saturate for all t ≥ tI , and the state trajectory will converge to zero. This problem is equivalent to (n ) (1) the one of finding u f (t), . . . , u f f (t) that make the ellipsoid
T x x x ∈ R2nx : ≤ νf E (P, ν f ) = P (30) xc xc xc contractively invariant, and such that it is contained within L (u, α), for t ≥ tI .
The solution proposed in this paper relies on LPV virtual actuators, with the following structure, as depicted by Rotondo et al. (2015b) (h) (h) (h) (h) x˙v (t) = A (θ (t)) + B∗ M (h) (θ (t)) xv (t)+ B − B∗ uc (t) (31) (h) (h) u f (t) = N (h) uc (t) − M (h) (θ (t)) xv (t) (32)
Proof. Due to a basic property of matrices (Horn and Johnson, 1990), any linear combination of (17) with non-negative coefficients, of which at least one different from zero, is negative definite. Hence, using the coefficients µi (θ (t)), taking into account (16), and the fact that the same property holds for the negative and positive definiteness of (18)-(19), respectively, (9)(11) are obtained.
(h)
where h = 1, . . . , n f , xv are the virtual actuators states with (h) xv (tI ) = 0, M (h) (θ (t)) ∈ Rnu ×nx are the virtual actuators gains (h) to be designed, and the matrices N (h) and B∗ are given by † (h) B (33) N (h) = B f † (h) (h) (h) (h) B (34) B∗ = B f N (h) = B f B f
3. DESIGN OF THE VIRTUAL ACTUATOR Let us consider the following LPV system subject to actuator saturations x(t) ˙ = A (θ (t)) x(t) + B(t)sat (u(t)) (23)
t < tf B (n f ) (1) t ≥ tf Bf ∈ Bf ,...,Bf
where uc (t) is the output of the nominal controller (4)-(5) designed using the results presented in the previous section, and tI ∈ R+ , tI ≥ t f , is the fault isolation time, such that if the following condition holds x(tI ) T T (29) x(tI ) xc (tI ) P x (t ) ≤ ν f c I
j=1
B(t) =
(h)
Hereafter, it is solved the problem of designing the control law uc (t) t < tI (1) (1) u f (t) t ≥ tI , B(t) = B f u(t) = (28) .. .. . . u(n f ) (t) t ≥ t , B(t) = B(n f ) I f f
and vertex controller gains calculated as −1 Ac( j) Bc( j) Z XB −Y O (21) = Ξ( j) O I CY I Cc( j) Dc( j) −(A( j) + BL( j)C)T − XA( j)Y F( j) Ξ( j) = (22) K( j) L( j) with Z defined as in (13) is such that, for the closed-loop system obtained with u(t) = uc (t), E (P, 1) ⊆ S and E (P, 1) ⊆ L (u, α) with P defined as in (14).
(26)
are stabilizable, ∀θ ∈ Θ and ∀ h = 1, . . . , n f .
Then, the controller (4)-(5) with N Ac( j) Bc( j) Ac (θ (t)) Bc (θ (t)) (20) = ∑ µ j (θ (t)) Cc (θ (t)) Dc (θ (t)) Cc( j) Dc( j)
with
∈ Rnx ×nu are such that (h) rank B f < rank (B) A(θ ), B f
and let X,Y ∈ Snx ×nx , F( j) ∈ Rnx ×ny , K( j) ∈ Rnu ×nx and L( j) ∈ Rnu ×ny , j = 1, . . . , N, be such that (17) He XA( j) + F( j)C ≺ O He A( j)Y + BK( j) ≺ O (18) T X I CLi( j) Y Ki(T j) O i = 1, . . . , nu (19) I Li( j)C Ki( j) αi2
y(t) = Cx(t)
(n f )
Bf ,...,Bf
Also, in order to obtain the fault-hiding characteristic, the output equation (24) is slightly changed after tI , as follows (h) (h) y(t) = C x(t) + xv (t) t ≥ tI , B(t) = B f (35)
(24) (25)
The conditions for designing the virtual actuator gains M (h) (h) such that the control law (28) with u f , h = 1, . . . , n f , defined by (32), makes the ellipsoid E (P, ν f ), defined in (30), contractively invariant after tI , are given by the following Theorem, that constitutes an extension of a similar theorem provided for the LTI case in Rotondo et al. (2015b). −1 ∈ Snx ×nx and Γ(h) (θ ) ∈ Rnu ×nx , h = Theorem 2. Let Xva 1, . . . , n f be such that
and the saturation function sat (u(t)) defined as in (3). Before the fault occurrence time t f ∈ R+ , the matrix B(t) is the nominal input matrix B ∈ Rnx ×nu , and the corresponding LPV system, i.e. (1)-(2), is the nominal system. On the other hand, after t f , the matrix B(t) becomes the faulty input matrix B f ∈ Rnx ×nu , and the corresponding LPV system obtained from 20
SAFEPROCESS 2015 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
ν f Acl (θ )P−1 O2nx ×nx ≺O (h) (h) −1 ν f A∗ (θ )P−1 A(θ )Xva + B∗ Γ(h) (θ ) (h) −1 Xva Γk (θ )T 2 i= 1, .. . , nu α i O (h) (h) − µ f Ni = 0 Ni (h) Γk (θ ) (h) nu˜ hold ∀θ ∈ Θ, where A(θ ) + BDc (θ )C BCc (θ ) Acl (θ ) = Bc (θ )C Ac (θ ) (h) (h) (h) A∗ (θ ) = B − B∗ Dc (θ )C B − B∗ Cc (θ ) He
µ f = max uc E (P,ν f )
(h) nu˜
To this aim, since (32) is equivalent to (h) (h) (h) uc (t) − M (h) (θ )xv (t) u f ,i (t) = Ni
(36)
(h)
(37)
(38)
that leads to (h) (h) Ni µ f + M (h) (θ )xv (t) ≤ αi
(39)
At the expense of introducing conservativeness, it is possible to transform (52), whose left-hand side concerns the norm of a vector, into a condition about the norms of scalars. This is done by taking advantage of the fact that only the rows of M (h) (θ ) (h) corresponding to non-zero elements of Ni will contribute to (h) (h) u f ,i in (49). By denoting these rows as Mk (θ ), and the number
Proof. When the system is working in the region L (u, α), there exists a similarity transformation that transforms the closed-loop system made up by the system (23), with output equation (35) and control law (28), the nominal controller (4)(5), and the virtual actuator (31)-(32), in an equivalent blocktriangular form (h) (h) Acl (θ ) O2nx ×nx xcl (t) x˙cl (t) = (41) (h) (h) (h) (h) A∗ (θ ) Av (θ ) x˙v (t) xv (t) where (h) (h) (42) xcl (t) = xw (t) xc (t) (h)
(43)
(h)
(h)
(44)
Av (θ ) = A(θ ) + B∗ M(θ )
(51)
(h) For values of i such that Ni = 0, (51) is obviously satisfied. (h) On the other hand, when Ni = 0, (51) becomes αi (h) (h) (52) M (θ )xv ≤ (h) − µ f Ni
(40)
(h)
(h)
(49)
where u f ,i , i = 1, . . . , nu , denotes the ith input, the condition of non-saturation can be written as (h) (h) (h) uc (t) − M (h) (θ )xv (t) u f ,i (t) = Ni (h) (h) (50) ≤ Ni uc (t) + M (h) (θ )xv (t) (h) (h) (h) ≤ Ni µ f + M (θ )xv (t) ≤ αi
is the number of non-zero elements in Ni , and k in (37) takes values corresponding to the indices of the non-zero (h) elements in Ni . Then, if the virtual actuators gains M (h) (θ ) in (31)-(32) are calculated as M (h) (θ ) = Γ(h) (θ )Xva , E (P, ν f ) is contractively invariant for the system (23)-(24) with control law (28), and E (P, ν f ) ⊆ L (u, α), ∀t ≥ tI .
xw (t) = x(t) + xv (t)
21
(h)
(h)
of non-zero elements of Ni as nu˜ , (52) can be replaced by nu αi (h) (h) 2 − µf (53) ∑ Mk (θ )xv ≤ (h) (h) N i k=1,N =0 ik
that holds if
(h) (h) Mk (θ )xv ≤
By considering T P (h) (h) O x (t) xcl (t) 2n ×n x x cl νf (45) V2 (t) = (h) (h) xv (t) xv (t) Onx ×2nx Xva with Xva 0 to assess the stability of (41), the following Lyapunov inequality is obtained P (θ ) O A O 2nx ×nx cl 2nx ×nx ≺ 0 (46) He ν f (h) (h) O A (θ ) A (θ ) v ∗ Xva nx ×2nx that is equivalent to its dual version (Goebel et al., 2006) Acl (θ ) O2nx ×nx ν f P−1 O2nx ×nx He ≺ 0 (47) (h) (h) −1 Onx ×2nx Xva A∗ (θ ) Av (θ ) that can be brought to the LMI form (36) by considering −1 . Γ(h) (θ ) = M (h) (θ )Xva
αi (h) Ni
− µf
(h)
nu˜
k = 1, . . . , nu (h) Nik = 0
(54)
Given two vectors m and x, the existence of Q 0 such that Q−1 Q−1 m 0 (55) mT Q−1 γ 2 T implies that m x ≤ γ ∀x ∈ E (Q, 1) (Nguyen and Jabbari, 2000), with E (Q, 1) defined similarly to (6).
As a consequence, the existence of Q O such that T (h) Q−1 Q−1 Mk (θ ) 2 k = 1, . . . , nu O αi − µ f (h) (h) Nik = 0 Ni (h) Mk (θ )Q−1 (h) nu˜ (56) (h) implies that (54) holds ∀xv ∈ E (Q, 1). By choosing Q = Xva , −1 the and applying the change of variable Γ(h) (θ ) = M (h) (θ )Xva LMIs (37) are obtained.
Provided that, if (36) holds, then the convergence of the closedloop trajectories of (41) to zero is assured as long as the inputs u do not saturate, the remaining of the proof will demonstrate that, if the LMIs (37) hold and
T ∈ E P, ν f (48) x(tI )T xc (tI )T then the additional effort brought by the virtual actuator will not cause the saturation of the control inputs u.
Finally, it is needed to demonstrate that if (48) holds, then (h) xv (t) ∈ E (Xva , 1) ∀t ≥ tI . This is straightforward, since (48) corresponds to (29), that is equivalent to
P x(tI ) T T ≤1 (57) x(tI ) xc (tI ) ν f xc (tI ) 21
SAFEPROCESS 2015 22 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
(h)
and, since xv (tI ) = 0 (see Eq. (31)), x(tI ) in (57) can be (h) replaced with xw (tI ), thus obtaining that V2 (tI ) ≤ 1, where V2 (t) is defined in (45). Due to the fact that V˙2 (t) < 0 ∀t ≥ tI , it (h) follows that xv (t) ∈ E (Xva , 1).
5 x with FTC
M
N
(θ (t)) =
∑
j=1
(h) µ j (θ (t))M( j)
x without FTC
3
x2 with FTC
1
x2 without FTC
2 system states
Also in this case, under the assumption that A (θ (t)) and the controller matrices Ac (θ (t)), Bc (θ (t)), Cc (θ (t)), Dc (θ (t)) are polytopic as in (15) and (20), with the coefficients µ j (θ (t)) satisfying (16), it is possible to use a polytopic virtual actuator gain (h)
1
4
1 0 −1 −2
(58)
−3 −4
and apply the following corollary to perform the design.
−5
(h)
−1 ∈ Snx ×nx and Γ Corollary 2. Let Xva ∈ Rnu ×nx , h = 1, . . . , n f , ( j) j = 1, . . . , N, be such that O2nx ×nx ν f Acl( j) P−1 ≺O (59) He (h) (h) (h) −1 ν f A∗( j) P−1 A( j) Xva + B∗ Γ( j) (h) T −1 Γk( j) Xva 2 i= 1, .. . , nu (60) O (h) αi − µ f (h) Ni = 0 Ni (h) Γk( j) (h) nu˜ hold ∀ j = 1, . . . , N, where A( j) + BDc( j)C BCc( j) Acl( j) = (61) Bc( j)C Ac( j) (h) (h) (h) (62) B − B∗ Dc( j)C B − B∗ Cc( j) A∗( j) =
0
2
4
6
8
10
time [s]
Fig. 1. State trajectory, t f = 0.5 s,tI = 0.8 s.
10 u
1
u2 5
0
−5
−10
(h)
µ f is defined as in (40), nu˜ is the number of non-zero elements (h) in Ni , and k in (60) takes values corresponding to the indices (h) of the non-zero elements in Ni . Then, if the vertex virtual ac(h) (h) tuators gains in (58) are calculated as M( j) = Γ( j) Xva , E (P, ν f ) is contractively invariant for the system (23)-(24) with control law (28), and E (P, ν f ) ⊆ L (u, α), ∀t ≥ tI .
0
2
4
6
8
10
time [s]
Fig. 2. Control inputs with FTC, t f = 0.5 s,tI = 0.8 s. Then, Corollary 2 is applied with a value ν f = 0.05, that corresponds to a value of µ f = 1.8610. The LPV virtual actuator gains are obtained using a feasible solution of LMIs (59)-(60).
Proof. It follows the reasoning provided in Corollary 1 and thus it is omitted.
Let us consider a simulation that lasts 10 s with x(0) = T T ( 1 0 ) , xc (0) = ( 0 0 ) , t f = 0.5 s and tI = 0.8 s. Since T ∈ E (P, 1), the state trajectory will converge x(0)T xc (0)T towards the origin, and the control input will not saturate, in the time interval [0,t f ], as shown in Fig. 1 (before the first vertical dotted black line) and Fig. 2, respectively.
4. EXAMPLE Let us consider an open-loop unstable LPV system subject to actuator saturations as in (23)-(24), with 2 + θ (t) 0 A (θ (t)) = θ ∈ [−1, 1] 1 1.5 20 t < tf B= 0 1 10 C= B(t) = 01 20 Bf = 0 0 t ≥ tf
At the fault occurrence time t f , the evolution of the Lyapunov function is such that V (t f ) = 0.032 < ν f = 0.05. However, due to the fault occurrence, the state trajectory will diverge during the time interval between t f and tI (see between the two vertical dotted black line in Fig. 1). Anyway, at the fault isolation time tI , V (tI ) = 0.044 < ν f = 0.05. Hence, according to the theory, the activation of the virtual actuator guarantees that the system trajectory will converge to zero with non-saturating control inputs despite the change in the input matrix from B to B f (see blue and red lines after the second vertical dotted black line in Fig. 1). In contrast, if the proposed FTC strategy was not used, the state trajectory would continue diverging (see cyan and magenta lines after the second vertical dotted black line in Fig. 1). To complete the analysis, the reconfiguration of the
and sat(u) defined as in (3) with αi = 10, i = 1, 2. By choosing X = I in order to guarantee that, if xc (0) = 0 and x1 (0)2 + x2 (0)2 ≤ 1, the control input will never saturate and the state trajectory will converge to zero, Corollary 1 is applied to the polytopic LPV system obtained using a bounding box approach, and the LMIs (17)-(19) are solved using the YALMIP toolbox (L¨ofberg, 2004) with SeDuMi solver (Sturm, 1999).
22
SAFEPROCESS 2015 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
23
J. L¨ofberg. YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference, pages pp. 284–289, 2004. J. Lunze and T. Steffen. Control reconfiguration after actuator failures using disturbance decoupling methods. IEEE Transactions on Automatic Control, vol. 51(10):pp. 1590–1601, 2006. P. Mhaskar, C. McFall, A. Gani, P. D. Christofides, and J. F. Davis. Isolation and handling of actuator faults in nonlinear systems. Automatica, vol. 44(1):pp. 53–62, 2008. E. F. Mulder, M. V. Kothare, and M. Morari. Multivariable antiwindup controller synthesis using linear matrix inequalities. Automatica, vol. 37(9):pp. 1407–1416, 2001. T. Nguyen and F. Jabbari. Output feedback controllers for disturbance attenuation with actuator amplitude and rate saturation. Automatica, vol. 36:pp. 1339–1346, 2000. H. Noura, D. Theilliol, J.-C. Ponsart, and A. Chamseddine. Fault-tolerant control systems: Design and Practical Applications. Springer-Verlag London Limited, 2009. X. Qi, D. Theilliol, J. Qi, Y. Zhang, L. Wang, and J. Han. Self healing control method against unmanned helicopter actuator stuck faults. In Proceedings of the International Conference on Unmanned Aircraft Systems (ICUAS), pages pp. 842–847, 2014. J. H. Richter, W. P. M. H. Heemels, N. van de Wouw, and J. Lunze. Reconfigurable control of piecewise affine systems with actuator and sensor faults: stability and tracking. Automatica, vol. 47:pp. 678–691, 2011. D. Rotondo, F. Nejjari, V. Puig, and J. Blesa. Model reference FTC for LPV systems using virtual actuators and set-membership fault estimation. International Journal of Robust and Nonlinear Control, 25(5):735–760, 2015a. D. Rotondo, J.-C. Ponsart, D. Theilliol, F. Nejjari, and V. Puig. A virtual actuator approach for the fault tolerant control of unstable linear systems subject to actuator saturation and fault isolation delay. Annual Reviews in Control, 39:68–80, 2015b. M. M. Seron, J. A. De Don´a, and J. Richter. Fault tolerant control using virtual actuators and set-separation detection principles. International Journal of Robust and Nonlinear Control, vol. 22:pp. 709–742, 2012. T. Steffen. Control reconfiguration of dynamical systems: Linear approaches and structural tests, volume 230 of Lecture Notes in Control and Information Sciences. Springer, 2005. J. Stoustrup and H. Niemann. Fault tolerant control for unstable systems: a linear time varying approach. In Proceedings of the American Control Conference, pages 1794–1798, 2004. J. F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization methods and software, vol. 11-12:pp. 625–653, 1999. H. J. Sussmann, E. D. Sontag, and Y. Yang. A general result on the stabilisation of linear systems using bounded controls. IEEE Transactions on Automatic Control, vol. 39:pp. 2411– 2425, 1994. S. Tarbouriech and M. Turner. Anti-windup design: an overview of some recent advances and open problems. IET Control Theory and Applications, vol. 3(1):pp. 1–19, 2009. Z. Weng, R. Patton, and P. Cui. Active fault-tolerant control for unstable systems. In Proceedings of the 4th IFAC Symposium on Mechatronic Systems, pages 79–84, 2006. Z. Zuo, D. W. C. Ho, and Y. Wang. Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation. Automatica, vol. 46(3):pp. 569–576, 2010.
control inputs brought by the change in the control law from uc (t) to u f (t) is depicted in Fig. 2. 5. CONCLUSIONS This paper has considered the design of an FTC strategy for unstable LPV systems subject to actuator saturations. At first, the nominal controller has been designed taking into account the saturations using a direct approach. Later, a virtual actuator has been added to the control scheme for reconfiguring the faulty plant in the event of an actuator fault. In this way, the property of fault tolerance is added to the control scheme without affecting the other properties, e.g. stability and performance, already attained by the controller in the nominal situation. The design conditions through which the virtual actuator gains are obtained are such that, if at the fault isolation time the closedloop system state is inside a region defined by a value of the Lyapunov function, the state trajectory will converge to zero despite the fault and, moreover, the inputs will not saturate at any time. A numerical example has shown the effectiveness of the proposed strategy. REFERENCES P. Apkarian, P. Gahinet, and G. Becker. Self-scheduled H∞ control of linear parameter-varying systems: a design example. Automatica, 31(9):1251–1261, 1995. M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki. Diagnosis and Fault-Tolerant Control. Springer-Verlag Berlin Heidelberg, 2006. J. Blesa, D. Rotondo, V. Puig, and F. Nejjari. FDI and FTC of wind turbines using the interval observer approach and virtual actuators/sensors. Control Engineering Practice, vol. 24:pp. 138–155, 2014. M. Chilali and P. Gahinet. H∞ design with pole placement constraints: an LMI approach. IEEE Transactions on Automatic Control, 41(3):358–367, 1996. Ł. Dziekan, M. Witczak, and J. Korbicz. Active fault-tolerant control design for Takagi-Sugeno fuzzy systems. Bulletin of the Polish Academy of Sciences, vol. 59(1):pp. 93–102, 2011. J. H. Fan, Y. M. Zhang, and Z. Q. Zheng. Robust fault-tolerant control against time-varying actuator faults and saturation. IET Control Theory and Applications, vol. 6(14):pp. 2198– 2208, 2012. R. Goebel, T. Hu, and A. R. Teel. Dual matrix inequalities in stability and performance analysis of linear differential/difference inclusions. Current Trends in Nonlinear Systems and Control Systems and Control: Foundations and Applications, pages pp. 103–122, 2006. R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, 1990. T. Hu, Z. Lin, and B. M. Chen. An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica, vol. 38:pp. 351–359, 2002. I. Hwang, S. Kim, Y. Kim, and C. E. Seah. A survey of fault detection, isolation, and reconfiguration methods. IEEE Transactions on Control Systems Technology, 18(3):636– 653, 2010. R. Isermann. Model-based fault-detection and diagnosis status and applications. Annual Reviews in Control, 29(1):71–85, 2005. T. Iwasaki and M. Fu. Regional H2 performance synthesis. In V. Kapila and K. M. Grigoriadis, editors, Actuator Saturation Control. 2002. 23
ScienceDirect
IFAC-PapersOnLine 48-21 (2015) 018–023
Fault tolerant control of unstable LPV systems Fault tolerant control of unstable LPV systems Fault tolerant control of unstable LPV systems Fault tolerant control of unstable LPV systems subject to actuator saturations using virtual subject to actuator saturations using virtual subject to actuator saturations using virtual subject to actuator saturations using virtual actuators actuators actuators actuators
∗ Jean-Christophe Ponsart ∗∗ Didier Theilliol ∗∗ Damiano Rotondo ∗∗ Jean-Christophe Ponsart ∗∗ ∗∗ ∗∗ Didier ∗∗ Damiano Rotondo Theilliol ∗ Vicenc ∗,∗∗∗ Damiano Rotondo Ponsart Theilliol ∗ Jean-Christophe ∗∗ Didier ∗∗ Fatiha Nejjari ¸ Puig Damiano Rotondo Jean-Christophe Ponsart Didier Theilliol ∗ ∗,∗∗∗ ∗ ∗,∗∗∗ Fatiha Nejjari Nejjari ∗ Vicenc Vicenc¸¸ Puig Puig ∗,∗∗∗ Fatiha Fatiha Nejjari Vicenc¸ Puig ∗ Automatic Control Department, Universitat Polit` eecnica de Catalunya ∗∗ Automatic Control Department, Universitat Polit` de Catalunya Control Department, Universitat Polit` eecnica cnica de Catalunya ∗ Automatic (UPC), Rambla de Sant Nebridi, 11, 08222 Terrassa, Spain Automatic Control Department, Universitat Polit` cnica de Catalunya (UPC), Rambla de Sant Nebridi, 11, 08222 Terrassa, Spain ∗∗ Centre (UPC), Rambla de Sant Nebridi, 11, 08222 Terrassa, Spain de Recherche en Automatique de Nancy (CRAN), Universit´ ee de (UPC), Rambla de Sant Nebridi, 11, 08222 Terrassa, Spain ∗∗ ∗∗ Centre de Recherche en Automatique de Nancy (CRAN), Universit´ de Recherche en Automatique de Nancy (CRAN), Universit´ ee de de ∗∗ Centre Lorraine, CNRS UMR 7039, 54506 Vandoeuvre-les-Nancy, France Centre de Recherche en Automatique de Nancy (CRAN), Universit´ de Lorraine, CNRS UMR 7039, 54506 Vandoeuvre-les-Nancy, France ∗∗∗ Institut Lorraine, CNRS UMR 7039, 54506 Vandoeuvre-les-Nancy, France de Robotica i Informatica Industrial (IRI), UPC-CSIC Lorraine, CNRS UMR 7039, 54506 Vandoeuvre-les-Nancy, France ∗∗∗ ∗∗∗ Institut de Robotica i Informatica Industrial (IRI), UPC-CSIC de Robotica Informatica Industrial (IRI), ∗∗∗ Institut Carrer Llorens ii iiArtigas, 4-6, 08028 Barcelona, Spain Institut de Robotica Informatica Industrial (IRI), UPC-CSIC UPC-CSIC Carrer de Llorens Artigas, 4-6, 08028 Barcelona, Spain Carrer de Llorens i Artigas, 4-6, 08028 Barcelona, Carrer de Llorens i Artigas, 4-6, 08028 Barcelona, Spain Spain
Abstract: An An active active fault fault tolerant tolerant control control (FTC) (FTC) strategy strategy for for open-loop open-loop unstable unstable linear linear parameter parameter varying varying Abstract: Abstract: An active fault tolerant control (FTC) strategy for open-loop unstable linear parameter varying (LPV) systems subject to actuator saturations and faults is proposed in this paper. At first, the nominal Abstract: An active fault tolerant control (FTC) strategy for open-loop unstable linear parameter varying (LPV) systems subject to actuator saturations and faults is proposed in this paper. At first, the nominal (LPV) systems subject to saturations proposed this paper. first, the controller is designed designed using direct approach and that faults takes is into accountin the actuator limits. Later, virtual (LPV) systems subjectusing to actuator actuator saturations and faults is proposed inthe thisactuator paper. At At first,Later, the nominal nominal controller is aaa direct approach that takes into account limits. virtual controller is designed using direct approach that takes into account the actuator limits. Later, virtual actuators are added to the control scheme, such that the faulty plant is reconfigured, and the fault controller is designed using a direct approach that takes into account the actuator limits. Later, virtual actuators are added to the control scheme, such that the faulty plant is reconfigured, and the fault actuators are added to the control scheme, such that the faulty plant is reconfigured, and the fault is hidden from the controller point of view. Conditions for designing the virtual actuator gains are actuators are added to the control scheme, such that the faulty plant is reconfigured, and the fault is hidden from the controller point of view. Conditions for designing the virtual actuator gains are is hidden from the controller point of view. Conditions for designing the virtual actuator gains are designed, aiming at guaranteeing that if at the fault isolation time the closed-loop system state is inside is hidden from the controller point of view. Conditions for designing the virtual actuator gains are designed, aiming aiming at at guaranteeing guaranteeing that that if if at at the the fault fault isolation isolation time time the the closed-loop closed-loop system system state state is is inside inside aaa designed, region defined by a value of the Lyapunov function, the state trajectory will converge to zero despite the designed, aiming at guaranteeing that if at the fault isolation time the closed-loop system state is inside a region defined by aa value of the Lyapunov function, the state trajectory will converge to zero despite the region defined by value of the Lyapunov function, the state trajectory will converge to zero despite the appearance of faults within a predefined set. The theoretical results are demonstrated using a numerical region defined by a value of the Lyapunov function, the state trajectory will converge to zero despite the appearance of faults within aa predefined set. The theoretical results are demonstrated using aa numerical appearance example. of appearance of faults faults within within a predefined predefined set. set. The The theoretical theoretical results results are are demonstrated demonstrated using using a numerical numerical example. example. example. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Linear Linear parameter parameter varying varying (LPV) (LPV) systems, systems, actuator actuator saturation, saturation, unstable unstable open open loop loop systems, systems, Keywords: Keywords: Linear parameter varying (LPV) systems, actuator saturation, unstable openactuators. loop systems, systems, linear matrix inequalities (LMIs), active fault tolerant control system (AFTCS), virtual Keywords: Linear parameter varying (LPV) systems, actuator saturation, unstable open loop linear matrix inequalities (LMIs), active fault tolerant control system (AFTCS), virtual actuators. linear matrix inequalities (LMIs), active fault tolerant control system (AFTCS), virtual actuators. linear matrix inequalities (LMIs), active fault tolerant control system (AFTCS), virtual actuators. 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION 1. In many many applications, applications, aa significant significant control control action action is is necessary necessary In In many many applications, adesired significant control action is necessary necessary in order to perform a task with a high performance. In applications, a significant control action is in perform aa desired task with aa high performance. in order order to tothe perform desired task withthe high performance. However, actuators which deliver control actions to in order to perform a desired task with a high performance. However, the which deliver the actions However, the actuators actuators which deliver the control control actions to to the controlled system are limited in magnitude (Tarbouriech However, the actuators which deliver the control actions to the controlled system are limited in magnitude (Tarbouriech the controlled system are limited in magnitude (Tarbouriech and controlled Turner, 2009). 2009). If these these limits are are not taken taken into into account, the system are limited in magnitude (Tarbouriech and Turner, limits account, Turner, 2009). If Ifofthese these limits are are not not taken into into and account, aaand strong degradation the closed-loop performance, even and Turner, 2009). If limits not taken account, of the closed-loop performance, and a strong strong degradation degradation ofFor thethis closed-loop performance, and even even instability, may occur. reason, approaches to deal with ainstability, strong degradation of the closed-loop performance, and even may occur. For this reason, approaches to deal with instability, may occur. For this reason, approaches to deal with this problem have been proposed, such as the anti-windup cominstability, may occur. For this reason, approaches to deal with this problem have been proposed, such as the anti-windup comthis problem have been proposed, such as the anti-windup compensation, where part proposed, of the controller dedicated to achieve this problem haveaabeen such asis the anti-windup compensation, where the controller is to achieve pensation, where a part part of ofand thethe controller is dedicated dedicated tothe achieve the desired performance other part deals with conpensation, where a part of the controller is dedicated to achieve the performance and the part deals the the desired desired performance and and the other other part approach, deals with withwhere the conconstraints (Mulder et al., 2001), the direct the the desired performance and the other part deals with the constraints (Mulder et al., 2001), and the direct approach, where the straints (Mulder et al., 2001), and the direct approach, where the controller is designed taking into account both desired perforstraints (Mulder et al., 2001), and the direct approach, where the controller is designed taking into account both desired perforcontroller is designed taking into account both desired performance and andissaturation saturation the account same time time (Sussmann et al., al., controller designed limits takingatinto both desired performance mance and and saturation saturation limits limits at at the the same same time time (Sussmann (Sussmann et et al., al., 1994). mance limits at the same (Sussmann et 1994). 1994). 1994). When actuator actuator faults faults are are considered, considered, mitigating mitigating the the effects effects of of When When actuator faults are considered, mitigating the effects of saturations becomes of utmost importance. In fact, in these When actuator faults of areutmost considered, mitigating the effects of saturations becomes importance. In fact, in these saturations becomes of utmost importance. In fact, in these cases, fault fault becomes tolerant control control (FTC) techniquesInare are usually apsaturations of utmost importance. fact, in these cases, tolerant (FTC) techniques usually cases, with fault the tolerant control (FTC) techniques are degradation usually apapplied aim of reducing the performance cases, fault tolerant control (FTC) techniques are usually applied with the aim of reducing the performance degradation plied with the aim of reducing the performance degradation and preserving the closed-loop stability in the event of faults plied with the aim of reducing stability the performance degradation and the in event and preserving preserving the closed-loop closed-loop stability in the thetechniques event of of faults faults (Blanke et al., al., 2006, 2006, Noura et et al., al.,stability 2009). These These usuand preserving the closed-loop in the event of faults (Blanke et Noura 2009). techniques usu(Blanke et al., 2006, Noura et al., 2009). These techniques usually redistribute, in some way, the control effort corresponding (Blanke et al., 2006, Noura et al., 2009). These techniques usually redistribute, in some way, the control effort corresponding ally redistribute, in some way, the control effort corresponding to the theredistribute, faulty actuators actuators among the healthy ones. This This ally in some way,the theremaining control effort corresponding to among healthy ones. to the the faulty faulty actuators actuators among the remaining remaining healthy ones. This This redistribution may lead to saturation of both the faulty and to faulty among the remaining healthy ones. redistribution redistribution may may lead lead to to saturation saturation of of both both the the faulty faulty and and redistribution may lead to saturation of both the faulty and This work has been funded by the Spanish Ministry of Science and Technol This work has been by Spanish Ministry of This work the has projects been funded funded by the the Spanish Ministry of Science Science and and TechnolTechnol ogy through CICYT ECOCIS (Ref. DPI2013-48243-C2-1-R) and This work the has projects been funded by the Spanish Ministry of Science and Technology through CICYT ECOCIS (Ref. DPI2013-48243-C2-1-R) and ogy through the projects CICYT ECOCIS (Ref. DPI2013-48243-C2-1-R) HARCRICS (Ref. DPI2014-58104-R), by (Ref. AGAUR through the contract and FIogy through the projects CICYT ECOCIS DPI2013-48243-C2-1-R) and HARCRICS (Ref. DPI2014-58104-R), by through the contract FIHARCRICS (Ref. DPI2014-58104-R), bytheAGAUR AGAUR through the de contract FIDGR 2014 (ref. 2014FI B1 00172) and by DGR of Generalitat Catalunya HARCRICS (Ref. DPI2014-58104-R), bytheAGAUR through the de contract FIDGR 2014 (ref. 2014FI B1 00172) and by DGR of Generalitat Catalunya DGR 2014 (ref. 2014FI B1 00172) and by the DGR of Generalitat de Catalunya (SAC 2014 group(ref. Ref.2014FI 2014/SGR/374). e-mail: [email protected] DGR B1 00172) and by the DGR of Generalitat de Catalunya (SAC group Ref. 2014/SGR/374). e-mail: [email protected] (SAC group Ref. 2014/SGR/374). e-mail: [email protected] (SAC group Ref. 2014/SGR/374). e-mail: [email protected]
the healthy actuators. It has been shown that neglecting this the healthy actuators. It has been shown that neglecting this the healthy actuators. It has been shown that neglecting this fact may lead to severe performance degradation and instability the healthy actuators. It has been shown that neglecting this fact may lead to severe performance degradation and instability fact may lead to severe performance degradation and instability (Fan et al., 2012). Even though several solutions for designing fact may lead to severe performance degradation and instability (Fan et et al., al., 2012). Even Even though though several several solutions solutions for for designing designing (Fan FTC systems subject to actuator have been proposed (Fan et al., 2012). 2012). Even though saturations several solutions for designing FTC systems subject to actuator saturations have been proposed FTC systems subject to actuator saturations have been proposed for the open-loop stable case, e.g. (Zuo et al., 2010, Mhaskar FTC systems subject to actuator saturations have been proposed for the open-loop stable case, e.g. (Zuo et al., 2010, Mhaskar for the open-loop stable case, e.g. (Zuo et al., 2010, Mhaskar et al., 2008), only a few works have considered this problem for the open-loop stable case, e.g. (Zuo et al., 2010, Mhaskar et al., 2008), only aa few works have considered this problem et al., 2008), only few works have considered this problem for open-loop unstable systems (Stoustrup and Niemann, 2004, et al., 2008), only a few works have considered this problem for open-loop open-loop unstable unstable systems systems (Stoustrup (Stoustrup and and Niemann, Niemann, 2004, for 2004, Weng et al., 2006, Qi et al., 2014). for open-loop unstable systems (Stoustrup and Niemann, 2004, Weng et al., 2006, Qi et al., 2014). Weng et al., 2006, Qi et al., 2014). Weng et al., 2006, Qi et al., 2014). The main contribution of this paper consists in designing an The main contribution of this paper consists in designing an The main contribution of this paper consists in designing an FTC strategy for open-loop unstable linear parameter varying The main contribution of this paper consists in designing an FTC strategy for open-loop unstable linear parameter varying FTC strategy for open-loop unstable linear parameter varying (LPV) systems subject to actuator saturations and faults. In FTC strategy for open-loop unstable linear parameter varying (LPV) systems subject to actuator saturations and faults. In (LPV) systems subject to actuator saturations and faults. In particular, the faults considered in this paper are undesired (LPV) systems subject to actuator saturations and faults. In particular, the the faults faults considered considered in in this this paper paper are are undesired undesired particular, changes in the input matrix that cause its loss of rank, e.g. particular, the faults considered in this paper are undesired changes in the input matrix that cause its loss of rank, e.g. changes in the input that its loss rank, total losses LPV systems are aa standard changes in of theactuators. input matrix matrix that cause cause its loss of of formalism rank, e.g. e.g. total losses of actuators. LPV systems are standard formalism total losses of actuators. LPV systems are a standard formalism that allows performing gain scheduling of nonlinear systems total losses of actuators. LPV systems are a standard formalism that allows performing gain scheduling of systems that allows performing gain scheduling of nonlinear nonlinear systems using an of invariant techniques. In that performing gaintime scheduling nonlinear systems usingallows an extension extension of linear linear time invariantof(LTI) (LTI) techniques. In using an extension of linear time invariant (LTI) techniques. In particular, this paper extends some results obtained in Rotondo using an extension of linear time invariant (LTI) techniques. In particular, this paper extends some results obtained in Rotondo particular, this paper extends some results obtained in Rotondo et al. (2015b) for the LTI case. It is worth highlighting that in particular, this paper extends some results obtained in Rotondo et al. (2015b) for the LTI case. It is worth highlighting that in et al. (2015b) for the LTI case. It is worth highlighting that in this work, it is assumed that the information about the fault is et al. (2015b) for the LTI case. It is worth highlighting that in this work, it is assumed that the information about the fault is this work, it is assumed that the information about the fault is provided by a fault detection and isolation (FDI) module. The this work, it is assumed that the information about the fault is provided by aa fault detection and isolation (FDI) module. The provided by fault detection and isolation (FDI) module. The design of this module is not considered in this work, due to provided by a fault detection and isolation (FDI) module. The design of this module is not considered in this work, due to design of this module is considered this to the fact theoretical is independent from design ofthat thisthe module is not notdevelopment considered in in this work, work, due due to the fact that the theoretical development is independent from the fact that the theoretical development is independent from strategy used to perform the FDI. The interested reader the fact that the theoretical development is independent from the strategy used to perform the FDI. The interested reader the strategy used to FDI. The reader could find useful about designing FDI strategies the strategy used information to perform perform the the FDI. The interested interested reader could find useful information about designing FDI strategies could find useful information about designing FDI strategies in Isermann (2005), Hwang et al. (2010), and the references could find useful information about designing FDI strategies in Isermann (2005), Hwang et al. (2010), and the references in Isermann therein. in Isermann (2005), (2005), Hwang Hwang et et al. al. (2010), (2010), and and the the references references therein. therein. therein. At first, aa nominal controller is designed using aa direct apAt first, controller is designed using apAt first, aa nominal nominal controller is designed using aa direct direct approach, i.e. taking into account the saturations. Then, a reconAt first, nominal controller is designed using direct approach, i.e. taking into account the saturations. Then, a reconproach, i.e. taking into account the saturations. Then, a reconfiguration block is inserted in the control scheme, such that proach, i.e. taking into account saturations. Then, a reconfiguration block block is inserted inserted in the the control scheme, scheme, such that that figuration figuration block is is inserted in in the control control scheme, such such that
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 IFAC 18 Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 18 Copyright ©under 2015 IFAC IFAC 18 Control. Peer review© of International Federation of Automatic Copyright © 2015 responsibility IFAC 18 10.1016/j.ifacol.2015.09.498
SAFEPROCESS 2015 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
the property of fault tolerance is added without affecting the other properties, e.g. stability and performance, already attained by the controller under nominal situation. This is achieved through a fault-hiding paradigm (Steffen, 2005), since the reconfiguration block hides the fault from the controller point of view. In case of actuator faults, the reconfiguration block is named virtual actuator, as initially proposed by Lunze and Steffen (2006), and further investigated by Dziekan et al. (2011), Richter et al. (2011), Seron et al. (2012), Rotondo et al. (2015a) and Blesa et al. (2014).
E (P, 1) =
x xc
∈R
2nx
:
x xc
T x ≤1 P xc
with P O is an ellipsoid, and x0 2nx ∈ R : lim ψ(t, x0 , xc0 ) = 0 S = xc0 t→∞
(6)
(7)
is the domain of attraction of the origin, with ψ(t, x0 , xc0 ) denoting the state trajectory of the closed-loop system given by (1)-(2) plus (4)-(5), obtained with x(0) = x0 and xc (0) = xc0 .
In other words, it is wished to design the controller (4)-(5) for the system (1)-(2) such that if the initial condition satisfies x(0) (8) x(0)T xc (0)T P x (0) ≤ 1 c then the control input will not saturate, and the closed-loop state trajectory ψ will converge to the origin.
The paper is structured as follows. Section 2 presents the design of the nominal controller. Section 3 describes the virtual actuator strategy. The theoretical results are illustrated using a numerical example in Section 4. Finally, Section 5 outlines the main conclusions. NOTATION Symbol Mi Mi j ∗ He {M} M ∈ Sn×n I O MO M≺O ⊗ †
19
In order to achieve this objective, the following Theorem is proposed, obtained as an extension of a similar theorem presented for the LTI case in Iwasaki and Fu (2002). Theorem 1. (LPV output feedback controller design) Let X,Y ∈ Snx ×nx , F(θ ) ∈ Rnx ×ny , K(θ ) ∈ Rnu ×nx and L(θ ) ∈ Rnu ×ny be such that He {XA(θ ) + F(θ )C} ≺ O (9) He {A(θ )Y + BK(θ )} ≺ O (10) X I CLi (θ )T I (11) Y Ki (θ )T O i = 1, . . . , nu 2 Li (θ )C Ki (θ ) αi ∀θ ∈ Θ. Then, the controller (4)-(5), with matrices calculated as −1 Z XB Ac (θ ) Bc (θ ) = ··· O I Cc (θ ) Dc (θ ) −1 T −Y O − XA(θ )Y F(θ ) − (A(θ ) + BL(θ )C) ··· CY I K(θ ) L(θ ) (12) Z = X −Y −1 (13) is such that, for the closed-loop system obtained with u(t) = uc (t), E (P, 1) ⊆ S and E (P, 1) ⊆ L (u, α) where
Description ith row of a matrix M Element i, j of a matrix M Symmetric element in a matrix Shorthand for M + M T Shorthand for a symmetric matrix M ∈ Rn×n Identity matrix of appropriate dimensions Zero matrix of appropriate dimensions Shorthand for a positive definite matrix Shorthand for a negative definite matrix Kronecker product Moore-Penrose pseudoinverse
2. DESIGN OF THE NOMINAL CONTROLLER Let us consider the following linear parameter varying (LPV) system subject to actuator saturations x(t) ˙ = A (θ (t)) x(t) + Bsat (u(t)) (1) y(t) = Cx(t) (2) n n n x u y where x ∈ R is the state, u ∈ R is the control input, y ∈ R is the measured output, A (θ (t)) ∈ Rnx ×nx is the parameter varying state matrix, whose values depend on the vector θ (t) ∈ Θ ⊂ Rnθ , B ∈ Rnx ×nu is the input matrix, C ∈ Rny ×nx is the output matrix, and sat : Rnu → Rnu is the saturation function, defined as sat1 (u1 ) .. αi (ui > αi ) . ui (|ui | ≤ αi ) sat(u) = sati (ui ) , sati (ui ) = .. −αi (ui < −αi ) . satnu (unu ) (3) where α = (α1 , . . . , αnu )T ∈ Rnu is a given vector with positive entries. For an output feedback law u(t) = g (y(t)) = g (Cx(t)), let us define L (u, α) the region of the state space in which the actuators are not saturated.
P=
X Z Z Z
(14)
Proof. The proof follows the reasoning developed by Iwasaki and Fu (2002) in the case of LTI systems, and is based on demonstrating that if (9)-(11) hold and the controller matrices are calculated as in (12), then E (P, 1) is contractively invariant, i.e. by defining the quadratic Lyapunov function V (x(t)) = x(t)T Px(t), it is obtained that V˙ (x(t)) < 0 for all x ∈ E (P, ρ)\ {0}, if the controller matrices are chosen as (12). Since E (P, 1) is contractively invariant, it is inside the domain of attraction S (Hu et al., 2002), such that the stability is guaranteed over the whole set of possible values of θ . The complete proof is omitted due to lack of space.
The problem solved hereafter is to design an LPV dynamic output feedback controller for the system (1)-(2) x˙c (t) = Ac (θ (t)) xc (t) + Bc (θ (t)) y(t) (4) uc (t) = Cc (θ (t)) xc (t) + Dc (θ (t)) y(t) (5) where xc ∈ Rnx is the controller state and uc ∈ Rnu is the controller output, such that when u(t) = uc (t), then E (P, 1) ⊆ S and E (P, 1) ⊆ L (u, α), where
The conditions provided by Theorem 1 rely on the satisfaction of infinite constraints, due to the fact that (9)-(11) should hold for all the possible values of θ . However, by considering a polytopic approach, as proposed by Apkarian et al. (1995) and Chilali and Gahinet (1996), (9)-(11) can be transformed in a finite number of linear matrix inequalities (LMIs), as shown by the following Corollary. 19
SAFEPROCESS 2015 20 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
(23) is the faulty system. It is assumed that the n f matrices
Corollary 1. (Polytopic LPV output feedback controller design) Assume that the LPV system (1) is polytopic, i.e. the matrix A (θ (t)) can be written as A (θ (t)) =
N
∑ µ j (θ (t))A( j)
(1)
(15)
and the pairs
j=1
where µ j , j = 1, . . . , N, are the coefficients of the polytopic decomposition, such that N
∑ µ j (θ (t)) = 1,
j=1
µ j (θ (t)) ≥ 0
∀θ ∈ Θ
(16)
(h) †
Bf
B
(27)
then the control input will not saturate for all t ≥ tI , and the state trajectory will converge to zero. This problem is equivalent to (n ) (1) the one of finding u f (t), . . . , u f f (t) that make the ellipsoid
T x x x ∈ R2nx : ≤ νf E (P, ν f ) = P (30) xc xc xc contractively invariant, and such that it is contained within L (u, α), for t ≥ tI .
The solution proposed in this paper relies on LPV virtual actuators, with the following structure, as depicted by Rotondo et al. (2015b) (h) (h) (h) (h) x˙v (t) = A (θ (t)) + B∗ M (h) (θ (t)) xv (t)+ B − B∗ uc (t) (31) (h) (h) u f (t) = N (h) uc (t) − M (h) (θ (t)) xv (t) (32)
Proof. Due to a basic property of matrices (Horn and Johnson, 1990), any linear combination of (17) with non-negative coefficients, of which at least one different from zero, is negative definite. Hence, using the coefficients µi (θ (t)), taking into account (16), and the fact that the same property holds for the negative and positive definiteness of (18)-(19), respectively, (9)(11) are obtained.
(h)
where h = 1, . . . , n f , xv are the virtual actuators states with (h) xv (tI ) = 0, M (h) (θ (t)) ∈ Rnu ×nx are the virtual actuators gains (h) to be designed, and the matrices N (h) and B∗ are given by † (h) B (33) N (h) = B f † (h) (h) (h) (h) B (34) B∗ = B f N (h) = B f B f
3. DESIGN OF THE VIRTUAL ACTUATOR Let us consider the following LPV system subject to actuator saturations x(t) ˙ = A (θ (t)) x(t) + B(t)sat (u(t)) (23)
t < tf B (n f ) (1) t ≥ tf Bf ∈ Bf ,...,Bf
where uc (t) is the output of the nominal controller (4)-(5) designed using the results presented in the previous section, and tI ∈ R+ , tI ≥ t f , is the fault isolation time, such that if the following condition holds x(tI ) T T (29) x(tI ) xc (tI ) P x (t ) ≤ ν f c I
j=1
B(t) =
(h)
Hereafter, it is solved the problem of designing the control law uc (t) t < tI (1) (1) u f (t) t ≥ tI , B(t) = B f u(t) = (28) .. .. . . u(n f ) (t) t ≥ t , B(t) = B(n f ) I f f
and vertex controller gains calculated as −1 Ac( j) Bc( j) Z XB −Y O (21) = Ξ( j) O I CY I Cc( j) Dc( j) −(A( j) + BL( j)C)T − XA( j)Y F( j) Ξ( j) = (22) K( j) L( j) with Z defined as in (13) is such that, for the closed-loop system obtained with u(t) = uc (t), E (P, 1) ⊆ S and E (P, 1) ⊆ L (u, α) with P defined as in (14).
(26)
are stabilizable, ∀θ ∈ Θ and ∀ h = 1, . . . , n f .
Then, the controller (4)-(5) with N Ac( j) Bc( j) Ac (θ (t)) Bc (θ (t)) (20) = ∑ µ j (θ (t)) Cc (θ (t)) Dc (θ (t)) Cc( j) Dc( j)
with
∈ Rnx ×nu are such that (h) rank B f < rank (B) A(θ ), B f
and let X,Y ∈ Snx ×nx , F( j) ∈ Rnx ×ny , K( j) ∈ Rnu ×nx and L( j) ∈ Rnu ×ny , j = 1, . . . , N, be such that (17) He XA( j) + F( j)C ≺ O He A( j)Y + BK( j) ≺ O (18) T X I CLi( j) Y Ki(T j) O i = 1, . . . , nu (19) I Li( j)C Ki( j) αi2
y(t) = Cx(t)
(n f )
Bf ,...,Bf
Also, in order to obtain the fault-hiding characteristic, the output equation (24) is slightly changed after tI , as follows (h) (h) y(t) = C x(t) + xv (t) t ≥ tI , B(t) = B f (35)
(24) (25)
The conditions for designing the virtual actuator gains M (h) (h) such that the control law (28) with u f , h = 1, . . . , n f , defined by (32), makes the ellipsoid E (P, ν f ), defined in (30), contractively invariant after tI , are given by the following Theorem, that constitutes an extension of a similar theorem provided for the LTI case in Rotondo et al. (2015b). −1 ∈ Snx ×nx and Γ(h) (θ ) ∈ Rnu ×nx , h = Theorem 2. Let Xva 1, . . . , n f be such that
and the saturation function sat (u(t)) defined as in (3). Before the fault occurrence time t f ∈ R+ , the matrix B(t) is the nominal input matrix B ∈ Rnx ×nu , and the corresponding LPV system, i.e. (1)-(2), is the nominal system. On the other hand, after t f , the matrix B(t) becomes the faulty input matrix B f ∈ Rnx ×nu , and the corresponding LPV system obtained from 20
SAFEPROCESS 2015 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
ν f Acl (θ )P−1 O2nx ×nx ≺O (h) (h) −1 ν f A∗ (θ )P−1 A(θ )Xva + B∗ Γ(h) (θ ) (h) −1 Xva Γk (θ )T 2 i= 1, .. . , nu α i O (h) (h) − µ f Ni = 0 Ni (h) Γk (θ ) (h) nu˜ hold ∀θ ∈ Θ, where A(θ ) + BDc (θ )C BCc (θ ) Acl (θ ) = Bc (θ )C Ac (θ ) (h) (h) (h) A∗ (θ ) = B − B∗ Dc (θ )C B − B∗ Cc (θ ) He
µ f = max uc E (P,ν f )
(h) nu˜
To this aim, since (32) is equivalent to (h) (h) (h) uc (t) − M (h) (θ )xv (t) u f ,i (t) = Ni
(36)
(h)
(37)
(38)
that leads to (h) (h) Ni µ f + M (h) (θ )xv (t) ≤ αi
(39)
At the expense of introducing conservativeness, it is possible to transform (52), whose left-hand side concerns the norm of a vector, into a condition about the norms of scalars. This is done by taking advantage of the fact that only the rows of M (h) (θ ) (h) corresponding to non-zero elements of Ni will contribute to (h) (h) u f ,i in (49). By denoting these rows as Mk (θ ), and the number
Proof. When the system is working in the region L (u, α), there exists a similarity transformation that transforms the closed-loop system made up by the system (23), with output equation (35) and control law (28), the nominal controller (4)(5), and the virtual actuator (31)-(32), in an equivalent blocktriangular form (h) (h) Acl (θ ) O2nx ×nx xcl (t) x˙cl (t) = (41) (h) (h) (h) (h) A∗ (θ ) Av (θ ) x˙v (t) xv (t) where (h) (h) (42) xcl (t) = xw (t) xc (t) (h)
(43)
(h)
(h)
(44)
Av (θ ) = A(θ ) + B∗ M(θ )
(51)
(h) For values of i such that Ni = 0, (51) is obviously satisfied. (h) On the other hand, when Ni = 0, (51) becomes αi (h) (h) (52) M (θ )xv ≤ (h) − µ f Ni
(40)
(h)
(h)
(49)
where u f ,i , i = 1, . . . , nu , denotes the ith input, the condition of non-saturation can be written as (h) (h) (h) uc (t) − M (h) (θ )xv (t) u f ,i (t) = Ni (h) (h) (50) ≤ Ni uc (t) + M (h) (θ )xv (t) (h) (h) (h) ≤ Ni µ f + M (θ )xv (t) ≤ αi
is the number of non-zero elements in Ni , and k in (37) takes values corresponding to the indices of the non-zero (h) elements in Ni . Then, if the virtual actuators gains M (h) (θ ) in (31)-(32) are calculated as M (h) (θ ) = Γ(h) (θ )Xva , E (P, ν f ) is contractively invariant for the system (23)-(24) with control law (28), and E (P, ν f ) ⊆ L (u, α), ∀t ≥ tI .
xw (t) = x(t) + xv (t)
21
(h)
(h)
of non-zero elements of Ni as nu˜ , (52) can be replaced by nu αi (h) (h) 2 − µf (53) ∑ Mk (θ )xv ≤ (h) (h) N i k=1,N =0 ik
that holds if
(h) (h) Mk (θ )xv ≤
By considering T P (h) (h) O x (t) xcl (t) 2n ×n x x cl νf (45) V2 (t) = (h) (h) xv (t) xv (t) Onx ×2nx Xva with Xva 0 to assess the stability of (41), the following Lyapunov inequality is obtained P (θ ) O A O 2nx ×nx cl 2nx ×nx ≺ 0 (46) He ν f (h) (h) O A (θ ) A (θ ) v ∗ Xva nx ×2nx that is equivalent to its dual version (Goebel et al., 2006) Acl (θ ) O2nx ×nx ν f P−1 O2nx ×nx He ≺ 0 (47) (h) (h) −1 Onx ×2nx Xva A∗ (θ ) Av (θ ) that can be brought to the LMI form (36) by considering −1 . Γ(h) (θ ) = M (h) (θ )Xva
αi (h) Ni
− µf
(h)
nu˜
k = 1, . . . , nu (h) Nik = 0
(54)
Given two vectors m and x, the existence of Q 0 such that Q−1 Q−1 m 0 (55) mT Q−1 γ 2 T implies that m x ≤ γ ∀x ∈ E (Q, 1) (Nguyen and Jabbari, 2000), with E (Q, 1) defined similarly to (6).
As a consequence, the existence of Q O such that T (h) Q−1 Q−1 Mk (θ ) 2 k = 1, . . . , nu O αi − µ f (h) (h) Nik = 0 Ni (h) Mk (θ )Q−1 (h) nu˜ (56) (h) implies that (54) holds ∀xv ∈ E (Q, 1). By choosing Q = Xva , −1 the and applying the change of variable Γ(h) (θ ) = M (h) (θ )Xva LMIs (37) are obtained.
Provided that, if (36) holds, then the convergence of the closedloop trajectories of (41) to zero is assured as long as the inputs u do not saturate, the remaining of the proof will demonstrate that, if the LMIs (37) hold and
T ∈ E P, ν f (48) x(tI )T xc (tI )T then the additional effort brought by the virtual actuator will not cause the saturation of the control inputs u.
Finally, it is needed to demonstrate that if (48) holds, then (h) xv (t) ∈ E (Xva , 1) ∀t ≥ tI . This is straightforward, since (48) corresponds to (29), that is equivalent to
P x(tI ) T T ≤1 (57) x(tI ) xc (tI ) ν f xc (tI ) 21
SAFEPROCESS 2015 22 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
(h)
and, since xv (tI ) = 0 (see Eq. (31)), x(tI ) in (57) can be (h) replaced with xw (tI ), thus obtaining that V2 (tI ) ≤ 1, where V2 (t) is defined in (45). Due to the fact that V˙2 (t) < 0 ∀t ≥ tI , it (h) follows that xv (t) ∈ E (Xva , 1).
5 x with FTC
M
N
(θ (t)) =
∑
j=1
(h) µ j (θ (t))M( j)
x without FTC
3
x2 with FTC
1
x2 without FTC
2 system states
Also in this case, under the assumption that A (θ (t)) and the controller matrices Ac (θ (t)), Bc (θ (t)), Cc (θ (t)), Dc (θ (t)) are polytopic as in (15) and (20), with the coefficients µ j (θ (t)) satisfying (16), it is possible to use a polytopic virtual actuator gain (h)
1
4
1 0 −1 −2
(58)
−3 −4
and apply the following corollary to perform the design.
−5
(h)
−1 ∈ Snx ×nx and Γ Corollary 2. Let Xva ∈ Rnu ×nx , h = 1, . . . , n f , ( j) j = 1, . . . , N, be such that O2nx ×nx ν f Acl( j) P−1 ≺O (59) He (h) (h) (h) −1 ν f A∗( j) P−1 A( j) Xva + B∗ Γ( j) (h) T −1 Γk( j) Xva 2 i= 1, .. . , nu (60) O (h) αi − µ f (h) Ni = 0 Ni (h) Γk( j) (h) nu˜ hold ∀ j = 1, . . . , N, where A( j) + BDc( j)C BCc( j) Acl( j) = (61) Bc( j)C Ac( j) (h) (h) (h) (62) B − B∗ Dc( j)C B − B∗ Cc( j) A∗( j) =
0
2
4
6
8
10
time [s]
Fig. 1. State trajectory, t f = 0.5 s,tI = 0.8 s.
10 u
1
u2 5
0
−5
−10
(h)
µ f is defined as in (40), nu˜ is the number of non-zero elements (h) in Ni , and k in (60) takes values corresponding to the indices (h) of the non-zero elements in Ni . Then, if the vertex virtual ac(h) (h) tuators gains in (58) are calculated as M( j) = Γ( j) Xva , E (P, ν f ) is contractively invariant for the system (23)-(24) with control law (28), and E (P, ν f ) ⊆ L (u, α), ∀t ≥ tI .
0
2
4
6
8
10
time [s]
Fig. 2. Control inputs with FTC, t f = 0.5 s,tI = 0.8 s. Then, Corollary 2 is applied with a value ν f = 0.05, that corresponds to a value of µ f = 1.8610. The LPV virtual actuator gains are obtained using a feasible solution of LMIs (59)-(60).
Proof. It follows the reasoning provided in Corollary 1 and thus it is omitted.
Let us consider a simulation that lasts 10 s with x(0) = T T ( 1 0 ) , xc (0) = ( 0 0 ) , t f = 0.5 s and tI = 0.8 s. Since T ∈ E (P, 1), the state trajectory will converge x(0)T xc (0)T towards the origin, and the control input will not saturate, in the time interval [0,t f ], as shown in Fig. 1 (before the first vertical dotted black line) and Fig. 2, respectively.
4. EXAMPLE Let us consider an open-loop unstable LPV system subject to actuator saturations as in (23)-(24), with 2 + θ (t) 0 A (θ (t)) = θ ∈ [−1, 1] 1 1.5 20 t < tf B= 0 1 10 C= B(t) = 01 20 Bf = 0 0 t ≥ tf
At the fault occurrence time t f , the evolution of the Lyapunov function is such that V (t f ) = 0.032 < ν f = 0.05. However, due to the fault occurrence, the state trajectory will diverge during the time interval between t f and tI (see between the two vertical dotted black line in Fig. 1). Anyway, at the fault isolation time tI , V (tI ) = 0.044 < ν f = 0.05. Hence, according to the theory, the activation of the virtual actuator guarantees that the system trajectory will converge to zero with non-saturating control inputs despite the change in the input matrix from B to B f (see blue and red lines after the second vertical dotted black line in Fig. 1). In contrast, if the proposed FTC strategy was not used, the state trajectory would continue diverging (see cyan and magenta lines after the second vertical dotted black line in Fig. 1). To complete the analysis, the reconfiguration of the
and sat(u) defined as in (3) with αi = 10, i = 1, 2. By choosing X = I in order to guarantee that, if xc (0) = 0 and x1 (0)2 + x2 (0)2 ≤ 1, the control input will never saturate and the state trajectory will converge to zero, Corollary 1 is applied to the polytopic LPV system obtained using a bounding box approach, and the LMIs (17)-(19) are solved using the YALMIP toolbox (L¨ofberg, 2004) with SeDuMi solver (Sturm, 1999).
22
SAFEPROCESS 2015 September 2-4, 2015. Paris, France
Damiano Rotondo et al. / IFAC-PapersOnLine 48-21 (2015) 018–023
23
J. L¨ofberg. YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference, pages pp. 284–289, 2004. J. Lunze and T. Steffen. Control reconfiguration after actuator failures using disturbance decoupling methods. IEEE Transactions on Automatic Control, vol. 51(10):pp. 1590–1601, 2006. P. Mhaskar, C. McFall, A. Gani, P. D. Christofides, and J. F. Davis. Isolation and handling of actuator faults in nonlinear systems. Automatica, vol. 44(1):pp. 53–62, 2008. E. F. Mulder, M. V. Kothare, and M. Morari. Multivariable antiwindup controller synthesis using linear matrix inequalities. Automatica, vol. 37(9):pp. 1407–1416, 2001. T. Nguyen and F. Jabbari. Output feedback controllers for disturbance attenuation with actuator amplitude and rate saturation. Automatica, vol. 36:pp. 1339–1346, 2000. H. Noura, D. Theilliol, J.-C. Ponsart, and A. Chamseddine. Fault-tolerant control systems: Design and Practical Applications. Springer-Verlag London Limited, 2009. X. Qi, D. Theilliol, J. Qi, Y. Zhang, L. Wang, and J. Han. Self healing control method against unmanned helicopter actuator stuck faults. In Proceedings of the International Conference on Unmanned Aircraft Systems (ICUAS), pages pp. 842–847, 2014. J. H. Richter, W. P. M. H. Heemels, N. van de Wouw, and J. Lunze. Reconfigurable control of piecewise affine systems with actuator and sensor faults: stability and tracking. Automatica, vol. 47:pp. 678–691, 2011. D. Rotondo, F. Nejjari, V. Puig, and J. Blesa. Model reference FTC for LPV systems using virtual actuators and set-membership fault estimation. International Journal of Robust and Nonlinear Control, 25(5):735–760, 2015a. D. Rotondo, J.-C. Ponsart, D. Theilliol, F. Nejjari, and V. Puig. A virtual actuator approach for the fault tolerant control of unstable linear systems subject to actuator saturation and fault isolation delay. Annual Reviews in Control, 39:68–80, 2015b. M. M. Seron, J. A. De Don´a, and J. Richter. Fault tolerant control using virtual actuators and set-separation detection principles. International Journal of Robust and Nonlinear Control, vol. 22:pp. 709–742, 2012. T. Steffen. Control reconfiguration of dynamical systems: Linear approaches and structural tests, volume 230 of Lecture Notes in Control and Information Sciences. Springer, 2005. J. Stoustrup and H. Niemann. Fault tolerant control for unstable systems: a linear time varying approach. In Proceedings of the American Control Conference, pages 1794–1798, 2004. J. F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization methods and software, vol. 11-12:pp. 625–653, 1999. H. J. Sussmann, E. D. Sontag, and Y. Yang. A general result on the stabilisation of linear systems using bounded controls. IEEE Transactions on Automatic Control, vol. 39:pp. 2411– 2425, 1994. S. Tarbouriech and M. Turner. Anti-windup design: an overview of some recent advances and open problems. IET Control Theory and Applications, vol. 3(1):pp. 1–19, 2009. Z. Weng, R. Patton, and P. Cui. Active fault-tolerant control for unstable systems. In Proceedings of the 4th IFAC Symposium on Mechatronic Systems, pages 79–84, 2006. Z. Zuo, D. W. C. Ho, and Y. Wang. Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation. Automatica, vol. 46(3):pp. 569–576, 2010.
control inputs brought by the change in the control law from uc (t) to u f (t) is depicted in Fig. 2. 5. CONCLUSIONS This paper has considered the design of an FTC strategy for unstable LPV systems subject to actuator saturations. At first, the nominal controller has been designed taking into account the saturations using a direct approach. Later, a virtual actuator has been added to the control scheme for reconfiguring the faulty plant in the event of an actuator fault. In this way, the property of fault tolerance is added to the control scheme without affecting the other properties, e.g. stability and performance, already attained by the controller in the nominal situation. The design conditions through which the virtual actuator gains are obtained are such that, if at the fault isolation time the closedloop system state is inside a region defined by a value of the Lyapunov function, the state trajectory will converge to zero despite the fault and, moreover, the inputs will not saturate at any time. A numerical example has shown the effectiveness of the proposed strategy. REFERENCES P. Apkarian, P. Gahinet, and G. Becker. Self-scheduled H∞ control of linear parameter-varying systems: a design example. Automatica, 31(9):1251–1261, 1995. M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki. Diagnosis and Fault-Tolerant Control. Springer-Verlag Berlin Heidelberg, 2006. J. Blesa, D. Rotondo, V. Puig, and F. Nejjari. FDI and FTC of wind turbines using the interval observer approach and virtual actuators/sensors. Control Engineering Practice, vol. 24:pp. 138–155, 2014. M. Chilali and P. Gahinet. H∞ design with pole placement constraints: an LMI approach. IEEE Transactions on Automatic Control, 41(3):358–367, 1996. Ł. Dziekan, M. Witczak, and J. Korbicz. Active fault-tolerant control design for Takagi-Sugeno fuzzy systems. Bulletin of the Polish Academy of Sciences, vol. 59(1):pp. 93–102, 2011. J. H. Fan, Y. M. Zhang, and Z. Q. Zheng. Robust fault-tolerant control against time-varying actuator faults and saturation. IET Control Theory and Applications, vol. 6(14):pp. 2198– 2208, 2012. R. Goebel, T. Hu, and A. R. Teel. Dual matrix inequalities in stability and performance analysis of linear differential/difference inclusions. Current Trends in Nonlinear Systems and Control Systems and Control: Foundations and Applications, pages pp. 103–122, 2006. R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, 1990. T. Hu, Z. Lin, and B. M. Chen. An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica, vol. 38:pp. 351–359, 2002. I. Hwang, S. Kim, Y. Kim, and C. E. Seah. A survey of fault detection, isolation, and reconfiguration methods. IEEE Transactions on Control Systems Technology, 18(3):636– 653, 2010. R. Isermann. Model-based fault-detection and diagnosis status and applications. Annual Reviews in Control, 29(1):71–85, 2005. T. Iwasaki and M. Fu. Regional H2 performance synthesis. In V. Kapila and K. M. Grigoriadis, editors, Actuator Saturation Control. 2002. 23