Fault-tolerant Control of Linear Singularly Perturbed Systems with Applications to Hypersonic Vehicles⋆

Fault-tolerant Control of Linear Singularly Perturbed Systems with Applications to Hypersonic Vehicles⋆

10th IFAC Symposium on Fault Detection, 10th IFAC Symposium on Fault Detection, Supervision and Safetyon for Technical Processes 10th IFAC Symposium D...

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10th IFAC Symposium on Fault Detection, 10th IFAC Symposium on Fault Detection, Supervision and Safetyon for Technical Processes 10th IFAC Symposium Detection, Supervision and Safety forFault Technical Processes Available online at www.sciencedirect.com 10th IFACPoland, Symposium on Fault Detection, Warsaw, August 29-31, 2018 Supervision and Safety for Technical Processes Warsaw, Poland, August 29-31, 2018 Supervision and Safety Technical Warsaw, Poland, Augustfor 29-31, 2018 Processes Warsaw, Poland, August 29-31, 2018

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IFAC PapersOnLine 51-24 (2018) 268–273

Fault-tolerant Control of Linear Singularly Fault-tolerant Control of Linear Singularly Fault-tolerant Control of Linear Singularly Perturbed Systems with Applications to Fault-tolerant Control of Linear Singularly Perturbed Systems with Applications to Perturbed Systems with Applications to ⋆ ⋆ Hypersonic Vehicles PerturbedHypersonic Systems with Applications to Vehicles ⋆ Hypersonic Vehicles ⋆ Hypersonic Vehicles Wenjing Ren, Bin Jiang, Hao Yang, Liyan Wen

Wenjing Ren, Bin Jiang, Hao Yang, Liyan Wen Wenjing Ren, Bin Jiang, Hao Yang, Liyan Wen Wenjing Ren,Engineering, Bin Jiang,Nanjing Hao Yang, LiyanofWen College of Automation University Aeronautics College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China of Aeronautics College (e-mail: of Automation Engineering, Nanjing University and Astronautics, Nanjing 211106, China [email protected], [email protected], (e-mail:and [email protected], [email protected], Astronautics, Nanjing 211106, China [email protected], [email protected]). (e-mail: [email protected], [email protected], [email protected], [email protected]). (e-mail: [email protected], [email protected], [email protected], [email protected]). [email protected], [email protected]). Abstract: Abstract: This This paper paper studies studies the the fault-tolerant fault-tolerant control control (FTC) (FTC) problem problem of of linear linear singular singular perturbed systems with component faults. Based on the analysis of two decomposed lowerAbstract: This paper studies the fault-tolerant control (FTC) problem of linear singular perturbed systems with component faults. Based on the analysis of two decomposed lowerAbstract: This paper studies the fault-tolerant control (FTC) problem of linear stabilize singular order subsystems, a composite fault-tolerant controller is designed to asymptotically perturbed systems with component faults. Based on the analysis of two decomposed lowerorder subsystems, a composite fault-tolerant controller is designed to asymptotically stabilize perturbed systems component faults. Based on the analysis of decomposed lowerorder subsystems, awith composite fault-tolerant controller is designed to two asymptotically stabilize the original impaired system through cooperative regulation of the control gains. An application the original impaired system through cooperative regulation of the control gains. An application order subsystems, a control composite fault-tolerant controller is designed to asymptotically stabilize to the longitudinal system of a hypersonic vehicle with actuator faults is given. the original impaired system through cooperative regulation of the control gains. An application to longitudinal control system ofcooperative a hypersonic vehicle of with actuator faultsAn is application given. The The thethe original impairedcontrol systemof through regulation theactuator control gains. singularly perturbed model such a system is derived and a composite fault-tolerant controller to the longitudinal system of a hypersonic vehicle with faults is given. The singularly perturbed model of such a system is derived and a composite fault-tolerant controller to designed, the longitudinal control system a hypersonic vehicle actuator faults is given. The is which the effectiveness the method. singularly model of such aofsystem is of derived and awith composite controller is designed,perturbed which illustrates illustrates effectiveness the proposed proposed method. fault-tolerant singularly model ofthe such a system is of derived and a composite is designed,perturbed which illustrates the effectiveness of the proposed method. fault-tolerant controller © designed, 2018, IFACwhich (International Federation of Automatic Hostingmethod. by Elsevier Ltd. All rights reserved. is illustrates the effectiveness of Control) the proposed Keywords: Keywords: Fault-tolerant Fault-tolerant control, control, singular singular perturbation, perturbation, linear linear system, system, hypersonic hypersonic vehicle vehicle Keywords: Fault-tolerant control, singular perturbation, linear system, hypersonic vehicle Keywords: Fault-tolerant control, singular perturbation, linear system, hypersonic vehicle 1. vehicles in Naidu (2010). In Zhu et al. (2000), aa singular 1. INTRODUCTION INTRODUCTION vehicles in Naidu (2010). In Zhu et al. (2000), singular 1. INTRODUCTION perturbation approach is applied to the attitude control vehicles in Naidu (2010). Zhu et a singular perturbation approach is In applied toal. the(2000), attitude control 1. INTRODUCTION vehicles in Naidu (2010). In Zhu by et al. athe singular of the X-33 ascent flightismission inner perturbation approach applied toconsidering the(2000), attitude control Singular perturbation method emerges from the boundary Singular perturbation method emerges from the boundary of the X-33 ascent flight mission by considering the inner approach applied toconsidering thetoattitude of thedynamics X-33 ascent flightismission by thecontrol inner loop as a singular perturbation the outer loop. Singular perturbation the boundary layer theory in fluid method systems emerges at the from beginning of 20th perturbation loop dynamics as a singular perturbation to the outer loop. layer theory in fluid systems at the beginning of 20th of the X-33 ascent flight mission by considering the inner Singular perturbation method emerges from the boundary loop dynamics as a singular perturbation to the outer loop. Singular perturbation theory is also used in rotorcraft layer theory in fluid systems at the to beginning of 20th century. Its applications have spread many other ar- Singular perturbation theory is also used in rotorcraft century. Its applications have spread to many other arloop dynamics as a singular perturbation to the outer loop. layer theory in fluid systems at the beginning of 20th control, see Kim et al. (2014); Xu et al. (2017). Results on century. Its applications have spread to many other arSingular perturbation theory is also used in rotorcraft eas of mathematical physics and engineering. In control control, see Kim et al. (2014); Xu et al. (2017). Results on eas of physics and engineering. In control perturbation theory is et also used in Results rotorcraft century. Its applications have spread to manyof ar- Singular the singular perturbation approaches in aerospace control eas of mathematical mathematical physics and engineering. Inother control control, see Kim et al. (2014); Xu al. (2017). on systems, the boundary layer is a characteristic systems the singular perturbation approaches in aerospace control systems, the boundary layer is a characteristic of systems control, see Kim et al. (2014); Xu et al. Results on eas ofthe mathematical Inphenomcontrol the singular perturbation approaches in(2017). aerospace control can be found in the references of the survey papers, Naidu systems, the boundaryphysics layer isand a characteristic of systems with simultaneous occurrence ofengineering. slow and fast can be foundperturbation in the references of the survey papers,control Naidu with the simultaneous occurrence of slow and fast phenomthe singular approaches in aerospace systems, the boundary layer is a characteristic of systems can be found in the references of the survey papers, Naidu and Calise (2001); Ramnath (2010). with the simultaneous occurrence of slow and fast phenomena which also called the two-time-scale behavior. This and Calise (2001); Ramnath (2010). ena also called the two-time-scale behavior. This found in theRamnath references(2010). of the survey papers, Naidu with which the simultaneous occurrence slow and fast phenomena which the to two-time-scale behavior. This and be Calise (2001); behavior is also verycalled common mostof physical systems with can The FTC considered in this paper requires that the system behavior is very common to most physical systems with and Calise (2001); Ramnath (2010). ena which the to two-time-scale behavior. This behavior is also verycalled common most time physical systems with The FTC considered in this paper requires that the system parasitic parameters, such small constants, inertia, parasitic parameters, such as as small time constants, inertia, functions well in the presence of impaired The FTC considered in this paper thatcomponents the system behavior is very common to most physical systems with functions well in the presence of impaired components parasitic parameters, suchcapacitances, as small timeetc.. constants, inertia, The FTC considered in this paper requires resistances, inductances, The standard requires that thesystems, system (Patton (2015); Zhang and Jiang (2008)). In FTC functions well in the presence of impaired components resistances, inductances, capacitances, etc.. The standard parasitic parameters, suchcapacitances, asofsmall timeetc.. constants, inertia, (2015); Zhang and Jiang of (2008)). In FTC systems, resistances, inductances, The is standard singularly perturbed model a dynamic system a state (Patton functions well in the presence impaired components the controller either designed aa priori to be to (Patton (2015);is Zhang Jiang (2008)). systems, singularly perturbed model of system aa state resistances, inductances, capacitances, etc.. The is standard the controller eitherand designed priori In toFTC be robust robust to singularly perturbed model of a aofdynamic dynamic system is state model where the some the are (Patton (2015);is Zhang and Jiang (2008)). In systems, controller is either designed aal.priori toFTC be robust to model where the derivatives derivatives ofdynamic some of ofsystem the states states are the presumed faults (see Xu et (2015)), or reconfigsingularly perturbed model of a is a state presumed faults (see Xu et al. (2015)), or reconfigmodel where the derivatives of some of the states are the multiplied by a small perturbation parameter ε. Setting the controller is either designed aal.priori to be robust to multiplied by a small perturbation parameter ε. Setting ured when fault occurs to compensate the influence of the presumed faults (see Xu et (2015)), or reconfigderivatives of some of states are ured when fault occurs toXucompensate the influence of multiplied by the a to small perturbation ε. Setting εmodel = 0 iswhere helpful reduce the dimension of the the system. parameter the presumed faults (see et al. Zhao (2015)), or reconfigthe fault (see Yang et al. (2015); et al. (2016)). ured when fault occurs to compensate the influence of εεmultiplied = 0 is helpful to reduce the dimension of the system. by a to small perturbation parameter ε. Setting the fault (see Yang et al. (2015); Zhao et al. (2016)). = 0 is helpful reduce the dimension of the system. ured when(see fault occurs to (2015); compensate the influence of The FTC of hypersonic vehicles has been studied in Xu the fault Yang et al. Zhao et al. (2016)). The stability analysis and stabilization problems have ε = 0stability is helpfulanalysis to reduceand thestabilization dimension of problems the system. FTC (see of hypersonic vehicles hasZhao been et studied in Xu The have The the fault Yang et al. (2015); al. (2016)). The FTC of hypersonic vehicles has been studied in Xu al. (2015); Jiang et al. (2010). However, the FTC of The been stability studied deeply. have published, analysisMany and results stabilization problems have et deeply. Many have been been published, al.FTC (2015); Jiang et al. (2010).has However, the FTC of been studied The of hypersonic vehicles been studied in Xu The Kokotovi´ stability analysis and results stabilization problems have et singularly perturbed is addressed in been studied cdeeply. results published, et al. (2015); Jiang etsystems al. (2010). However, the FTC of see et al. Many (1999); Naiduhave andbeen Calise (2001); singularly perturbed systems is barely barely addressed in the the see Kokotovi´ c et al. (1999); Naidu and Calise (2001); et al. (2015); Jiang et al. (2010). However, the FTC of beenKokotovi´ studied Many results have published, current literature. Thesystems fault estimation is studied see cdeeply. et al. (1999); Naidu andbeen Calise (2001); singularly perturbed is barelyproblem addressed in the Khalil (2002); Ren et al. (2016). A representative idea is Khalil (2002);c Ren et al. (2016). A representative idea is current literature. Thesystems fault estimation problem is studied singularly perturbed is barely addressed in the see Kokotovi´ et al. (1999); Naidu and Calise (2001); for Lipschitz nonlinear singularly perturbed systems with Khalil (2002);the Ren et al. (2016). A representative idea is current literature. The fault estimation problem is studied decomposing singularly perturbed system for Lipschitz nonlinear singularly perturbed systems with decomposing the singularly perturbed system into into aa slow slow current literature. The fault estimation problem is FTC studied Khalil (2002); Ren etone, al. (2016). A representative is for Lipschitz nonlinear singularly perturbed systems with respect to sensor faults in Liu et al. (2016). The of decomposing singularly perturbed system into idea a slow subsystem andthe a fast and analyzing the stability and respect to sensor faults in Liu et al. (2016). The FTC of subsystem and a fast one, and analyzing the stability and for Lipschitz nonlinear singularly perturbed systems with decomposing the singularly perturbed system into a slow nonlinear singularly perturbed system with the actuator subsystem and a fast one, and analyzing the stability and respect to sensor faults in Liu et al. (2016). The FTC of stabilizing problems of each subsystem with the perturba- nonlinear singularly perturbed system with the actuator stabilizing problems of each subsystem with the perturbarespect to sensor faults in Liu et al. (2016). The FTC of subsystem and aε fast and analyzing thethe stability and nonlinear affected bysingularly additive perturbed faults is considered, which imposes stabilizing problems ofone, each subsystem with perturbasystem with the actuator tion parameter small enough. tion parameter ε small enough. affected by additive faults is considered, which imposes nonlinear singularly perturbed system with the actuator stabilizing problems of each subsystem with the perturbaaffected by additive faults is considered, which imposes strong constraint on the type of fault. Therefore, the FTC tion parameter ε small enough. constraint on type fault. Therefore, the FTC In the field of εaerospace control systems, singular per- strong affected by additive faults imposes tionthe parameter small enough. strong constraint on the the typeisof ofconsidered, fault. Therefore, FTC of singularly perturbed systems is still an which open the problem In field of aerospace control systems, singular persingularly perturbed systems is still an open problem In the field of aerospace control systems, singular per- of turbation method has been widely used due to the timestrong constraint on the systems type of fault. Therefore, the FTC which demands further investigation. In this paper, we of singularly perturbed is still an open problem turbation method has been widely used due to the timedemands further systems investigation. In we In theproperty field ofofaerospace per- which turbation method has systems. beencontrol widely used duesingular to the timescale such Thesystems, singular perturbation of singularly perturbed is stillperturbed anthis openpaper, problem address the FTC of linear singularly systems which demands further investigation. In this paper, we scale property of such systems. The singular perturbation address the FTC of linear singularly perturbed systems turbation method has been widely used due to the timescale property of such systems. Thenon-dimensional singular perturbation parameter is identified by using forms address which demands investigation. In this paper, we the FTCfurther of linear singularly perturbed systems with component faults. The main contributions of this parameter is identified by using non-dimensional forms component faults. Thesingularly main contributions of this scalenonlinear property of such systems. Thenon-dimensional singular perturbation parameter is identified byequation using forms with for dynamical described hypersonic address the FTC of linear perturbed systems with component faults. The main contributions of this paper are as follows: for nonlinear dynamical described hypersonic parameter is identified using non-dimensional forms with papercomponent are as follows: for nonlinear dynamicalbyequation equation described hypersonic faults. The main contributions of this paper are as follows: for nonlinear dynamical equation described hypersonic ⋆ •• A novel composite work is supported by National Natural Science Foundation of paper are as follows: ⋆ This A novel composite fault-tolerant fault-tolerant controller controller is is designed designed This work is supported by National Natural Science Foundation of • by A novel composite fault-tolerant controller is designed ⋆ regulating the control gains of the two decomposed China (61533009, 61773201, 61622304), Natural Science Foundation This work is supported by National Natural Science Foundation of by regulating the control gains of the two decomposed China (61533009, 61773201, 61622304), Natural Science Foundation ⋆ • A novel composite fault-tolerant controller is designed work is supported by National Natural Science Foundation of of This Jiangsu province (BK20160035), Fundamental Research Funds for subsystems cooperatively according ε. Such by regulating the control gains of theto two China (61533009, 61622304), Natural Science Foundation of Jiangsu province61773201, (BK20160035), Fundamental Research Funds for subsystems cooperatively according to ε. decomposed Such aa conconby regulating the control gains of the two decomposed China (61533009, 61773201, 61622304), Natural Science Foundation theJiangsu Central Universities (NE2014202, NE2015002), and theFunds Funding of (BK20160035), Fundamental Research for troller can maintain the asymptotical the subsystems cooperatively according tostability ε. Such aofconthe Centralprovince Universities (NE2014202, NE2015002), and the Funding troller can maintain the asymptotical stability of the of Jiangsu (BK20160035), Research Funds for subsystems cooperatively according tostability ε. Such aofconJiangsu Innovation Program for Fundamental Graduate Education (KYCX17the Centralprovince Universities (NE2014202, NE2015002), and the Funding troller can maintain the asymptotical the faulty system. of Jiangsu Innovation Program for Graduate Education (KYCX17faulty system. the CentralInnovation Universities (NE2014202, NE2015002), and the Funding 0268). of Jiangsu Program for Graduate Education (KYCX17troller can maintain the asymptotical stability of the faulty system. 0268). of Jiangsu Innovation Program for Graduate Education (KYCX170268). faulty system.

0268). 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 268 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 268 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 268 10.1016/j.ifacol.2018.09.587 Copyright © 2018 IFAC 268

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ε0 =

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1

√ −1 ∥A−1 ∥(∥A ∥ + ∥A ∥∥A A ∥ + 2 ∥A0 ∥∥A12 ∥∥A−1 0 12 21 22 22 22 A21 ∥)

• A singularly perturbed model of the longitudinal control system of an air-breathing hypersonic vehicle with actuator fault is developed. And the theoretical results are applied successfully which achieves the FTC goal of the hypersonic vehicle. The remainder of this paper is organized as follows. Some preliminary results on singular perturbation theory and the problem intended to be solved are presented in Section 2. An FTC strategy for linear singularly perturbed system with component faults is developed in Section 3. Then, an application on a hypersonic vehicle is illustrated in Section 4, which followed by some concluding remarks of this paper in Section 5. n

Notations: ℜ and ℜ respectively denote the real number field and the n-dimensional real vector space. The symbol ∥ · ∥ represents the Euclidian norm of a matrix. The notation λ(·) means the eigenvalue of a n × n dimensional matrix, and Reλ(·) is the real part of the eigenvalue of the matrix. 2. PRELIMINARIES AND PROBLEM FORMULATION In this section, we introduce you the model of the linear singularly perturbed system and some basic related lemmas. After that, the control problem we need to solve is formulated. 2.1 Preliminaries Consider an autonomous singularly perturbed system x(t) ˙ = A11 x(t) + A12 z(t), x(t0 ) = x0 ,

(1)

εz(t) ˙ = A21 x(t) + A22 z(t), z(t0 ) = z0 ,

(2)

where ε is a small parameter, x(t) ∈ ℜm and z(t) ∈ ℜn are states. By setting ε = 0 and assuming that A22 is invertible, the system (1)-(2) can be decomposed into a slow system and a fast system. The mth-order slow system takes the following form x˙ s (t) = A0 xs (t), xs (t0 ) = x0 , zs (t) = −A−1 22 A21 xs (t), −1 A11 − A12 A22 A21 , the vectors

(3) (4)

where A0 = xs (t), zs (t) are the slow parts of the corresponding variables x(t), z(t) in the original system (1)-(2). And the nth-order fast system is εz˙f (t) = A22 zf (t), zf (t0 ) = z0 + A−1 22 A21 x0 ,

269

(8)

the states of the original system (1)-(2) starting from any bounded initial conditions x0 and z0 , are approximated for all t ∈ [t0 , ∞) by x(t) = xs (t) + O(ε),

(6)

−A−1 22 A21 xs (t)

+ zf (t) + O(ε), (7) z(t) = where xs (t) and zf (t) are the states of the slow model (3) and the fast model (5), respectively. The parameter ε0 is given as eq. (8) at the top of this page. 2.2 Problem Formulation In this paper, we study the FTC of singularly perturbed systems. It is assumed that all the faults considered enters into the model in the parameter form. Some fault parameters enter as physical parameters when the model is formed based on the physical laws. Some enter as coefficients when the model is identified by experimental means. Others enter as dimensionless scaling factors which indicate the degree of the abnormality of some particular components. Define an impairment parameter vector θ whose elements are all fault parameters. In the presence of faults, the singularly perturbed system can then be described as follows. x(t) ˙ = A11 (θ)x(t) + A12 (θ)z(t) + B1 (θ)u(t),

(9)

εz(t) ˙ = A21 (θ)x(t) + A22 (θ)z(t) + B2 (θ)u(t), (10) where x(t) ∈ ℜm and z(t) ∈ ℜn are states whose initial conditions are also x(t0 ) = x0 and z(t0 ) = z0 , u(t) ∈ ℜp is the control input, ε is a small positive scalar, θ is a vector of dimension N that changes the system matrices A11 , A12 , A21 , A22 and the control matrices B1 , B2 . It is supposed that the faults considered in this paper have been diagnosed. For fault diagnosis schemes, the interested readers may refer to Patton et al. (2013); Jiang et al. (2006); Blanke et al. (2015) and the references therein. Obviously, the dynamic of the system changes when fault occurs. According to Lemma 1, the condition ε ∈ (0, ε0 ] may be violated in this situation. Hence, we need to design a fault-tolerant control law to recover this defect. Consider the composite control law of the following form u(t) = K1 (θ)x(t) + K2 (θ)z(t).

(11)

Objective. The goal of FTC is to design a controller in the form of (11) such that the closed-loop system (9)-(10) is asymptotically stable at the origin in the presence of faults, i.e., limt→∞ ∥x(t)∥ = 0 and limt→∞ ∥z(t)∥ = 0. 3. MAIN RESULTS

(5)

where the vector zf (t) = z(t) − zs (t) is the fast part of the corresponding variables of system (2). Lemma 1. Kokotovi´c et al. (1999) If A0 and A22 are Hurwitz, there exists an ε0 > 0 such that, for all ε ∈ (0, ε0 ], 269

This section provides the main results of this paper. We first decompose the system into two subsystems, and then design control scheme to ensure the accuracy of the decomposed system and the asymptotical stability of the original system.

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1 −1 ∥F22 (θ)∥(∥F0 (θ)∥

+

−1 ∥F12 (θ)∥∥F22 (θ)F21 (θ)∥

+2

with

√ −1 ∥F0 (θ)∥∥F12 (θ)∥∥F22 (θ)F21 (θ)∥)

(25)

F11 (θ) = A11 (θ) + B1 (θ)K1 (θ), F12 (θ) = A12 (θ) + B1 (θ)K2 (θ), F21 (θ) = A21 (θ) + B2 (θ)K1 (θ), F22 (θ) = A22 (θ) + B2 (θ)K2 (θ), −1 (θ)F21 (θ). F0 (θ) = F11 (θ) − F12 (θ)F22

3.1 System Decomposition Firstly, the faulty system (9)-(10) needs to be decomposed. By setting ε = 0 and assuming that A22 (θ) is invertible, it can be derived from (10) that zs (t) = −A−1 22 (θ)(A21 (θ)xs (t) + B2 (θ)us (t)),

(12)

where the vectors zs (t) and us (t) represent the slow parts of the corresponding vectors z(t) and u(t) in the system (9)-(10). Substituting (12) into (9) yields the mth-order slow system x˙ s (t) = A0 (θ)xs (t) + B0 (θ)us (t), xs (t0 ) = x0 , (13) where

zs (t) = −A−1 22 (θ)A21 (θ)xs (t) + B2 (θ)us (t)),

u(t) = K0 (θ)x(t) + Kf (θ) ] [ × z(t) + A−1 22 (θ)(A21 (θ)x(t) + B2 (θ)Ks x(t)) = K1 (θ)x(t) + K2 (θ)z(t),

K1 (θ) = (I + Kf (θ)A−1 22 (θ)B2 (θ))K0 (θ) +Kf (θ)A−1 22 (θ)A21 (θ).

(14)

B0 (θ) = B1 (θ) − A12 (θ)A−1 22 (θ)B2 (θ),

(15) (16)

and the vector xs (t) is the slow part of vector x(t) in the system (9)-(10). In order to derive the fast subsystem, it is assumed that the slow variables stay still during the fast transient, i.e., x˙ s (t) = 0 and z˙s (t) = 0. Denoting the fast variable by zf (t) := z(t) − zs (t),

Reλ(A0 (θ) + B0 (θ)K0 ) < 0,

(22)

Reλ(A22 (θ) + B2 (θ)K2 ) < 0,

(23)

and substituting it into (10), one can obtain the nth-order fast system εz(t) ˙ = A22 (θ)zf (t) + B2 (θ)uf (t), zf (t0 ) = z0 −zs (t0 ).(18) 3.2 Fault-tolerant control design After decomposition, it is reasonable to design two state feedback control law us (t) = Ks (θ)xs (t) and uf (t) = Kf (θ)xf (t) for the decomposed slow subsystem (13) and fast subsystem (47), respectively. Hence, a composite controller for the whole system (9)-(10) can be possibly chosen as us (t) + uf (t) = K0 (θ)xs (t) + K2 (θ)zf (t).

ε > ε,

(19)

However, the above composite controller us (t) + uf (t) is not realizable since xs (t) and zf (t) do not appear in the system (9)-(10) explicitly. To solve this problem, we replace xs (t) by x(t) and zf (t) by z(t) − zs (t). A realizable controller then takes the following form 270

(24)

where the parameter ε⋆ is shown in (25) at the top of this page. Moreover, under the control law (20), the states of the closed-loop system (9)-(10) are approximated for all t ∈ [t0 , ∞) by x(t) = xs (t) + O(ε), z(t) =

(17)

(21)

Now we give the first result of this paper. Theorem 1. For any bounded initial conditions x0 and z0 , the FTC goal of the system (9)-(10) can be achieved by the fault-tolerant controller (20) if



A0 (θ) = A11 (θ) − A12 (θ)A−1 22 (θ)A21 (θ),

(20)

where

−A−1 22 (θ)(A21 (θ)

(26) + B2 (θ)K0 (θ))xs (t)

+zf (t) + O(ε).

(27)

Proof. The closed-loop system (9)-(10) with the controller (20) can be written as x(t) ˙ = F11 (θ)x(t) + F12 (θ)z(t), x(t0 ) = x0 ,

(28)

εz(t) ˙ = F21 (θ)x(t) + F22 (θ)z(t), z(t0 ) = z0 .

(29)

The proof concludes two steps: the first step is to obtain an accurate decomposed system, the second is the control scheme analysis. Step 1: In order to derive the accurate decomposed system of system (28)-(29), we introduce two transformations: η(t) = z(t) + Lx(t),

(30)

ξ(t) = x(t) − εHη(t),

(31)

where L and H are matrices related with ε in proper dimensions. (a). Applying the transformation (30), the closed-loop system can be written as

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x(t) ˙ = (F11 (θ) − F12 (θ)L)x(t) + F12 (θ)η(t),

(32)

εz(t) ˙ = R(L, ε, θ)x(t) + (F22 (θ) + εLF12 (θ))η(t), (33) where

271

˙ = (As (θ) + Bs (θ)Ks (θ))ξ(t), ξ(t)

(46)

εη(t) ˙ = (Af (θ) + Bf (θ)K2 (θ))η(t),

(47)

where

R(L, ε, θ) = F21 (θ)−F22 (θ)L+εLF11 (θ)−εLF12 (θ)L. (34) We can see that when R(L, ε, θ) = 0, a separated fast subsystem can be derived as

As (θ) = A0 (θ) − εA12 (θ)A−1 22 (θ)L(A11 (θ) − A12 (θ)L),(48) Bs (θ) = B0 (θ) − εA12 (θ)A−1 22 (θ)LB1 (θ),

(49)

Af (θ) = A22 (θ) + εLA12 (θ),

(50)

εz(t) ˙ = (F22 (θ) + εLF12 (θ))η(t), η(t0 ) = z0 + Lx0 .(35) Then we need to show that there exists a solution L to equation R(L, ε, θ) = 0 when ε ∈ [0, ε⋆ ].

Bf (θ) = B2 (θ) + εLB1 (θ).

(51)

When ε = 0, the unique solution of R(L, ε, θ) = 0 is

−1 (θ)F21 (θ) := L0 . L(0) = F22 Differentiating R(L, ε, θ) = 0 w.r.t. ε yields that  dL  −1 = F22 (θ)L0 (F11 (θ) − F12 (θ)L0 ) dε ε=0

(36)

Let Ks (θ) = K0 (θ), and use (A0 (θ), B0 (θ)) to approximate (As (θ), Bs (θ)), (A22 (θ), B2 (θ)) to approximate (Af (θ), Bf (θ)). Due to Reλ(A0 (θ) + B0 (θ)K0 ) < 0 and Reλ(A22 (θ) + B2 (θ)K2 ) < 0, The closed-loop system (46)(47) is asymptotically stable. Thus, the closed-loop system (9)-(10) is also asymptotically stable since the coordinate transformation (30)-(31) is nonsingular. According to (44) and (45), K1 (θ) can be approximated by

−1 = F22 (θ)F21 (θ)F0 (θ). (37) By introducing C = L − L0 , the equation R(L, ε, θ) = 0 can be written as −1 (θ)(L0 F0 (θ)+CF0 (θ)−L0 F12 (θ)C −CF12 (θ)C) C = εF22

:= f (C). (38) One can derive that f (C) is a contraction mapping when ε ∈ [0, ε⋆ ] Hence, there exists a solution L approximated by

−1 −1 (θ)F21 (θ) + εF22 (θ)F21 (θ)F0 (θ) + O(ε). (39) L = F22

(b). Applying the transformation (31), the closed-loop subsystem (32) can be written as

K1 (θ) = K0 (θ) + K2 (θ)A−1 22 (θ) ×(A21 (θ) + B2 (θ)K0 (θ)) + O(ε), which is identical with the K1 (θ) in (20).

(52)

The state approximations (26)-(27) can be derived according to Lemma 1. This completes the proof.  Remark 1. The conditions (22) and (23) make sure that the decomposed slow subsystem and fast subsystem are both stable at the origin. To ensure the accuracy of the approximated state variables, the control gains Ks (θ) and Kf (θ) need to be regulated cooperatively to realize the condition (24). Corollary 1. The closed-loop system (9)-(10) is asymptotically stable with u(t) = K0 (θ)x(t) if

˙ = (F11 (θ) − F12 (θ)L)ξ(t) + S(H, ε, θ)η(t), (40) ξ(t) where

Reλ(A22 (θ22 )) < 0,

(53)

Reλ(A0 (θ) + B0 (θ)Ks ) < 0,

(54)

ε⋆0

S(H, ε, θ)η(t) = ε(F11 (θ) − F12 (θ)L)H

−H(F22 (θ) + εLF12 (θ)) + F12 (θ). (41) Then, when S(H, ε, θ) = 0, a separated slow subsystem can be derived as ˙ = (F11 (θ) − F12 (θ)L)ξ(t), ξ(t0 ) = x0 − εHη0 . (42) ξ(t) Since (41) is linear, there exists a unique solution H to S(L, ε, θ) = 0 which is approximated for all ε ∈ [0, ε⋆ ] by −1 (θ) + O(ε). H = F12 (θ)F22 Step 2: We introduce a matrix Ks (θ), and let

(43)

K1 (θ) := Ks (θ) + K2 (θ)L. Substituting it into R(L, ε, θ) = 0 yields

(44)

> ε, (55) ⋆ where is similar to ε described in (25) with K2 = 0. Remark 2. The above corollary can be directly obtained from Theorem 1 by setting the control gain K2 (θ) = 0. The control design procedure are simplified when A22 (θ) is Hurwitz. For sufficient small ε, the state response of the original closed-loop system (9)-(10) is infinitely approximated by that of the reduced order system (13)(14). This explains why the parasitic parameters are neglected in some dynamic control systems. ε⋆0

4. AN APPLICATION TO A HYPERSONIC VEHICLE

−1 L = A−1 22 (θ)(A21 (θ) + B2 (θ)Ks (θ)) + εA22 (θ)L

(45) ×[(A11 (θ) + B1 (θ)Ks (θ)) − A12 (θ)L]. Substituting (45) into the exact fast system (35) and slow system (42), one can obtain the accurate closed-loop system 271

In this section, we use the proposed method to design an FTC scheme for the longitudinal control system of an airbreathing hypersonic vehicle with actuator faults. First of all, we give the nomenclatures that will be used in the longitudinal model of hypersonic vehicles. V − velocity, ft/s h − altitude, ft

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γ − flight path angle, deg

α − angle of attack, deg q − pitch rate, deg/s

ϕc − throttle setting, 1

δe − elevator deflection, deg

4.1 Singular perturbation modeling Consider the longitudinal model linearized at the trimmed cruise condition V = 7702f t/s, h = 85000f t, γ = 0deg, α = 1.5153deg, and q = 0deg/s provided in Levin et al. (2008). ˙ X(t) = AX(t) + B(θ)u(t), (56) ⊤ ⊤ where X := [V h γ α q] , u := [ϕc δe ] , θ := [θ1 , θ2 ]⊤ is the impairment parameter vector, with θ1 , θ2 ∈ [0, 1] being the actuator effectiveness factors which enter the model in the form of B(θ) = [b1 θ1 b2 θ2 ], b1 and b2 are the columns of matrix B in the nominal system. The matrices A and B(θ) are given as follows

A=





−0.05173 −31.55 0.01076 −67.48 0 −7 0 1.026 × 10−8 −0.001406 0   5.558 × 10  0 7702 0 0 0 ,   −5.558 × 10−7 0 −1.026 × 10−8 0.001406 1 −6 −7 0 −4.365 × 10 0.01487 0 2.098 × 10



(57) εz(t) ˙ = A21 x(t) + A22 z(t) + B2 (θ)u(t), where the system and control matrices are as follows:   −0.05173 −31.55 0.01076 A11 =  5.558 × 10−7 0 1.026 × 10−8  , 0 7702 0 [ ] −67.48 0 A12 = −0.001406 0 , 0 0 ] [ −1.6118 × 10−7 0 −2.9754 × 10−9 , A21 = 6.0842 × 10−7 0 −1.2658 × 10−7 [ ] 4.0774 × 10−4 0.29 A22 = , 4.3123 × 10−3 0   8.295θ2 27.56θ1 B1 (θ) =  −7.457 × 10−5 θ1 0.00568θ2  , 0 0 ] [ −5 2.1625 × 10 θ1 −1.6472 × 10−3 θ2 .(58) B2 (θ) = −2.0982 × 10−3 θ1 −6.5424 × 10−3 θ2 After decomposition, the slow subsystem takes the following form x˙ s (t) = A0 xs (t) + B0 (θ)us (t), where



27.56θ1 8.295θ2  −7.457 × 10−5 θ1 0.00568θ2    0 0 B(θ) =  .  7.457 × 10−5 θ −0.00568θ  1 2 −0.007235θ1 −0.02256θ2 One finds that the above model is not in the singularly perturbed form. We need to select proper time scales and transform it into singularly perturbed form. The mostly used methods are classified into three ways (Naidu and Calise (2001); Kim et al. (2014)): • directly identify the small parameters, such as small time constants, capacitances, high Reynolds number, and other similar “parasitic” parameters. • transformation of the state equations. • linearization of the state equations.

The system (56) is a linearized model, for such systems, the eigenvalues of the system matrix are examined for time scale separation, see Syrcos and Sannuti (1983). The open-loop dynamics of system (56) has eigenvalues of 0.12051, −0.12988, 0.022955, −0.031956 ± 0.025926i. Obviously, the eigenvalues can be divided into two distinct sets, Λs := {λs1 , λs2 , λs3 } = {0.022955, −0.031956 ± 0.025926i} and Λf := {λf 1 , λf 2 } = {0.12051, −0.12988} which characterize the slow and fast modes, respectively. The small perturbation parameter ε can be chosen as ε := maxi∈{1,2,3} |λsi |/ minj∈{1,2} |λf j |. The eigenvalues in the set Λs represent the dynamic of V , h and γ, and the eigenvalues in the set Λf indicate the dynamic of α and q. By choosing x := [V h γ]⊤ , z := [α q]⊤ , the system (56) can be transformed into the following singularly perturbed form. x(t) ˙ = A11 x(t) + A12 z(t) + B1 (θ)u(t), 272

 −0.042209 −0.3155 8.7792 × 10−3 A0 =  7.5417 × 10−7 0 −3.1011 × 10−8  , 0 7702 0   94.082θ2 5.2732θ1 B0 (θ) =  −7.5868 × 10−4 θ1 3.5469 × 10−3 θ2  . 0 0 

And the fast subsystem is

εz˙f (t) = A22 zf (t) + B2 (θ)uf (t).

(59)

4.2 Controller design Now, the system (56) is decomposed into a slow subsystem and a fast one. When the actuator effectiveness factors are θ1 = 0.6 and θ2 = 0.8, we can design two state feedback controllers us (t) = K0 (θ)xs (t) and uf (t) = K2 (θ)zf (t) for the two decomposed subsystems, respectively. According to Theorem 1, by regulating the control gains Ks and Kf cooperatively, a composite fault-tolerant controller u(t) in the form of (20) can be derived with [ ] −0.0005 −71.8667 −0.0005 , K0 (θ) = 0.0001 4.3634 −0.0001 [ ] 726.1784 321.3378 K2 (θ) = . −233.5503 −171.8377

Choosing the initial condition as X(0) = [7740 84900 0 1.5153 0]⊤ and applying the fault-tolerant controller u(t) to system (56) yields the state responses shown in Fig. 1 and Fig. 2. The two figures reveal that the system (56) is asymptotically stable at the equilibrium point [7702 85000 0 1.5153 0]⊤ , which indicates that the FTC goal is achieved under the control law u(t).

IFAC SAFEPROCESS 2018 Warsaw, Poland, August 29-31, 2018

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7750 7700 7650

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Fig. 2. The angle of attack and pitch rate responses 5. CONCLUSION This paper has developed an FTC scheme for linear singularly perturbed systems with component impairments. The asymptotical stability of such systems can be ensured. The effectiveness of the proposed method has been demonstrated by a hypersonic vehicle. Further work will focus on the FTC of nonlinear singularly perturbed system without decomposing the original system. REFERENCES Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M. (2015). Diagnosis and Fault-Tolerant Control. Springer Verlag, Berlin. Jiang, B., Gao, Z.F., Shi, P., and Xu, Y.F. (2010). Adaptive fault-tolerant tracking control of near space vehicle using takagi-sugeno fuzzy models. IEEE Transactions on Fuzzy Systems, 18(5), 1000–1007. Jiang, B., Staroswiecki, M., and Cocquempot, V. (2006). Fault accommodation for nonlinear dynamic systems. IEEE Transactions on Automatic Control, 51(9), 1578– 1583. 273

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