reconfigurable control of input affine nonlinear systems
TRANSCRIPT
8/19/2019 Reconfigurable Control of Input Affine Nonlinear Systems
http://slidepdf.com/reader/full/reconfigurable-control-of-input-affine-nonlinear-systems 1/8
Recongurable Control of Input AffineNonlinear Systems under Actuator Fault
S. Mojtaba Tabatabaeipour ∗ Roberto Galeazzi ∗∗
∗System Operation Analysis, Energinet.dk, Tonne Kjærsvej 65,Fredericia, Denmark, (e-mail: [email protected])
∗∗Automation and Control, Department of Electrical Engineering,Technical University of Denmark, Kgs. Lyngby, Denmark, (e-mail:
Abstract: This paper proposes a fault tolerant control method for input-affine nonlinearsystems using a nonlinear reconguration block (RB). The basic idea of the method is to insertthe RB between the plant and the nominal controller such that fault tolerance is achievedwithout re-designing the nominal controller. The role of the RB is twofold: on one hand it
transforms the output of the faulty system such that its behaviour is similar to that of thenominal one from the controller’s viewpoint; on the other hand it modies the control input tothe faulty system such that the stability of the recongured loop is preserved. The RB is realizedby a virtual actuator and a reference model. Using notions of incremental and input-to-statestability (ISS), it is shown that ISS of the closed-loop recongured system can be achieved bythe separate design of the virtual actuator. The proposed method does not need any knowledgeof the nominal controller and only assumes that the nominal closed-loop system is ISS. Themethod is demonstrated on a dynamic positioning system for an offshore supply vessel, wherethe virtual actuator is designed using backstepping.
1. INTRODUCTION
With the ever-increasing requirement on safety, reliability,availability, and performance of industrial systems, it isessential to design controllers that can tolerate occurrenceof some faults without interrupting the operation whilepreserving the system stability, functionality, and simul-taneously providing acceptable performance. Such con-trollers are called fault-tolerant. The area of fault-tolerantcontrol (FTC) has attracted a lot of attention in the pasttwo decades, see e.g. the review papers (Blanke et al., 1997;Patton, 1997; Jiang, 2005) and books (Isermann, 2006;Blanke et al., 2006).
FTC methods can broadly be divided into two classes:passive (PFTC) and active (AFTC). In PFTC, the struc-ture and the parameters of the controller are xed andpre-designed such that during operations the system isrobust towards occurrence of a given set of faults. Hence,a PFTC solution is a common solution to the controlproblem for the nominal and the given set of faulty sys-tems. However a common solution may not always exist,specially if severe faults are considered; when it exists itmay be too conservative and result in low performancefor the nominal operation. On the other hand, in AFTCthe controller is not xed and reacts to the occurrenceof faults during operations by adjusting its parametersor structure. A fault detection and identication (FDI)module is designed to detect and identify the occurred
fault and then based on this information, the controlleris modied for the identied faulty system. Consequently,AFTC can usually provide better performance.
Most of the AFTC methods available in the literaturerely on a batch of controllers designed for each consideredfaulty case. When the occurred fault is identied thenominal controller is replaced by the controller specicallydesigned for this faulty scenario.The idea proposed in this paper is to keep the nominalcontroller in the loop and design a reconguration block,which is inserted between the faulty system and the nomi-nal controller to guarantee the stability of the reconguredclosed-loop system. This idea, depicted in Fig. 1, is knownas reconguration through fault-hiding. The goal of thereconguration block is twofold: it transforms the outputof the faulty system such that from the viewpoint of thecontroller it has a similar behaviour to that of the nominalsystem; it changes the input from the nominal controllersuch that the stability of the recongured loop is guaran-teed. The reconguration block is respectively realized bya virtual sensor, a virtual actuator, or a series connectionof both of them in case of a sensor fault, an actuator fault,or a simultaneous sensor and actuator fault.
The idea of control reconguration using virtual sensorsand actuators was proposed by Steffen (2005) for linearsystems. Lunze and Steffen (2006) showed that controlreconguration of a linear system after an actuator faultis equivalent to disturbance decoupling. Control recong-uration methods using virtual actuators and sensors forpiecewise affine systems and Hammerstein-Wiener systemswere proposed in Richter et al. (2008), Richter and Lunze(2008), Richter et al. (2011) and Richter (2011).
AFTC for Lur’e systems with Lipschitz continuous non-linearity subject to actuator fault using a virtual actuatorwas presented in Richter et al. (2012), where it was as-
8/19/2019 Reconfigurable Control of Input Affine Nonlinear Systems
http://slidepdf.com/reader/full/reconfigurable-control-of-input-affine-nonlinear-systems 2/8
dz
Nominal
Controllerr
u c
y
Nominal
Plant
(a)
dz f
Nominal
Controllerr
u f
yf
Faulty
Plant
(b)
dzf
Nominal
Controllerr
u cyc
Reconguration
Block
u f yf
Faulty
Plant
(c)
Fig. 1. Fault-tolerant control using a reconguration block: (a) nominal loop, (b) faulty plant with nominal controller,(c) recongured plant with nominal controller
sumed that the state of the faulty system is measurable.
AFTC for a system with additive Lipschitz nonlinearitysubject to actuator faults using a virtual actuator waspresented in Khosrowjerdi and Barzegary (2013). Pedersenet al. (2014) proposed a new design method for the virtualactuator based on absolute stability theory, which wastested for the reconguration of power systems subject tofaults in local controllers in emergency situations.
Fault tolerant control of polytopic linear parameter vary-ing (LPV) systems subject to sensor faults using virtualsensor was proposed in de Oca and Puig (2010), wherethe structure of the nominal controller was assumed tobe known. It was further assumed that the nominal con-troller consists of a state feedback combined with an
LPV observer. Tabatabaeipour et al. (2012) consideredthe problem of control reconguration for continuous-time LPV systems with both sensor and actuator faultsand without any assumptions about the structure of thenominal controller. In this context input-to-state stabilityproperties of the recongured system were investigated.In Tabatabaeipour et al. (2014) the control recongura-tion for discrete-time LPV systems with both sensor andactuator faults were considered and both stability andperformance of the reconguration block was investigated.
In this paper we extend the idea of recongurable con-trol design using a reconguration block to input-affinenonlinear dynamical systems. Only actuator faults are
considered, and the reconguration block is realized bya nonlinear virtual actuator. Using incremental stabilityproperties, it is shown how to design the nonlinear virtualactuator independent of the nominal controller to achieveISS of the recongured closed-loop system. The main con-tributions are given in Theorems 12, 13 and Corollary 14.The proposed method does not require any informationabout the nominal controller and only assumes that thenominal closed-loop system is ISS. The design of a fault-tolerant dynamic positioning system for an offshore supplyvessel is utilised as case study. The design of the nonlinearvirtual actuator is demonstrated using backstepping con-trol. The simulation results show the effectiveness of theproposed method.
2. PRELIMINARIES
The eld of real numbers and the set of nonnegative realsare respectively denoted by R , and R ≥ 0 . For any vectorx ∈ R n , x T stands for its transpose and x = √ x T xdenotes its Euclidean norm. Also, the i-th entry of x isdenoted by x i . The innity norm of x denoted by x ∞ isgiven by max i |x i |. Given a measurable function u : R ≥ 0 →R n , its (essential) supremum is denoted by u ∞ which isdened as: u ∞ := (ess)sup { u (t ) , t ≥0}. The functionu is essentially bounded if u ∞ < ∞.
The function α : R ≥ 0 → R ≥ 0 is called a class K functiondenoted by α ∈ K if it is continuous, strictly increasing,unbounded and satises α (0) = 0. The function β : R ≥ 0
×R ≥ 0 → R ≥ 0 is called a class KL function denoted byβ ∈KL if β (·, t )∈K and β (r, t ) →0 as t → ∞.Consider the following nonlinear system
Σ :x = f (x (t ), u (t)) , x(0) = x 0 ,y (t) = h (x (t)) ,
(1)
where x(t) ∈ R n is the state, u (t) ∈ R m is the input,y (t) ∈ R q is the output. We use the following stabilitydenitions.Denition 1. 0-global asymptotic stability (Sontag(2008)) The system (1) with u (t) = 0 , ∀t ∈ R ≥ 0 is called0-globally asymptotically stable (0-GAS) if there exists a
function β ∈KLsuch that for all t0
and x (t0
), the solutionof the system satisesx (t) ≤β ( x (t 0 ) , t ). (2)
Denition 2. Input-to-state stability (Sontag (2008))The system (1) is called input-to-state stable (ISS) withrespect to (w.r.t.) the input u (t ) if there exist some β ∈KLand some γ ∈Ksuch that for all t0 and x (t 0 ) and all inputsu (t ), all solutions of the system satisfy
x (t) ≤β ( x (t 0 ) , t ) + γ ( u (t ) ∞ ). (3)Denition 3. Input-to-output stability (Sontag (2008))The system (1) is called input-to-ouput stable (IOS) w.r.t.the input u (t) and the output y (t) if there exist someβ
∈KL and some γ
∈K such that for all t 0 and x (t 0 ) and
all inputs u (t ), the output of the system satisesy (t) ≤β ( x (t 0 ) , t ) + γ ( u (t) ∞ ). (4)
8/19/2019 Reconfigurable Control of Input Affine Nonlinear Systems
http://slidepdf.com/reader/full/reconfigurable-control-of-input-affine-nonlinear-systems 3/8
Theorem 4. IOS of interconnected systems (Jianget al. (1994)) Consider the following interconnected sys-tems:
x 2 = f 2 (x 2 (t), y 1 (t), u (t))y 2 = h 2 (x 2 (t ), y 1 (t ), u (t))x 1 = f 1 (x 1 (t), u (t ))y 1 = h 1 (x 1 (t ), u (t))
(5)
Assume that the rst system is IOS w.r.t. the input uand the output y 1 and the second system is IOS w.r.t.the input ( y 1 , u ) and output y 2 . Then the interconnectedsystem is IOS w.r.t. the input u and outputs ( y 1 , y 2 ).
Next, we recall the denition of incremental stability. In-cremental stability considers the stability and convergenceof the trajectories with respect to each other rather thanto an equilibrium point.Denition 5. [Zamani and Tabuada (2011)] The nonlinearsystem (1) is incrementally globally asymptotically stable(δ -GAS) if there exist a metric d and a class
KL function
β such that for all locally essentially bounded u , allinitial conditions ξ0 , η 0 ∈ R n and all t ≥ 0 the followinginequality is satised
d(x (t, ξ 0 , u ), x (t, η 0 , u )) ≤β (d(ξ 0 , η 0 ), t ) (6)
If the origin is an equilibrium point for (1), then δ -GASimplies 0-GAS.Denition 6. [Zamani and Tabuada (2011)] The nonlinearsystem (1) is incrementally input-to-state stable ( δ -ISS)if there exist a metric d, a class KL function β and aclass K∞ function γ such that for all inputs u 1 , u 2 , allinitial conditions ξ0 , η 0 ∈ R n and all t ≥ 0 the followinginequality holds true
d(x (t, ξ 0 , u 1 ), x (t, η 0 , u 2 )) ≤β (d(ξ 0 , η 0 ), t )+ γ ( u 1 −u 2 ∞ ). (7)
From (6) and (7) it is straightforward to see that δ -ISSimplies δ -GAS, but the converse is not true in general.Moreover it is concluded that if the origin is an equilibriumpoint for the system (1) then δ -ISS implies ISS.Remark 7. Denitions 5 and 6 are invariant under changesof coordinates because they are based on the existenceof a generic metric d, not necessarily Euclidean. Thiscoordinate invariance was not included in the formerdenitions provided by Angeli (2002) where only the
Euclidean metric was considered.Lyapunov characterizations of δ -GAS and δ -ISS were rstpresented in (Angeli, 2002), however those were not in-variant under change of coordinates. Later, Zamani andMajumdar (2011) proposed the following Lyapunov char-acterizations that are coordinate independent.Theorem 8. (Zamani and Majumdar (2011)). The nonlin-ear system (1) is δ -GAS if there exist a function V (x 1 , x 2 ),a metric d, and class K∞ functions α 1 and α 2 such that
α 1 (d(x 1 , x 2 )) ≤V (x 1 , x 2 ) ≤α 2 (d(x 1 , x 2 )) , (8)and for any x 1 and x 2 ∈R n and any u
∈U it is hold that∂V ∂ x 1 f (x 1 , u ) +
∂V ∂ x 2 f (x 2 , u ) ≤ −κV (x 1 , x 2 ), (9)
with κ > 0.
In case V is a quadratic function, the system is calledquadratically incrementally stable ( δ -QS). Similarly for δ -ISS we have the following Lyapunov characterization.Theorem 9. (Zamani and Majumdar (2011)). The nonlin-ear system (1) is said to be δ -ISS if there exist a functionV (x 1 , x 2 ), a metric d and class K∞ functions α 1 , α 2 , and
ρ such thatα 1 (d(x 1 , x 2 )) ≤V (x 1 , x 2 ) ≤α 2 (d(x 1 , x 2 )) , (10)and for any u 1 , u 2 ∈U and x 1 , x 2 ∈R n :
∂V ∂ x 1
f (x 1 , u 1 ) + ∂V ∂ x 2
f (x 2 , u 2 ) ≤ −κV (x 1 , x 2 )
+ ρ( u 1 −u 2 ), (11)with κ > 0.
3. NOMINAL AND FAULTY NONLINEAR SYSTEM
Consider the fault-free plant Σ P
ΣP
:x = f (x ) + Bu c + B d d , x(t 0 ) = x 0
y = Cxz = C z x
(12)
where x ∈ R n is the state, u c ∈ U ⊂ R m is the controlinput, d
∈D ⊂R k is the input disturbance, y ∈R q is themeasured output, and z
∈R p is the controlled output.
The vector eld f (·) : R n → R n is continuously dif-ferentiable and locally Lipschitz in a domain X ⊂ R n .This guarantees existence and uniqueness of the solutionx (t, x 0 , u c , d ) for any initial condition x 0 and for all t ≥ t 0 .The state of the system (12) is assumed to be fully accessi-ble to the measurement; hence the output matrix C = I n ,where I n is the n-dimensional identity matrix.
The nominal nonlinear dynamical state feedback controllerΣ C is
Σ C : x c = f c (x c , x ) , x c (t 0 ) = x c, 0 ,u c = α (x c , x ) (13)
where x c ∈R n is the controller state, f c (·, ·) : R n ×R n →R n is locally Lipschitz, and α (·, ·) : R n ×R n → U ⊂ R m
is a smooth mapping.Assumption 10. ISS of the nominal closed-loop sys-tem: Let d (t ) be a bounded input disturbance, i.e.
d (t) ∞ ≤ d. The nominal fault-free closed-loop systemΣ L = (Σ P , Σ C ) composed by the fault-free nonlinear plant(12) and the nonlinear controller (13) is input-to-statestable with respect to the disturbance d (t).
Let now assume that at time t f > t 0 an actuator fault oc-curs in the nominal nonlinear plant (12). As a consequencethe input matrix B changes to
B f = B θ , (14)where θ = diag( θ1 , θ2 , . . . , θ m ) with θi ∈ (0, 1] being anunknown parameter giving the level of control authorityof each actuator after a fault has occurred.
The faulty nonlinear plant dynamics Σ P f initialized atx f (t f ) = x (t f ) is given by
Σ P f :x f = f (x f ) + B f u f + B d dy f = x f
z f = C z x f
(15)
where x f ∈R n is the state of the faulty system, u f ∈U ⊂R m is the faulty control input, y f ∈ R q and zf ∈ R p are
8/19/2019 Reconfigurable Control of Input Affine Nonlinear Systems
http://slidepdf.com/reader/full/reconfigurable-control-of-input-affine-nonlinear-systems 4/8
the measured and controlled outputs of the faulty system,respectively.
4. RECONFIGURATION PROBLEM
After occurrence it is assumed that the actuator fault isdetected, isolated and its magnitude estimated within atime td by an FDI unit. Once detection and isolation isachieved, in the proposed method for AFTC the nominalcontroller is kept in the loop and a reconguration blockis inserted between it and the faulty plant. The RB is adynamical system that receives the output of the faultyplant y f and the output of the nominal controller u c asits input, and produces the input to the faulty system u f and the input to the nominal controller y c , see Fig. 1.
The nonlinear dynamics of the reconguration block canin general be represented as
Σ R :
x r = f (x r , u c , y f ),
u f = h r (x r , u c ),y c = h ry (z , y f ). (16)
where x r is an internal state. The RB must be designedsuch that the overall closed-loop system (Σ P f , Σ R , Σ C )is stable, and its performance fulls some requirements.Different goals can be considered in the design of thereconguration block. In this paper, the focus is on thestability recovery problem .Problem 11. Consider the nominal nonlinear system Σ P and the faulty nonlinear system Σ P f . Design, if possible,a reconguration block Σ R such that for all nominalcontrollers Σ C that render Σ L ISS w.r.t. d , the closed-looprecongured system (Σ P f , Σ R , Σ C ) is ISS w.r.t. d .
5. STABILITY RECOVERY THROUGH FAULTHIDING
In this work, the RB is realized by a virtual actuator. Aftera fault has occurred the nominal nonlinear controller (13)may not be able to guarantee closed-loop stability and/orperformance, hence a reconguration of the controller isneeded. The reconguration is based on the design of avirtual actuator , i.e. an intermediate system that interfacesboth with the faulty plant Σ P f and the nominal controllerΣ C such that control system keeps seeing the output of thenominal fault-free plant Σ P , and the input to the faultyplant is compensated for the presence of the fault. Thestructure of the recongured loop with the virtual actuatoris depicted in Fig. 2.
The nonlinear virtual actuator is given by
Σ A :˙x = f (x ) + Bu c , x (t 0 ) = x 0
u f = α ∆ (x ) −α ∆ (x f ) + Nu c
y c = x c
(17)
where x ∈ R n is the state of a reference model providingthe trajectory of the fault-free plant Σ P in the absence of input disturbances, d (t) = 0, and N is a gain matrix thatfeed-forwards the control input u c to the plant input u f .
To analyse the stability of the recongured system weintroduce the difference state x ∆ = x −x f . The dynamicsof the difference state is then given by
x f = f (x f ) + B f u f + B d d
α ∆ (·)+ +
α ∆ (·)N
Σ P : Reference Model
u cΣ C : Controller
−u ∆
x f
u f
d
x
Fig. 2. Structure of the Reconguration Block
Σ C Σ P
Σ ∆
d
x ∆
u c x
Fig. 3. Closed-loop recongured system as series intercon-nection of (Σ C , Σ P ) and Σ ∆
x ∆ = ˙x − x f
= f (x ) + Bu c −[f (x f ) + B f u f + B d d ]= f (x ) −B f α ∆ (x ) −[f (x f ) −B f α ∆ (x f )]
+ ( B
−B f N )u c
−B d d
= f (x ) −B f α ∆ (x ) −[f (x −x ∆ ) −B f α ∆ (x −x ∆ )]+ ( B −B f N )u c −B d d
= κ ∆ (x ) −κ ∆ (x −x ∆ ) + ( B −B f N )u c −B d d (18)
where κ ∆ (ξ ) f (ξ ) −B f α ∆ (ξ ).
In the following, we show the conditions for ISS of thedifference system and we show that if the virtual actuatoris designed independently such that the difference systemis ISS, then the recongured closed-loop system is also ISS.Theorem 12. (Recongured system stability ) Con-sider the recongured closed-loop system (Σ P f , Σ A , Σ C ). If the nominal closed-loop system Σ L is ISS and the virtualactuator Σ A is designed such that the difference system Σ ∆
is ISS, then the recongured closed-loop system is ISS.
Proof. Introducing the new variable x ∆ the dynamics of the closed-loop recongured system (Σ P f , Σ A , Σ C ) in newvariables is re-written by:
Σ P : ˙x = f (x ) + Bu c
y c = x c
Σ C : x c = f c (x c , y c )u c = α (x c , y c ) (19)
Σ ∆ : x ∆ = κ ∆ (x ) −κ ∆ (x −x ∆ )
+ ( B −B f N )u c −B d dwhich is graphically depicted in Figure 3.
By Assumption 10 the nominal closed-loop system Σ L isISS, hence also the closed-loop system (Σ P , Σ C ) is ISS
8/19/2019 Reconfigurable Control of Input Affine Nonlinear Systems
http://slidepdf.com/reader/full/reconfigurable-control-of-input-affine-nonlinear-systems 5/8
since the reference model Σ P is a copy of the nominalopen loop system Σ P . By designing the virtual actuatorΣ A such that it renders the dynamics of the differencesystem Σ ∆ ISS, then the recongured closed-loop systemis ISS by Theorem 4.
5.1 Stability Analysis of the Difference System
The stability analysis of the difference system (18) iscarried out within the framework of incremental stabilitytheory.
Consider the systemξ = κ ∆ (ξ ) + u 1 = f (ξ ) −B f α ∆ (ξ ) + u 1 (20)
where ξ ∈ R n is the state, f (·) : R n → R n is thevector eld dening the fault-free and the faulty plant,α ∆ (·) : R n → R n is a nonlinear function of the state ξto be designed, and u 1 = ϑ[(B − B f N )u c − B d d ] with0 < ϑ < 1 .Theorem 13. [ISS of the difference system ] Considerthe faulty nonlinear plant (15). If there exists a nonlinearstabilizing function α ∆ (·) such that the ξ-dynamics isδ -ISS then the difference system (18) is ISS w.r.t. thenominal controller input u c and the disturbance d .
Proof. If the ξ -dynamics is δ -ISS then there exist β ∈KLand γ ∈ K∞ such that for any t ≥ 0, any pair of initialconditions ( ξ 0 , η 0 ), and any pair of input signals ( u 1 , u 2 )the following inequality is satised
ξ (t, ξ 0 , u 1 ) −η (t, η 0 , u 2 ) ≤β ( ξ 0 −η 0 , t )+ γ ( u 1 −u 2 ∞ ) (21)
Let ξ0 = x 0 and η 0 = x 0 − x ∆ ,0 . Then the x ∆ -dynamics (18) is given by the linear combination of thesolutions of the ξ-dynamics (20) for the two initial condi-tions ξ0 , η 0 and the two inputs u 1 , u 2 , with u 2 = (1 −ϑ)[(B −B f N )u c −B d d ]. Therefore (21) can be rewrittenas
x ∆ (t, x ∆ ,0 ) = ξ (t, ξ 0 , u 1 ) −η (t, η 0 , u 2 )
≤β ( ξ 0 −η 0 , t ) + γ ( u 1 −u 2 ∞ )= β ( x ∆ ,0 , t ) + γ ( u ∞ ) (22)
that is the x ∆ -dynamics is ISS with respect to u = ( B −B f N )u c −B d d .Corollary 14. Consider the faulty nonlinear plant (15). If there exist a nonlinear function α ∆ (·) such that the ξ-dynamics with zero input, is δ -QS then the difference
system (18) is ISS w.r.t. the disturbance d .Proof. If the ξ-dynamics is δ -QS then there exist aquadratic function V ξ (ξ , η ) and K∞ functions α 1 , α 2 , andκ > 0 such that for any pair of system trajectories ( ξ , η )
α 1 ( ξ −η ) ≤V ξ (ξ , η ) ≤α 2 ( ξ −η ) (23)∂V ξ∂ ξ
κ ∆ (ξ ) + ∂V ξ∂ η
κ ∆ (η ) ≤ −κV (ξ , η ) (24)
Consider then the quadratic Lyapunov functionV ∆ (x , x −x ∆ ) = x T
∆ Px ∆ = ( x −(x −x ∆ )) T P (x −(x −x ∆ )) ,(25)
with P = P T > 0. Note that:∂V ∆∂ x κ ∆ (x ) +
∂V ∆
∂ (x −x ∆ ) κ ∆ (x −x ∆ ) =x T
∆ P (κ ∆ (x ) −κ ∆ (x −x ∆ )) ≤κx T∆ Px ∆ (26)
The derivative of V ∆ along the trajectories of the differencesystem (18) satisesV ∆ = x T
∆ P (κ ∆ (x ) −κ ∆ (x −x ∆ ) + ( B −B f N )u c −B d d )
≤ −κx T ∆ Px ∆ + x ∆ P (B −B f N ) u c
+ x ∆ PB d d
≤ −b x ∆
2
+ x ∆ P (B −B f N ) u c+ x ∆ PB d d
≤ −(1 −ϑ)b x ∆2 (27)
for x ∆ ≥ 1
ϑb ( P (B −B f N ) u c + PB d d ), whereb = κλ min (P ), which proves ISS of the system w.r.t. d .
6. STUDY CASE - DYNAMIC POSITIONINGSYSTEM
The effectiveness of the proposed nonlinear virtual actu-ator reconguration strategy is evaluated on the dynamicpositioning (DP) system of an offshore supply vessel. DPsystems are control systems that can maintain the positionand orientation of a marine craft in the vicinity of anoperating point exclusively by means of thrusters despitethe presence of environmental disturbances such as wind,waves, and currents (DNV, 1990).
6.1 Vessel Model for DP Operations
For the problem at hand the analysis of the vessel motion isrestricted to the horizontal plane neglecting the dynamicsassociated with the heave, roll and pitch motions. Theinterested reader can nd details about notation andmodelling of marine crafts in (Fossen, 2011).
Let η [N,E,ψ ]T be the position-orientation vectorin the North-East-Down (NED) inertial frame, and ν [u,v,r ]T be the velocity vector in the body frame. Thevessel dynamics can be described by the following nonlin-ear model
η = R (ψ)ν (28)M ν + N (ν )ν = τ (29)
where M = M T > 0 is the mass-inertia matrix thataccounts for the rigid body and hydrodynamics effects;N (ν )ν = C (ν )ν + D (ν )ν accounts for rigid-body andhydrodynamic Coriolis-centripetal forces ( C (ν )ν ), anddissipative forces due to hull-water interaction ( D (ν )ν ).For low-speed operation, like dynamic positioning, the
quadratic velocity terms due to nonlinear damping andctitious forces can be neglected and only linear dampingis considered ( N (ν )ν = D ν ). R (ψ) is a rotational matrixfunction of the ship heading angle φ, which transformsa vector from the body frame to the NED frame. Thevector of generalized forces and moments τ = τ t + τ wtakes into account the actions of the thrusters τ t , and theenvironmental disturbances such as wind τ w .
The vessel is assumed to be equipped with two azimuththrusters, one close to the bow and one close to thestern, and one tunnel thruster, positioned between thebow thruster and midship. Let u c = [n az, 1 , n tu , n az, 2 ]T bethe vector of thrusters’ shaft speeds, then the considered
actuators conguration gives rise to control forces andmoments according toτ t = B 1 u c = KT (ϕ )u c . (30)
8/19/2019 Reconfigurable Control of Input Affine Nonlinear Systems
http://slidepdf.com/reader/full/reconfigurable-control-of-input-affine-nonlinear-systems 6/8
K = diag {K 1 , K 2 , K 3 } is the thrust coefficient matrix,and T (ϕ ) is the actuator conguration matrix
T (ϕ ) cosϕ1 0 cosϕ2
sin ϕ1 1 sinϕ2
l1 sinϕ1 l2 −l3 sinϕ2
where li (i
∈ {1, 2, 3
}) are the moment arms in yaw with
respect to the ship centre of gravity, and ϕj ( j ∈ {1, 2})are the angles of rotation of the azimuth thrusters w.r.tthe fore-aft direction.
6.2 Nominal DP Controller
The general DP control objective is to track a time-varyingreference trajectory η d ∈C
2 , which is bounded ( η d ∞ ≤ηd < ∞) with bounded derivatives. If ηd is constant thendynamic positioning reduces to station keeping, which isa set-point regulation problem.
In order to solve the DP control problem a change of coordinates is introduced. Let η η
−η d be the tracking
error in the NED frame, and s ˙η −Λ η be an additionalmeasure of tracking, with Λ > 0 a diagonal design matrix.It can be shown (Fossen and Strand, 1999) that the vesseldynamics can be rewritten as
˙η = R (ψ)ν − η d (31)M η (η )s = −D η (η )( s + η d −Λ η ) + R (ψ)(B 1 u c + τ w )
−M η (η )( η d −Λ (R (ψ)ν − η d )) (32)
Let x [η T , sT ]T be the state vector, u c the control input,and d = τ w the wind generated disturbance. Then thefault-free plant Σ P is given by
Σ P :x = f (x ) + B (η )u c + B d (η )d , x(t 0 ) = x 0
y = xz = x 1
(33)
where
f (x ) = R (ψ)ν − η d
−M − 1η (η )D η (η )(s + η d −Λ η ) −(η d −Λ ˙η )
B (η ) = 0
M − 1η (η )R (ψ)B 1
, B d (η ) = 0
M − 1η (η )R (ψ)
The nominal controller for the system (33) under theassumption that d = 0 has been implemented based onthe design by Fossen and Strand (1999), who have used thenonlinear MIMO backstepping technique to obtain a DPcontrol system that guarantees global exponential stability
of the tracking error dynamics.The DP backstepping nominal control law is given by (Fos-sen and Strand, 1999)
φ = R (ψ)ν −Λ η + ˙η d (34)u c
B − 11 R T (ψ) D η (η )( η d −Λ η ) −K p η −K d s
+ M η (η )( η d −Λ (R (ψ)ν − η d )) (35)where φ is the virtual control that stabilizes the η dynam-ics in the rst step; K p > 0, and K d > 0 are diagonaldesign matrices. The nominal closed-loop dynamics reads
˙η = −Λ η + s (36)s = −M − 1
η (η )K p η −M − 1η (η )(D η (η ) + K d )s (37)
For a detailed overview of the design strategy and of thestability properties of the origin of (36)-(37) the reader isaddressed to (Fossen and Strand, 1999).
The control law (34)-(35) has been selected because back-stepping controllers are known to guarantee ISS with re-spect to input disturbances (Krstic et al., 1995).
6.3 Virtual Actuator DP Controller
Actuator faults in the system (33) appears as the reductionof one or more coefficients of the matrix K from the nom-inal values. Therefore the faulty input matrix is given byB 1 ,f
K f T (ϕ ), where K f = diag {θ1 K 1 , θ2 K 2 , θ3 K 3 }.In this study case we focus on partial reduction of thethrust coefficients, that is the scaling factors θi ∈ (0, 1](i∈ {1, 2, 3}) specically cannot assume value zero.
The virtual actuator DP control law is
α ∆ (z ) :
φ ∆ = R (ψ)ν −Λz 1 + ˙η d
u ∆ B − 11 ,f R T (ψ) D η (η )( η d −Λz 1 )
+ M η (η )( η d −Λ (R (ψ)ν − η d ))
−K pz 1
−K d z 2
(38)
where z = [z T1 , z T
2 ]T is a dummy vector representing eitherthe state of the faulty system or the state of the referencemodel. The virtual actuator exploits the knowledge of themagnitude of the fault through the input matrix B 1 ,f .
Let ξ = [η , s]T be the state of the closed-loop system, thenits dynamics reads
ξ = A (η )ξ + B d (η )d (39)where
A (η ) = −Λ I
−M − 1η (η )K p −M − 1
η (η )(D η (η ) + K d )
Consider the Lyapunov function candidateV (ξ 1 , ξ 2 ) = ( ξ 1 −ξ 2 )T P (η )(ξ 1 −ξ 2 ) (40)
whereP (η ) = K p 0
0 M η (η ) = P T (η ) > 0
which satises that∂V ∂ ξ 1
ξ 1 + ∂V ∂ ξ 2
ξ 2 = ( ξ 1 −ξ 2 )T (P (η )A (η )
+ A T (η )P (η ))( ξ 1 −ξ 2 )+ ( ξ 1 −ξ 2 )T P (η )B d (η )(d 1 −d 2 )
= −(ξ 1 −ξ 2 )T Q (η )(ξ 1 −ξ 2 )
+ ( ξ 1 −ξ 2 )T
P (η )B d (η )(d 1 −d 2 )≤κV (ξ 1 , ξ 2 ) + γ ( d 1 −d 2 ) (41)
where
Q (η ) = K pΛ 00 D η (η ) + K d
= Q T (η ) > 0 .
Therefore according to Theorem 9 the closed-loop sys-tem (39) is δ -ISS.
6.4 Simulation Results
The DP backstepping controller with virtual actuatorhas been tested on a model of an offshore supply vessel
subject to wind disturbances. The numerical values of theparameters of the ship and of the nominal controller aregiven in Table 1.
8/19/2019 Reconfigurable Control of Input Affine Nonlinear Systems
http://slidepdf.com/reader/full/reconfigurable-control-of-input-affine-nonlinear-systems 7/8
Table 1. Vessel & Controller Parameters
Quantity Symbol Value
Length overall LOA 76.2 [m]Beam B 18 [m]Centre of gravity [ x G , y G , z G ]T [42, 0, 0]T [m]Moment arms [ l1 , l 2 , l 3 ]T [27.4, 17.2, − 10.5]T [m]Angle of rotations [ ϕ1 ,ϕ 2 ]T [30, 30]T [deg]Mass-inertia matrix M diag {5.3e6, 8.3e6, 3.7e9}
Damping matrix D5.0e4 0 0
0 2.7e5 − 4.4e60 − 4.4e6 4.2e8
Thrust coeff. matrix K diag {1.4e5, 1.4e5, 1.4e5}Proportional gain K p diag {2.5, 2.5, 2.5}Derivative gain K d diag {2.5, 2.5, 2.5}
Λ diag {1, 1, 1}Time constants T a diag {25, 25, 25} [sec]
−3 −2 −1 00
100
200
300
400
500
North [m]
T i m e
[ s e c ]
0 100 200 300 400 500−1
−0.5
0
E a s t [ m
]
Time [sec]
− 0. 8 − 0. 7 − 0. 6 − 0. 5 − 0. 4 − 0. 3 − 0. 2 − 0. 1 0 0. 1−2.5
−2
−1.5
−1
−0.5
0
0.5
N o r t h
[ m ]
East [m]
0 200 400 60045
45.5
46
46.5
ψ
[ d e g
]
Time [sec]
−0.494 −0.493 −0.492−1.12
−1.1
−1.08
Fig. 4. North-East trajectory of the supply vessel affectedby faults in both azimuth thrusters while subject to aconstant wind. The green diamond is the desired op-erating point, and the dots represent the end positionof the vessel. The blue trajectory shows the behaviourof the fault-free vessel; the red trajectory shows thefaulty ship without reconguration, while the dashedblack trajectories represent the system after beingrecongured.
In order to obtain realistic time responses to step changesof the set-point or of the disturbance the thrusters havebeen model as rst order systems with rate and magnitudesaturation
T a u e = u c −u e (42)where u c is the vector of commanded shafts speed by thecontrol law, u e is the vector of delivered shafts speed, andT a = diag {τ az, 1 , τ tu , τ az, 2 } is the actuator time constantmatrix. This has requested to extend the nominal controllaw (34)-(35) by backstepping once more through theactuator dynamics. The implemented solution is basedon Fossen and Berge (1997).
The vessel is subject to a constant wind disturbance withspeed V w = 20 m/s and direction β w = 30 degrees withrespect to the North. At time tf = 50 seconds both
azimuth thrusters are subject to faults of equal magnitudewhich reduce the respective thrust coefficients of 50%, i.e.K f = diag {0.5K 1 , K 2 , 0.5K 3 } ∀t ≥ tf . It is assumed
0 50 100 150 200 250 300 350 400 450 5000
1
2
3x 10
4
τ e X
[ N ]
0 50 100 150 200 250 300 350 400 450 500−6
−4
−2
0x 10
4
τ e Y [ N ]
0 50 100 150 200 250 300 350 400 450 500−15
−10
−5
0x 10
5
τ e N
[ N m
]
Time [sec]
Fig. 5. Forces and moments in surge ( τ X ), sway ( τ Y ), andyaw (τ N ) without (red lines) and with (black dashedlines) reconguration.
50 100 150 200 250 300−6
−4
−2
0
2
4
6
8
t f = 50 sec
t d = 55 sec
Time [sec]
[ -
]
N ∆ [m]E ∆ [m]ψ∆ [deg]
Fig. 6. Behaviour of the position-orientation componentsof the difference state x ∆ : after the reconguration of the control system at t = t d with the virtual actuatorthe components converge to bounded values close tozero.
that within 5 seconds from faults occurrence an FDI
module has detected and isolated the faults, and that thereconguration of the control system has taken place.
Figures 4-6 show the performance of the DP control sys-tem without and with the virtual actuator recongura-tion. Although stability of the closed-loop system is notcompromised it is evident the benecial action of thecontrol system reconguration. The presence of the virtualactuator allows the recongured vessel (black dashed line)to remain in very close proximity of the desired operat-ing point (green diamond at ( N, E ) = (0 , 0) in Fig. 4),with performance extremely close to those of the nominalsystem (blue dashed-dotted line). Conversely the faultyship (red line) shows deviations from the desired operating
point approximately 50% larger in both direction com-pared to the recongured system. Figure 6 clearly showsthe input-to-state stable behaviour of the difference state
8/19/2019 Reconfigurable Control of Input Affine Nonlinear Systems
http://slidepdf.com/reader/full/reconfigurable-control-of-input-affine-nonlinear-systems 8/8
x ∆ : after the reconguration has taken place at t = tdits components converge to a neighbourhood of the origin,whose size is obviously a function of the magnitude of thedisturbance.
7. CONCLUSIONS
In this paper a new method for FTC of nonlinear systemssubject to actuator faults using a nonlinear recongurationblock was proposed. The main idea of the method is toachieve fault-tolerance without re-designing the nominalcontroller by inserting the reconguration block betweenthe faulty system and the nominal controller. The pro-posed method does not need any knowledge of the nominalcontroller and it is only assumed that the nominal closed-loop system is input-to-state stable. It was shown that if the virtual actuator is designed separately such that thedifference system is δ -ISS, then the recongured closed-loop system is ISS. The effectiveness of the method isshown on a case study of dynamic positioning system of
an offshore supply vessel, where the virtual actuator isdesigned using the backstepping control technique.
REFERENCESAngeli, D. (2002). A lyapunov approach to incremental
stability properties. IEEE Transactions on Automatic Control , 47(3), 410–421.
Blanke, M., Izadi-Zamanabadi, R., Bogh, S.A., and Lunau,C.P. (1997). Fault-tolerant control systems-a holisticview. Control Engineering Practice , 5(5), 693–702.
Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M.(2006). Diagnosis and Fault-Tolerant Control . Springer-Verlag.
de Oca, S. and Puig, V. (2010). Fault-tolerant controldesign using a virtual sensor for LPV systems. InProceedings of the 2010 Conference on Control and Fault-Tolerant Systems , 88–93.
DNV (1990). Rules for classication of steel ships: Dy-namic positioning systems. Technical report, Det NorskeVeritas.
Fossen, T.I. (2011). Handbook of Marine Craft Hydrody-namics and Motion Control . Wiley.
Fossen, T.I. and Berge, S.P. (1997). Nonlinear vectorialbackstepping design for global exponential tracking of marine vessels in the presence of actuator dynamics. InProceedings of the 36th IEEE Conference on Decision and Control .
Fossen, T.I. and Strand, J.P. (1999). Tutorial on nonlinearbackstepping: Applications to ship control. Modeling,Identication and Control , 20(2), 83 – 134.
Isermann, R. (2006). Fault-diagnosis systems . SpringerVerlag.
Jiang, J. (2005). Fault-tolerant control systems-an intro-ductory overview. Acta Automatica Sinica , 31(1), 161–174.
Jiang, Z., Teel, A., and Praly, L. (1994). Small-gaintheorem for iss systems and applications. Mathematics of Control, Signals, and Systems , 7(2), 95–120.
Khosrowjerdi, M.J. and Barzegary, S. (2013). Fault tol-erant control using virtual actuator for continuous-
time lipschitz nonlinear systems. International Journal of Robust and Nonlinear Control . doi:http://dx.doi.org/10.1002/rnc.3002.
Krstic, M., Kanellakopoulos, I., and Kokotovic, P. (1995).Nonlinear and Adaptive Control Design . John Wiley &Sons.
Lunze, J. and Steffen, T. (2006). Control recongura-tion after actuator failures using disturbance decouplingmethods. IEEE Transactions on Automatic Control ,51(10), 1590–1601.
Patton, R.J. (1997). Fault-tolerant control systems: The1997 situation. In Proceedings of the 3rd IFAC Sym-posium on Fault Detection, Supervision, and Safety for Technical Processes , volume 3, 1033–1054.
Pedersen, A.S., Richter, J.H., Tabatabaeipour, S.M., Jo-hansson, H., and Blanke, M. (2014). Stabiliser faultemergency control using reconguration to preservepower system stability. In Proceedings of the 19th IFAC World Congress . Cape Town, South Africa.
Richter, J., Seron, M., and De Dona, J.A. (2012). Virtual
actuator for Lure systems with lipschitz-continuous non-linearity. In Proceedings of the 8th IFAC Symposium on Fault Detection, Supervision, and Safety for Technical Processes , volume 8, 222–227. Mexico City, Mexico.
Richter, J.H. (2011). Recongurable Control of Nonlinear Dynamical Systems . Springer.
Richter, J.H., Heemels, W., van de Wouw, N., and Lunze,J. (2008). Recongurable control of PWA systems withactuator and sensor faults: stability. In Proceedings of the 47th IEEE Conference on Decision and Control ,1060–1065.
Richter, J., Heemels, W., van de Wouw, N., and Lunze,J. (2011). Recongurable control of piecewise affinesystems with actuator and sensor faults: stability andtracking. Automatica , 47(4), 678–691.
Richter, J. and Lunze, J. (2008). Recongurable controlof hammerstein systems after actuator faults. In Pro-ceedings of the 17th IFAC World Congress , 3210–3215.
Sontag, E. (2008). Input to state stability: Basic conceptsand results. Nonlinear and Optimal Control Theory ,163–220.
Steffen, T. (2005). Control reconguration of dynam-ical systems: linear approaches and structural tests .Springer.
Tabatabaeipour, S.M., Stoustrup, J., and Bak, T. (2012).Control reconguration of LPV systems using virtualsensor virtual actuator. In Proceedings of the 8th IFAC Symposium on Fault Detection, Supervision, and Safety for Technical Processes , volume 8, 222–227. Mexico City,Mexico.
Tabatabaeipour, S.M., Stoustrup, J., and Bak, T. (2014).Fault-tolerant control of discrete-time lpv systems usingvirtual actuators and sensors. International Journal of Robust and Nonlinear Control . doi:10.1002/rnc.3194.
Zamani, M. and Majumdar, R. (2011). A Lyapunovapproach in incremental stability. In Proceedings of the 50th IEEE Conference on Decision and Control .
Zamani, M. and Tabuada, P. (2011). Backstepping designfor incremental stability. IEEE Transactions on Auto-matic Control , 56(9), 2184–2189.
All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.