International Journal ofObserver-based Control, Automation, and Systems, vol. no. 6, pp. 707-711, Fault Tolerant Control for5,Constrained SwitchedDecember Systems 2007
707
Observer-based Fault Tolerant Control for Constrained Switched Systems Hao Yang, Bin Jiang*, and Vincent Cocquempot Abstract: An observer-based fault tolerant control (FTC) method is proposed for constrained switched systems (CSS) with input constraints. A family of Lyapunov-based bounded controllers are designed to ensure that, whenever actuator faults occur at the dwell time period of each continuous mode, the mode is always within its corresponding stability region. A set of switching laws are designed to guarantee the asymptotic stability of the overall CSS. The fixed stability regions on which the FTC method is based are also relaxed by the proposed variable stability regions. An example of CPU processing illustrates the effectiveness of proposed method. Keywords: Constrained switched system, fault tolerant control, observer.
1. INTRODUCTION Many tools have been developed for stability analysis of switched systems [1]. Control action of the majority of practical switched system is often subject to hard actuator constraints, the general synthesis of control for CSS is based on concepts of stability region and multiple Lyapunov functions (MLFs) e.g., [2]. Most of related methods only consider the CSS in the fault-free case and with full state measurements. Fault may lead to an unacceptable anomaly in the system performance. Fault detection and diagnosis (FDD) and fault tolerant control (FTC) procedures are designed to guarantee that the system goal is still achieved in spite of the faults [3-5]. This paper focuses on designing a FTC strategy for CSS with actuator faults and without full state measurements. The novelty of this work are twofold: 1) to design a family of Lyapunov-based bounded controllers for __________ Manuscript received February 18, 2006; revised February 13, 2007 and June 1, 2007; accepted August 28, 2007. Recommended by Editorial Board member Guang-Hong Yang under the direction of Editor Tae-Woong Yoon. This work is partially supported by National Natural Science Foundation of China (60574083), National “863” program of China (2006AA12A108) and Natural Science Foundation of Jiangsu Province (BK2007195). Hao Yang is with the College of Automation Engineering (CAE), Nanjing University of Aeronautics and Astronautics (NUAA), 29 YuDao Street, Nanjing, 210016, China and LAGIS-CNRS, UMR 8146, Université des Sciences et Technologies de Lille (USTL), 59655 Villeneuve d'Ascq cedex, France (e-mail:
[email protected]). Bin Jiang is with the College of Automation Engineering, Nanjing Univ. of Aeronautics and Astronautics (NUAA), Nanjing, 210016, China (e-mail:
[email protected]). Vincent Cocquempot is with LAGIS-CNRS, UMR 8146, USTL, 59655 Villeneuve d'Ascq cedex, France (e-mail:
[email protected]). * Corresponding author.
ensuring that, whenever actuator faults occur at the dwell time period of each mode, the mode is always in its corresponding stability region; 2) to design a set of switching laws based on MLFs, to guarantee the asymptotical stability of the overall CSS.
2. PRELIMINARIES Consider the following switched system with input constraints: x(t ) = f σ(t ) ( x(t )) + Gσ(t ) ( x(t ))uσ(t ) (t ),
uσ(t ) ≤ uσmax , σ(t ) : [t0 ,∞) → M = {1, 2,…, N }, (1) where x ∈ ℜn is the state vector, uσ denotes the input vector taking values in the nonempty compact subset U σ := {uσ ∈ ℜm : uσ ≤ uσmax }. uσmax > 0 is
the magnitude of the input constraints, fσ ( x) and Gσ ( x) are sufficiently smooth. σ : [t0 , ∞) → M is a switching signal, which is assumed to be a piecewise constant function that is continuous from the right. The switching laws are defined as: mode k switches to mode ( k + 1), if the dwell period of mode k is ∆tkj , where k ∈ M ,
j = 1, 2,….
∆tkj belongs to a series of dwell time periods for mode k when it is activated for the j th time. Also assume that the switching sequence is fixed and the initial mode is random. x is continuous everywhere. Consider system (1) with a fixed σ(t ) = k for some k ∈ M , for which a control Lyapunov function [2] Vk exists, using the results in [2], the following continuous bounded control law can be constructed uk ( x) = − K k ( L∗fk Vk ( x), x)( LGk Vk )T ( x) bk ( x), (2)
708
Hao Yang, Bin Jiang, and Vincent Cocquempot T uk ( x) = − K k ( L∗∗ f k Vk ( x ), x )( LGk Vk ) ( x) b k ( x ) (7)
with K k ( L∗f k Vk , x)
=
L∗f k Vk + ( L∗f k Vk )2 + (ukmax ( LGk Vk )T )4 T 2
( LGk Vk )
for ( LGk Vk ) ≠ 0, and T
[1 + 1 + (ukmax ( LGk Vk )T ) 2 ]
K k ( L∗fk Vk , x)
(3) = 0 for ( LGk Vk )T
= 0, where L∗fk Vk = L fk Vk + ρ kVk and ρ k > 0 . One
can show that for all initial states of k th mode within the stability region described by the set Φ k = {x ∈ ℜn : L∗fk Vk ( x) < ukmax ( LGk Vk )T ( x) } (4)
the controller (2) respects the constraints and ensures that the states of the k th mode remain within the region Φ k and converge to origin asymptotically. An estimate of Φ k
is described by Ω k = {x ∈ ℜn :
Vk ( x) ≤ ckmax }, where Ω k is expected to be the largest invariant set of Φ k ,
ckmax is the largest
number for which Ω k \ {0} ⊆ Φ k .
3. FTC FOR CSS 3.1. FTC with full state measurement Consider the linear form of (1) with σ(t ) = k for some k ∈ M , under actuator faulty conditions:
x(t ) = Ak x(t ) + Bk uk (t ) + Ek f ka (t ), uk ≤ ukmax , (5) where x is measurable, ( Ak , Bk ) is a controllable pair. The actuator fault vector is modelled by
f ka (t ) ∈ ℜq ,
assume that
f ka
≤ f k , where f k > 0.
Consider a Lyapunov function candidate Vk = T
x Pk x, where Pk is a positive definite symmetric matrix that satisfies the Riccati equation AkT Pk + Pk Ak − Pk Bk BkT Pk = −Qk for some positive
definite matrix Qk . Define n ∗∗ max ( LGk Vk )T }, Φ k = {x ∈ ℜ : L fk Vk < uk
(6)
where L∗∗fk Vk = L fk Vk + ρkVk + LEk Vk f k , with L fk Vk = xT ( AkT Pk + Pk Ak ) x, ( LGk Vk )T = 2 BkT Pk x, ρk > 0.
the states remain within the region Ω k , and the origin of the k th mode is asymptotically stable, whenever the fault occurs in [tkj , tkj + ∆tkj ), where j denotes the
j th time that the k th mode is
switched in, ∆tkj is the dwell period. Proof: (sketch) From the time-derivative of Vk along the closed-loop trajectories (omitted due to space), we have that whenever L∗∗ f k Vk ( x ) < ukmax ( LGk Vk )T ( x) , Vk < −ρ kVk . Since Ω k is the
largest invariant set, the inequality (6) holds for all x ≠ 0, then, the origin of the system is asymptotically stable. Theorem 1: Consider the switched system (5) under a family of bounded controllers {uk ( x) = b k ( x) : k ∈ M }, with the initial states x(t0 ) ∈ Ω k. If, at any time instant T , the following conditions hold: x(T ) ∈ Ω k +1,
(8)
Vk +1 ( x(T )) < Vk +1 ( x(t( k +1)( j −1) )), j > 0,
(9)
then, choosing ∆tkj ≥ T − tkj and setting σ(t ) = k + 1 for the j th time at t = tkj + ∆tkj , guarantees that the origin of the overall CSS is asymptotically stable. Proof: (sketch) From Lemma 1, V σ(t ) < 0, ∀σ = k . From (9), we have that for any admissible switching time tkj ,Vk +1 ( x(t( k +1) j )) < Vk +1 ( x(t( k +1)( j −1) )), MLFs
Theorem [1] can be applied to conclude that the origin of the switched system is Lyapunov stable. Also note that for each switching time tkj , j = 1, 2,… such that σ(tkj+ ) = k , the sequence Vσ(tkj ) is decreasing and
positive, so there exists a class K function α such that 0 = lim j →∞ [Vk +1 ( x(t( k +1)( j +1) )) − Vk +1 ( x(t(k +1) j ))] ≤ lim j →∞ [−α( x(t( k +1) j )] ≤ 0. This means that x(t )
converges to the origin, which together with Lyapunov stability, leads to the asymptotical stability of the origin. 3.2. FTC without full state measurement Consider the system (5)
Denote Ω k = {x ∈ ℜn : Vk ( x) ≤ c max k } as a set of Vk ,
x(t ) = Ak x(t ) + Bk uk (t ) + Ek f ka (t ),
> 0. completely contained in Φ k for some c max k Lemma 1: Consider system (5) with any initial condition x(tkj ) ∈ Ω k , under the continuous bounded
y = Ck x(t ) = [Ck1 0]x(t ),
controller
uk ≤ ukmax ,
(10)
where y ∈ ℜr is the output vector, with q < r , where q is the dimension of f ka . Ck1 is an r × r
Observer-based Fault Tolerant Control for Constrained Switched Systems
nonsingular matrix, and (Ck , Ak ) is an observable pair. Denoting xˆ (t ) as the state estimate and e(t ) x − xˆ. The following Lemma 2 can be obtained based on [2]. Lemma 2: Consider system (10) with x(tkj ) ∈ Ω k ,
assume (Gk 33 , Gk13 )
under the controller uk = b k ( xˆ ) in (7). There exists a positive real number eu ,k related to uk , such that if
ζ k Gk13 ) stable.
e(t ) ≤ eu ,k , ∀t ∈ [tkj , tkj + ∆tkj ), then the states remain
within the region Ω k , and the origin of the k th mode is asymptotically stable. Assumption 1: Rank (Ck Ek ) = q. Define a transformation x = N k−1 z , where N k = Ck1 0
0 . The system (10) can be transformed into I
where A k = N k Ak N k−1 , B k = N k Bk , E k = N k Ek .
(11)
(12) z
can be represented as [ z1 z2 z3 ]T = [ y1 y2 z3 ]T , where z3 ∈ ℜ
n−r
. We just need to estimate z3 . Define
I − E k1E −k 12 0 Sk = 0 I 0 . −1 0 − E k 3E k 2 I
y1 − E k1E −k 12 y 2 − Gk11 y1 − Gk12 y2 − H k1uk To estimate z3 , an observer can be designed as zˆ 3 = Gk 33 zˆ 3 + s + ζ k (v − Gk13 zˆ 3)
(15)
is an observable pair, the
observer gain ζ k can be chosen to make (Gk 33 − From the above discussion, we let xˆ = N k−1 zˆ =
N k−1[ y1 y2 zˆ 3]T . Denote z 3 = z3 − zˆ 3 , then e(t ) ≤ µ(λ∗k ) z 3(tkj ) exp(−λ∗k (t − tkj )) ,
(16)
where λ∗k is such that all eigenvalues of (Gk 33 − ζ k Gk13 ) satisfy λ k ≤ −λ∗k , µ(λ∗k ) > 0 is polynomial
z = A k z + B kuk + E k f ka A k1 B k1 E k1 = A k 2 z + B k 2 uk + E k 2 f ka , A k 3 B k 3 E k 3 I ( r − q )×( r − q ) 0 0 y = C kz = z, 0 I q 0
709
(13)
in λ∗k . From the second block row in (14), the fault a estimate fˆ k can be obtained as a −1 fˆ k = E k 2( y 2 − A k 21y1 − A k 22 y2 − A k 23 zˆ 3 − B k 2uk ). (17) The above method can provide accurate state and fault estimates. For CSS, the observer (15) is switched according to the current mode at each switching time. The initial states of the current observer are chosen as a the final states of the previous observer. fˆ are k
always obtained from (17) for each mode. Theorem 2: Consider the switched system (10) under a family of bounded controllers {uk = b k ( xˆ ) : k ∈ M }, with the initial states x(t0 ) ∈ Ω k and xˆ(t0 ) such that µ(λ∗k ) e(t0+ ) ≤ eu,k . If, at any time instant T e(T ) ≤ e u ,k +1,
Pre-multiplying (13) into (11), we have
y − E k1E −k 12 y A k1 − E k1E −k 12 A k 2 2 1 (14) y2 Ak 2 = z −1 −1 z 3 − E k 3E k 2 y 2 A k 3 − E k 3E k 2 A k 2 B k1 − E k1E −k 12 B k 2 0 a + Bk2 uk + E k 2 f k . −1 0 B k 3 − E k 3E k 2 B k 2 Define Gki = A ki − E ki E −k 12 A k 2, H ki = B ki − E ki E −k 12 B k 2
(18)
xˆ(T ) ∈ Ψ k +1 , Vk +1 ( xˆ (T )) + 2M k +1 < Vk +1 ( xˆ (t( k +1)( j −1) )),
(19) (20)
∗ + e u ,k +1 is such that µ(λ k +1 ) e k +1 (T ) ≤ eu ,k +1 , Ψ k +1 is such that xˆ ∈ Ψ k +1 → x ∈ Ω k +1
where for
e ≤ eu ,k +1 ,
M k +1 is such that
Vk +1 ( x) − Vk +1 ( xˆ ) ≤ M k +1 ,
T − tkj
e ≤ eu ,k +1 →
then, choosing
∆tkj ≥
and setting σ(t ) = k + 1 at t = tkj + ∆tkj ,
for i = 1, 3, and partitioning Gki as [Gki1Gki 2Gki 3 ], i = 1, 3, then the first and third block rows of system (14) can be written as z 3 = Gk 33 z3 + s, v = Gk13 z3 ,
guarantees that the origin of the overall CSS is asymptotically stable. Proof: (sketch) Based on Lemmas 1, 2, if (18) and (19) hold at any time instant T , we set σ(t ) = k + 1
where s = Gk 33 y1 + Gk 32 y2 + E k 3 E −k 12 y 2 + H k 3uk , v =
at t = tkj + ∆tkj , then the origin of the (k + 1) th
710
Hao Yang, Bin Jiang, and Vincent Cocquempot
mode is asymptotically stable. Due to the continuity of Vk +1 (⋅), there exists a positive real number M k +1 , such that if e ≤ eu ,k +1 , then Vk +1 ( x) − Vk +1 ( xˆ ) ≤ M k +1 ,
which, together with (20), leads to
Vk +1 ( x(T )) < Vk +1 ( x(t( k +1)( j −1) )). Similar to Theorem
is input-to-state stable w.r.t. e for t ∈ [tkjf , tkj + ∆tkj ), the result follows. Theorem 3: Consider switched system (10) under a family of bounded controllers {uk◊ ( xˆ ), k ∈ M }, the initial states x(t0 ) ∈ Ω k , and xˆ (t0 ) are such that
1, we can conclude the result.
µ(λ∗k ) e(t0+ ) ≤ eu ,k .
3.3. Relaxation of the stability region The region Φ k in (6) is based on a fixed norm bound of faults, which could be relaxed. Define the a fault detection threshold as fˆ = E k−12 A k 23e(t ) ,
euf,k ∀k ∈ M , and if at any time instant T ≥ tkjf
k
and define
a fˆ k
> 0 such that
a fˆ k
≤
a fˆ k .
A variable
stability region is designed as
(21) a L◊fk Vk = L fk Vk + ρ kVk + LEk Vk fˆ k + LEk Vk
−1 E k 2 A k 23 eu,k , and eu ,k
is related to uk = bk ( xˆ ).
a The region (21) is variable according to different fˆ k . This region is less conservative than (6). Let’s similarly define Ω k , fˆ = {x ∈ ℜn : Vk ( x)
≤ c maxˆ}. Ω k , fˆ is a set of Vk , completely contained k, f in Φ k , fˆ
for some c maxˆ > 0, and define Ψ k , fˆ k, f
such that if e ≤ eu,k , then xˆ ∈ Ψ k , fˆ → x ∈ Ω k , fˆ . Lemma 3: Consider system (10) under control law uk = bk ( xˆ ), x(tkj ) ∈ Ω k , and xˆ(tkj ) is such that
µ(λ∗k ).
e(tkj+ ) ≤ eu ,k . Assume the faults occur at ˆ
t = tkjf . If xˆ(tkjf ) ∈ Ψ k , fˆ , then there exists euf,k > 0,
ˆ
e(T ) ≤ e u ,k +1, xˆ (T ) ∈ Ψ k +1 ,
(23)
Vk +1 ( xˆ (T )) + 2M k +1 < Vk +1 ( xˆ (t( k +1)( j −1) )),
(25)
(24)
at t = tkj + ∆tkj , guarantees that the origin of the overall CSS is asymptotically stable.
4. CPU PROCESSING CONTROL SYSTEM A simplified CPU processing control system illustrates our approach. The system have two modes: Mode 1: The amount of CPU tasks is large while CPU temperature is not too high. Mode 2: The amount of CPU tasks is not large and more energy is used for decreasing the temperature. Three states are respectively the amount of CPU tasks π, the temperature ρ, and angular velocity of a cooling fan ω. c ∈ ℜ and v ∈ ℜ are the clock frequency and the voltage input of a cooling fan. The system model is omitted due to the page limit. In Mode 1, c ≤ 5,
v ≤ 10,
in Section 3.2. Choosing x(0) = x(t11 ) = [8 9.5 9]T ,
ˆ
bounded controller (22)
where b k ( xˆ ) − K k ( L◊f k Vk ( xˆ ), xˆ )( LGk Vk )T ( xˆ ), makes the origin of the k th mode asymptotically stable. Proof: (sketch) Lemma 2 ensures the system is stable for t ∈ [tkj , tkjf ). At t = tkjf , the faults occur, from the time-derivative of Vk along the closed-loop trajectories, we obtain that for any x(tkjf ) ∈ Ω k , fˆ , x
f1a ≤ 2.5. In Mode 2,
c ≤ 2, v ≤ 5, f 2a ≤ 1. We only illustrate the method
such that for all e ≤ euf,k ∀t ∈ [tkjf , tkj + ∆tkj ), the
bk ( xˆ ), t ∈ [tkj , tkjf ) uk◊ ( xˆ ) = f b k ( xˆ ), t ∈ [tkj , tkj + ∆tkj ),
e(tkjf ) ≤
then, choosing ∆tkj ≥ T − tkj and setting σ(t ) = k + 1
n ◊ max ( LGk Vk )T ( x) }, Φ k , fˆ = {x ∈ ℜ : L fk Vk ( x) < uk
where
If xˆ (tkjf ) ∈ Ψ k , f and
Fig. 1. State trajectories.
Observer-based Fault Tolerant Control for Constrained Switched Systems
which is in Ω1. Only f1a is considered: f1a = 0, for 0 s ≤ t < t11f , and f1a = 2 + 0.2sin(5t ), for t11f ≤ t ≤ 0.7 s, with E1 = [2 − 0.2 0]T . The parameters are omitted. We switch the system to Mode 2 at t = t21 = 0.7 s, Fig. 1 shows that the origin of CSS is asymptotical stable.
5. CONCLUSIONS In this work, the FTC problem for CSS with input constraints is investigated. The proposed method will be also extended to more general nonlinear CSS with application to real systems in the future. [1]
[2]
[3]
[4]
[5]
REFERENCES R. A. Decarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, “Perspectives and results on the stability and stabilizability of hybrid systems,” Proc. of the IEEE, vol. 88, no. 7, pp. 1069-1082, 2000. N. H. EI-Farra, P. Mhaskar, and P. D. Christofides, “Output feedback control of switched nonlinear systems using multiple lyapunov functions,” Systems and Control Letters, vol. 54, no. 12, pp. 1163-1182, 2005. M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and Fault-Tolerant Control, Springer, Verlag Berlin Heidelberg, 2003. B. Jiang, M. Staroswiecki, and V. Cocquempot, “Fault accommodation for a class of nonlinear dynamic systems,” IEEE Trans. on Automatic Control, vol. 51, no. 9, pp. 1578-1583, 2006. H. Yang, B. Jiang, and M. Staroswiecki, “Observer based fault tolerant control for a class of switched nonlinear systems,” IET Control Theory and Applications, vol. 1, no. 5, pp. 15231532, 2007.
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Hao Yang was born in Nanjing, China, in 1982. He is currently a Ph.D. candidate in Nanjing University of Aeronautics and Astronautics, China, and Université des Sciences et Technologies de Lille, France. His research interests include fault tolerant control of hybrid systems. Bin Jiang was born in Jiangxi, China, in 1966. He obtained the Ph.D. in Automatic Control from Northeastern University, Shenyang, China, in 1995. Currently he is a Full Professor and Department Head of Automatic Control in Nanjing University of Aeronautics and Astronautics. He serves as Associate Editor for International Journal of System Science, International Journal of Control, Automation and Systems, etc. His research interests include fault diagnosis and fault tolerant control. Vincent Cocquempot was born in France, in 1966. He received the Ph.D. degree in Automatic Control from the Lille University of Sciences and Technologies, in 1993. He is currently a Professor in automatic control and computer science at the Institut Universitaire de Technologies de Lille, France. He is Head of Research in the LAGIS-CNRS UMR 8146, at Université des Sciences et Technologies de Lille, France. His research interests include robust on-line fault diagnosis for uncertain dynamical nonlinear systems, Fault Detection and Isolation (FDI) and Fault Tolerant Control (FTC) for Hybrid Dynamical Systems.
