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Reliable Control of Nonlinear Systems via Variable Structure Scheme
Yew-Wen Liang and Sheng-Dong Xu
Abstract—This study proposes a class of variable structure stabilizing laws which make the closed-loop system be capable of tolerating the abnormal operation of actuators within a pre-specified subset of actuators. The ranges of acceptable change in control gain magnitude that preserves system’s stability are estimated for the whole set of actuators. These ranges are shown to be able to be made larger than those obtained by linear quadratic regulator (LQR) reliable design (Veillette, 1995, and Liang et al.,
Manuscript received April 16, 2004; revised November 8, 2004, July 28, 2005, and December 30, 2005. Recommended by Associate Editor D. Nesic. This work was supported by the National Science Council, Taiwan, R.O.C., under Grants NSC 92-2213-E-009-124 and NSC 93-2218-E-009-037.
The authors are with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, R.O.C. (e-mail: ywliang@cc.nctu.edu.tw).
Color versions of Figs. 1–3 are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2006.880802
2000) by the choice of control parameters. Besides, this approach doesnot need the solution of Hamilton-Jacobi (HJ) equation or inequality, which is essential for optimal approaches such as LQR and reliable designs. As a matter of fact, this approach can also relax the computational burden for solving the HJ equation or inequality.
Index Terms—Hamilton–Jacobi (HJ) equation, nonlinear systems, reli-able control, varireli-able structure control (VSC).
I. INTRODUCTION
The study of reliable control has recently attracted lots of atten-tion (see, e.g., [1]–[3], [5], [8]–[10], and [12]–[15]). The objective of this study is to design an appropriate controller such that the closed-loop system can tolerate the abnormal operation of some specific con-trol components and retain an overall system stability with acceptable system performance. An abnormal operation may include degradation, amplification and partial outage. From the approach viewpoint, in gen-eral, reliable control systems can be classified as active [1]–[3], [5] and passive [8]–[10], [12]–[15]. In this note, we consider only the passive issues. In an active reliable control system, faults are to be detected and identified by a fault detection and diagnosis (FDD) mechanism. Then the controllers are reconfigured according to the online detec-tion results in real time. On the other hand, the passive approach ex-ploits system’s inherent redundancies to design a fixed controller so that the closed-loop system can achieve an acceptable performance not only during normal operation but also under various components fail without the need of FDD and controllers’ reconfiguration. Although the performances of the active reliable control which uses controllers’ reconfiguration are generally superior to those of passive one under various faulty situations, the active approach needs a reliable FDD but the passive one doesnot. This is important when the available reaction time is short after the occurrence of faults.
Several approaches for passive reliable control have been proposed, for example, linear matrix inequality (LMI)-based approach [10], al-gebraic Riccatti equation (ARE)-based approach [12], [13], coprime factorization approach [14] and Hamilton–Jacobi (HJ)-based approach [9], [15]. Although the HJ-based approach is mainly for nonlinear sys-tems, its reliable controllers need a solution of an HJ equation or in-equality, which is known not easy to obtain. A power series method [6] may alleviate the difficulty of solving the HJ equation or inequality through computer calculation. However, the obtained solution is only an approximate one and, when system is complicated, the computa-tional load grows fast as the order of the approximated solution in-creases. Due to these potential drawbacks of the HJ-based approach, this note investigates the reliability issues from the variable structure control (VSC) viewpoint, which is known to have the advantages of fast response and low sensitivity to model uncertainties and/or external disturbances (see, e.g., [4], [7], and [11]). In this note, we propose a VSC design that is shown to be able to tolerate the abnormal operation of actuators within a prespecified subset of actuators. The regions of acceptable change in control gain magnitude that preserves system’s stability are also estimated. These regions are shown to be able to be made larger than those obtained in [9] and [13] by suitable choice of control parameters. Besides, the VSC approach needs not the solution of HJ equation or inequality. Thus, this approach can also alleviate the computational burden for solving the HJ equation or inequality.
This note is organized as follows. The reliable control problem and the main goal of the note are given in Section II. This is followed by de-signing the VSC controllers and analyzing their reliability. An example is also given in this section to demonstrate the use and the benefits of the design. Finally, Section IV gives concluding remarks.
1722 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10, OCTOBER 2006
II. PROBLEMSTATEMENT
Consider a class of nonlinear control systems as described by
_x1= f1(x) + G (x)u (1) and
_x2= f2(x) + G (x)u+ G(x)u + d: (2)
Here, x1 2 IRn ; x2 2 IRn , and x = (xT1; xT2)T denotes the system states, u 2 IRm and u 2 IRn are the control inputs, d = (d1; . . . ; dn )T denotes possible model uncertainties and/or
external disturbances, and ( 1 )T denotes transpose of a matrix or a vector.f1(x) 2 IRn ; f2(x) 2 IRn ; G (x) 2 IRn 2m ; G (x) 2 IRn 2m , andG
(x) 2 IRn 2n are smooth functions. For the
interest of study, we assume thatf1(0) = 0 and f2(0) = 0. Note that we have divided the control inputs into two disjoint groups and 0 within which the abnormal operation of actuators in the set must be tolerated. We also note that system (1)–(2) might come from a general nonlinear affine system _x = f(x) + G(x)u through a diffeomorphic transformation. Whenn1 = n2; f1(x) = x2 and G (x) = 0, system (1)–(2) reduces to an important class of second
order dynamical systems.
If all the actuators in the set fail to operate, (1)–(2) becomes
_x1= f1(x) (3)
and
_x2= f2(x) + G(x)u + d: (4)
In practical applications, the number of susceptible actuators in may be selected to be as many as possible. In addition, we assume that system (3)–(4) is in regular form. That is,G(x) is a nonsingular ma-trix, as described in Assumption 1. This assumption is necessary for the existence of the equivalent control (see, e.g., [4]).
Assumption 1: The origin of system (3)–(4) is locally asymptotically
stabilizable andG(x) is a nonsingular matrix.
In addition to Assumption 1, we also impose the next two assump-tions.
Assumption 2: Suppose there exists a functionx2 = (x1) such
that the reduced-order system _x1 = f1(x1; (x1)) has an
asymptoti-cally stable (AS) equilibrium point at the originx1 = 0.
Assumption 3: There exist functionsi(x; t) 0; i = 1; . . . ; n2, such thatjdij i(x; t).
The objective of this study is then to organizeu anduso that the origin of the closed-loop system is AS even when the actuators in the set experience abnormal operation. The susceptible actuators in this design are used to improve system performance when they are available.
III. MAINRESULTS
To achieve the objective of the note as stated previously, in this sec-tion we first employ the VSC technique to design the reliable con-trollers. This is followed by analyzing the overall reliability of the de-signed system. Finally, we present an illustrative example to demon-strate the benefits of the design.
A. Design of Reliable Controllers
The idea is first to organize a VSC lawu as ifuis unavailable. Then the remaining controlsuare designed to promote the overall
system performances. Suppose now that all the actuators in are un-available. Then, by Assumption 2, we choose the sliding surface to be
s = x20 (x1) = 0: (5)
It follows from (3)–(4) that
_s = f2(x) + G(x)u 0 @ @x
1 1 f1(x) + d: (6)
Following the VSC design procedure [11], the VSC law for actuators in0is designed as
u3
= G01 (x) @ @x
1 1 f1(x) 0 f2(x) 0 3 1 sgn(s)
(7) where3 = diag(1; . . . ; n ) with i> i(x; t) + riandri> 0
for alli = 1; . . . ; n2; sgn( 1 ) denotes the sign function and sgn(s) :=
(sgn(s1); . . . ; sgn(sn ))T. By direct calculation sT_s 0 n
i=1
ri1 jsij (8) wheresi denotes theith entry of the sliding vector s. Equation (8) implies that the system states will reach the sliding surface in a finite time and remain there [11]. Then, by Assumption 2, the reduced-order dynamics on sliding surface makes the states slide toward the origin.
In addition to the design of actuators in0, as described by (7), we now suppose that actuators in are also available. The governing equa-tion in this case is given by (1)–(2). From (1)–(2), and (5),u = u3 given in (7), and (8) we have
sT_s 0 n i=1
ri1 jsij + sT0(x)u (9)
where0(x) := G (x)0((@ )=(@x1))1G (x). Clearly, an intuitive
candidate ofuto makesTs more negative than the case of u= 0 has the form
u3
= 031 sgn(0T(x)s) (10) where3:= diag(1; . . . ; m ) and i 0 for all i = 1; . . . ; m1. In practical applications, actuators might experience a change in control gain magnitude which covers the cases of normal operation, degrada-tion, amplification and outage. Therefore,uhas the form of (11)
u= Nu3 (11)
whereN2 IRm 2m is a nonsingular diagonal matrix which denotes the change in gain magnitude ofu3. Clearly, the two casesN = I andN= 0 correspond to the situations where all actuators in are
in normal operation and are in outage, respectively. It follows from (9)–(11) that sT_s 0 n i=1 ri1 jsij 0 m j=1 (N)jj1 j1 j(0T(x)s)jj (12)
Fig. 1. Relation between and in the ROS estimation of VSC reliable design.
where ( 1 )j and ( 1 )ij denote the jth entry of a vector and the (i; j)-entry of a matrix, respectively. Equation (12) implies that, when some or all of actuators in are healthy and u3 has been chosen in the form of (7), system states towards the sliding surface are faster than the case when all actuators in fail to operate. These lead to the next result.
Theorem 1: Suppose that Assumptions 1–3 hold. Then the origin of
system (1)–(2) is locally asymptotically stable (AS) under the controls (7) and (10) even when some or all of actuators in experience ab-normal operation in the sense of (11).
B. Reliability Analysis
System (1)–(2) with controls given by (7) and (10) discussed above has been shown to be able to tolerate any abnormal operation of actu-ators in. In this section, we will also estimate the allowable changes in control gain magnitude of actuators in0that still guarantee asymp-totic stability performance of the system with control (7) and (10). For this purpose, we suppose thatG(x) is a diagonal matrix and the ac-tual effective controls in0have the form
u = Nu3 (13)
whereN 2 IRn 2n is a nonnegative diagonal matrix which denotes the change in gain magnitude ofu3. Under the effective controls (11) and (13), the overall system becomes
_x1= f1(x) + G (x)Nu3 (14)
and
_x2= f2(x) + G (x)Nu3+ G(x)Nu3 + d: (15)
Since bothG(x) and N are diagonal matrices, we then have sT_s = sT (N 0 I) @ @x 1f1(x) 0 f2(x) 0 N3sgn(s) + d 0 m j=1 j(N)jjj(0(x)Ts)jj: (16)
Similar to the derivation of robust controllers in [11], we have the next result, which addresses the reliability of the design.
Theorem 2: Suppose that Assumptions 1–3 hold andG(x) is a
diagonal matrix. Then, the origin of system (1)–(2) is locally AS under the effective controls given by (11) and (13) if
i(x; t) + (N 0 I) @ @x
1 1 f1(x) 0 f2(x) i
< (N3)ii; for all i = 1; . . . ; n2: (17)
To compare the result with those given in [9] and [13], we consider the special case whered = 0 and N I=2. The latter implies that j(N0I)iij (N)ii. Condition (17) can then be simplified as (18). Corollary 1: Suppose thatN I=2; d = 0; G(x) is a
diag-onal matrix, and Assumptions 1 and 2 hold. Then, the origin of system (1)–(2) is locally AS under the control given by (13) if
@
@x1f1(x) 0 f2(x) i < (3)ii;
for alli = 1; . . . ; n2: (18)
Remark 1: If all states are available for feedback, then the
1724 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10, OCTOBER 2006
Fig. 2. Norm of states, sliding vector, and control inputs for = 0.
asj(((@ )=(@x1))f1(x) 0 f2(x))ij + cifor some positive constantci. Otherwise, they can be tuned as large as possible to increase the re-sponse but fulfill the maximum control magnitude constraint. Clearly, (18) will establish a region of stability (ROS) that depends on the choice ofi.
Remark 2: From Corollary 1, we know that, when the control
pa-rameters(3)ii(ori) fori = 1; . . . ; n2are selected to satisfy (18), the rangesN I=2 and N 0 are sufficient to guarantee the
asymptotic stability performance of the closed-loop system. Since this note only deals with passive reliable control (i.e., without requiring fault information), the conditions N I=2 and N 0 then characterize the reliability level of the closed-loop system. That is, the asymptotic stability is preserved even when the actuators in experi-ence abnormal operation in any order and in any combination. A larger region forN may also be allowed if3 is chosen to satisfy (17). Thus, the acceptable regions ofNandN for system’s stability can be made larger than those given by [9] and [13]. However, the enlarge-ment of the gain magnitude3 might come at the price of increased chatter in the sliding mode.
Remark 3: It is noted thatNu3= 0 if system states keep staying
on sliding surface. This implies that the actuators in have no effect on reduced-order dynamics no matter whether or not they are in normal operation. In order to promote system performance on sliding surface and keep the same reliability level as in Theorem 2 when the actua-tors in are available, u3can be modified asu33 given by (19) if a Lyapunov functionV (x1) of the reduced-order system given in
As-sumption 2 is known u33 = 031 sgn(0T(x)s) 0 K1 GT (x1; (x1)) @V@x 1 T (19) whereK = diag(k1; . . . ; km ) and ki 0 for all i = 1; . . . ; m1. The derivatives of V (x1) along the reduced-order system _x1 =
f1(x1; (x1)) + G (x1; (x1))Nu with u = u33 and
u = u3 are found to be _V ju =u = ((@V )=(@x1)) 1
f1(x1; (x1))0 mi=1ki1 (N)i1 (((@V )=(@x1))G (x1; (x1)))i2
and _V ju =u = ((@V )=(@x1)) 1 f1(x1; (x1)), respectively, since Nu3 = 0 whenever system states keep staying on sliding surface.
Clearly, _V ju _V ju < 0. This implies that the convergence speed of the reduced-order system withu = u33 is faster than that with
u= u3. Next, we investigate the reliability of (1)–(2) under controls
(13) withu3 being replaced by u33. In this case, sT_s is modified from (16) as sT_s = sT (N 0 I) @ @x 1f1(x) 0 f2(x) 0N3 1 sgn(s) + d 0 m j=1 (N)j j1 j(0(x)Ts)jj 0kj @x@V 1G (x1; (x1)) j 0 T(x)s j : (20)
To guarantee the same reliability level as those given by Theorem 2 and Corollary 1, the control parametersj andkj should be selected to satisfy
kj1 @V@x
1G (x1; (x1)) j j; for j = 1; . . . ; m1 (21)
in addition to (17). In particular, if0(x) = 0, then (16) and (20) be-come the same. This implies that the requirement of (21) can be re-moved without affecting the reliability level.
Fig. 3. Norm of states, sliding vector, and control inputs for = 2.
C. An Illustrative Example
Consider a nonlinear control system _y = f(y) + u1 1 g1(y) + u21 g2(y) from [9] with f(y) = (0y13; 0y2+ y3y4; 0y3+ y24; y4+
y3y4)T; g1(y) = (0; y1; 0; 1)T andg2(y) = (1; 0; y3; 0)T. It was
pointed out in [9] that(f; g2) is not a stabilizable pair, while (f; g1) is
asymptotically stabilizable. This means that the first actuator can not be taken as the susceptible input. Thus, in this example, we consider 0= u
1and = u2. A change of coordinatex = T (y) = (y1; y20
y1y4; y3; y4)T withx = (xT1; x2T)T; x1 = (x1; x2; x3)T andx2 =
x4leads to the form of (1)–(2) withf1(x) = (0x31; x31x40 2x1x4+
x3x40 x1x3x40 x2; x240 x3)T; f2(x) = x4 + x3x4; G (x) =
(1; 0x4; x3)T; G (x) = 0 and G(x) = 1. Clearly, the function
x2 = (x1) = 0 fulfills the requirement of Assumption 2, and a
Lyapunov function for the reduced-order system _x1 = f1(x1; 0) is
found to beV (x1) = (x21+ x22+ x23)=2. According to (5), the sliding
surface has the forms = x4= 0, and the VSC laws given by (7) have the formu31= 0x40x3x401sgn(x4). Since in this example 0(x) =
G (x)0((@ )=(@x1))1G (x) = 0, it follows from Remark 3 that
u2 may be selected asu332 = 0k 1 GT (x1; 0) 1 ((@V )=(@x1))T =
0k(x1+ x23); k 0, to promote system stability on sliding surface
and maintain the same reliability level as those of Theorem 2 without the need of (21). The overall control in terms of original variablesy then has the form
u = 0 y4+ y3y4+ 1 sgn(y4); ky1+ ky23 T: (22)
In caseu1is healthy, the dynamics ofy4decouples from the others as _y4 = 0 1 sgn(y4). This implies that y4approaches zero (i.e., the states reach the sliding surface) in a finite timey4(0)=. After reaching
the sliding surface we havef2(x) = 0. Condition (18) is then fulfilled since (x1) = 0. However, the state might move out of its ROS before
reaching the sliding surface. As a matter of fact, it still needs the infor-mation of ROS, depending on, for stability. An estimation of the ROS
foru1being healthy can be derived asD= fyjy21+ y23+ y42< rg
from the analysis given in [9] with slight modifications, whereris the solution of (23)
minimize y21+y23+y24
subject to 0y410 y32+y3y240jy4j =0 & (y1; y3; y4)6=(0; 0; 0): (23) Clearly,r is a function of, and the relation between r and is described in Fig. 1.
To compare the performances between VSC and LQR reli-able designs, the LQR relireli-able laws are recalled from [9] as u = (0ky4; 0y23)T with k > p2 + 1. However, under these
laws, the closed-loop dynamics ofy1is short of linear terms. It follows thaty1 will converge slowly for smally1even whenu2 is available. To promote the convergence speed ofy1, we modify the LQR reliable laws as (24) below:
u = 0ky4; 0y10 y32 T withk >
p
2 + 1: (24) Clearly, these modified laws result in exponential convergence ofy1
near the origin whenu2 is available, and they are also the LQR reli-able laws associated with the class of positive semidefinite solutions V (x) = y2
1+ y23+ ky42; k >p2 + 1, of the same HJ-inequality given
in [9].
In this example, the LQR and VSC reliable laws are adopted from (24) withk = 3 and (22) with k = 1, respectively. The initial states are selected asy0 = (0:1; 1:2; 0:7; 0:9)T. To emphasize the relation between control magnitude and speed of response, the value of control parameter in the VSC law is set to be 0.7 and 2.2. Clearly, y0is inside the estimated ROS for = 0:7 and 2:2. Furthermore, the sign function is replaced by saturation function with boundary layer width 0.01 to avoid chattering. To examine the influences of the change in control gain magnitude, we also consider the two cases of whichN2= 0 and
1726 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10, OCTOBER 2006
2. These two cases correspond to the second loop gain being broken and amplified, respectively.
Numerical simulations are given in Figs. 2–3, which correspond to N2 = 0 and 2, respectively. In these figures, the dashed, solid and
dash-dotted lines denote the timing responses of norms of states and controls by LQR and VSC designs with = 0:7 and 2:2, respectively. The dotted line and starred line denote the norm of the sliding vector by VSC design with = 0:7 and 2:2, respectively. From these figures, system states are observed convergent to zero for all the two cases, which agree with the theoretic results. Whenu2 fails to operate, the system is found to have a linear uncontrollable mode = 0 and the associated closed-loop dynamics ofy1 decouples from the others as _y1= 0y13. It means thaty1will approach zero but the convergence rate will be progressively smaller asjy1j gets smaller. This is why the norm
of system states in Fig. 2 converges to zero very slowly. In addition, since in this example0(x) = 0; u31 is then the main force to make system states reach the sliding surface. It follows that the first reaching time of system states to the sliding surface depends only on the choice of control parameter. The first reaching time observed from jjsjj in Figs. 2–3 for = 0:7 and = 2:2 are around treach 1:2 and
0.4. This implies that the larger the value of is, the shorter the first reaching timetreach is. These phenomena can also be told from the abrupt change of the control magnitude, where the VSC reliable system is driven mainly by the equivalent part ofu after the first reaching time. This example verifies that the control parameters of the VSC reliable design can be tuned as large as possible to increase the response while fulfilling the maximum control magnitude constraint.
IV. CONCLUDINGREMARKS
Variable structure type stabilizing control laws have been proposed in this note to study reliable control issues. This approach can alleviate the computational burden for solving the HJ equation or inequality. In addition, the regions of acceptable change in control gain magnitude that guarantees system’s stability can be made larger than those ob-tained by LQR reliable design by the choice of control parameters. As a matter of fact, the control parameters can be tuned as large as pos-sible in practical applications to promote the responding performances while fulfilling the maximum control magnitude constraint.
ACKNOWLEDGMENT
The authors would like to thank Prof. D. Nesic for his assistance and the anonymous reviewers for their constructive comments and helpful suggestions which have led to substantial improvements to this note.
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Comments and Remarks on “On Improved Delay-Dependent Robust Control for Uncertain
Time-Delay Systems”
Qing-Long Han
Abstract—The purpose of this note is to correct some statements and nu-merical examples’ results in the above paper. A few remarks are also given to clarify the facts that for systems with small delay, the results in Han et al. (2003) are much less conservative than those in Kwon and Park (2004); for systems with nonsmall delay, the criterion in Kwon and Park (2004) fails to make any conclusion, while the criterion in Han et al. (2003) can be appli-cable to these kinds of systems.
I. COMMENTS ANDREMARKS
Consider the following system [1]:
_x(t)=(A+1A)x(t)+(A1+1A1)x(t0h)+(B+1B)u(t)
x(s) = (s); s 2 [0h; 0] (1)
where1A, 1A1, and1B are uncertain matrices satisfying 1A = D1F1(t)E1; 1A1= D2F2(t)E2; 1B = D3F3(t)E3
Manuscript received April 17, 2005; revised December 20, 2005 and April 19, 2006. Recommended by Associate Editor L. Magni. This work was sup-ported in part by Central Queensland University for the Research Advancement Awards Scheme Project “Robust Fault Detection, Filtering and Control for Un-certain Systems with Time-Varying Delay” (January 2006–December 2008) and the Strategic Research Project “Delay Effects: Analysis, Synthesis and Appli-cations” (2003–2006).
The author is with the School of Information Technology, Faculty of Busi-ness and Informatics, Central Queensland University, Rockhampton QLD 4702, Australia (e-mail: q.han@cqu.edu.au).
Digital Object Identifier 10.1109/TAC.2006.883032 0018-9286/$20.00 © 2006 IEEE
