controllers can be obtained to attenuate external perturbations which are bounded to a ball and have a uniform distribution. Similar results are expected in robust control, maximum likelihood parameter estima-tion, etc.
ACKNOWLEDGMENT
The authors would like to express their gratitude to the anonymous referees, whose comments and suggestions contributed much to the quality of this paper. In particular, a more direct and simpler proof of Theorem 1 was hinted at by one of the referees.
REFERENCES
[1] J. Ackermann, Robust Control: Systems with Uncertain Physical
Param-eters. New York: Springer-Verlag, 1993.
[2] B. R. Barmish, New Tools for Robustness of Linear Systems. New York: Macmillan, 1994.
[3] B. R. Barmish, C. M. Lagoa, and R. Tempo, “Radially truncated uniform distribution for probabilistic robustness of control systems,” in Proc.
Amer. Control Conf., Albuquerque, NM, 1997.
[4] B. R. Barmish and B. T. Polyak, “A new approach to open robustness problems based on probabilistic prediction formulae,” in Proc. 13th
IFAC World Congress, vol. H, San Francisco, CA, July 1996, pp. 1–6.
[5] S. P. Bhattacharyya, H. Chapellat, and L. H. Keel, Robust Control: The
Parametric Approach. Upper Saddle River, NJ: Prentice-Hall, 1995. [6] G. Calafiore, F. Dabbene, and R. Tempo, “Uniform sample generation
of vectors inl balls for probabilistic robustness analysis,” in Proc. 37th
CDC, Tampa, FL, 1998.
[7] X. Chen and K. Zhou, “‘On the probabilistic characterization of model uncertainty and robustness,” in Proc. 36th CDC, vol. 4, San Diego, CA, 1997, pp. 3816–3825.
[8] F. L. Chernous’ko, State Estimation for Dynamic Systems. Boca Raton, FL: CRC Press, 1994.
[9] X. Zhu, Y. Huang, and J. Doyle, “Soft vs. hard bounds in probabilistic robustness analysis,” in Proc. 35th CDC, Kobe, Japan, 1996. [10] A. B. Kurzhanskii and I. Valyi, Ellipsoidal Calculus for Estimation and
Control. Basel: Birkhäuser, 1997.
[11] M. E. Muller, “A note on a method for generating random points uni-formly onn-dimensional spheres,” Commun. ACM, vol. 2, pp. 19–20, 1959.
[12] K. Pearson, Tables of the Incomplete Beta Function. Cambridge, U.K.: Cambridge Univ. Press, 1934.
[13] L. R. Ray and R. F. Stengel, “A Monte Carlo approach to the analysis of control systems robustness,” Automatica, vol. 29, no. 1, pp. 229–236, 1993.
[14] F. C. Schweppe, Uncertain Dynamic Systems. Englewood Cliffs, NJ: Prentice-Hall, 1973.
[15] C. B. Soh, C. S. Berger, and K. P. Dabke, “On the stability properties of polynomials with perturbed coefficients,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 1033–1036, Oct. 1985.
[16] S. Wilks, Mathematical Statistics. New York: Wiley, 1963.
Stability Analysis and Bang–Bang Sliding Control of a Class of Single-Input Bilinear Systems
Yon-Ping Chen, Jeang-Lin Chang, and Kuo-Ming Lai
Abstract—This paper introduces a novel bang–bang sliding control of a class of single-input bilinear systems. The sliding function is chosen via the well-known pole-assignment method for linear time-invariant systems. Im-portantly, the bang–bang sliding control generates a reaching-and-sliding region and a stable-sliding region, each expressed by a set of linear inequal-ities. Both regions comprise the equilibrium point, shown to be asymptoti-cally stable. However, the stability analysis is processed under the limitation that the system state should be initially located in the reaching-and-sliding region. Two numeric examples are used for demonstration.
Index Terms—Bang–bang sliding control, bilinear systems, stability analysis.
I. INTRODUCTION
In general, bilinear systems are expressed by a state differential equa-tion, which is linear in control and linear in state but not jointly linear in state and control. Bilinear systems have been found in diverse pro-cesses and fields (see [1] and [2] for an excellent introduction) and many control strategies have developed, such as the quadratic feed-back control [3], [4] and the sliding-mode control [5], [6]. Here, we will focus on a class of “bang–bang” single-input bilinear systems with controller designed by the sliding-mode theory [7], [8]. Note that the term “bang–bang” means the single input only switches between two fixed values.
Further assume the bilinear system is time invariant and controllable while the multiplicative terms of the single input and state variables are omitted, i.e., the bilinear system can be treated as a linear time-in-variant (LTI) system by neglecting all these multiplicative terms. Based on this LTI system, a novel design technique of sliding function, called the pole-assignment-based method, is presented by directly using the prevailing pole-assignment method [9]. Actually, there are many other techniques for the sliding function design, such as the eigenstructure-assignment method [10] and the Lyapunov-based method [11]. For an LTI system, the Lyapunov-based method is simple and quite straight-forward to derive a sliding function, but it is not suitable for a bilinear system due to the fact that the existing multiplicative terms still dis-turb the system behavior and thus, complicate the process of stability analysis. As for the eigenstructure-assignment method, its main idea is to generate a desired eigenstructure of the sliding mode, just like the pole-assignment-based method introduced in this paper. Therefore, the eigenstructure-assignment method can be also found useful for a bi-linear system; however, it is often difficult to achieve an appropriate eigenstructure via the eigenstructure-assignment method. The pole-as-signment-based method is then proposed to make it easier to determine the eigenstructure of the sliding mode. In addition, with the help of well-developed tools such as The MATLAB software, the process of the sliding function design is highly simplified in this method.
Since the bang–bang control only can switch between two finite values, the system should be finally restricted to a bounded area. In fact, this area relates to two important regions, called the
reaching-and-Manuscript received March 19, 1998; revised December 1, 1998, September 20, 1999, and May 20, 2000. Recommended by Associate Editor, G. Tao.
Y.-P. Chen and K.-M. Lai are with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan.
J.-L. Chang is with the Department of Electrical Engineering, Oriental Insti-tute of Technology, Pan Chiao, Taipei County 220, Taiwan.
Publisher Item Identifier S 0018-9286(00)10016-9. 0018–9286/00$10.00 © 2000 IEEE
sliding (RAS) region and the stable-sliding (SS) region. Each region is formulated by a set of linear inequalities. When a system is initially located in the RAS-region, its trajectory will reach the SS-region and then move toward the origin.
In Section II, the bang–bang sliding control is introduced. Also dis-cussed is the stabilizability to the origin by means of reachability con-ditions to sliding mode and stability on the designed manifold. Sec-tion III uses two numeric examples of second-order bilinear systems to demonstrate the bang–bang sliding control in simulations [12]. Finally, concluding remarks are given in Section IV.
II. DESIGN OFBANG–BANGSLIDINGCONTROL ANDSTABILITY ANALYSIS
General bilinear systems are mathematically expressed by _x = Ax + Bu + m
k=1
Nkxuk (1)
wherex = [x1 x2 1 1 1 xn]T is the state,u = [u1 u2 1 1 1 um]T is the control, andA(2 Rn2n), B(2 Rn2m), Nk(2 Rn2n) are the system matrices. This paper will focus on the case of single input, de-scribed by
_x = Ax + bu + Nxu (2)
whereb 2 Rn andu 2 f1; 01g. Note that u switches between 1 and 01 depending on a switching condition concerning the system statex. Besides, the system is assumed controllable when Nxu is ne-glected. Hence, a gain vectork(2 Rn) can be uniquely obtained from the pole-assignment method by assigningn eigenvalues to A 0 bkT. The objective is to design a sliding controller for the single-input bi-linear system (2).
Now, let us introduce the pole-assignment-based method to design the sliding function. First, then eigenvalues i,i = 1; 2; 1 1 1 ; n, of A 0 bkT are selected to satisfy:
C1) All the eigenvalues are real and negative, i.e.,i < 0, i = 1; 2; 1 1 1 ; n;
C2) All the eigenvalues are distinct, i.e.,i6= jfori 6= j; C3) Thenth eigenvalue nis not in the spectrum ofA. Based on these conditions, we have
(A 0 bkT) [W
n01 wn] = [Wn01 wn] 30n01T 0
n (3)
where
3n01= diag[1 1 1 1 n01]; Wn01= [w1 1 1 1 wn01] and wi is the ith right-eigenvector corresponding to i, for i = 1; 2; 1 1 1 ; n. Let VT n01 vT n = [Wn01 wn] 01 and V n01= [v1 1 1 1 vn01] then (3) becomes VT n01 vT n (A 0 bk T) = 3n01 0 0T n VT n01 vT n (4)
wherevTi is theith left-eigenvector. Rearranging the second row of (4) leads to
vT
n(A 0 nIn) = vTnb kT (5) whereInis then 2 n identity matrix. From C3), A 0 nInis nonsin-gular and thus,vTnb 6= 0. It results in
vT nb
01 vT
n = kT(A 0 nIn)01: (6)
Let the sliding function be chosen as s = cTx = kT(A 0
nIn)01x (7)
i.e.,cT = kT(A 0 nIn)01, which also results in
cTA = ncT+ kT: (8)
From (6) we havecTb = 1. Then the derivative of s with respect to time becomes
_s = ns + kTx + 1 + cTNx u: (9) It is reasonable to assume1 + cTNx > 0 for the system state x near the origin. Let the control bang–bang switch between 1 and01 as
u = 0sgn(s) (10) then s _s = ns20 jsj 1 + cTNx 1 0 sgn(s) k Tx 1 + cTNx : (11) It is evident that if 1 + cTNx > kTx (12)
thens _s < 0 for s 6= 0, i.e., the reaching and sliding condition is guaranteed [7]. In other words, the system satisfying (12) is able to reach and slide along the sliding mode. For convenience, we call (12) the reaching-and-sliding region or RAS-region in brief. It is equivalent to1 + cTNx > kTx > 0(1 + cTNx) or
RAS-region: 1 + c
TN 0 kT x > 0
1 + cTN + kT x > 0: (13) Intuitively, there exists a subregionkxk bounded by (13). The maximal value of is defined as the RAS-index, expressed by
1= min kcTN + k1 Tk; kcTN 0 k1 Tk : (14) A largerI implies a larger RAS-region. Later it will be found that a largerIis accompanied with a slower convergence rate to the origin. Clearly, the design of sliding function is a compromise between the RAS-index and the system convergence rate.
Actually, the RAS-region only ensures the reaching and sliding be-havior. For the system stability, a subregion in (13) subject tos = 0 should be further decided, which is called the stable-sliding region or SS-region in brief. To effectively describe the sliding motions = 0, let us define yn01 yn = 3n01 0 0T n VT n01 vT n x: (15)
From (6) and (7), ifs = 0 then vTnx = 0, i.e., yn = 0. Therefore, rearranging (15) yields x = [Wn01 wn] 3 01 n01 0 0T 01 n yn01 yn = Wn01301n01yn01: (16)
Clearly,yn01= 0 leads to x = 0 for s = 0. A candidate of Lyapunov function is then given as for s=0
L =1
From (2), (4), and (15), its derivative with respect to time becomes _L = yT n013n01VTn01(Ax + bueq+ Nxueq) = yT n013n01VTn01((A 0 bkT)x + bkTx + bueq+ Nxueq) = yT n013n01yn01+ yTn013n01Vn01T (bkTx+ bueq+ Nxueq): (18) Note that based on the concept of equivalent control [7], the bang–bang inputu has been replaced by the continuous equivalent control ueq, expressed by
ueq= 0 k Tx
1 + cTNx (19)
which is derived from (9) by lettings = 0 and _s = 0. From (16) and (19), we have
kTx = 0 1 + cTNW
n01301n01yn01 ueq (20) and then (18) is changed into
_L = yT n013n01yn010 yTn01Hyn01 k Tx 1 + cTNx (21) where H = 3n01VTn01(In0 bcT)NWn01301n01:
If _L < 0, then (17) is really a Lyapunov function, which means the system ins = 0 performs a stable sliding motion converging to the origin. Hence, the SS-region is defined as
SS-region:
(1 + cTNx)yTn013n01yn010 (kTx)yTn01Hyn01< 0 (22) where the truth of1 + cTNx > 0 is adopted. Interestingly, if N = bdT thenH = 0 for arbitrary d. In this case, the SS-region is not needed to be considered since from C1)yTn013n01yn01 < 0. Only the RAS-region (13) is required.
For a second-order bilinear system, (22) is reduced to SS-region forn = 2: 1 + cTN 0 h
1k
T x > 0 (23)
whereh = vT1(I20 bcT)Nw1. While for a higher order system, the expression becomes
1 + cTNx yT
n013n01yn010 kTx yTn01Hyn01 < 0 jjmin 1 + cTNx 0 kTx kHk kyn01k2 (24) where jjmin = minf j1j; j2j; 1 1 1 ; jn01jg. Obviously, if jjmin(1 + cTNx) > jkTxj kHk then _L < 0. Thus, the SS-region can be further restricted tojjmin(1 + cTNx) > jkTxj kHk. It is equivalent to SS-region forn 3: 1 + cTN 0 kHk jjmin k T x > 0 1 + cTN + kHk jjmin k T x > 0: (25)
Significantly, the RAS-region (13) and the SS-region (23) forn = 2 and (25) forn 3 are all in the form of linear inequalities. This indeed simplifies the stability analysis. Most significantly, the neighborhood
near the origin satisfies all these inequalities; in other words, the origin is an asymptotically stable equilibrium point. Before getting into the numeric examples, a property concerning the convergence rate to the origin durings = 0 should be emphasized here. From (17), (21), and (24), we have d dt kyn01k2< 02 jjmin0 jkTxj 1 + cTNx kHk kyn01k2: (26) Clearly, ifs = 0 then jjmin = minf j1j; j2j; 1 1 1 ; jn01j g is re-lated to the convergence rate ofkyn01k2, or the convergence rate to the originx = 0. It is easy to conclude that the larger jjmin, the faster the convergence rate.
Besides, one limitation related to the system initial condition should be addressed here: The initial system statex(0) has to be located within the RAS-region. In practice, it is difficult to locate the initial system statex(0) precisely. In case that x(0) is not in the RAS-region, it is required to use other control algorithms to drive the system into the RAS-region first. For example, a conventional linear state-feedback control with high gain may be adopted to control a system of (1) pos-sessing smallNk. In the next section, to appropriately demonstrate the bang–bang sliding control, the system state is assumed to be initially located in the RAS-region, as mentioned.
III. NUMERICEXAMPLES OFSECOND-ORDERBILINEARSYSTEMS Consider the following second-order bilinear systems, governed by:
_x = Ax + bu + Nxu (27)
wherex = [x1 x2]T,u 2 f1; 01g and the system matrices A = aa11 a12 21 a22 ; b = b1 b2 ; N = n11 n12 n21 n22 : There are two numeric examples discussed. The first one adopts the work of Longchamp [12] to control the continuously stirred tank reactor with a first-order irreversible exothermic reaction under nonisothermal conditions. In this example, the conditionN = bdT is satisfied, so only the RAS-region is required. Further, in order to demonstrate the case ofN 6= bdT, the second example is introduced by directly modifying theN matrix in the first example.
Based on the conditions C1)–C3) the gain vectorkT = [k1 k2] is obtained by assigning eigenvalues1and2forA0bkT. Note that1 is related to the convergence rate to the origin during the sliding mode s = 0, as shown in (26). The sliding function is then determined by s = cTx with cT = kT(A 0
2I2)01 = [c1c2]. Now, the RAS-region (13) becomes
1 + (c1n11c2n210 k1)x1+ (c1n12c2n220 k2)x2 > 0 1 + (c1n11c2n21+ k1)x1+ (c1n12c2n22+ k2)x2 > 0 (28) and the SS-region (23) is rewritten as
1 + c1n11+ c2n210 hk1 1 x1 + c1n12+ c2n220 hk2
1 x2 > 0 (29) whereh = vT1(I20 bcT)Nw1. Here,vT1 andw1are the left- and right eigenvectors corresponding to1. Note that the RAS-region (28) applies to both examples, while the SS-region (29) is only used for the second example becauseN 6= bdT. Finally, as mentioned before, the originx = 0 is asymptotically stable.
TABLE I RAS-INDEX OFEXAMPLE1
Fig. 1. RAS-region (shadowed) of Example 1.
Example-1: Consider the continuously stirred tank reactor with a first-order irreversible exothermic reaction under nonisothermal con-ditions, expressed as (27) with
A = 06:25 024:25 1 ; b = 00:250 ; N = 01 00 0 : (30)
The spectrum ofA is f3; 00:75g and N = bdT withdT = [4 0]. Hence, only the RAS-region (27) is needed for the stability analysis. Let2= 00:7, which is not in the spectrum of A, and the convergence rate1range from00.1 to 03.0. Table I lists the RAS-index Iand shows that a faster convergence rate results in a smallerI. The case of1= 01:0 and 2= 00:7 are selected for demonstration and then the feedback gain is obtained askT= [015:8 03:168]. Therefore, the sliding function in (7) is set ass = 04x10 0:64x2. Fig. 1 depicts the RAS-region in the phase plane with1= 0:0499. Let the system be initially located atxT(0) = [0:05 0:05] in the RAS-region. Then the system trajectory shown in Fig. 2 is driven tos = 0 and performs the sliding motion. The state variablesx1andx2are given in Fig. 3. After reaching the sliding mode at timetr = 0:31 s, these two state variables converge to the originx = 0 with a convergence rate 1= 01:0, not affected byN. It can be seen from (26) with H = 0 since N = bdT. Obviously, the origin is asymptotically stable.
For comparison, the stability region in the work of Longchamp [12] is found aboutkxk 0:045 in the similar sense of RAS-index. It is approximate to the regionkxk Ias given in Table I. However, the stability analysis introduced in this paper is simpler due to the fact that only linear inequalities are processed.
Fig. 2. The system trajectory in the phase plane of Example 1.
Fig. 3. The state variables of Example 1. TABLE II RAS-INDEX OFEXAMPLE2
Example 2: To demonstrate the bang–bang sliding controller is still useful for the case ofN 6= bdT, this example employs the same model in (30) except that
N = 01 10 1 :
Clearly,N 6= bdT and the SS-region (29) should be included for the stability analysis. The eigenvalue2 is also set as00.7, not in the spectrum ofA, and the eigenvalue 1ranges from00.1 to 03.0. Table II lists the RAS-index and shows that a faster convergence rate results in a smaller RAS-index. Let1= 01:0 and 2= 00:7, then kT = [015:8 03:168] and s = 04x
10 0:64x2, same as those in the Example 1. Fig. 4 illustrates the RAS-region and the SS-region in the phase plane. Figs. 5 and 6 are obtained by setting the initial condition
Fig. 4. RAS-region (shadowed) and SS-region (bold line) of example 2.
Fig. 5. The system trajectory in the phase plane of example 2.
asxT(0) = [0:05 0:05], which is located in the RAS-region. From Fig. 5, the system trajectory is first steered tos = 0 and then restricted in the sliding mode. Fig. 6 shows the success of controllingx1andx2 to the originx = 0. The state variables x1andx2are given in Fig. 6. After reaching the sliding mode at timetr = 0:32 s, these two state variables converge to the originx = 0 with convergence rates affected by1= 01:0 and H 6= 0 since N 6= bdT. It is evident the origin is asymptotically stable.
Note that the success of the bang–bang sliding controller is demonstrated from the above simulation results. Most importantly, it is quite straightforward to design the sliding function by the pole-as-signment-based method. Besides, it is simple to analyze the system stability from the RAS-region and the SS-region, each expressed by a set of linear inequalities.
IV. CONCLUSIONS
A new design approach for bang–bang sliding control of a class of single-input bilinear systems is introduced. The sliding function is de-termined by the pole-assignment-based method. Three conditions are required to choose the eigenvalues, which are related to the system
Fig. 6. The state variables of example 2.
stability and the convergence rate in the sliding mode. The origin is found to be asymptotically stable. The stability region depends on the RAS-region and the SS-region, each expressed by a set of linear in-equalities. Besides, an RAS-index is produced to show whether the RAS-region is reasonable or not. It is noticed that the sliding func-tion design is a compromise of the RAS-index and the convergence rate to the origin. Finally, simulation results of two second-order bi-linear systems are adopted for demonstration. Investigations to extend the approach to multi-input bilinear systems are in progress.
REFERENCES
[1] R. R. Mohler, Nonlinear Systems, Volume I, Dynamics and
Con-trol. Englewood Cliffs, NJ: Prentice-Hall, 1991.
[2] , Nonlinear Systems, Volume II, Applications to Bilinear
Con-trol. Englewood Cliffs, NJ: Prentice-Hall, 1991.
[3] J. P. Quinn, “Stabilization of bilinear systems by quadratic feedback con-trollers,” J. Math. Anal. Appl., vol. 75, pp. 66–80, 1980.
[4] P. O. Gutman, “Stabilizing controllers for bilinear systems,” IEEE Trans.
Automat. Contr., vol. AC-26, pp. 917–922, 1981.
[5] H. Sira-Ramirez, “Sliding motions in bilinear switched networks,” IEEE
Trans. Circuits Syst., vol. CAS-34, pp. 919–933, 1987.
[6] H. Sira-Ramirez, R. Ortega, R. Perez-Moreno, and M. Garcia-Esteban, “A sliding mode controller-observer for dc-to-dc power converters: A passivity approach,” in Proc. 34th Conf. Decision Contr., New Orleans, LA, 1995.
[7] V. I. Utkin, Sliding Regimes and Their Applications in Variable Structure
Systems. Moscow: MIR, 1978.
[8] U. Itkis, Control System of Variable Structure. New York: Wiley, 1976. [9] C. T. Chen, Linear System Theory and Design: Winston, 1984. [10] O. M. E. Elghezawi, A. S. I. Zinober, and S. A. Billings, “Analysis and
design of variable structure systems using a geometric approach,” Int. J.
Contr., vol. 38, pp. 657–671, 1983.
[11] W. C. Su, S. V. Drakunov, and U. Ozguner, “Constructing discontinuity surfaces for variable structure system: A Lyapunov approach,”
Auto-matica, vol. 32, pp. 925–928, 1996.
[12] R. Longchamp, “Stable feedback control of bilinear systems,” IEEE
