Ultimate boundedness control of linear systems with
band-bounded nonlinear actuators and additive
measurement noise
Chih-Chin Hsu, I-Kong Fong
∗Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, ROC Received 20 June 2000; received in revised form 14 November 2000
Abstract
For linear systems driven by band-bounded nonlinear actuators, a set of linear matrix inequality (LMI) based su3cient conditions are derived for 4nding state feedback controllers which assure ultimate boundedness of state trajectories. Besides actuator nonlinearity, it is assumed that additive noise exists when state variables are measured for feedback. The purpose is to minimize the ultimate boundedness region while tolerating noise of the largest magnitude. When a state feedback controller is determined for a given system by solving the LMI conditions or by any other means, a less conservative LMI condition is given for further examination of the resultant ultimate boundedness region and tolerable noise magnitude.
c
2001 Elsevier Science B.V. All rights reserved.
Keywords: Nonlinear actuator; Ultimate boundedness control; Linear matrix inequality; Measurement noise
1. Introduction
Actuators in control systems inevitably have nonlinear characteristics. For example, no actuators have an in4nite operation range. Thus saturation is the most often discussed nonlinearity for actuators [1,2]. However, depending on the underlying physical principles and manufacturing precision, most actuators have nonlinear characteristics other than saturation. Designing control systems without taking these characteristics into account may produce an undesirable system response. The sector-bounded nonlinearity [12] has long been used to represent a quite general class of actuator characteristics, but with only a few exceptions, such as [10]; related works seem to focus more on the analysis problems, such as the circle and Popov criteria for the famous Lur’e problem. This may be an indication of the di3culty of the controller design problem in the presence of nonlinear actuators. As to still other actuator nonlinearities like deadzone, hysteresis, backlash, and quantization which cannot be accommodated by the sector-bounded nonlinearity, general discussions are rarely seen. Individual results usually cover speci4c types of nonlinearity or use approximations. For instance [15] treats the hysteresis characteristic by converting it into a passive operator, and [5] uses the describing function method to approximate the nonlinearity in a controller synthesis problem. Here we intend to deal
This work is supported by the National Science Council of the Republic of China under Grant NSC 89-2213-E-002-088. ∗Corresponding author. Fax: +886-2-23660449.
E-mail address: [email protected] (I-K. Fong).
0167-6911/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.
enter an ultimate boundedness region, and to minimize the “size” of the region. In the mean time, it is hoped that the closed-loop systems can tolerate the measurement noise of the largest magnitude. In the following sections, we start by presenting a rigorous formulation of the problem, and proceed with the help of the linear matrix inequality (LMI) approach. The 4rst result is a set of LMI based su3cient conditions for the existence of the feasible state feedback control law. These LMI conditions can be augmented by an appropriate objec-tive function to form a convex optimization problem, which reNects our quest of state feedback controllers that minimize the ultimate boundedness region and tolerate the largest measurement noise. After solving the very tractable convex optimization problem, we propose a generalized eigenvalue problem (GEVP) to further examine the optimal state feedback controller, because it is derived based on a set of su3cient conditions. The examination may produce less conservative estimates of the ultimate boundedness region “size” and tolerable measurement noise magnitude. In fact, the examination procedure may also be applied to any state feedback controller designed by other methods. All proposed methods are applied to a numerical example so that all application results may be compared on the same basis.
2. Problem formulation
Before we start, some notations are de4ned 4rst. For any positive integer k and M; N ∈ Rk×k; M ¿ N
means that M; N are symmetric and M − N is positive semi-de4nite. Similar notations apply to symmetric positive=negative de4nite matrices. The notation Ik represents the k × k identity matrix, max(M) denotes the
largest eigenvalue of the symmetric matrix M, and diag(M1; : : : ; Mk) is a block diagonal matrix with the blocks
M1; : : : ; Mk on the diagonal position.
Let us consider the feedback control system in Fig. 1, which has the mathematical model ˙x(t) = Ax(t) + Bu(t); x(0) = xo;
˜u(t) = K[x(t) + n(t)];
u(t) = [1(t; ˜u1(t)) · · · m(t; ˜um(t))]T:
(1)
Fig. 2. Actuator input–output characteristic.
In (1) the 4rst linear state equation describes the plant to be controlled, where x(t) ∈ R‘ is the state vector,
and u(t) ∈ Rmis the control input vector. The second equation represents the state feedback controller, where
n(t) ∈ R‘ is the additive measurement noise vector, bounded by n(t)Tn(t) 6 2 for all t ¿ 0 and some
¿ 0; K ∈ Rmב is the state feedback gain matrix, and ˜u(t) ∈ Rm is the controller output vector. Finally, the
third equation stands for the nonlinear characteristics of the actuators, where ˜ui(t) is the ith component of
˜u(t), and i(·; ·) : [0; ∞) × R → R denotes the ith nonlinear characteristic, illustrated in Fig. 2. Note that the
dotted curve (or curves, as the relationship may be time-varying) of i(·; ·) is only known to be con4ned to the
band between the two parallel dashed lines. Hence we call it the band-bounded nonlinearity. Mathematically, it is equivalent to say that ui(t) and ˜ui(t) satisfy the algebraic inequality
[ui(t) − i˜ui(t) + i][ui(t) − i˜ui(t) − i] 6 0; (2)
for t ¿ 0 and i = 1; 2; : : : ; m, where ui(t) is the ith component of u(t), and i; i are positive constants.
Clearly the band-bounded nonlinearity can accommodate characteristics such as backlash, hysteresis, dead-zone, and quantization, which are not included by the sector-bounded nonlinearity [12]. As to the measurement noise n(t), our formulation allows one to handle the sensor-introduced noise with only the worst case infor-mation, or the quantization error introduced in the analog-to-digital conversion process [3]. Facing these nonlinearity and noise factors, we want to 4nd state feedback controllers to reach the ultimate boundedness control [6,9,14] goal, which basically means making every state trajectories of (1) enter a neighborhood (the ultimate boundedness region) of the origin of the state-space eventually. To be more speci4c, we consider the ellipsoidal neighborhood Ec= {xTPx 6 c} for some P ¿ 0 and c ¿ 0. Of course, it is best if the “size”
(with any acceptable de4nition) of Ec can be guaranteed to the smallest while is as large as possible.
We end this Section by presenting two useful lemmas subsequently. Lemma 1. For any ‘ × ‘ matrix Q ¿ 0 andscalar ¿ 0, − Q26 (1= )I
‘− 2Q.
Proof.
−(√ Q − (1=√ )I‘)T(√ Q − (1=√ )I‘) 6 0 ⇔ − Q26 (1= )I‘− 2Q:
Lemma 2. For any F ∈ Rp×p, G ∈ Rq×q; H ∈ Rp×q, and R; S of appropriate dimensions, where p; q are
positive integers, we have F − HG−1HT R RT S ¡ 0; G ¡ 0 ⇔ RF R HT S 0 HT 0 G ¡ 0: (3)
xT(ATP + PA)x + xTPBu + uTBTPx −m i=1
[ i(ui− i˜ui+ i) · (ui− i˜ui− i)]
− m+1(& − xTx) − m+2(nTn − 2) ¡ 0; (5)
for all x; n∈R‘ and u∈Rm. Letting '=diag(1; 2; : : : ; m) ¿ 0; T =diag( 1; 2; : : : ; m) ¿ 0; =[1 2 · · · m]T,
and ˆ& = m+1& ¿ 0, we can re-arrange (5) as
xT(ATP + PA + m+1I‘− KT'T'K)x + xT(PB + KT'T)u +uT(BTP + T'K)x − nTKT'T'Kx − xTKT'T'Kn − uTTu +nTKT'Tu + uTT'Kn − nT(KT'T'K + m+2I‘)n + m+22+ TT − ˆ& =[xT nT uT 1]) 0 0 [xT nT uT 1]T¡ 0; (6) where ) = ATP + PA + m+1I‘ −KT'T'K −KT'T'K PB + KT'T −KT'T'K −KT'T'K − m+2I‘ KT'T BTP + T'K T'K −T (7) and = m+22+ TT − ˆ&: (8)
Therefore a su3cient condition for ultimate boundedness control is that for some diagonal T ¿ 0; m+1¿ 0;
m+2¿ 0 and ˆ&¿ 0, the inequalities ) ¡ 0 and ¡ 0 hold. The remaining task is to convert the condition
to LMIs with respect to suitable variables.
First of all, multiply ) in (7) on the left- and right-hand sides by the nonsingular matrix diag(P−1; P−1; I
m),
and introduce the new variable matrices Q = P−1¿ 0; Y = KQ. The result can be used to transform )¡0
into QAT+ AQ + m+1Q2 −YT'T'Y −YT'T'Y B + YT'T
−YT'T'Y −YT'T'Y −
m+2Q2 YT'T BT+ T'Y T'Y −T ¡ 0: (9)
It follows from the Schur complement [4] pivoting on the third diagonal block −T that when T ¿ 0, (9) is equivalent to QAT+ AQ + BT−1BT+ YT'BT+ B'Y + m+1Q2 B'Y YT'BT − m+2Q2 ¡ 0: (10)
With the help of Lemma 2 twice, (10) may be changed to QAT+ AQ + YT'BT+ B'Y B'Y Q B YT'BT − m+2Q2 0 0 Q 0 − −1 m+1I‘ 0 BT 0 0 −T ¡ 0; (11) which is implied by QAT+ AQ + YT'BT+ B'Y B'Y Q B YT'BT −1 m+2I‘− 2Q 0 0 Q 0 − −1m+1I‘ 0 BT 0 0 −T ¡ 0 (12)
due to Lemma 1. As to in (8), usage of the Schur complement shows that when m+2¿ 0 the inequality
¡ 0 is identical to TT − ˆ& − −1m+2 ¡ 0: (13)
Thus, we have obtained two LMIs (12) and (13) with respect to the variables Q ¿ 0; Y; T ¿ 0; ¿ 0; ˆ&¿ 0; −1m+1¿ 0, and −1m+2¿ 0. This means that given the plant information {A; B} and actuator nonlinearity infor-mation {'; }, we can search for state feedback controllers to reach the ultimate boundedness control goal. Feasible solutions of the two LMIs not only give the state feedback gain (via Q and Y ), but also the ultimate boundedness region Ec (via Q, m+1 and ˆ&), and the tolerable noise magnitude , which may also be speci4ed
beforehand.
Naturally, it would be best if we could set an objective function on top of the two LMIs (12) and (13) to form an optimization problem, enabling us to 4nd the state feedback controller which results in the smallest ultimate boundedness region while tolerating the largest measurement noise. However, compromises must be made in the choice of the objective function because some quantity, such as &, directly related to the “size” of the ultimate boundedness region is not a variable in (12) and (13), and because it is advantageous to form convex objective functions compatible with the LMI formulation. After making some trade-oT, we choose the following optimization problem:
min
Q¿0; Y; T¿0; −1
m+1¿0; −1m+2¿0; ¿0; ˆ&¿0
ˆ& + −1m+1+ max(Q) −
subject to (12) and (13): (14)
In the objective function, the 4rst two terms represent an indirect eTort to minimize the length of the shortest principal axis of Ec, which is
√ & = −1 m+1ˆ& 6 ˆ& + −1 m+1 2 : (15)
The third term represents the attempt to minimize the length of the longest principal axis of Ec, which is
proportional to max(Q). Finally, the last term reNects our desire to tolerate noise of the largest magnitude. Of
course, suitable weighting factors can be multiplied to each term if necessary. This kind of an optimization problem may be solved easily by using software tools like [8].
Example. Consider the system (1) with A = 0:2 −1 1 0:2 ; B = 0:5 1 ; 1= 1; 1= 0:4;
Fig. 3. Ultimate boundedness regions of the Example.
which is open-loop unstable. By solving the convex optimization problem (14), we obtain (hereafter the superscript∗ means the optimal value)
K∗= [−0:0715 − 1:963] ; Q∗= 0:233 0:071 0:071 0:208 ; ∗ 1= 7:437 ( ∗ 2)−1= 0:874; ( ∗3)−1= 4:211 × 10−4; &∗= 1:042 and ∗= 6:612 × 10−4:
The corresponding ultimate boundedness region Ec∗ is the interior of the ellipse depicted with a solid line in Fig. 3, where the ball B&∗ is also shown with a dotted line. The numerical results show that & is of the same order as the width of the nonlinearity band (21= 0:8), but the tolerable noise magnitude seems very
small. As a remedy, we multiply a weighting factor w ¿ 1 to in (14) and solve the modi4ed optimization problem. It is found that for w 6 5, the eTect is mild only. Corresponding to ultimate boundedness regions that are compromised marginally in size, the resultant ’s do become larger, though not signi4cantly. For instance, when w = 5 the bound is about twice as large as the original ∗, and the corresponding ultimate
boundedness region Ec is not so diTerent from Ec∗. For w¿5, the growth of is more evident, but the expansion of Ec is equally noticeable. In other words, meaningful gain in the tolerable noise bound has to be
obtained at the expense of accepting a larger Ec.
4. A GEVP for conservativeness reduction
From the example in Section 3 it is suspected that there is conservativeness in the estimation of the tolerable noise magnitude. Furthermore, a comparison of (11) and(12) reveals that the usage of Lemma 1 may require that −1
m+2 be unnecessarily small, which in turn limits in (13) too much. Therefore we seek to reduce this
potential conservativeness. It is observed from (7) that ) ¡ 0 is an LMI with respect to the variables P ¿ 0, T ¿ 0, m+1¿ 0, and m+2¿ 0 when K is known. Also, ¡ 0 can be written as
TT − ˆ&¡ − 2
m+2; (16)
which together with ) ¡ 0 forms a GEVP with respect to the variables P ¿ 0; T ¿ 0; m+1¿ 0; m+2¿ 0;
ˆ&¿ 0, and the “generalized eigenvalue” variable −2. Substituting the optimal solution K∗ of the problem
(14) into ) in (7) and solving the GEVP with the help of [8] will give new estimations of B&, Ec and
. In fact, any state feedback gain K such that A + BK is strictly Hurwitz may be tested by the GEVP to estimate the corresponding B&, Ec and . However, there are two details to notice. The 4rst one is that if
{P; T; m+1; m+2; ˆ&; } is a set of the solution of the GEVP, then {-P; -T; - m+1; - m+2; - ˆ&; } is also a solution
set, where - is an arbitrary positive constant. Theoretically this nonuniqueness says that scaling the Lyapunov function %(x) = xTPx by - does not aTect the analysis results , B&, and Ec, since & = ˆ&= m+1 and c = &max(P),
[8] always tries to 4nd the solutions of the GEVP with extremely small magnitudes. Fortunately, this di3culty may be overcome easily by adding constraints like P ¿ I‘ or similar ones which 4x a lower bound for the
variables. The second detail is that the GEVP, due to its nature, tends to give very large as its answer, which is accompanied by very large B& and Ec, and means that in the trade-oT between tolerating large noise
and getting small ultimate boundedness region, it is biased to the former. When the GEVP is used to evaluate the eTectiveness of a given state feedback controller, extra constraints to be introduced in the sequel may be needed to prevent this preference.
Suppose a state feedback gain K∗, along with its corresponding E
c∗ and B&∗, are found by solving (14). To use the above GEVP to evaluate its noise tolerance capability, it is suggested that the following two LMIs be added:
ˆ& 6 &∗
m+1; (17)
P∗
max(P∗) 6 P 6 I‘; (18)
where P∗= (Q∗)−1. The purpose is to make sure that when a new estimation of is obtained, it is not
obtained at the expense of a larger Ec, since (17) and (18) imply max(P) = 1 and P∗=&∗max(P∗) 6 P=&,
which in turn imply that Ec⊆ Ec∗.
Example. For the system and state feedback controller obtained in the example of Section 3, we may apply the above procedure to derive a new estimation of its noise tolerance capability. Substituting K∗ into ) in
(7), placing &∗ and P∗= (Q∗)−1 into (17) and (18), respectively, and solving the GEVP which consists of
(16), ) ¡ 0, (17), and (18), we obtain Pnew= 0:710 −0:243 −0:243 0:797 ; new 1 = 0:548; new2 = 0:170; new
3 = 2:085; &new= 1:042 and new= 0:208:
Within our computational precision, it is seen that the newly estimated ultimate boundedness region Enew c can
be regarded as equal to Ec∗ obtained previously, and Bnew& = B&∗ in the same sense. However, here a much less conservative estimation new=0:208 is derived, compared with ∗=6:612×10−4 obtained in the example
of Section 3.
Next, assume that there is an alternative state feedback gain Ka for the system (1), which is designed to
make A + BKa strictly Hurwitz by using, for example, the eigenvalue assignment method. Then the GEVP
consisting of (16), ) ¡ 0 with K in (7) replaced by Ka, P ¿ I
‘, and ˆ& 6 &◦ m+1 may be utilized to 4nd
and Ec corresponding to Ka and a pre-determined B&◦.
Example. Let us continue to consider the above Example. Suppose a Ka= [ − 0:608 − 2:096] is chosen to
place eigenvalues of A + BKa at {−1; −1}, which have about the same magnitude as those of A + BK∗. If in
the last GEVP &◦ is set to &new= 1:042, the solution is P = 2:070 −1:390 −1:390 5:322 ; 1= 3:983; 2= 1:368; 3= 15:309; & = 1:042 and = 0:227:
Note that although the tolerable noise magnitude is larger ( = 0:227 ¿ new= 0:208) in this case, the
corre-sponding Ec with approximately the same short axis length as Enewc (& = 1:042 = &new) is larger in area. This
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