A two-step LMI approach to robust dynamic output feedback control

for the MIMO aircraft model F -18

Lei Song and Jianying Yang

Abstract- This work is concerned with robust dynamic output feedback controller design for a simplified aircraft model F-18 which can be rewritten as a LPV system. A general LMI-based two-step approach is proposed to tackle this problem. Such a separation procedure overcomes the limitation of traditional approach that can not be applied to the linear parameter varying (LPV) system. Numerical results show the proposed controller can ensure simultaneous quadratic stability with Hoc performance and satisfactory dynamic performance for a selected flight envelop containing some operating points.

I. INTRODUCTION

The development of modem automatic control systems has played an important role in the growth of civil and military aviation. Modem aircraft, such as F-18, include a variety of automatic control systems that aid the flight crew in navigation, flight management, and augmenting the stability characteristics of the airplane. The stability and the analysis and synthesis of F-18 aircraft model have been investigated in literatures [1][2]. In [1], a simplified aircraft model F-18 is studied by using sliding-mode control method. Based on variable structure control theory and Lyapunov V­function method, two types of robust feedback controllers for uncertain multi-input multi-output (MIMO) systems are considered. In [2], the authors designed a state feedback robust H 00 controller for a linearized F-18 aircraft model. Both methods are using state feedback technique. Although state feedback is usually more efficient, there are significant difficulties for implementing such a control structure in flight systems. The major one comes from the fact that some of the states are referred to a common reference. The estimation of this reference is very complicated and may introduce errors in the feedback control scheme. For this reason, an output feedback structure is more desirable for the problem in study.

In this paper, we study the problem of robust Hoo per­formance analysis with pole placement and dynamic output feedback synthesis for a simplified F-18 longitudinal model. Since the system parameters of this kind of supersonic aircraft dependent on flight velocity that changes at a sig­nificantly rapid rate, the system considered is time-varying. A good choice to solve this problem is to regard the time­varying system as a LPV system. Here, a novel two-step LMI approach is proposed for robust dynamic output feedback

This research was supported by the National Natural Science Foundation of China (90916003) and scientific research key project fund of Ministry of Education of China (107110).

L. Song and J.Y. Yang are with Faculty of State Key Lab for Turbulence and Complex Systems and Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, P.R.China songlei@pku.edu.en (L. Song), jianyingy2000@163.eom(J.Y. Yang) 978-1-4244-6044-1110/$26.00 ©2010 IEEE 74

controller design of LPV systems. In the first step, matrix variables of the output feedback controller are calculated and fixed by state feedback control. In the second step, nonlinear terms of the nonlinear matrix inequality of control performance constraints for dynamic output feedback control can be transformed into linear terms since the fixed and known matrix variable of the controller matrix variables simplified the nonlinear terms. Such a separation procedure overcomes the limitation of traditional approach that can not be applied to the LPV system.

II. F-18 LONGITUDINAL DYNAMICS

The aircraft model described in this paper is based upon a modified version of the F-18 aircraft and has been taken from [3]. The linearized state dynamical equations of longitudinal motion of the F-18 aircraft are given by [3]

[�] = [ZOI Zq] [a] + [ZOE ZOPTV] [ 8E ] (1) q MOl Mq q MoE MoPTV 8PTV

In this model, a and q represent angle of attack and pitch rate, respectively; 8 E and 8 PTV represent symmetric elevator position and symmetric pitch thrust velocity nozzle position, respectively. Jet aircrafts typically have multiple operating flight conditions that correspond to the convex combination of the given operating points.

Denoting x = [a q]T, u = [8E 8PTVV, we can rewrite the system as {X(t) = A(O)x(t) + Bu(O)u(t) + Bw(O)w(t)

z(t) = G(O)x(t) + Du(O)u(t) + Dw(O)w(t) y(t) = Hx(t)

(2)

where x(t) E Rn is the state vector; z(t) is the output to be regulated; y(t) is the measured output; w(t) E L 2[0, 00) is the exogenous disturbance signal; H is a constant matrix; (A(O),Bu(O),Bw(O),G(O),Du(O),Dw(O)) are the system matrices belonged to a poly topic uncertainties domain 01 and can be written as a convex combination of the vertices, that is,

(A(O), Bu(O), Bw(O), G(O), Du(O), Dw(O)) r

= L Oi(Ai, Bui, Bwi, Gi, Dui, Dwi) E Ob i=1

(3)

where uncertain vector 0 = [Ob O2, • • • ,0r]T E Rr is a fixed but unknown parameters satisfying

r o E S � {O E Rr: L Oi = 1,0i > O}. (4)

i=1

Throughout the paper, the following definitions and lemma will be of great help to derive the main results of this present study. Definition 1.(the LMI region)[4]: Suppose V is a subset of a complex plane. If there exists a symmetric matrix a =

[akd E Rmxm and a matrix (3 = [(3kd E Rmxm such that V = {z E C : fv(z) < O}, where Jv(z) := a+z(3+z(3T =

[akl + (3klz + (3kIZh::;k,l::;m is the characteristic function of V and takes value in the m x m Hermitian matrix space, then V is called a LMI region. Definition 2.(quadratic V stability)[4]: Suppose that for the

LPV x = A(O(t))x with respect to 0, when O(t) is a fixed value, its pole location in the LMI region V can be described in the following: Mv[A(O(t)), X] = [aklX +(3kIA(O(t))X + (3kIXA(O(t))Th::;k,l::;m where X is a positive definite matrix, and Mv[A(O(t)),X] and fv(z) can be related by the fol­lowing substitution [X,A(O(t))X,XA(O(t))T] � (l,z,z). Then, the matrix A( O( t)) is quadratic V stable if and only if there exists a symmetric positive definite matrix X such that Mv[A(O(t)),X] < 0 for all admissible values of the parameter O(t). Definition 3.(quadratic Hoo performance)[5]: The unforced

LPV system (2) has quadratic Hoo performance "I if and only if there exists a positive definite matrix X > 0 such that

SO [A(9),B", (9),C(9),V", (9)] [XAT(O) + A(O)X Bw(O) XCT(O)] (5) := BT(O) -"II D'E(O) < 0

C(O)X Dw(O) -"II for all admissible values of the parameter O. Lemma 1. (vertex property): For the unforced poly topic

LPV system (2) satisfying (3) and (4), the following three statements are equivalent:

(1) The system (2) is quadratic V stable with quadratic H 00 performance "I.

(2) There exists a positive definite matrix X > 0 such that for all 0,

Mv(A(O), X) < 0, SrA(9),B", (9),c(9),V", (9)] (X, "I) < O. (3) There exists a positive definite matrix X > 0 satisfying

the following LMIs

Mv(Ai, X) < 0,

SrAi,B",i,ci,V",i] (X, "I) < 0, i = 1,2, ... , T.

The objective of this paper is to find a dynamic output feedback controller u(s) = G(s)y(s) such that:

i) The closed-loop system is quadratic V stable. ii) For the resulting closed-loop system, the Hoo perfor­

mance Ilz(t)11 2 ::; "I llw(t)11 2 holds for all nonzero w(t) E L 2[0, (0) , where "I > 0 is a prescribed scalar.

III. TWO-STEP LMI APPROACH TO ROBUST DY NAMIC

OUT P UT FEEDBACK CONTROL WITH POLE PLACEMENT

The dynamic output feedback controller is designed using the two-step LMI approach proposed in this Section. It is assumed that the following state space form of the output

75

feedback controller (with subscript c denoting that this is a controller) suitable for the LMI approach can be constructed:

{Xc = Aexe + BeY u=Kxe

(6)

where Xc E Rn. Here, this controller structure assumes two implicit facts. First, the controller transfer function G (s) is always strictly proper, which can be relaxed at the expense of much more involved calculations. Second, the dimension of the controller is the same as the one of the controlled system (2), which is necessary for the following developments.

The LMI-based two-step approach is to determine matrix variables K, Ae and Be of the output feedback controller (6), which includes two steps: determination of matrix variable K, and determination of matrix variables Ae, Be. In the first step, determination of the variable K is reduced to the problem of finding a state feedback controller satisfying all design objectives described in Section II via LMI. In the second step, determination of matrix variables Ae, Be is reduced to the problem of finding an output feedback controller. The output of the output feedback controller is regulated to comparable with that of the state feedback controller designed in the first step. This two-step based LMI approach overcomes the design difficulty of traditional output feedback controller caused by the nonlinearity of controller matrix variables.

A. First step: determination of matrix variable K The determination of matrix variable K is reduced to

a state feedback control problem since the state feedback control problem can easily turn into the LMI problem [4]. Various closed-loop objectives for the state feedback control can be easily transformed into LMI constraints.

Consider the LPV system (2) with the uncertainties do­main 01. The state feedback controller can be written as

u(t) = Kx(t). (7)

Thus, the corresponding closed-loop system with the con­troller (7) is as follows:

{X(t) = Aau(O)x(t) + Bw(O)w(t) (8)

z(t) = Cau(O)x(t) + Dw(O)w(t)

where Aau(O) = A(O) + Bu(O)K,cau(O) C(O) + Du(O)K, i = 1,2, ... , T. Suppose D is an LMI region defined as

D = {z E C: fv(z) := a + z(3 + z(3T < O}. It is seen that the closed-loop system (8) has a poly topic

structure. To achieve the two main objectives described in Section II, according to Lemma 1, as long as all the vertices satisfy

Mv(Aau,i, X) < 0 (9)

srAau,i,B",i,Cau,i,V",i] (X, "I) < 0, i = 1,2, ... , T, (10)

where X > 0, the state feedback controller (7) guarantees that the closed-loop system (8) is quadratic V stable with

quadratic Hoc performance "I between w(t) and z(t) for all admissible values of the varying parameter ().

To solve the above LMIs, we can rewrite (9) and (10) as

Mv(Aau,i, X) = a @ X + f3 @ (Aau,iX) + f3T @ (X A�u,i) = a@X + f3@ [(Ai + BuiK)X ]

+ f3T @ [X(Ai + BUiK)T] < 0 (11)

XCiiU i] T' DWi

-"II X(Ci + !luiK)T]

DWi < 0 -"II

(12)

where A = X(Ai + BuiK)T + (Ai + BuiK)X, i = 1,2, ... ,r; @ denotes the Kronecker product of matrices. To overcome the difficulty caused by nonlinear terms of the above LMI, a new variable P is introduced as

P=KX Then, LMI (11) and (12) can be transformed into the follow­ing LMIs with respect to matrix variables X, P, respectively:

Mv(Aau,i, X) = a @ X + f3 @ (AiX + BuiP) + f3T @ (XA[ + pTB�) < 0

(13)

< 0 (14)

X>O (15)

where � = (XA[ + pTB'Ei) + (AiX + BuiP),i = 1,2, ... ,r. Up to now, the LMIs (13)-(15) can be solved over matrix variables X and P by using LMI tool box in MATLAB. The corresponding state feedback controller gain K is given by

K = p*X*-l (16)

In the first step, the full state feedback controller satisfying the design objectives can be found for LPV system (2). This state feedback method overcomes the limitation of output feedback controller that results in the constraints of the nonlinear matrix inequalities for the LPV system. Moreover, the LMI based state feedback design method can consider multiobjective design problem such as pole placement, Hoc performance.

B. Second step: determination a/matrix variable Ae and Be Assume that a full state feedback controller has been

found in the first step for the LPV system (2) without consideration of the feedback signal y(t). In this step, the problem of determining matrix variables Ae and Be becomes how to find a dynamic output feedback controller to replace the state feedback controller designed in the first step so

76

that the closed-loop performances of the system under both controllers are comparable with or equivalent to each other.

As stated in the first step, a full state feedback controller satisfying the design objectives can be found for the LPV system (2) as

u(t) = K x(t). (17)

The problem here is to find an output feedback controller u (s) = G(s)y(s) which can replace (17). The state space form of the output feedback controller (6) is rewritten as

{Xc = Aexe + BeY u=Kxe

(18)

where K in (18) takes the same value as that of K in (17). The corresponding closed-loop state-space equations with the dynamic output feedback controller (18) are as follows:

{Xcl(t) = Acl(())Xcl(t) + Bcl(())W(t) (19) z(t) = Ccl(())Xcl(()) + Dcl(())W(t)

where Xcl(()) = [x xejT, Acl(()) = [��� But;K] , Bcl(()) = [Bw(()) of, Ccl(()) = [C(()) Du(())K] Dcl(()) = Dw(()).

The dynamic output feedback controller (18) can be equal to or comparable with the state feedback controller (17) if

lim Xe(t) = x(t). t-++oc

(20)

Based on this consideration, variable changes are introduced as follows:

x = [x x -Xe]T = T [x Xe]T (21)

where T = [� �I]. Thus, the closed-loop system with

respect to this linear transformation is changed into

where

{:i(t) = 1cl(())x(t) + �cl(())W(t) (22) z(t) = Ccl(())X(()) + Dcl(())W(t)

Acl(()) = T Acl(())T-1

[ A(()) + Bu(())K -Bu(())K ] (23) - A(()) + Bu(())K - BeH -Ae Ae - Bu(())K

Bcl(()) = TBcl(()) = [Bw(()) Bw(())]T, A 1 [ ] Ccl(()) = Ccl(())T- = C(()) + Du(())K -Du(())K ,

Dcl(()) = Dcl(()) = Dw(()). The dynamic system (22) is equivalent to the dynamic

closed-loop system (19). For parameter K, the system design objectives have been guaranteed by the first-step design. In the second step, the controller matrix variables Ae and Be need to be determined so that system (22) satisfies all the design objectives, and obviously, (20) will be satisfied simultaneously.

Similar to the design procedure of the first step, the fact that the resulting closed-loop system (22) satisfies all the

design objectives is equivalent to the existence of a solution Y>OtoLMIs

that is,

Mv(Acl,i' Y) < 0

BO[ • • • • 1 (Y, 1) < 0, Acl,i ,Bel,i ,Cel,i ,Del,i

(24)

(25)

(26)

(27)

where i = 1,2, ... , r. As it can be seen from (26) and (27) that the term Y Acl,i and its transpose are nonlinear with respect to the variables Y, Ae and Be. To remove the nonlinear terms, the Lyapunov matrix Y is selected to have a structure of

(28)

where Q = QT, R = RT. Replacing the Lyapunov matrix Y with (28), we obtain

(29)

where r1 = QAi+QBuiK +RAi+RBuiK -RBeH -RAe, r2 = -QBuiK +RAe-RBuiK, r3 = RT Ai+RTBuiK + RAi + RBuiK - RBeH - RAe, r 4 = RAe - RBuiK -RT BUiK. Applying the following change of variables for a purpose of linearization:

M=RAe, N=RBe·

Thus, (29) represents a linear relationship in terms of the new variables Q, R, M, N.

The LMI formulation for the closed-loop LPV system (22) satisfying all the design objectives are given as follows:

[ r,Hi * * J r3+rr r4+rT * <0 (31) B'EiQ + B'EiR 2B'EiR -11

Ci + DUiK _KTDT. DWi W� Q = QT,R = RT (32)

[� �] > 0 (33)

where i = 1,2, ... ,r. Denoting the feasible solution by Q*, R*, M*, N*, the corresponding controller matrices of the output feedback controller (18) are given by

Ae = R*-lM*,

Be = R*-lN*.

(34)

(35)

To make the whole procedure of two-step approach clearer, the design process of the dynamic output feedback controller (6) for the LPV system (2) satisfying the

77

objectives of quadratic V stability with Hoo performance 1 is summarized as follows:

• Determination of matrix variable K by LMI-based state feedback design, solving the LMIs (13)-(15);

• Determination of matrix variables Ae and Be by LMI­based output feedback controller design, solving the LMIs (30)-(33).

IV. LONGITUDINAL FLIGHT CONTROL

According to the design procedure of dynamic output feed­back controller in Section Ill, we consider the longitudinal flight control for F-18 which has two control inputs and two output variables. This is a typical plant model of F-18 [3]. Our theoretical development of previous sections is implemented on this model.

Here, we only consider three operating points of the lon­gitudinal motions of F-18 model. For each of these operating points, the values of the system matrices parameters are given in the following table.

TABLE I

SYSTEM MATRICES PARAMETERS OF THE THREE OPERATING POINTS

Operating points 1 2 3 Z", -1.175 -2.328 -2.452 Zq 0.9871 0.9831 0.9856

M", -0.5485 -30.44 -38.61 Mq -0.8776 -1.493 -1.34 ZoE -0.194 -0.3012 -0.2757

ZoPTV -0.0359 -0.0587 -0.0523 MoE -19.29 -38.43 -37.36

MoPTv -3.803 -7.815 -7.247

The above three operating points are adjacent design points. Then, system (1) can be rewritten as {X(t) = A(O)x(t) + Bu(O)u(t) + Bww(t)

z(t) = Cx(t) + Duu(t) + Dww(t) y(t) = Hx(t)

(36)

where Bw = Dw = 1, C = [El/2 oJ, Du = [0 Fl/2] with E = 0.02 and F = 100, H = [1 0], A(O) and Bu(O) are the system matrices defined as

3 (A(O), Bu(O)) = L Oi(Ai, Bui)

i=l with the parameters of Ai, Bui, i = 1,2,3 take the corre­sponding values in Table I. Equation (36) approximately de­scribes the flight dynamics in the flight envelope containing the three operating points as apexes.

The design problem considered in this section is to find a output feedback controller such that the closed-loop system (36) is quadratic V stable with quadratic Hoo performance 1 = 10. To obtain satisfied dynamic performance, the closed­loop poles are required to be placed in the region 8(0:,0) in Figure 1. Note that confining the closed-loop poles to

1m

R.

Fig. l. The system pole placement region S(a, 8).

this region ensures a minimum decay rate a and a minimum damping ratio ( = cosB. This in turn bounds the maximum overshoot, the frequency of oscillatory modes, the delay time, the rise time, and the settling time [4]. As can be seen from Definition 1, region S(a, B) is an LMI region. Here, we select a = 1, B = 7r /3.

Next, the dynamic output feedback controller (6) for system (36) is designed using the two-step LMI approach described in Section III. Two steps are needed to determine the controller state matrices K, Ae and Be.

At the first step, since the region S( a, B) is an LMI region, according to the results in [4], inequality (13) and (14) are equivalent to the existence of a solution of X > 0 to the

LMIs Aau,iX + X A�U,i + 2aX < 0,

[sinB(Aau'iX + XA�u,i) cosB(Aau,iX - XA�U'i)] cosB(Aau,iX - XArU,i) sinB(Aau,iX + XArU,i)

(37)

< 0 (38)

< 0 (39)

(40)

where Aau,i = Ai + BUiK, i = 1,2,3. With introducing a new variable P = K X, the solution X*, P* is obtained by solving LMIs (37)-(40), the matrix variable K can be computed via (16) and is given as

K = [ 1.1704 0.7396 ] -0.2631 -0.1279 .

At the second step, as the LMI region is selected as the region S( a, B), similar to the above process, (30)-(33) can be rewritten as follows:

A AT Y Acl,i + Acl,iY + 2aY < 0,

[T 31 2 31 3 �1 41 322 323 '::'24 < 0 * 333 334 * * 344

[ r,+ri * * J r3+rf r4 +rr *

BEiQ + BEiR 2BEiR -,,(I Ci + DUiK _KTDT DWi wt

(41)

(42)

< 0 (43)

78

where

Q = QT,R = RT

[� �] > 0

31 1 = sinB(r 1 + rf)

31 2 = sinB(r 2 + rD, 31 3 = cosB(r 1 -rf)

31 4 = COSB(r2 -rD, 322 = sinB(r 4 + rn

323 = cosB(r 3 -rD, 324 = cosB(r 4 -rn

(44)

(45)

and i = 1,2,3. Solving the LMI problem of (41)-(45) over variables Q, R, M, N, we obtain the feasible solution. The matrix variables Ae, Be are then computed, respectively, by (34) and (35):

[ -739 Ae = -11533 -79 ] 1312

[ 1260 ] Be = 19433 .

With state matrices K, Ae and Be known, the dynamic output feedback controller (6) is obtained.

A. Quadratic V stability analysis For robust stability, response on different operating points

is shown in Figure 2, where the exogenous disturbance signal w(t) is selected as a stochastic vector lies in the region of (-0.1,0.1). From Figure 2, it is obvious that attack of angle a and pitch rate q converge to eqUilibrium by the designed controller on given operating points. This result has shown that the proposed dynamic output feedback controller ensures robust stability of the whole flight envelop containing the three operating points as apexes in spite of exogenous disturbance in the F-18 model. In addition, responses of the F-18 longitudinal closed-loop model are shown in Figures 3-5, as well as the dynamic error between x and Xc where x and Xc denote the state vectors of the original model and the dynamic output feedback controller, respectively. We can conclude that the designed output feedback controller is comparable with the state feedback controller in terms of the control input signal to original open-loop system.

Furthermore, The eigenvalue analysis is carried out in Table II for the nominal F-18 model with the dynamic output feedback controller, which gives the decay rate and

0.2

v; o· � :a: -0.2

i -0.4

-0.6 ,

Time/s 3 0

Fig. 2. Response of longitudinal dynamics on different operating points.

TABLE II

EIGENVALUE ANALYSIS

Operating points B Closed-loop eigenvalues

I 2 3 4 5 6

I 0 0

0.5 0.3 0.8

0 0 -1.4, -6.8, -24.4, -2021 I 0 -4.4, -2020.3, -15.3±8. l i 0 I -5.8, -2020.4, -14.5±7.5i

0.5 0 -3.6, -11.2, -19.0, -2020.6 0.3 0.4 -4.8, -14.7±3. l i, -2020.6 0.1 0.1 -2.7, -7.5, -22.8, -2020.9

Fig. 3. Response and system vector error on operating point I.

Fig. 4. Response and system vector error on operating point 2.

damping ratio of the open-loop system and resulted closed­loop system for all the operating points and some other randomly selected points. As can be seen that with the designed controller, the closed-loop poles are all placed in the prescribed region, and moreover, both the decay rate and damping ratio of the closed-loop F-18 system are enhanced. From the results in Table 1, it is clear that the dynamic output feedback controller obtained from this proposed two-step approach provides satisfactory decay rate and damping ratio for all the points containing in the described flight envelop.

B. Hoo performance analysis With state feedback controller, the actually achieved values

of Hoo performance "I presented in [2] is 5.5952, 8.5811, 10.9078 on different operating points, respectively. For the sake of comparison, under the same conditions, the smallest value of Hoo performance "I this proposed output feedback controller can arrive at is 5.432. Note that the best Hoo performance using state feedback controller in [2] is 5.5952, which is larger than that of using output feedback controller in this study. It can be concluded that this dynamic output

Open loop Closed loop decay rate Q damping ratio ( decay rate Q damping ratio (

79

1.02 0.818 1.91 0.331 1.89 0.295 1.47 0.353 1.64 0.316 1.20 0.409

0.2

1.4 I 4.4 0.884 5.8 0.888 3.6 I 4.8 0.978 2.7 I

! o:� _______ --1 _osLI --7"""�---;--7"""-! o 3

Time/.

3 Timels

Fig. 5. Response and system vector error on operating point 3.

feedback controller obtained by the two-step approach can provide better H 00 performance than the state feedback con­troller, at the expense of more complicated control technique containing two more freedom parameters Ae and Be, which is also an advantage of this presented method.

V. CONCLUSIONS A robust dynamic output feedback controller was designed

for a simplified F-18 model via a two-step approach. In the first step, a state feedback controller satisfies all the constraints was designed. In the second step, with one of the matrix variables of output feedback controller replaced by the known state feedback gain obtained in the first step, the nonlinear terms of the nonlinear matrix inequality of control performance constraints for dynamical output feedback con­trol can be transformed into linear terms. This novel approach overcomes the difficulties caused by parameterizations for

LPV system under output feedback control, and most im­portantly, can be applied to general LPV systems. Numerical simulations have shown the effectiveness and superiority of this proposed method.

REFERENCES

[I] E.M. Jafarov and R. Tasaltin, "Robust Sliding Mode Control for the Uncertain MIMO Aircraft Model F-18," IEEE Trans. Aerosp. Electron. Syst., vol. 36, pp. 1127-1141, 2000.

[2] S.L. Dai, G.M. Dimirovski and J. Zhao, "A descriptor system approach to robust H 00 control and its application to flight control," in Proc. Amer. Contr. Con!, Minneapolis Minnesota, 2006. pp. 1068-1073.

[3] lA. Richard, lM. Buffington, A.G. Sparks and S.S. Banda, Robust Multivariable Flight Control, Springer-Verlag,London, 1994.

[4] M. Chilali and P. Gahinet, "Hoo design with pole placement con­straints: An LMI approach," IEEE Trans. Automat. Contr., vol. 41, pp. 358-367, 1996.

[5] P. Apkarian, P. Gahinet and G. Becker, "Self-scheduled Hoo control of linear parameter-varying systems: A design example," Automatica, vol. 31, pp. 1251-1261, 1995.