Uniform Estimates of Attracting Sets of Extended Lurie Systems Using ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10, OCTOBER 2006
Uniform Estimates of Attracting Sets of Extended Lurie Systems Using LMIs
1675
II. LURIE PROBLEM Consider the nonlinear systems of the form
André C. P. Martins, Luís F. C. Alberto, and Newton G. Bretas
Abstract—This research presents a systematic procedure to obtain estimates, via extended Lyapunov functions, of attracting sets of a class of nonlinear systems, as well as an estimate of their stability regions. The considered class of nonlinear systems, called in this note the extended Lurie system, consists of nonlinear systems like those of the Lurie problem where one of the nonlinear functions can violate the sector conditions of the Lurie problem around the origin. In case of nonautonomous systems the concept of absolute stability is extended and uniform estimates of the attracting set are obtained. Two classical nonlinear systems, the forced duffing equation and the Van der Pol system, are analyzed with the proposed procedure. Index Terms—Attracting sets, linear matrix inequality (LMI), Lurie problem, Lyapunov methods, nonlinear systems.
x_ = Ax 0 Bh(t; y) y = Cx
(1)
where the matrices have suitable dimension, the matrix A is Hurwitz, the nonlinearity h(t; y) is decentralized, that is h(t; y) = [h1 (t; y1 ); . . . ; hp (t; yp )]T , and satisfies a sector condition, defined by h(t; y)T [h(t; y) 0 Ky] 0, in a region 0 = fy 2 Rp : ai yi bi g that contains the origin, with K = diag(ki ) > 0. The sector condition guarantees that the origin is an equilibrium point of system (1). The Lurie problem consists in determining if the origin stays asymptotically stable for any nonlinearity satisfying the sector condition, [8]. When this is true the system is said to be absolutely stable with a finite domain. In the case of nonautonomous systems, the absolute stability of the origin can be analyzed by means of Lyapunov methods using the quadratic function (2). In the case of autonomous systems, the Lurie-type Lyapunov function (3) is used to verify the absolute stability of the origin, where P = P T > 0 and
N = diag(i ) 0
I. INTRODUCTION
V (x) = xT P x The stability analysis of nonlinear systems is present in several engineering fields. Usually, the concern is the determination of stable equilibrium points and their associated stability regions, [8]. However, in nonlinear systems, attracting sets may be more complicated than isolated equilibrium points. Very often one finds limit cycles, strange attractors, etc. Although there are some classical methods that provide information about attracting sets, [10], [12] and [6], finding these sets is, in general, very difficult, [2] and [3]. Lyapunov’s ideas and LaSalle’s Invariance Principle are the main practical tools to provide estimates of attracting sets and stability regions. However, usually, these methods involve a nonsystematic search for auxiliary functions called Lyapunov functions, [8]. Due to the enormous difficulty of finding Lyapunov functions, researchers tried to find general Lyapunov functions that are suitable for a class of problems. Lurie-type system is one of these classes of nonlinear systems formed by a linear term plus a combination of nonlinear functions satisfying certain sector conditions. In order to verify the stability of these systems, general Lyapunov functions were proposed for this class of systems (Lurie-type Lyapunov functions), [5] and [8]. In this research, these concepts are extended for a larger class of problems. The extended Lurie system, defined in this note, consists of nonlinear systems, like those of the Lurie problem, where one of the nonlinear functions may violate the sector conditions around the origin. In this case, the attracting set may be more complex than single equilibrium points. A systematic procedure, based on the extension of LaSalle’s invariance principle, [11], and linear matrix inequality (LMI) techniques, to obtain estimates of the attracting set and of the stability region of this class of nonlinear systems is proposed.
Manuscript received May 10, 2005; revised June 6, 2006. Recommended by Associate Editor M.-Q. Xiao. This work was supported by the FAPESP. A. C. P. Martins is with the Universidade Estadual Paulista, Bauru-SP 17033360, Brazil (e-mail: [email protected]). L. F. C. Alberto and N. G. Bretas are with the Universidade de São Paulo, São Carlos-SP 13566-590, Brazil (e-mail: [email protected]; ngbretas@sel. eesc.usp.br). Digital Object Identifier 10.1109/TAC.2006.883063
V (x) = xT P x + 2
Cx
h(y)T Ndy:
(2) (3)
0
III. DEVELOPED PROCEDURE The extended Lurie system consists of nonlinear systems that may be written like (1), where the nonlinearity violates the sector condition around the origin. In this case the attracting set may be more complex than single equilibrium points. This section presents the developed procedure to estimate the attracting set and the stability region of the extended Lurie system. When the system is non-autonomous the procedure provides a generalization of absolute stability for the attracting sets. Autonomous and nonautonomous systems are analyzed in different ways. For saving space only the single-input–single-output (SISO) case is presented, although the multiple-input–multiple-output (MIMO) case is quite direct, the resulting LMI equation is very extensive. A. Nonautonomous Systems Consider a nonlinear system in the form of the extended Lurie system, where A 2 Rn2n , B; c 2 R and h : R 2 R ! R is a C 1 function
x_ = Ax 0 Bh(t; y) : y = cT x
(4)
The nonlinearity h(t; y), shown in Fig. 1, satisfies the sector condition 0 h(t; y) Ky , for t 0, 0 y b, and Ky h(t; y) 0, for t 0, a y 0. Moreover, around the origin, the nonlinearity violates the sector condition but it is bounded, so
h(t; y)K 01 [h(t; y) 0 Ky] 0 jh(t; y)j m
8t; 8y 2f[ ; a]; [ ; b]g 8t; y
(5) (6)
In contrast to the Lurie problem, the origin of this system may not be an equilibrium point. However, since that nonlinearity violates the
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Fig. 1. Nonlinearity characteristic of the extended Lurie system.
sector condition only around the origin, this system can have an attracting set in the neighborhood of this point. Also, if the states in this set are sufficiently close to the origin, so they can be considered acceptable operating points, the origin is said to be a practical stability point of the system, [9]. The next theorem presents sufficient conditions to guarantee the existence of an attracting set in the neighborhood of the origin. It is worth noting that these estimates can only be obtained using extended Lyapunov functions. That is, the function in (2) is not a classical Lyapunov function, since it presents regions where the time derivative is greater than zero. Theorem 1: Consider the nonlinear system (4) satisfying conditions (5) and (6). Assume that there are a symmetric matrix P > 0 and a
A P + P A 0P B + c 0, such that 0B P + c 02 K 01 < 0. Let fc1 ; c2 ; . . . ; c 01g be a set of orthonormal vectors each one orthogonal to vector c, < 0 the largest eigenvalue of the matrix A P + P A, L = min(a2 =c P 01 c; b2 =c P 01 c), l = max 2 (z P z), and = 2(mkP B k2 =j j). The set Z consist of the 2 solutions of linear system [c c1 . . . c 01 ] z = d , where each vector d is a vector from f ; g 2 f0; + g 01 . Assume that L > l. Then, given an initial condition (t0 ; x0 ), where x0 2 = fx 2 R : x P x < Lg, there is t1 t0 such that the trajectories of the system are contained in the set = fx 2 R : x P x lg for all t t1 . T
constant
T
Fig. 2. Region of state–space where V_ (t; x)
T
min = max fA P + P Ag P >0 A P + P A 0P B + c s:t: 0B P + c 02 K 01 < 0 0:
T
M
V
T
n
Z n
k
n
k
n
L
n
t
T
T
z
T
T
Proof: The main idea of the proof is to show that the set fx 2 Rn : xT P x = Lg is the greatest level curve contained in the region where we can analytically establish that V_ (t; x) < 0 8t 0. Moreover, the set fx 2 Rn : xT P x = lg is a level curve that contains the region where the sign of V_ (t; x) can not be analytically verified, Fig. 2. Consider the quadratic function V (x) = xT P x. Its derivative along the trajectories of the system is V_ (t; x) = xT (P A + AT P )x 0 2xT P Bh(t; cT x). However, in the intervals [a; ] and [ ; b], the sector condition is satisfied, so, 02h(t; cT x)K 01 [h(t; cT x) 0 KcT x] 0 8(t; x) 2 R 2 00 where 00 = fx 2 Rn : a cT x ; cT x bg. So, using the S -procedure with some 0, [1], it is obtained V_ (t; x)
T
M
T
T
0.
on the hyperplanes cT x = a and cT x = b can be calculated as, respectively, a2 =cT P 01 c and b2 =cT P 01 c. So, the sets l and L can be defined, where L > l. The application of the theorem of uniform dissipativity [4] completes the proof. It is worth noting that the estimates l and L works to any nonlinearity h(t; y) satisfying the conditions (5) and (6). The Theorem 1 provides a sufficient condition for V_ (t; x) < 0 8x 2 L n l , the matrix P , satisfying the hypothesis of Theorem 1, can be obtained solving the optimization problem
n
M
0 and constants 0 and 0, such that
0P B + c x 02 K 01 h(t; c x) . However, from Theorem 3 (Appendix A) V_ (t; x) x [(P A + A P ) + ((0PB + c)(0B P + c )=2 K 01 )]x < 0 8x 2 00 . In the region 0+ = fx 2 R : c x gjh(t; c x)j m 8t 0, so V_ (t; x) kxk22 + 2mkxk2 kPB k2 . Since is a negative real number, V_ (t; x) < 0 whenever kxk2 > = 2mkPB k2 =j j. So, in the region fx 2 0+ : kxk2 g one cannot assure that V_ (t; x) < 0. This region is contained in PA + A P 0P B + A c + c =1 ... 01 fx 2 0+ : 0 c x g because, by construction, each hyperplane c x = 6 is tangent to M = 0B P + c A + c 02 K 01 0 c B 0 B c < 0. = the hyper sphere kxk2 = . Due the convexity of that set and function Let V (x), the largest value of V (x) inside that set is at their vertices, m((kPB 0 A ck2 + kP B 0 A ck22 + 2j kc B j)=j j), [x h(t; c x)] T
T
A P + PA T
0B P + c T
T
T
T
T
T
T
n
T
T
M
M
M
k
;
;n
T k
T
T k
T
T
T
Z
V
. Moreover, using the Lagrange multipliers, the minimum of V (x)
where
M
l such that L = fx 2 Rn : V (x) < Lg f00 0+ g and
l = fx 2 Rn : V (x) lg fx 2 0+ : kxk2 g. Then, each solution of the system starting in the set L converges to the set l , when t ! 1. Moreover, each solution of the system starting in the set l stays in l , 8t 0.
Proof: The idea of the proof is the same of Theorem 1. Consider c x the function V (x) = xT P x + 2 0 h(u)du. Its time derivative is
V_ (x) = xT (P A + AT P )x 0 2xT P Bh(cT x)+2h(cT x)cT [Ax 0 Bh(cT x)]. However, in the intervals [a; ] and [ ; b] the sector condition is satisfied, so 02h(cT x)K 01 [h(cT x) 0 KcT x] 0 8x 2 00 , where 00 = fx 2 Rn : a cT x ; cT x bg. So, using the S -procedure with 0, [1], and by Theorem 1, V_ (x) xT [(P A + AT P )+((0PB + AT c + c)(0B T P + cT A + cT )=2 K 01 + cT B + B T c)]x, so the hypothesis on the matrix M guarantees that V_ (t; x) < 0 8x 2 00 . In the2 region 0+ , the nonlinearity is bounded, so V_ (x) M kxk2 + 2mkxk2 kPB 0 AT ck2 + 2m2 jcT B j. Since M < 0 then AT P + P A < 0, so M < 0 and V_ (x) < 0 for all x sufficiently large such that kxk2 > = m((kPB 0 AT ck2 + kP B 0 AT ck22 + 2jM jcT B j)=jM j). Suppose the existence of sets l and L , then the application of the
Fig. 3. Estimate of limit cycle and its stability region.
extended LaSalle invariance principle [11] completes the proof. In general, function V (x) is not convex. So, the analytical computation of constants L and l is not possible in general and we have to use numerical methods to obtain these constants. However, the matrix P and the constant , satisfying the conditions of the previous Theorem 2, can be obtained solving the optimization problem in (8) using LMI techniques:
min M = max fAT P + P Ag P >0 0 s:t: M < 0 0:
(8) Fig. 4. Estimate of limit cycle and its stability region for
K = 30.
IV. NUMERICAL RESULTS In this section, the results proved in the previous section are applied to estimate the attracting set of the forced Duffing equation and the Van der Pol System. A. Forced Duffing Equation Consider the dynamic system M z + kc z_ + km z + km 2 z 3 = ku cos(!t); making x1 = z and x2 = z_ , and choosing km =M = 3, kc =M = 3, km 2 =M = 0:1, and ku =M = 2e ! = 1, this system can
0 1 , be written like an extended Lurie system, where A = 0 3 03 0 1 3 B = 1 , c = 0 and h(t; y) = 0:1y 0 2 cos(t). Simulations in-
dicate that this system has a stable limit cycle around the origin. Theorem 1 will be used to estimates this attracting set. Due the cubic term in h(t; y), the set 00 depends on the sector condition, i.e., the value of K . For K = 15, we have = 0 = 01:2599, a = 0b = 012:1802, and m = 4. In this case, it is easy to see that c1 = [0 1]T . The resolution of the optimization problem, (7), provides
9:2804 1:2684 P = 2 108 , = 2:3608 2 108 and 1:2684 1:0069 M = 02:3573 2 108 . Therefore, using Theorem 1, we have = 5:4948, L = 1:1398 2 1011 and l = 6:2697 2 109 . Fig. 3 shows the trajectories for several initial conditions and the sets L and l . It is worth nothing that these estimates are uniform to any nonlinearity that satisfies the same given conditions.
B. Van der Pol System
Consider the autonomous dynamic system v + "f 0 (v)v_ + v = 0, where f(v) = 0v + (1=3)v 3 . Making x1 = 0"01 v_ 0 f(v), x2 = v and using the parameter , the system can be written like an extended Lurie system, where A =
0 "01 0 0 0" 0" , B = " , c = 1
and h(y) =
0(1+)y+(1=3)y3 . Simulations indicate that this system has a stable
limit cycle around the origin. Choosing " = 1, = 1 and K = 30 one have m = 1:8856, = 0 = 02:4495, and a = 0b = 09:7980. Solving the optimization problem (8), one have = 4:6257:108 , M = 04:13902108 , the extended Lyapunov function (9) and = 5:3022. According to Theorem 2, one can calculate L = 1:9236 2 1011 and l = 2:0280 2 1010 . Fig. 4 shows the trajectories for several initial conditions, the sets L and l , and the region where V_ 0 (gray region)
V (x1 ; x2 ) =
x1 x2
T
6:2117 2:0707 2:0707 4:1426 +2:3688
x1 x2
2 108
0x22 + 121 x42 2 108 :
(9)
In this case, the hyperplane cT x = a bounds the stability region estimate, the set L ; the constant K of the sector condition determines the value of a. So, to expand the set L it is necessary to increase
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the value of K . For the Van der Pol system, the optimization problem . In this case, (8) remains feasible even if K ! 1, i.e., K 01 the solution of problem (8) provides a slightly poor estimate of the : . However, the stability region is the whole attracting set, R2 , so that the limit cycle is globally stable.
An Improved ILMI Method for Static Output Feedback Control With Application to Multivariable PID Control
V. CONCLUSION
Abstract—An improved iterative linear matrix inequality (ILMI) algorithm for static output feedback (SOF) stabilization problem without introducing any additional variables is proposed in this note. The proposed controller design ILMI algorithm is also extended to solve the SOF problem. They are applied to the multivariable PID controllers. Numerical examples show that the proposed algorithms yield better results and faster convergence than the existing ones.
=0
= 6 4912
This note presents a systematic procedure for estimating the attracting set of a class of nonlinear systems called the extended Lurie system. The extended Lurie system consists of systems that can be written like the Lurie problem, but with just one nonlinearity that violates the sector condition. The procedure was applied to two classical nonlinear systems: the forced Duffing equation and the Van der Pol system. The numerical results show that the proposed procedure provides good estimates of the attracting sets. Moreover, for the Van der Pol system the attracting set was proved to be the whole R2 . The calculus of matrix P , using the LMI problems in (7) and (8), tends to improve the stability region estimate since the negative term to be minimized is on the denominator of . However, that improvement is bounded since the norm of matrix P is on the numerator of . APPENDIX
: [a; b] 2 D R 2 R ! R be a C function 1 8 1 8 , where and W (t; x) = [x h(t; x) ] 8 6 80 6 is symmetric negative definite. Then, W (t; x) x [1 0 86 8 ]x 8(t; x) 2 [a; b] 2 D, where [1 0 860 8 ] is symmetric negative Theorem 3: Let h T
n
T
p
1
x
h(t;x)
T
T
T
1
1
T
T
definite.
REFERENCES [1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [2] M. Dellnitz and A. Hohmann, “A subdivision algorithm for the computation of unstable manifolds and global attractors,” Numerische Mathematik, vol. 75, pp. 293–317, 1997. [3] J.-Y. Dieudelot and P. Borne, “An estimation of the stability domain of a fuzzy controled pendulum using overlapping attractors and vector norms,” in Proc. IEEE Int. Conf. Systems, Man, and Cybernetics, 2001, vol. 4, pp. 2245–2249. [4] M. F. Gameiro and H. M. Rodrigues, “Applications of robust synchronization to communication systems,” Appl. Anal., vol. 79, pp. 21–45, 2001. [5] W. M. Haddad and V. Kapila, “Absolute stability criteria for multiple slope-restricted monotonic nonlinearities,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 361–365, Feb. 1995. [6] S. Heidari and C. L. Nikias, “Characterizing chaotic attractors using fourth-order off-diagonal cumulant slices,” in Proc. 27th Asilomar Conf. Signal, Systems, and Computers, 1993, vol. 1, pp. 466–470. [7] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge Univ. Press, 1996. [8] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ: PrenticeHall, 1992. [9] J. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method With Applications. New York: Academic, 1991. [10] R. K. Miller, A. N. Michel, and G. S. Krens, “Stability analysis of limit cycles in nonlinear feedback systems using describing functions: improved results,” IEEE Trans. Circuit Syst., vol. CAS-31, no. 6, pp. 561–567, Jun. 1984. [11] H. M. Rodrigues, L. F. C. Alberto, and N. G. Bretas, “On the invariance principle: generalizations and applications to synchronization,” IEEE Trans. Circuits Syst. I, vol. 47, no. 5, pp. 730–739, May 2000. [12] J. Argyris, G. Faust, and M. Haase, An Exploration of Chaos. Amsterdan, The Netherlands: North-Holland, 1994.
Yong He and Qing-Guo Wang
control, linear matrix inequality (LMI), multivariIndex Terms— able PID control, stabilization, static output feedback (SOF).
I. INTRODUCTION The static output feedback (SOF) plays a very important role in control theory and applications. Recently, it has attracted considerable attention (see, e.g., [1]–[7] and the references therein). Yet, it is still left with some open problems. Unlike the state feedback case, a SOF gain which stabilizes the system is not easy to find. Linear matrix inequality (LMI) [8] is one of the most effective and efficient tools in controller design and a great deal of LMI-based design methods of SOF design have been proposed over the last decade [9]–[19]. Among these methods, an iterative linear matrix inequality (ILMI) method was proposed by Cao et al. [13] and later employed to solve some multivariable PID controller design problems [20], [21]. In this context, a new additional variable was introduced such that the stability condition becomes a sufficient one. The iterative algorithm in [13] tried to find a sequence of the additional variables such that the sufficient condition is close to the necessary and sufficient one. The similar idea is used in the so-called substitutive LMI method by [19]. Both of them set the additional matrix variables at the current step with the matrices derived in the preceding step. With additional variables, the dimensions of the LMIs become higher. We observe that it is possible that the matrices derived in the preceding step can be used in the next one directly without introducing the additional variables and the dimensions of the LMIs need not be increased. In addition, we can develop some efficient ways to get suitable initial values in the iterative procedure, which the existing approaches have not been thought of. In this note, a new ILMI algorithm is proposed for SOF stabilization problem without introducing any additional variables, and assisted with a separate ILMI algorithm to find good initial variables. The algorithms for SOF stabilization are also extended to solve the SOF H1 control problem. They are applied to multivariable PID control. Numerical examples show the effectiveness and an improvement of the algorithms over the existing methods.
Manuscript received April 3, 2005; revised January 12, 2006 and April 23, 2006. Recommended by Associate Editor M. Kothare. The work with Y. He was supported in part by the National Science Foundation of China under Grant 60574014 and in part by the Doctor Subject Foundation of China under Grant 20050533015. Y. He is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore. He is also with the School of Information Science and Engineering, Central South University, Changsha 410083, China. Q.-G. Wang is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2006.883029
0018-9286/$20.00 © 2006 IEEE
Uniform Estimates of Attracting Sets of Extended Lurie Systems Using LMIs
1675
II. LURIE PROBLEM Consider the nonlinear systems of the form
André C. P. Martins, Luís F. C. Alberto, and Newton G. Bretas
Abstract—This research presents a systematic procedure to obtain estimates, via extended Lyapunov functions, of attracting sets of a class of nonlinear systems, as well as an estimate of their stability regions. The considered class of nonlinear systems, called in this note the extended Lurie system, consists of nonlinear systems like those of the Lurie problem where one of the nonlinear functions can violate the sector conditions of the Lurie problem around the origin. In case of nonautonomous systems the concept of absolute stability is extended and uniform estimates of the attracting set are obtained. Two classical nonlinear systems, the forced duffing equation and the Van der Pol system, are analyzed with the proposed procedure. Index Terms—Attracting sets, linear matrix inequality (LMI), Lurie problem, Lyapunov methods, nonlinear systems.
x_ = Ax 0 Bh(t; y) y = Cx
(1)
where the matrices have suitable dimension, the matrix A is Hurwitz, the nonlinearity h(t; y) is decentralized, that is h(t; y) = [h1 (t; y1 ); . . . ; hp (t; yp )]T , and satisfies a sector condition, defined by h(t; y)T [h(t; y) 0 Ky] 0, in a region 0 = fy 2 Rp : ai yi bi g that contains the origin, with K = diag(ki ) > 0. The sector condition guarantees that the origin is an equilibrium point of system (1). The Lurie problem consists in determining if the origin stays asymptotically stable for any nonlinearity satisfying the sector condition, [8]. When this is true the system is said to be absolutely stable with a finite domain. In the case of nonautonomous systems, the absolute stability of the origin can be analyzed by means of Lyapunov methods using the quadratic function (2). In the case of autonomous systems, the Lurie-type Lyapunov function (3) is used to verify the absolute stability of the origin, where P = P T > 0 and
N = diag(i ) 0
I. INTRODUCTION
V (x) = xT P x The stability analysis of nonlinear systems is present in several engineering fields. Usually, the concern is the determination of stable equilibrium points and their associated stability regions, [8]. However, in nonlinear systems, attracting sets may be more complicated than isolated equilibrium points. Very often one finds limit cycles, strange attractors, etc. Although there are some classical methods that provide information about attracting sets, [10], [12] and [6], finding these sets is, in general, very difficult, [2] and [3]. Lyapunov’s ideas and LaSalle’s Invariance Principle are the main practical tools to provide estimates of attracting sets and stability regions. However, usually, these methods involve a nonsystematic search for auxiliary functions called Lyapunov functions, [8]. Due to the enormous difficulty of finding Lyapunov functions, researchers tried to find general Lyapunov functions that are suitable for a class of problems. Lurie-type system is one of these classes of nonlinear systems formed by a linear term plus a combination of nonlinear functions satisfying certain sector conditions. In order to verify the stability of these systems, general Lyapunov functions were proposed for this class of systems (Lurie-type Lyapunov functions), [5] and [8]. In this research, these concepts are extended for a larger class of problems. The extended Lurie system, defined in this note, consists of nonlinear systems, like those of the Lurie problem, where one of the nonlinear functions may violate the sector conditions around the origin. In this case, the attracting set may be more complex than single equilibrium points. A systematic procedure, based on the extension of LaSalle’s invariance principle, [11], and linear matrix inequality (LMI) techniques, to obtain estimates of the attracting set and of the stability region of this class of nonlinear systems is proposed.
Manuscript received May 10, 2005; revised June 6, 2006. Recommended by Associate Editor M.-Q. Xiao. This work was supported by the FAPESP. A. C. P. Martins is with the Universidade Estadual Paulista, Bauru-SP 17033360, Brazil (e-mail: [email protected]). L. F. C. Alberto and N. G. Bretas are with the Universidade de São Paulo, São Carlos-SP 13566-590, Brazil (e-mail: [email protected]; ngbretas@sel. eesc.usp.br). Digital Object Identifier 10.1109/TAC.2006.883063
V (x) = xT P x + 2
Cx
h(y)T Ndy:
(2) (3)
0
III. DEVELOPED PROCEDURE The extended Lurie system consists of nonlinear systems that may be written like (1), where the nonlinearity violates the sector condition around the origin. In this case the attracting set may be more complex than single equilibrium points. This section presents the developed procedure to estimate the attracting set and the stability region of the extended Lurie system. When the system is non-autonomous the procedure provides a generalization of absolute stability for the attracting sets. Autonomous and nonautonomous systems are analyzed in different ways. For saving space only the single-input–single-output (SISO) case is presented, although the multiple-input–multiple-output (MIMO) case is quite direct, the resulting LMI equation is very extensive. A. Nonautonomous Systems Consider a nonlinear system in the form of the extended Lurie system, where A 2 Rn2n , B; c 2 R and h : R 2 R ! R is a C 1 function
x_ = Ax 0 Bh(t; y) : y = cT x
(4)
The nonlinearity h(t; y), shown in Fig. 1, satisfies the sector condition 0 h(t; y) Ky , for t 0, 0 y b, and Ky h(t; y) 0, for t 0, a y 0. Moreover, around the origin, the nonlinearity violates the sector condition but it is bounded, so
h(t; y)K 01 [h(t; y) 0 Ky] 0 jh(t; y)j m
8t; 8y 2f[ ; a]; [ ; b]g 8t; y
(5) (6)
In contrast to the Lurie problem, the origin of this system may not be an equilibrium point. However, since that nonlinearity violates the
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Fig. 1. Nonlinearity characteristic of the extended Lurie system.
sector condition only around the origin, this system can have an attracting set in the neighborhood of this point. Also, if the states in this set are sufficiently close to the origin, so they can be considered acceptable operating points, the origin is said to be a practical stability point of the system, [9]. The next theorem presents sufficient conditions to guarantee the existence of an attracting set in the neighborhood of the origin. It is worth noting that these estimates can only be obtained using extended Lyapunov functions. That is, the function in (2) is not a classical Lyapunov function, since it presents regions where the time derivative is greater than zero. Theorem 1: Consider the nonlinear system (4) satisfying conditions (5) and (6). Assume that there are a symmetric matrix P > 0 and a
A P + P A 0P B + c 0, such that 0B P + c 02 K 01 < 0. Let fc1 ; c2 ; . . . ; c 01g be a set of orthonormal vectors each one orthogonal to vector c, < 0 the largest eigenvalue of the matrix A P + P A, L = min(a2 =c P 01 c; b2 =c P 01 c), l = max 2 (z P z), and = 2(mkP B k2 =j j). The set Z consist of the 2 solutions of linear system [c c1 . . . c 01 ] z = d , where each vector d is a vector from f ; g 2 f0; + g 01 . Assume that L > l. Then, given an initial condition (t0 ; x0 ), where x0 2 = fx 2 R : x P x < Lg, there is t1 t0 such that the trajectories of the system are contained in the set = fx 2 R : x P x lg for all t t1 . T
constant
T
Fig. 2. Region of state–space where V_ (t; x)
T
min = max fA P + P Ag P >0 A P + P A 0P B + c s:t: 0B P + c 02 K 01 < 0 0:
T
M
V
T
n
Z n
k
n
k
n
L
n
t
T
T
z
T
T
Proof: The main idea of the proof is to show that the set fx 2 Rn : xT P x = Lg is the greatest level curve contained in the region where we can analytically establish that V_ (t; x) < 0 8t 0. Moreover, the set fx 2 Rn : xT P x = lg is a level curve that contains the region where the sign of V_ (t; x) can not be analytically verified, Fig. 2. Consider the quadratic function V (x) = xT P x. Its derivative along the trajectories of the system is V_ (t; x) = xT (P A + AT P )x 0 2xT P Bh(t; cT x). However, in the intervals [a; ] and [ ; b], the sector condition is satisfied, so, 02h(t; cT x)K 01 [h(t; cT x) 0 KcT x] 0 8(t; x) 2 R 2 00 where 00 = fx 2 Rn : a cT x ; cT x bg. So, using the S -procedure with some 0, [1], it is obtained V_ (t; x)
T
M
T
T
0.
on the hyperplanes cT x = a and cT x = b can be calculated as, respectively, a2 =cT P 01 c and b2 =cT P 01 c. So, the sets l and L can be defined, where L > l. The application of the theorem of uniform dissipativity [4] completes the proof. It is worth noting that the estimates l and L works to any nonlinearity h(t; y) satisfying the conditions (5) and (6). The Theorem 1 provides a sufficient condition for V_ (t; x) < 0 8x 2 L n l , the matrix P , satisfying the hypothesis of Theorem 1, can be obtained solving the optimization problem
n
M
0 and constants 0 and 0, such that
0P B + c x 02 K 01 h(t; c x) . However, from Theorem 3 (Appendix A) V_ (t; x) x [(P A + A P ) + ((0PB + c)(0B P + c )=2 K 01 )]x < 0 8x 2 00 . In the region 0+ = fx 2 R : c x gjh(t; c x)j m 8t 0, so V_ (t; x) kxk22 + 2mkxk2 kPB k2 . Since is a negative real number, V_ (t; x) < 0 whenever kxk2 > = 2mkPB k2 =j j. So, in the region fx 2 0+ : kxk2 g one cannot assure that V_ (t; x) < 0. This region is contained in PA + A P 0P B + A c + c =1 ... 01 fx 2 0+ : 0 c x g because, by construction, each hyperplane c x = 6 is tangent to M = 0B P + c A + c 02 K 01 0 c B 0 B c < 0. = the hyper sphere kxk2 = . Due the convexity of that set and function Let V (x), the largest value of V (x) inside that set is at their vertices, m((kPB 0 A ck2 + kP B 0 A ck22 + 2j kc B j)=j j), [x h(t; c x)] T
T
A P + PA T
0B P + c T
T
T
T
T
T
T
n
T
T
M
M
M
k
;
;n
T k
T
T k
T
T
T
Z
V
. Moreover, using the Lagrange multipliers, the minimum of V (x)
where
M
l such that L = fx 2 Rn : V (x) < Lg f00 0+ g and
l = fx 2 Rn : V (x) lg fx 2 0+ : kxk2 g. Then, each solution of the system starting in the set L converges to the set l , when t ! 1. Moreover, each solution of the system starting in the set l stays in l , 8t 0.
Proof: The idea of the proof is the same of Theorem 1. Consider c x the function V (x) = xT P x + 2 0 h(u)du. Its time derivative is
V_ (x) = xT (P A + AT P )x 0 2xT P Bh(cT x)+2h(cT x)cT [Ax 0 Bh(cT x)]. However, in the intervals [a; ] and [ ; b] the sector condition is satisfied, so 02h(cT x)K 01 [h(cT x) 0 KcT x] 0 8x 2 00 , where 00 = fx 2 Rn : a cT x ; cT x bg. So, using the S -procedure with 0, [1], and by Theorem 1, V_ (x) xT [(P A + AT P )+((0PB + AT c + c)(0B T P + cT A + cT )=2 K 01 + cT B + B T c)]x, so the hypothesis on the matrix M guarantees that V_ (t; x) < 0 8x 2 00 . In the2 region 0+ , the nonlinearity is bounded, so V_ (x) M kxk2 + 2mkxk2 kPB 0 AT ck2 + 2m2 jcT B j. Since M < 0 then AT P + P A < 0, so M < 0 and V_ (x) < 0 for all x sufficiently large such that kxk2 > = m((kPB 0 AT ck2 + kP B 0 AT ck22 + 2jM jcT B j)=jM j). Suppose the existence of sets l and L , then the application of the
Fig. 3. Estimate of limit cycle and its stability region.
extended LaSalle invariance principle [11] completes the proof. In general, function V (x) is not convex. So, the analytical computation of constants L and l is not possible in general and we have to use numerical methods to obtain these constants. However, the matrix P and the constant , satisfying the conditions of the previous Theorem 2, can be obtained solving the optimization problem in (8) using LMI techniques:
min M = max fAT P + P Ag P >0 0 s:t: M < 0 0:
(8) Fig. 4. Estimate of limit cycle and its stability region for
K = 30.
IV. NUMERICAL RESULTS In this section, the results proved in the previous section are applied to estimate the attracting set of the forced Duffing equation and the Van der Pol System. A. Forced Duffing Equation Consider the dynamic system M z + kc z_ + km z + km 2 z 3 = ku cos(!t); making x1 = z and x2 = z_ , and choosing km =M = 3, kc =M = 3, km 2 =M = 0:1, and ku =M = 2e ! = 1, this system can
0 1 , be written like an extended Lurie system, where A = 0 3 03 0 1 3 B = 1 , c = 0 and h(t; y) = 0:1y 0 2 cos(t). Simulations in-
dicate that this system has a stable limit cycle around the origin. Theorem 1 will be used to estimates this attracting set. Due the cubic term in h(t; y), the set 00 depends on the sector condition, i.e., the value of K . For K = 15, we have = 0 = 01:2599, a = 0b = 012:1802, and m = 4. In this case, it is easy to see that c1 = [0 1]T . The resolution of the optimization problem, (7), provides
9:2804 1:2684 P = 2 108 , = 2:3608 2 108 and 1:2684 1:0069 M = 02:3573 2 108 . Therefore, using Theorem 1, we have = 5:4948, L = 1:1398 2 1011 and l = 6:2697 2 109 . Fig. 3 shows the trajectories for several initial conditions and the sets L and l . It is worth nothing that these estimates are uniform to any nonlinearity that satisfies the same given conditions.
B. Van der Pol System
Consider the autonomous dynamic system v + "f 0 (v)v_ + v = 0, where f(v) = 0v + (1=3)v 3 . Making x1 = 0"01 v_ 0 f(v), x2 = v and using the parameter , the system can be written like an extended Lurie system, where A =
0 "01 0 0 0" 0" , B = " , c = 1
and h(y) =
0(1+)y+(1=3)y3 . Simulations indicate that this system has a stable
limit cycle around the origin. Choosing " = 1, = 1 and K = 30 one have m = 1:8856, = 0 = 02:4495, and a = 0b = 09:7980. Solving the optimization problem (8), one have = 4:6257:108 , M = 04:13902108 , the extended Lyapunov function (9) and = 5:3022. According to Theorem 2, one can calculate L = 1:9236 2 1011 and l = 2:0280 2 1010 . Fig. 4 shows the trajectories for several initial conditions, the sets L and l , and the region where V_ 0 (gray region)
V (x1 ; x2 ) =
x1 x2
T
6:2117 2:0707 2:0707 4:1426 +2:3688
x1 x2
2 108
0x22 + 121 x42 2 108 :
(9)
In this case, the hyperplane cT x = a bounds the stability region estimate, the set L ; the constant K of the sector condition determines the value of a. So, to expand the set L it is necessary to increase
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10, OCTOBER 2006
the value of K . For the Van der Pol system, the optimization problem . In this case, (8) remains feasible even if K ! 1, i.e., K 01 the solution of problem (8) provides a slightly poor estimate of the : . However, the stability region is the whole attracting set, R2 , so that the limit cycle is globally stable.
An Improved ILMI Method for Static Output Feedback Control With Application to Multivariable PID Control
V. CONCLUSION
Abstract—An improved iterative linear matrix inequality (ILMI) algorithm for static output feedback (SOF) stabilization problem without introducing any additional variables is proposed in this note. The proposed controller design ILMI algorithm is also extended to solve the SOF problem. They are applied to the multivariable PID controllers. Numerical examples show that the proposed algorithms yield better results and faster convergence than the existing ones.
=0
= 6 4912
This note presents a systematic procedure for estimating the attracting set of a class of nonlinear systems called the extended Lurie system. The extended Lurie system consists of systems that can be written like the Lurie problem, but with just one nonlinearity that violates the sector condition. The procedure was applied to two classical nonlinear systems: the forced Duffing equation and the Van der Pol system. The numerical results show that the proposed procedure provides good estimates of the attracting sets. Moreover, for the Van der Pol system the attracting set was proved to be the whole R2 . The calculus of matrix P , using the LMI problems in (7) and (8), tends to improve the stability region estimate since the negative term to be minimized is on the denominator of . However, that improvement is bounded since the norm of matrix P is on the numerator of . APPENDIX
: [a; b] 2 D R 2 R ! R be a C function 1 8 1 8 , where and W (t; x) = [x h(t; x) ] 8 6 80 6 is symmetric negative definite. Then, W (t; x) x [1 0 86 8 ]x 8(t; x) 2 [a; b] 2 D, where [1 0 860 8 ] is symmetric negative Theorem 3: Let h T
n
T
p
1
x
h(t;x)
T
T
T
1
1
T
T
definite.
REFERENCES [1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [2] M. Dellnitz and A. Hohmann, “A subdivision algorithm for the computation of unstable manifolds and global attractors,” Numerische Mathematik, vol. 75, pp. 293–317, 1997. [3] J.-Y. Dieudelot and P. Borne, “An estimation of the stability domain of a fuzzy controled pendulum using overlapping attractors and vector norms,” in Proc. IEEE Int. Conf. Systems, Man, and Cybernetics, 2001, vol. 4, pp. 2245–2249. [4] M. F. Gameiro and H. M. Rodrigues, “Applications of robust synchronization to communication systems,” Appl. Anal., vol. 79, pp. 21–45, 2001. [5] W. M. Haddad and V. Kapila, “Absolute stability criteria for multiple slope-restricted monotonic nonlinearities,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 361–365, Feb. 1995. [6] S. Heidari and C. L. Nikias, “Characterizing chaotic attractors using fourth-order off-diagonal cumulant slices,” in Proc. 27th Asilomar Conf. Signal, Systems, and Computers, 1993, vol. 1, pp. 466–470. [7] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge Univ. Press, 1996. [8] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ: PrenticeHall, 1992. [9] J. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method With Applications. New York: Academic, 1991. [10] R. K. Miller, A. N. Michel, and G. S. Krens, “Stability analysis of limit cycles in nonlinear feedback systems using describing functions: improved results,” IEEE Trans. Circuit Syst., vol. CAS-31, no. 6, pp. 561–567, Jun. 1984. [11] H. M. Rodrigues, L. F. C. Alberto, and N. G. Bretas, “On the invariance principle: generalizations and applications to synchronization,” IEEE Trans. Circuits Syst. I, vol. 47, no. 5, pp. 730–739, May 2000. [12] J. Argyris, G. Faust, and M. Haase, An Exploration of Chaos. Amsterdan, The Netherlands: North-Holland, 1994.
Yong He and Qing-Guo Wang
control, linear matrix inequality (LMI), multivariIndex Terms— able PID control, stabilization, static output feedback (SOF).
I. INTRODUCTION The static output feedback (SOF) plays a very important role in control theory and applications. Recently, it has attracted considerable attention (see, e.g., [1]–[7] and the references therein). Yet, it is still left with some open problems. Unlike the state feedback case, a SOF gain which stabilizes the system is not easy to find. Linear matrix inequality (LMI) [8] is one of the most effective and efficient tools in controller design and a great deal of LMI-based design methods of SOF design have been proposed over the last decade [9]–[19]. Among these methods, an iterative linear matrix inequality (ILMI) method was proposed by Cao et al. [13] and later employed to solve some multivariable PID controller design problems [20], [21]. In this context, a new additional variable was introduced such that the stability condition becomes a sufficient one. The iterative algorithm in [13] tried to find a sequence of the additional variables such that the sufficient condition is close to the necessary and sufficient one. The similar idea is used in the so-called substitutive LMI method by [19]. Both of them set the additional matrix variables at the current step with the matrices derived in the preceding step. With additional variables, the dimensions of the LMIs become higher. We observe that it is possible that the matrices derived in the preceding step can be used in the next one directly without introducing the additional variables and the dimensions of the LMIs need not be increased. In addition, we can develop some efficient ways to get suitable initial values in the iterative procedure, which the existing approaches have not been thought of. In this note, a new ILMI algorithm is proposed for SOF stabilization problem without introducing any additional variables, and assisted with a separate ILMI algorithm to find good initial variables. The algorithms for SOF stabilization are also extended to solve the SOF H1 control problem. They are applied to multivariable PID control. Numerical examples show the effectiveness and an improvement of the algorithms over the existing methods.
Manuscript received April 3, 2005; revised January 12, 2006 and April 23, 2006. Recommended by Associate Editor M. Kothare. The work with Y. He was supported in part by the National Science Foundation of China under Grant 60574014 and in part by the Doctor Subject Foundation of China under Grant 20050533015. Y. He is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore. He is also with the School of Information Science and Engineering, Central South University, Changsha 410083, China. Q.-G. Wang is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2006.883029
0018-9286/$20.00 © 2006 IEEE
