Boundedness without Absolute Stability in Systems with Stiffening Nonlinearities

European Journal of Control (2002)8:243±250 # 2002 EUCA

Boundedness without Absolute Stability in Systems with Stiffening Nonlinearities Murat Arcak1, Michael Larsen2,y and Petar KokotovicÂ3,z 1

Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590; Informations Systems Laboratories, Inc., 6370 Nancy Ridge Dr. Suite 101, San Diego, CA 92121; 3Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106-9560 2

If stability of an equilibrium cannot be achieved, the next desirable property is boundedness. In the framework of absolute stability, we determine boundedness from the zero dynamics of the feedforward linear block and from growth properties of the feedback nonlinearity. The linear block must be relative degree one or two, and minimum phase, so that it can be stabilized by high-gain feedback. The nonlinearity is required to grow faster than linear, that is, to be ``stiffening''. At large state magnitudes such a nonlinearity acts as a form of high-gain feedback. One of the results of the paper establishes boundedness in a system consisting of an unstable Mathieu equation in feedback with a stiffening nonlinearity. This phenomenon has been observed experimentally in several important classes of systems, including micro-electromechanical systems. Keywords: Absolute stability; Boundedness; Nonlinear Mathieu equation

area of research, as presented in Aizerman and Gantmacher [2], and Narendra and Taylor [13]. To characterize the stability of the system x_ ˆ Ax ‡ Bu, y ˆ Cx, uˆ

…1†

…y†,

which consists of a linear block G(s) ˆ C(sI A) 1B in feedback with a static nonlinearity (
) as in Fig. 1, Lurie and coworkers introduced a Lyapunov function made of a quadratic term and the integral of the nonlinearity. A major breakthrough came in 1960, when Popov [16] derived a stability criterion for (1) from the frequency response of the linear block G(s). Connections between Popov's criterion and the existence of a quadratic-plus-integral Lyapunov function were made by Yakubovich [21], and Kalman [7]. From

1. Introduction and Problem Formulation The absolute stability problem, formulated by Lurie and Postnikov [9], has been a well-studied and fruitful  Research supported in part by the National Science Foundation under grant ECS-9812346, and the Air Force Office of Scientific Research under grant F49620-00-1-0358. Correspondence and offprint requests to: Murat Arcak, Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590. Tel.: (518)-276-6535. E-mail: [email protected] y E-mail: [email protected] z E-mail: [email protected]

Fig. 1. Feedback interconnection of a linear system and a nonlinear element. Received 8 May 2001; Accepted 2 July 2001. Recommended by J. C. Willems and I. D. Landau.

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today's standpoint, the fundamental contribution of Popov's criterion is the introduction of the passivity concept in feedback control. This crucial property was made explicit by Popov [17] and, independently, by Brockett [3]. The passivity view also led to the circle criterion, and several other absolute stability criteria such as the multiplier methods, described in the book by Narendra and Taylor [13]. Some of the early conjectures about the stability of (1) were shown to be incorrect. Aizerman [1] conjectured that, if (1) is stable for all linear gains u ˆ ky in the interval k 2 [k1, k2], then it would be absolutely stable for nonlinearities satisfying the sector condition k1 

…y†  k2 : y

This was proved false by the counterexamples of Pliss [15], and others. Kalman's conjecture [8], which further restricted (
) to satisfy the incremental sector property k1 

d  k2 , dy

was also proved false by Fitts [4]. Despite extensive work on characterizing stability and instability, global boundedness of solutions have received very little attention. SÏiljak and Weissenberger [19] pursued a boundedness version of the circle criterion when the sector condition is satisfied outside a compact set. In this paper, we present a new boundedness mechanism for the solutions of (1), which makes use of a growth condition on the feedback nonlinearity (
). In this mechanism, the linear block G(s) satisfies structural conditions, such as minimum phase and relative degree one or two, which ensure that it is stabilizable by high-gain feedback. The nonlinearity (
) satisfies the following ``stiffening'' property, which allows it to play the role of a stabilizing high-gain feedback when the states grow large. Definition 1. We say that the nonlinearity (y) is stiffening if for every m > 0, there exists ` > 0 such that jyj  `

)

…y†  m: y

…2†

This means that (
) is to grow faster than linear, such p as (y) ˆ y3 which satisfies (2) with ` ˆ m. In Section 2, we analyze relative degree one and minimum phase systems, and prove their boundedness using the stiffening nonlinearity property. The proof for relative degree two systems, given in Section 3, is extended to a class of nonlinear time-varying systems

in feedback with a stiffening nonlinearity. Several examples are presented to illustrate the boundedness property and the emergence of limit cycles. Among them is the well-known Mathieu equation studied in Section 4, where we prove that unstable solutions remain bounded in the presence of a stiffening nonlinearity ± a phenomenon experimentally observed in several applications. In Section 5, conclusions are given and a generalization of our boundedness result is discussed. Throughout the paper, (
) is assumed to be locally Lipschitz to ensure the existence and uniqueness of solutions.

2. Relative Degree One We start our analysis with relative degree one systems, in which the boundedness effect of the stiffening nonlinearity is more transparent. Theorem 1. Suppose G(s) ˆ C(sI A) 1B is relative degree one and minimum phase (zeros are in the open left half-plane), and its high frequency gain kp is positive. If (y) is stiffening as in Definition 1, then the solutions of (1) are bounded. Proof. Because G(s) is relative degree one, upon a change of coordinates, (1) can be expressed in the form _ ˆ Q ‡ Ry, y_ ˆ S ‡ y

…3† kp …y†,

…4†

where kp > 0 is the high-frequency gain, the subsystem represents the zero dynamics of G(s), and Q is Hurwitz from the minimum phase property. We let P ˆ PT > 0 be such that QTP ‡ PQ ˆ I, and define the Lyapunov function V…, y† ˆ T P ‡ 12 y2 ,

…5†

which satisfies V_ 

kk2 ‡ T …2PR ‡ ST †y ‡ y2

kp y…y†: …6†

Completing squares, we can find a constant k such that T …2PR ‡ ST †y  12 kk2 ‡ k y2 ,

…7†

thus, V_ 

2 1 2 kk

1 2 2y

‡ …k ‡  ‡ 12†y2

kp y…y†: …8†

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Because (y) is stiffening as in (2), there exists an interval [ `, `], outside which the function
k ‡  ‡ 12 y2

kp y…y†

…9†

is negative. Denoting d :ˆ max …k ‡  ‡ 12†y2 y2‰ `;`Š


kp y…y† ,

…10†

we get V_ 

2 1 2 kk

1 2 2y

‡ d,

…11†

which implies that V_ is negative outside the compact set

:ˆ f…, y† : 12 kk2 ‡ 12 y2  dg:

…12†

Thus, the solutions ((t), y(t)) are bounded and converge to the smallest level set of V which includes . &

When  is increased from 1 to 10, the limit cycle shrinks because the nonlinearity (y) ˆ y3 becomes stiffer, which ensures that the value of d in (12) is smaller. & Theorem 1 proved boundedness using the stiffening property of (y). To interpret the boundedness mechanism from the root-locus of G(s), we note that the relative degree one, minimum phase, and kp > 0 assumptions ensure that G(s) is stabilizable by high gain, that is, its gain margin includes [k, 1) for some k > 0. Thanks to the stiffening property, for large states, the nonlinearity (y) provides the high-gain (y)/y required to bring the unstable poles to the left half-plane. This root-locus interpretation indicates that the sign condition kp > 0 plays a crucial role in Theorem 1. Indeed, if kp < 0, then (y) acts as a harmful positive feedback term and, as the states grow unbounded, the stiffening property makes it even more destabilizing, leading to finite escape time. Example 2. Consider the system x_ 1 ˆ x2 ,

Example 1. The feedback interconnection of

x_ 2 ˆ G…s† ˆ

s2

s‡1 s‡1

G…s† ˆ kp

1.5

1.5

1

1

0.5

0.5

0

–0.5

–1

–1

–1.5

–1.5

κ =1

0.5

1

1.5

s‡ ‡ as ‡ b

…15†

0

–0.5

0 x (1)

s2

and the nonlinearity (y) ˆ y3 as in Fig. 1. With kp > 0 and  > 0, all assumptions of Theorem 1 are satisfied and, hence, the solutions are bounded.

x(2)

x(2)

which is the feedback interconnection of

2

–0.5

kp …x1 ‡ x2 †3 ˆ: f2 …x1 , x2 †, …14†

2

–1

bx1

…13†

and the stiffening nonlinearity (y) ˆ y3,  > 0, satisfies the conditions of Theorem 1 and, therefore, its solutions are bounded. Moreover, because the only equilibrium is the unstable origin, the Poincare ± Bendixson Theorem predicts the existence of a limit cycle, as shown by numerical simulation in Fig. 2.

–2 –1.5

ax2

–2 –1.5

–1

–0.5

0 x (1)

0.5

1

1.5

κ = 10

Fig. 2. Limit cycles arising in the feedback interconnection of G…s† ˆ …s ‡ 1†=…s2 s ‡ 1† and the stiffening nonlinearity (y) ˆ y3. When  ˆ 1 (left) the limit cycle is larger. When  ˆ 10 (right), (y) becomes stiffer, and the limit cycle shrinks.

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However, when kp < 0, we can find positive constants , and , and an open bounded interval U such that, if x1 2 U, then x2 >

)

x32 < f2 …x1 , x2 † < x32 :

…16†

As proved by Mazenc et al. [11, Lemma 3], this implies that the system (14) exhibits finite escape time for & sufficiently large x2(0) > 0.

3. Relative Degree Two Relative degree two systems are stabilized by linear high-gain feedback if the zeros and the asymptote of the root-locus are contained in the open left halfplane, as illustrated in Fig. 3. This means that G(s) must be minimum phase, and a :ˆ sum of zeros

sum of poles of G…s†

1

Theorem 2. Suppose G(s) ˆ C(sI A) B is relative degree two and minimum phase, its high frequency gain is kp > 0, and a > 0 in (17). If (y) is stiffening as in Definition 1, then the solutions of (1) are bounded. Proof. Because G(s) is relative degree two, there exists a change of coordinates for which (1) becomes

kp …z1 † ‡ S,

V_ 2 

…18†

y ˆ z1 ,

 T 2 

v2 ‡ 22 y2 ,

…19†

for some 2 ˆ T2 > 0. We now show that a complementary small-gain condition holds for the nonlinear 1-subsystem when the states grow sufficiently large. To this end we first rewrite 1 as z_1 ˆ z2 , z_2 ˆ az2 y ˆ z1 ,

…17†

must be positive, because the asymptote intersects the real axis at a/2. We now prove that these conditions ensure boundedness in the presence of a stiffening nonlinearity:

_ ˆ Q ‡ Rz1 , z_1 ˆ z2 , z_2 ˆ az2 bz1

which is the feedback interconnection of the subsystems 1 and 2 shown in Fig. 4. The linear -subsystem 2 represents the zero dynamics of G(s), and because of the minimum phase assumption, it has a finite L2 -gain 2 from its input y to its output v. We let 2 > 2 , and conclude from the Bounded Real Lemma (see e.g. [5]) that there exists a positive definite storage function V2() ˆ TP2 satisfying

b z 1

#…z1 † ‡ v,

…20†

b †z1 ,

…21†

where #…z1 † :ˆ kp …z1 † ‡ …b

and b > 0 is to be selected. If #(z1) were absent, (20) would be a linear system with transfer function y…s† 1 ~ 1 …s† ˆ :ˆ  , v…s† s2 ‡ as ‡ b

…22†

~ 1 … j!†j, can in which the L2 -gain 1 , that is, sup!2R j be rendered arbitrarily small by selecting b sufficiently large. We let b > 0 be such that 1 < 1= 2 , and conclude from the Bounded Real Lemma that there exists a positive definite storage function V~1 …z† ˆ zT P1 z, and a matrix 1 ˆ T1 > 0, such that along the trajectories of the nonlinear system (20),   _V~  zT  z y2 ‡ 1 v2 2zT P 0 #…z †: 1 1 1 1 1

22 …23†

– a 2

Fig. 3. Root-locus of a relative degree two system. Because the zero and the asymptote are in the left half-plane, the system is stabilized with high-gain feedback.

Fig. 4. System (18) represented as the feedback interconnection of subsystems 1 and 2.

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Boundedness with Stiffening Nonlinearities

Rewriting P1 as   p q P1 ˆ , q r

…24†

where r > 0 because P1 > 0, and q > 0 because (23) implies  T   0 1 0 1 P1 ‡ P1 a a b b   2b q  ˆ < 0, …25†   we rewrite (23) as V_~1 

zT  1 z

y2 ‡

1 2 v

22

2…qz1 ‡ rz2 †#…z1 †: …26†

which implies Z Z z1 #…† d  min z1 2‰ `;`Š

0

z1

0

#…† d ˆ:

d0 : …33†

Because r > 0, the inequality (31) holds with d ˆ 2rd0 and, hence, V

1 T  P2  ‡ zT P1 z

22

d,

…34†

which proves that V is positive outside a compact set. To show that V_ is negative, we again use the stiffening property (32), and obtain z1 #…z1 †  min z1 #…z1 † ˆ:

d1 :

…35†

zT 1 z ‡ 2qd1 ,

…36†

z1 2‰ `; `Š

Substituting in (30), we get

To compensate for the sign-indefinite component 2rz2#(z1), we augment the quadratic storage function V~1 with the integral of #(z1) as in Popov's criterion, Z z1 #…† d, …27† V1 …z† ˆ V~1 …z† ‡ 2r

which proves that V_ is negative outside a compact set and, hence, the solutions (z(t), (t)) are bounded. &

and obtain

The following example shows that the condition a > 0 in Theorem 2 is tight:

0

V_ 1 

zT  1 z

y2 ‡

1 2 v

22

2qz1 #…z1 †:

…28†

1 V2 ‡ V1

22 Z z1 1 ˆ 2 T P2  ‡ zT P1 z ‡ 2r #…† d,

2 0

…29†

and using (28) and (19), we get zT 1 z

2qz1 #…z1 †:

…30†

To prove boundedness of (z(t), (t)), we show that V is positive and V_ is negative outside a compact set. We first show that Z z1 #…† d  d, …31† 2r 0

for some constant d > 0, which guarantees from (29) that V is positive outside a compact set, and radially unbounded. To prove (31), we note from the stiffening property of (
) that we can find ` > 0 such that jj  `

x_ 1 ˆ x2 ,

)

#…† ˆ kp …† ‡ …b

b †2  0,

ax2

x1

x31 ,

…37†

which is the feedback interconnection of

V :ˆ

1 T  2 

22

1 T  2 

22

Example 3. The system x_ 2 ˆ

Finally, defining

V_ 

V_ 

…32†

G…s† ˆ

1 s2 ‡ as ‡ 1

…38†

and the stiffening nonlinearity …y† ˆ y3 , has unbounded solutions when a < 0. To show this, we denote the right-hand side of (37) by f(x), and note that …@f1 =@x1 † ‡ …@f2 =@x2 † ˆ a 6ˆ 0 which, from Bendixson's criterion, implies that there are no closed orbits. Because the equilibrium (0, 0) is unstable, unboundedness follows from the Poincare ±Bendixson Theorem. & Our characterization of G(s) in Theorem 2 used the concepts of relative degree and minimum phase, which have direct nonlinear counterparts. Thus, it is not difficult to prove a boundedness result when the linear block G(s) is replaced with a nonlinear system satisfying similar relative degree and minimum phase conditions. In Theorem 3 below, this is achieved by restricting the relative degree to be two, and the nonlinear zero dynamics to have a finite L2 -gain as in the 2 subsystem in Fig. 4.

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Theorem 3. Consider the system

Thus, the right-hand side of

_ ˆ f…, z1 †, z_1 ˆ z2 , z_2 ˆ

az2

…39† bz1 ‡ …t, , z1 , z2 † ‡ h…†

…z1 †, …40†

which satisfies the following assumptions: (A1) a > 0; (A2) (t, z1, z2) is piecewise continuous in t, and satis®es for all t  0, z1 , z2 2 R, j …t, , z1 , z2 †j  …1 ‡ jz1 j†,

…41†

(A3) For the zero dynamics subsystem (39) with input y ˆ z1 and output v ˆ h(), there exists a constant

2, a positive de®nite and radially unbounded function 2() and a C1, radially unbounded storage function V2() satisfying 2 …†

v2 ‡ 22 y2 :

…42†

If the assumptions A1±A3 above hold, and (z1) is stiffening as in Definition 1, then the solutions of (39)±(40) are bounded. Proof. We represent the system (39)±(40) as in Fig. 4, where the 2-subsystem is now nonlinear. However, because of Assumption A3, we can still use the Lyapunov function V ˆ …1= 22 †V2 ‡ V1 as in the proof of Theorem 2. Because of the perturbation term …t, , z1 , z2 †, (30) is modified as V_ 

1 2 …† zT 1 z 2qz1 #…z1 †

22 ‡ 2…qz1 ‡ rz2 † …t, , z1 , z2 †:

…43†

Using (41) and completion of squares, we can find a constant c > 0 such that 1 2…qz1 ‡ rz2 † …t, , z1 , z2 †  zT 1 z ‡ c…1 ‡ z21 †: 2 …44† Substituting (44) in (43), we get V_ 

1 2 …†

22

1 T z 1 z 2

2qz1 #…z1 † ‡ cz21 ‡ c, …45†

where, because of the stiffening property, we can find a constant c > 0 such that 2qz1 #…z1 † ‡ cz21  c:

1 2 …†

22

1 T z 1 z ‡ c ‡ c 2

…47†

is negative outside a compact set, which proves boundedness of ((t), z(t)). &

4. Example: The Mathieu Equation with Stiffening Nonlinearity The Mathieu equation (Jordan and Smith [6], McLachlan [12], Nayfeh and Mook [14]) z_1 ˆ z2 ,

for some positive constant ;

V_ 2 

V_ 

…46†

z_2 ˆ

az2

…48†

…!20 ‡  cos…!t††z1 ,

arises in a variety of problems including wave propagation, parametric amplifiers, frequency modulation, and others surveyed by Ruby [18]. For the damped Mathieu equation, where a > 0, well-known parametric resonance occurs when  is sufficiently large, and the natural frequency !0 and the forcing frequency ! are related by n !0 ˆ ! 2

n ˆ 1, 2, 3, . . .

We now analyze a nonlinear version of the Mathieu equation, z_1 ˆ z2 , z_2 ˆ

az2

…!20 ‡  cos…!t††z1

…z1 †,

…49†

in which the stiffening nonlinearity (
) keeps the solutions from growing unbounded. As shown in Fig. 5 by simulations, the unstable solutions of the Mathieu equation remain bounded in the presence of the stiffening nonlinearity …z1 † ˆ z31 . This boundedness property has been observed experimentally by several authors, including Mandelstam and Papalexi [10] who redesigned an oscillating circuit to be nonlinear to limit the growth of oscillations. A recent micro electro mechanical systems (MEMS) application by Turner et al. [20] is particularly significant: A microelectromechanical probe, consisting of a cantilevered beam connected to a torsion bar, is parametrically excited via electrostatic actuation, and resonance is obtained for several values of n. However, the resulting torsional oscillations remain bounded, because the nonlinear restoring forces play the role of the stiffening (
) in (49). Since an analytical proof has not been given for this experimentally observed property, we note that the system (49) is of the form (40) with b ˆ !20 and …t, z1 † ˆ  cos…!t†z1 ,

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Boundedness with Stiffening Nonlinearities

(a)

z

6

(b) 6

4

4

2

2

0

0

–2

–2

–4

–4

–6

–6

–8

0

5

10

15

20

25 t

30

35

40

45

50

–8

0

5

10

15

20

25

t

30

35

40

45

50

Fig. 5. (a) Parametric resonance in the linear Mathieu equation (48) with a ˆ 1,  ˆ 2.5, !0 ˆ 1, ! ˆ 2. (b) In the presence of a cublic nonlinearity …z1 † ˆ z31 , the solutions of (49) remain bounded. (a)

1.5

(b)

5 4

1

3 2 Imag Axis

Imag Axis

0.5 0 –0.5

1 0 –1 –2 –3

–1

–4 –1.5 –1.5

–1

–0.5 0 Real Axis

0.5

1

–5 –0.04 –0.03 –0.02 –0.01 0 0.01 Real Axis

0.02

0.03

0.04

Fig. 6. Root-loci (a) the transfer function (51) in the van der Pol oscillator; (b) the transfer function (52) in the Fitts counterexample.

and …t, z1 † satisfies (41) with  ˆ . Thus, Theorem 3 establishes global boundedness for the nonlinear Mathieu equation (49), complementing a local analysis by Nayfeh [14, Section 5.7]. We emphasize that Theorem 3 can also be applied to higher order systems as in Fig. 4, where the Mathieu equation constitutes the 1-subsystem.

5. Discussion and Conclusion We have analyzed boundedness of relative degree one and relative degree two systems in feedback with a stiffening nonlinearity. The restriction on the relative degree is due to the high-gain nature of the feedback loop, in which the nonlinearity acts as a high-gain stabilizing feedback for large states, thus ensuring boundedness. For systems of relative degree three and

more, the high-gain effect of the nonlinearity is destabilizing. In addition to the relative degree restriction, we have required that the zeros of G(s) be in the open left half-plane. It may be possible to prove a similar result in the presence of zeros on the imaginary axis, because G(s) tolerates high-gain feedback when the root-locus approaches imaginary axis zeros from the left. Likewise, when a ˆ 0 for relative degree two systems, that is, when the asymptote in Fig. 3 coincides with the imaginary axis, boundedness may be possible if the root-locus approaches the asymptote from the left. Although these situations are not studied in this paper, we conjecture that their solutions would be bounded in the presence of a stiffening nonlinearity: Conjecture: Suppose the gain margin of G(s) includes [k, 1) for some k > 0. Then, the solutions of

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the feedback interconnection in Fig. 1 with a stiffening nonlinearity are bounded. Theorems 1 and 2 proved this conjecture for a significant class of G(s) satisfying the [k, 1) gain-margin property. For other classes, several examples corroborate our conjecture. The well-known van der Pol oscillator x_ 1 ˆ x2 , x_ 2 ˆ x1 ‡ x2

1 3 3 x2 ,

…50†

represented as in Fig. 1 with …y† ˆ 13 y3 and G…s† ˆ

s2

s s‡1

…51†

fails to satisfy the minimum phase condition of Theorem 1 because of the zero at zero. However, the boundedness of its solutions is consistent with the conjecture, because the root-locus of G(s) satisfies the [k, 1) gain margin condition as shown in Fig. 6(a). Likewise, Theorem 2 is not applicable to the Fitts counterexample [4] to Aizerman and Kalman conjectures, where (y) ˆ y3 and G…s† ˆ

10s…s ‡ 0:01† 2

……s ‡ 0:01† ‡ 0:92 †……s ‡ 0:01†2 ‡ 1:12 † …52†

has a zero at zero. Boundedness of its solutions, observed from simulations in [4], corroborates our conjecture because G(s) has an infinite gain-margin as shown in Fig. 6(b). A proof or a counterexample to our conjecture would be of interest.

Acknowledgements We thank Dr. Kimberly Turner for bringing to our attention the MEMS application of the Mathieu equation [20], discussed in Section 4.

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