Exponential stabilization of a class of unstable bilinear systems

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leads to the exponentially convergent observer dynamics m_^v = T 0 cD^vj^vj 0 kq(^v 0 _q)

I _^! = 0 k1^! + k2U 0 Q 0 k(^! 0 _)

hence exploiting and augmenting the natural contraction properties of the system.

REFERENCES

[1] V. I. Arnold, Mathematical Methods of Classical Mechanics. New York: Springer Verlag, 1978.

[2] R. Brockhaus, Flight Control. New York: Springer Verlag, 1994. [3] R. Dawkins, The Selfish Gene. New York: Penguin, 1979.

[4] H. Goldstein, Classical Mechanics. New York: Addison-Wesley, 1980.

[5] W. Hahn, Stability of Motion. New York: Springer Verlag, 1967. [6] P. Hartmann, Ordinary Differential Equations, 2nd ed. Boston, MA:

Birkhauser, 1982.

[7] A. Isidori, Nonlinear Control Systems, 3rd ed: Springer Verlag, 1995. [8] H. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, NJ:

Prentice-Hall, 1995.

[9] N. N. Krasovskii, Problems of the Theory of Stability of Motion (in Eng-lish translation by Stanford Univ. Press, 1963). Moscow, Russia: Mir, 1959.

[10] W. Lohmiller, “Contraction analysis for nonlinear systems,” Ph.D. dis-sertation, Dep. Mechanical Eng., M.I.T., 1998.

[11] W. Lohmiller and J. J. E. Slotine, “On contraction analysis for nonlinear systems,” Automatica, vol. 34, no. 6, 1998.

[12] , “Contraction analysis of mechanical systems,” M.I.T. Nonlinear Systems Laboratory, Rep. NSL-980 501, 1998.

[13] D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles. New York: Dover, 1989.

[14] D. G. Luenberger, Introduction to Dynamic Systems. New York: Wiley, 1979.

[15] R. Marino and T. Tomei, Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1995.

[16] S. Massaquoi and J. J. E. Slotine, “The intermediate cerebellum may function as a wave variable processor,” Neuroscience Lett., vol. 215, 1996.

[17] G. Niemeyer and J. J. E. Slotine, “Stable adaptive teleoperation,” IEEE J. Oceanic Eng., vol. 16, pp. 152–162, Jan. 1991.

[18] H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems. New York: Springer Verlag, 1990.

[19] V. M. Popov, Hyperstability of Control Systems. New York: Springer-Verlag, 1973.

[20] L. Schwartz, Analyze. Paris, France: Hermann, 1993.

[21] J. J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991.

[22] M. Vidyasagar, Nonlinear Systems Analysis, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1992.

[23] L. Withcomb and D. Yoerger, “Preliminary experiments in the dynam-ical control of marine thrusters—Part 1: Dynamdynam-ical modeling,” IEEE J. Oceanic Eng., vol. 23, 1998.

Exponential Stabilization of a Class of Unstable Bilinear Systems

Min-Shin Chen and Shia-Twu Tsao

Abstract—This paper considers the control design of bilinear systems with multiplicative control inputs. Previous control designs for such systems normally assume that the open-loop bilinear system is (neutrally) stable. In this paper, a new nonlinear control design is proposed for open-loop unstable bilinear systems. The new control stabilizes the bilinear system globally and exponentially if a sufficient stability condition, which can be checked by off-line computer simulations in advance of the control, is satisfied.

Index Terms—Bilinear system, exponential stability, global stability, multiplicative control, nonlinear control.

I. INTRODUCTION

Bilinear systems have been of great interest in recent years. This interest arises from the fact that many real-world systems can be ad-equately approximated by a bilinear model. Real-world examples in-clude engineering applications in nuclear, thermal, and chemical pro-cesses, and nonengineering applications in biology, socio-economics, immunology, and so on. Detailed reviews of bilinear systems and their control designs can be found in [1] and [2]. For a bilinear system whose control input is both multiplicative and additive [2], one can use linear state feedback control [3] to obtain local asymptotical stability. Other control designs, such as the bang-bang control [4] or the optimal con-trol [5], [6], obtain global asymptotic stability, but they all assume that the open-loop system is either stable or neutrally stable. When the open-loop system is unstable, it is difficult to obtain global asymptot-ical stability except when independent additive and multiplicative con-trol inputs [7] exist.

This paper considers the control design for bilinear systems with multiplicative control inputs only. For such bilinear systems, it has been shown that quadratic state feedback control [8]–[10] can achieve global asymptotical stabilization, and normalized quadratic state feed-back control [11] achieves global exponential stabilization. However, they also restrict the open-loop system to be stable or neutrally stable. In this paper, an attempt is made to find a nonlinear control, based on the normalized quadratic state feedback control design in [11], that can achieve global exponential stabilization for certain open-loop unstable bilinear systems. Our results show that the proposed new control will stabilize the system if a sufficient stability condition, which can be checked by off-line computer simulations in advance of the control, is satisfied.

II. NONLINEARCONTROL

Consider bilinear systems with multiplicative control inputs _x(t) = Ax(t) + u(t)Nx(t); x(0) = x0 (1) wherex(t) 2 Rnis the system state vector,u(t) is a scalar control input, andA 2 Rn2nandN 2 Rn2nare constant square matrices. For simplicity, only the single-input case is treated; the results in this paper, however, can be easily extended to the multi-input case.

Manuscript received October 9, 1997; revised August 31, 1998 and July 15, 1999. Recommended by Associate Editor, M. Di Benedetto.

The authors are with the Department of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan R.O.C. (e-mail: [email protected]).

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990 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000

Before introducing the control design, one first proposes the fol-lowing coordinate transformation:

x(t) = eA(t0kT )zk(t); t 2 [kT; kT + T ) (2) whereT > 0 is any chosen time interval length, zk(t) is the trans-formed state, and the transformation matrix is an open-loop funda-mental matrix8(t; kT ) = eA(t0kT )[12]. With the transformation (2), the governing equation of the transformed statezk(t) becomes

_zk(t) = u(t)Gk(t)zk(t)

Gk(t) = e0A(t0kT )NeA(t0kT ); t 2 [kT; kT + T ): (3) It should be noted that each transformed statezk(t) is defined only on a finite time interval[kT; (k + 1)T ):

To stabilize the system (3), a nonlinear control is proposed as fol-lows. Ife_z (t) denotes the normalized transformed state

ez (t) = z_kzk(t) k(t)k= zk(t) (zT k(t)zk(t))1=2 (4) the proposed control law is given by

u(t) = 0pk(t)eTz (t)Gk(t)ez (t); zk(t) 6= 0

0; zk(t) = 0;

t 2 [kT; kT + T ) (5)

wherepk(t) 2 R1is a positive time-varying, state-dependent gain defined by

_pk(t) = 0 p2k(t) eTz (t)Gk(t)ez (t) 2

pk(kT) = p0> 0; t 2 [kT; kT + T ) (6) Gk(t) is as in (3), and are two positive control design parameters satisfying

=2 > 0 (7)

and pk(t) is reset to be p0 at the beginning of every time interval [kT; kT + T ): Note that pk(t) is always positive and normally de-creasing. In fact, one can also choose = 0; that is, the gain p_k(t) becomes a positive constantp0for all time. In this case, however, one would need a different stability proof from the one given in this paper. Solving the differential equation (6), one can expresspk(t) as inte-gration ofez (); 2 [kT; t): pk(t) = p010 + t kT e T z ()Gk()ez () 2 d 01; t 2 [kT; kT + T ): (8)

III. STABILITYANALYSIS

It is assumed that the system (1) satisfies the following rank condi-tion [14]: an integerm exists such that

span adk_{(A; N)x}

0; k = 0; 1; 2; 1 1 1 ; m = Rn (9) for any nonzerox0inRn; where adk(A; N) is defined recursively as ad0_{(A; N) = N and ad}k+1_{(A; N) = A1ad}k_{(A; N)0ad}k_{(A; N)1A;}

k = 0; 1; 2; 1 1 1.

Lemma 1: If the system (1) satisfies the rank condition (9), and a constant vectore0exists such that

eT

0Gk(t)e0 0; 8 t 2 [kT; kT + T ) (10) whereGk(t) is defined in (3), then e0must be the null vector.

Proof: The proof is similar to that of Lemma 1 in [11].

Next, define a scalar function(1): Sn! R+ [ f0g; where Snis the unit sphere inRn; for the closed-loop system (1) and (5)

(ez (kT ))=1 (k+1)T kT e T z (t)Gk(t)ez (t) 2 dt: (11)

Given the matrixG_k(t) in (3), once the initial condition of e_z (t) at t = kT is given, the whole trajectory ez (t); t 2 [kT; kT + T ); is uniquely determined by the normalized closed-loop dynamics:

_ez (t) = I 0 ez (t)eTz (t) (A + u(t)N)ez (t) u(t) = 0 p01 0 + t kT e T z ()Gk()ez () 2 d 01 1 eT z (t)Gk(t)ez (t) (12)

according to (1), (5), and (8). Therefore,(1) in (11), which consists of the integration ofez (); 2 [kT; kT + T ); is defined as a function of the initial conditionez (kT ):

Lemma 2: A positive constant3exists such that inf

e (kT )2S (ez (kT )) =

3_{> 0} ₍₁₃₎

whereinf stands for the infimum taken over the unit sphere S_n: Proof: Note that by its definition, (1) must be nonnegative. Hence, to prove that(ez (kT )) > 0; one only needs to show that

(ez (kT )) is nonzero if ez (kT ) 2 Sn: A contradiction argument will be used to show this.

Assume that(e_z (kT )) = 0 for some e_z (kT ) 2 S_n: Then, fol-lowing (11), one has

eT

z (t)Gk(t)ez (t) 0; 8 t 2 [kT; kT + T ) (14) suggesting that the control input (5) is identically zero over the time interval[kT; kT + T ): From the transformed state (3), zero-control input implies that the transformed state will remain motionless during the entire time interval

z(t) = z(kT); 8 t 2 [kT; kT + T ) and so does the normalized state

ez (t) = ez (kT ); 8 t 2 [kT; kT + T ): (15) Substituting (15) into (14) gives

eT

z (kT )G(t)ez (kT ) 0; 8 t 2 [kT; kT + T ): Now, applying Lemma 1 to the above identity suggests thate_z (kT ) = 0, contradicting the fact that ez (kT ) 2 Sn: Therefore, (ez (kT )) must be nonzero and, hence, positive for anyez (kT ) 2 Sn:

Finally, note that(ez (kT )) in (11) depends continuously on its argumentez (kT ): Because the domain of (1); the unit sphere Sn; is compact, it follows from [13, Theorem 4.4.1] that a positive constant 3_{exists such that (13) holds.}

It will now be shown that under a sufficient stability condition, the controlled bilinear system is globally and exponentially stable.

Theorem: Consider the bilinear system (1) and the nonlinear control (5). If control design parameters; , and p0exist such that

keAT_k

(1 + p03)= < 1 (16) where3is given by (13), given any finite initial conditionx(0); the controlled system statex(t) converges to zero exponentially.

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Proof: First, from (6), one has dpk(t) p2 k(t) = 0 e T z (t)Gk(t)ez (t) 2 dt; 8 t 2 [kT; (k + 1)T ):

Integrating this equation gives p01 k (kT + T ) 0 p01k (kT ) = kT +T kT e T z (t)Gk(t)ez (t) 2 dt 3

where the inequality results from Lemma 2. Becausepk(kT) = p0; one actually has

pk(kT)

pk(kT + T ) 1 + 3_p

0: (17)

Second, define a positive scalar function

Vk(t) = p01k (t)kzk(t)k2; t 2 [kT; (k + 1)T ) (18)

wherepk(t) > 0 is from (6). Taking the time derivative of Vk(t) along (3), (5), and (6), one obtains

_Vk(t) = 0 (2 0 ) eTz (t)Gk(t)ez (t) 2 1 kzk(t)k2 = 0 (2 0 ) eT z (t)Gk(t)ez (t) 2 1 pk(t) 1 Vk(t):

Note that the choice =2 makes Vk(t) always nonincreasing. Substituting (6) into the above equation shows that

dVk(t) Vk(t) = 2 0 1 dp k(t) pk(t) and hence Vk(kT + T ) = pk(kT + T )_p k(kT) (2(= )01) 1 Vk(kT ): (19)

Using the definition ofV_k(t) in (18), one can further deduce from (19) that kzk((k + 1)T )k pk_p(kT + T ) k(kT) = kzk(kT)k _{(1 +
p}1 03)= kzk(kT)k (20) where the second inequality results from (17).

Finally, to check howkx(t)k varies, note from (2) that x(kT ) = zk(kT)

and

x((k + 1)T ) = eATzk((k + 1)T ): (21)

Taking the norm ofx((k + 1)T ) in (21), one obtains kx((k + 1)T )k keAT_{k 1 kz} k((k + 1)T )k _{(1 +
p}keATk 03)= 1 kzk(kT)k =_{(1 +
p}keATk 03)= 1 kx(kT )k (22)

where the second inequality results from (20) and the last equality from the first identity in (21). One can thus infer from the hypothesis (16) and the contraction mapping theorem [14] that given any finitex(0); the system statex(kT ) converges to zero exponentially.

Remark 1: When the bilinear system (1) is open-loop stable or neu-trally stable,T always exists such that keATk 1: In these cases, the stability condition (16) is trivially satisfied. In other words, the pro-posed control is guaranteed to stabilize an open-loop (neutrally) stable bilinear system (1).

Remark 2: When the bilinear system is open-loop unstable, one would have to check the condition (16) for global and exponential sta-bility. Note that in (16), the infimum3depends on various control design parameters:; ; p₀andT ; that is, 3= 3(; ; p₀; T ): The simulation experiences indicate that the normalized3=T is very much insensitive to the choice ofT except for very small T: Hence, different choices ofT do not significantly affect the performance of the con-trolled system.

The relationships between3and design parameters; , and p0are difficult to obtain analytically because of the complicated nonlinearity in the closed-loop dynamics. Their relationships, however, can still be found by off-line finite time computer simulations. The following pro-cedure demonstrates how to obtain the relationship between3andp₀ (or other design parameters).

1) Pick an initial design parametersp0; and fix two other design parameters and :

2) Pick an initial conditionz0(0) on the unit sphere Sn; and simu-late the closed-loop dynamics (3) and (5) withk = 0 to obtain z0(); 2 [0; T ): Calculate the normalized state ez (); 2

[0; T ); and the integration (11) to find out (ez (0)):

3) Repeat the above step for sufficiently many initial conditions z0(0)’s on the unit sphere Snso that an approximate of the in-fimum3in Lemma 2 can be obtained.

4) Increase the design parameterp0by a small amount, and then repeat Steps 2) and 3) to calculate the new3so a plot of3 versusp0can be obtained.

5) Create a plot of(1 + p03)= versusp0from the plot of3 versusp0and, finally, examine from the plot if the stability con-dition (16) is satisfied for certain ranges ofp₀:

A simulation example of the controlled bilinear system is presented below.

Example: Consider an open-loop unstable bilinear system (1) with A = 3 05₄ ₁ ; N = 1_{1 01}1

and the initial conditionxT(0) = [5; 04]: The open-loop system ma-trix has eigenvalues2 6p2i:

For the proposed control with = 0:2; p0 = 2; and T = 1 s, Fig. 1 shows the plot of(1 + p03)= versus; which indicates the stability condition (16) is satisfied as long as1:5 < < 3:6: When is chosen to be 2, the system state converges to zero in about 5 s, as is shown in Fig. 2. It should be noted that the stability condition (16) derived in the theorem is sufficient only. In practice, the range of, which results in a stable closed-loop system, is larger than as indicated by Fig. 1.

IV. OTHERCONTROLDESIGNS

It is possible to construct other control laws based on the design pre-sented above. For example, letF (1): R ! R be any piecewise contin-uous function such that

yF (y) > 0; 8 y 6= 0:

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992 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000

Fig. 1. Plot of(1 + p ) versus:

Fig. 2. Closed-loop state response.

The following control law then becomes a candidate to be considered for stabilization of the system (1):

u(t) = 0pk(t)F eTz (t)Gk(t)ez (t) ; zk(t) 6= 0 0; zk(t) = 0; t 2 [kT; kT + T ): (23) wherep_k(t) 2 R1is modified as _pk(t) = 0 p2k(t)ezT(t)Gk(t)ez (t) 1 F eT z (t)Gk(t)ez (t) ; t 2 [kT; kT + T )

in which all parameters are as in (5) and (6).

A list of commonly seen choices ofF (1) is given below.

1) F (y) = y; which corresponds to the normalized quadratic con-trol (5) in Section II;

2) F (y) = sign(y); which corresponds to the switching control; 3) F (y) = y=(+jyj), with being a small positive number, which

corresponds to the boundary-layer control [15];

4) F (y) = y=(+y2), with being a small positive number, which corresponds to the smooth division control becauseF (y) 1=y ifjyj :

The class of control in (23) will achieve the same goal as stated in the theorem in Section III: it will exponentially stabilize open-loop (neu-trally) stable bilinear system (1), and under the same condition (16) as in the theorem, stabilize open-loop unstable system (1). The only ex-ception is that3in the stability condition (16) is now different. For the control in (23),3is the infimum over the unit sphereSnof the following function(1): Sn! R+; (ez (kT ))=1 (k+1)T kT e T z (t)Gk(t)ez (t) 1 F eTz (t)Gk(t)ez (t) dt:

Because different choices of the functionF (1) produce different values of the infimum3; control laws with different F (1) may result in dif-ferent exponential convergent rates for the transformed statezk(t) in (3) (see (20)). Hence, the choice of F (1) will affect the stabilizing ability of the particular control design.

V. CONCLUSIONS

In this paper, a nonlinear control design is proposed for bilinear sys-tems with multiplicative control inputs. The bilinear system, if it is to be stabilized, must satisfy a closed-loop stability condition (16), which can be checked by off-line finite time computer simulations before the control is applied.

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