Nonlinear controls for a class of discrete-time bilinear systems

Nonlinear controls for a class of discrete-time bilinear systems

Min-Shin Chen

_{, Yean-Ren Hwang}

_{and Kun-Chi Huang}

1

Department of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C

2_{Department of Mechanical Engineering, National Central University, Chung-Li 310, Taiwan, R.O.C}

SUMMARY

For a discrete-time neutrally stable bilinear system, a nonlinear state feedback control based on the passivity design has been proposed to stabilize the system globally and asymptotically. This paper shows that the decay rate resulting from the passivity control is not exponential, and the system’s response speed becomes very sluggish asymptotically. A ‘normalized’ nonlinear control is therefore proposed to achieve exponential stability. The new exponentially stabilizing control not only improves the system’s response speed, but also enhances the system’s robustness against small parametric perturbations. Copyright # 2003 John Wiley & Sons, Ltd.

KEY WORDS: bilinear system; discrete-time system; exponential stability; asymptotical stability; robustness

1. INTRODUCTION

Bilinear systems comprise an important subclass of nonlinear dynamical systems because numerous real-world dynamical plants have bilinear structures References [1, 2]. Real-world examples include engineering applications in nuclear, thermal, chemical processes, and non-engineering applications in biology, social-economics, immunology and so on. Therefore, it is important to devise efficient control laws for such bilinear systems.

An early treatment of discrete time bilinear system control is a state feedback control design Reference [3] based on hyperstability, which ensures local asymptotical stability. In References [4,5], an output feedback control is proposed for general time-varying bilinear systems, which again ensures only local asymptotic stability. Later, based on the concept of passivity, a state feedback control Reference [6] is proposed for an open-loop stable bilinear system. The design achieves global asymptotic stabilization under a zero-state detectability assumption.

Received 7 September 2001 Revised 29 March 2002 Published online 3 April 2003

y

E-mail: [email protected]

n

Correspondence to: M Chen National Taiwan University, Taipei 106, Taiwan, R.O.C

}_{E-mail: [email protected]}

Contract/grant sponsor: National Science Council of the Republic of China; Contract/grant number: NSC89-2213-E-002-087

The objectives of this paper are as follows. First, using the Lyapunov function analysis, one can show that when the bilinear system is homogeneous (corresponding to the case D ¼ 0 in Reference [6]), the closed-loop decay rate for the passivity control in Reference [6] is not exponential. The norm of the state jjxðkÞjj decays as 1=pffiffiffik; as a result, the decay rate becomes sluggish as time goes on. Second, a ‘normalized’ nonlinear control is therefore proposed in this paper to achieve exponential stability of the closed-loop system. Such a new control not only substantially improves the system response speed, but also enhances robustness of the controlled system. In this paper, the analysis and control design are introduced in detail for single-input bilinear systems, and then roughly sketched for multi-input bilinear systems in the final section.

2. REFORMULATION OF ASYMPTOTICALLY STABILIZING CONTROL Considers a discrete-time homogeneous bilinear system

xðk þ1Þ ¼ AxðkÞ þ uðkÞNxðkÞ xð0Þ ¼ x0 ð1Þ

where xðtÞ 2 Rn is the system state vector, uðkÞ is a scalar control input, and A 2 Rn
n _and N 2 Rn
n are constant square matrices. The bilinear system (1) is assumed to satisfy the following two assumptions:

Assumption A1

There exists a positive definite matrix P > 0 such that

ATPA  P ¼0 ð2Þ

Assumption A1 restricts that the open-loop system be neutrally stable, and all the eigenvalues of

A fall on the unit circle. Assumption A2

There exists an integer m > 0 such that

rank½A1Nx; A2NAx; . . . ; AmNAm1x ¼ n ð3Þ for any non-zero x in Rn_{: Assumption A2 [7] guarantees that system (1) is controllable [8] at any}

non-zero x: Notice that under the Assumption A1, the zero-state detectability in Reference [6] is equivalent to the controllability Assumption A2 here.

One now presents the asymptotically stabilizing control introduced in Reference [6]; however, in this paper the control will be constructed based on the Lyapunov function theory instead of on the passivity theory as in Reference [6]. The advantage of using the Lyapunov analysis is to make decay rate analysis possible, as will be shown in Section 3. The asymptotically stabilizing control is as follows:

uðkÞ ¼ r ax 2bxþ e

; 05r42; e > 0 ð4Þ

where r is the control gain, normally chosen to be one, e is a small positive number, and axand bxare abbreviations of axðkÞ and bxðkÞ:

axðkÞ¼ 2xT_ðkÞAT_PNxðkÞ

in which P is the positive definite matrix in (2). Note that bxmay be either positive or zero. The

introduction of the small parameter eð> 0Þ in the control law (4) is to avoid possible division by zero in uðkÞ:

In this section, it will be shown that the bilinear system (1) with the control law (4) is asymptotically stable.

Lemma 1

Consider the bilinear system (1) with the control law (4). If the bilinear system satisfies Assumptions A1 and A2, and

axðkþiÞ¼ 2xðk þ iÞTATPNxðk þ iÞ ¼0; 8 i ¼0; 1; 2;. . . ; m  1 ð5Þ then xðkÞ must be the null vector.

Proof

Hypothesis (5) can be put into the following matrix form, using the system dynamics

xðk þ iÞ ¼ Ai_{xðkÞunder hypothesis (5),}

xTðkÞ Ah T_{PNxðkÞ; A} T 2_{PNAxðkÞ; . . . ; A} T m_PNAm1_xðkÞi_{¼ 0 ð6Þ} Since AT_{P ¼ PA}1 _{from Assumption A1, (6) becomes}

xTðkÞP A 1NxðkÞ; A2NAxðkÞ; . . . ; AmNAm1xðkÞ¼ 0 ð7Þ If xðkÞ is non-zero, according to the controllability assumption A2, the matrix in (7) is full rank; hence, xTðkÞP ¼ 0: Since P is positive, one concludes that xðkÞ ¼ 0: One then reaches a contradiction with the assumption that xðkÞ is non-zero, therefore, xðkÞ must be zero.

Theorem 1

The bilinear system (1) with the control law (4) is globally asymptotically stable. Proof

Choose a Lyapunov function candidate

V ðkÞ ¼ xTðkÞPxðkÞ50

where P is the positive definite matrix in (2). One can derive, along the trajectory of (1) and (4), DV ðkÞ ¼ V ðk þ 1Þ  V ðkÞ

¼ 2xTðkÞATPNxðkÞuðkÞ þ ½NxðkÞTP ½NxðkÞu2ðkÞ ¼  a2

x

r½ð2  rÞbxþ e

ð2bxþ eÞ2

40 ð8Þ

Since bx50 and 25r > 0; DV ðkÞ is always non-positive. From the Lyapunov stability theory [8],

the system state xðkÞ will remain uniformly bounded; that is,

xðkÞ 2 D ð9Þ

for some bounded region D in the state space.

To show that the state xðkÞ actually approaches zero asymptotically, notice from (8) that DV ðkÞ ¼ 0 implies axðkÞ¼ 0: Since the largest invariant set contained in fxjaxðkÞ 0g is f0g

according to Lemma 1, one concludes from the LaSalle theorem [8] that xðkÞ approaches zero asymptotically, and hence the controlled system is asymptotically stable. &

Remark

The passivity control (D ¼ 0) in Reference [6, Theorem 4] corresponds to control (4) in this paper with r ¼ 2 and e ¼ 4: However, these parameter values are not optimal as far as the decay rate of V ðkÞ is concerned. The reason is as follows. If one sets @DV ðkÞ=@uðkÞ ¼ 0 in the proof of Theorem 1, one obtains

un

ðkÞ ¼ ax 2bx

; bx50

which will maximize the decrease rate of V ðkÞ: Comparing the above un

ðkÞ with control (4) immediately suggests that for fast responses, one should choose r ¼ 1 and e sufficiently small in the control law (4). Simulation results do confirm such a choice. Note that e is added in the control (4) to avoid the possible division by zero in un

ðkÞ:

3. DECAY RATE ANALYSIS OF ASYMPTOTICALLY STABILIZING CONTROL The purpose of this section is to provide a decay rate analysis for the asymptotically stabilizing control (4), which is not available when the control was first proposed in Reference [6]. Define the notations:

ae¼4 ax

jjxjj2¼ 2eðkÞ

T_AT_{PNeðkÞ where eðkÞ ¼}4 xðkÞ

jjxðkÞjj ð10Þ

and eðkÞ is called the normalized state. Definition

Define a scalar real function B1ðeðkÞ; jjxðkÞjjÞ : S
I ! Rþ[ f0g for the controlled system (1) and (4) B₁ðeðkÞ; jjxðkÞjjÞ ¼ Pkþm1 i¼k aeðiÞ  2 ¼Pkþm_i¼k 1 2x T_ðiÞ jjxðiÞjjA T_PN xðiÞ jjxðkÞjj  2 ; xðkÞ=0 limjjxðkÞjj!0B1ðeðkÞ; jjðkÞjjÞ; xðkÞ ¼0 8 > < > : ð11Þ

where S is the unit sphere in Rn _{and I ¼ ½0; r; in which r ¼}4_max

ifjjxðiÞjj jxðiÞ 2 D inð9Þ; i ¼0; 1;. . . ; 1g:

From the knowledge of xðkÞ; one can determine consecutively xðk þ 1Þ; xðk þ 2Þ; . . . ; x
ðk þ m  1Þ; from the closed-loop dynamics (1) and (4):

xðk þ1Þ ¼ A  r 2x

T_ðkÞAT_PNxðkÞ ½NxðkÞTP ½NxðkÞ þeN

 

xðkÞ ð12Þ

The value of the function B1ð; Þ in (11) can then be exactly calculated; hence, the function B1ð; Þ is uniquely defined by eðkÞ and jjxðkÞjj: Note that B1ðeðkÞ; jjxðkÞjjÞ at the origin xðkÞ ¼ 0 is defined by a limiting process, for otherwise division by zero will take place at xðkÞ ¼ 0: The next lemma shows that the limit in (11) does exist.

Lemma 2

For the closed-loop system (1) and (4), there exists a real scalar function B0ðeðkÞÞ : S ! Rþ[ f0g such that

B₀ðeðkÞÞ ¼ limjjxðkÞjj!0B1ðeðkÞ; jjxðkÞjjÞ

Proof

When xðkÞ approaches the origin, the closed-loop dynamics (12) can be described by its linearized model:

xðk þ1Þ ¼ AxðkÞ as jjxðkÞjj ! 0

As a result, there exists a recursive relationship for the normalized state eðkÞ:

eðk þ1Þ ¼ AeðkÞ

jjAeðkÞjj as jjxðkÞjj ! 0: ð13Þ One can determine the value of B1ðeðkÞ; jjxðkÞjjÞ; a functions of eðiÞ; i ¼ k; k þ 1; . . . ; k þ m  1; using only the information of the initial normalized state eðkÞ and the recursive relationship (13). In other words, when xðkÞ is close to the origin, given any eðkÞ; the value of B1ðeðkÞ; jjxðkÞjjÞ is independent of jjxðkÞjj; and hence the limiting function in (11) is well defined. &

The following lemma shows that the controllability assumption (3) guarantees that the value of B1ðeðkÞ; jjxðkÞjjÞ is bounded above from zero on its domain S
I:

Lemma 3

For the closed-loop system (1) and (4), there exists some positive constant b such that infeðkÞ2S

jjxðkÞjj2I

fB1ðeðkÞ; jjxðkÞjjÞg ¼ b > 0 where inf denotes the infimum taken over all eðkÞ 2 S and all jjxðkÞjj 2 I: Proof

One first shows that B1ðeðkÞ; jjxðkÞjjÞ is positive for all eðkÞ 2 S and jjxðkÞjj 2 I \f0g (at every point in its domain except at xðkÞ ¼ 0), then shows that B1ðeðkÞ; jjxðkÞjjÞ is actually also positive at

xðkÞ ¼0:

First, consider the case jjxðkÞjj=0: By definition, B1ð; Þ must be non-negative. Hence, to prove that B1ð; Þ is positive, one only needs to show that B1ð; Þ is non-zero. Assume the contrary:

B1ðeðkÞ; jjxðkÞjjÞ is zero at some eðkÞ and some non-zero xðkÞ; Then,

xTðiÞAT_{PNxðiÞ 0; 8 i ¼ k; k þ 1; . . . ; k þ m  1}

Applying Lemma 1 to the above equation suggests that xðkÞ ¼ 0; contradicting the hypothesis of this case that xðkÞ=0: Therefore, B1ð; Þ must be positive at every point where xðkÞ=0:

Next, consider the case jjxðkÞjj ¼ 0: To prove that B1ðeðkÞ; jjxðkÞjjÞ is positive at jjxðkÞjj ¼ 0; recall from the proof of Lemma 2 that, given any eðkÞ; the value of B1ðeðkÞ; jjxðkÞjjÞ becomes a constant independent of jjxðkÞjj as it approaches zero. Since it is already proved in the first case of this lemma that B1ðeðkÞ; jjxðkÞjjÞ is positive no matter how small jjxðkÞjj is, this constant must be positive. Hence, from the limiting definition of B1ðeðkÞ; jjxðkÞjjÞ at jjxðkÞjj ¼ 0; the value of B1

eðkÞ; jjxðkÞjj

Finally, since B1ð; Þ is positive at every point in its domain from the above analysis, and the domain of B1ð; Þ; S
I; is compact, it follows from Theorem 4.4.1 in Reference [9] that the infimum of B1ð; Þ (¼ b) is also positive. &

One can now present a decay rate analysis for the Lyapunov function V ðkÞ: Theorem 2

Consider the Lyapunov function V ðkÞ ¼ xT_{ðkÞPxðkÞ for the closed-loop system (1) and (4). For} the integer m as in (3) and for some real positive constant p; the decay rate of V ðkÞ is given by

V ðmkÞ41 pk as k ! 1 Proof Rewrite (8) as V ðk þ1Þ  V ðkÞ ¼  ax jjxjj2  2 r½ð2  rÞbxþ e ð2bxþ eÞ2 jjxjj4 ðxT_PxÞ2V 2_ðkÞ 4  a2e r½ð2  rÞbxþ e ð2bxþ eÞ2 1 l2P V2ðxÞ ð14Þ

where lP is the maximum eigenvalue of P : Since xðkÞ is uniformly bounded from Theorem 1, one

can show that there exists a positive constant c1 such that for all xðkÞ 2 D in (9), r½ð2  rÞbxþ e

ð2bxþ eÞ2

5c1> 0

Equation (14) then becomes

V ðk þ1Þ  V ðkÞ4 c1 l2P a2_eðkÞV2ðkÞ ð15Þ Derive, using (15), 1 V ðk þ1Þ 1 V ðkÞ¼ V ðkÞ  V ðk þ1Þ V ðkÞV ðk þ1Þ 5c1 l2P a2_eðkÞ V ðkÞ V ðk þ1Þ 5c1 l2_Pa 2 eðkÞ

where the last inequality results from V ðk þ 1Þ4V ðkÞ: Summing the above equations from k to

k þ m 1 yields 1 V ðk þ mÞ 1 V ðkÞ5 c1 l2_P X kþm1 i¼k a2_eðiÞ5c1 l2_Pb; ð16Þ

where the second inequality results from Lemma 3. By setting k ¼ 0; m; 2m;. . . ; ð‘  1Þm in (16), and summing all the equations, one has

1 V ð‘mÞ 1 V ð0Þ5c1 b l2_P‘

therefore, V ð‘mÞ4V ð0Þ=ðc1ðb=l2PÞ‘V ð0Þ þ 1Þ; which, when ‘ is replaced by k and c1ðb=l2PÞ by p;

proves the claim of the theorem. & Remark

Theorem 2 shows that the passivity control (4) in Reference [6] results in a power-law decay rate. With the power-law decay rate, the system response slows down substantially as k approaches infinity, making control (4) unacceptable in real-world applications. This motivates a new control design that can produce a faster decay in the next section.

4. EXPONENTIALLY STABILIZING CONTROL

As a modification of the asymptotically stabilizing control (4), the following exponentially stabilizing control is proposed for system (1):

uðkÞ ¼ r ae 2beþ e

; 05r42; e > 0 ð17Þ

where r is the control gain, normally one, e is a small positive number to avoid division by zero, and

aeðkÞ¼ 2eT_ðkÞAT_PNeðkÞ

beðkÞ¼ ½NeðkÞT_{P ½NeðkÞ}

in which eðkÞ is the normalized state defined in (10). The above exponentially stabilizing control law (17) has exactly the same structure as the asymptotically stabilizing control law (4) studied in the previous section except that the system state xðkÞ in (4) is now replaced by the ‘‘normalized’’ state eðkÞ in the new control.

The following lemmas are required for the stability analysis of the proposed new control. Lemma 4

Consider the bilinear system (1) and (17). If

aeðkþiÞ¼ 2eðk þ iÞTATPNeðk þ iÞ ¼0; 8 i ¼0; 1; 2;. . . ; m  1 ð18Þ

where eðkÞ is defined in (10), then xðkÞ must be the null vector. Proof

Assume that xðkÞ is non-zero. Multiplying (18) by jjxðkÞjj2 results in (5). The rest of proof then follows that of Lemma 1.

Next, define a scalar function:

B₂ðeðkÞÞ ¼ X kþm1 i¼k aeðiÞ  2 ¼ X kþm1 i¼k 2eTðiÞATPNeðiÞ  2 ð19Þ

for the new closed-loop system (1) and (17): xðk þ1Þ ¼ A  r 2e T_ðkÞAT_PNeðkÞ ½NeðkÞT_{P ½NeðkÞ þe}N   xðkÞ ¼4f ðxðkÞÞ ð20Þ

Note that the right-hand side of the function B2ðÞ in (19) is almost the same as that of B1ð; Þ in (11), but the two functions are defined differently. First, B2ðÞ is not defined at xðkÞ ¼ 0; while

B1ð; Þ is. Second, B2ðÞ is a function of eðkÞ only, while B1ð; Þ a function of both jjxðkÞjj and eðkÞ: The reason why B2ðÞ in (19) is defined a function of eðkÞ only is as follows. Notice that f ðxðkÞÞ in (20) satisfies the so-called homogeneous property for all jjxðkÞjj (not just for jjxðkÞjj ! 0 as shown in Lemma 2 for B1ð; Þ); that is,

f ðrxðkÞÞ ¼ rf ðxðkÞÞ for all r > 0 ð21Þ

Based on (20) and (21), it can be deduced that there exists a continuous function F ðÞ: S ! S; where S is the unit sphere in Rn_{; such that}

eðk þ1Þ ¼ F ðeðkÞÞ; F ðeðkÞÞ ¼4 f ðeðkÞÞ

jjf ðeðkÞÞjj: ð22Þ Using relation (22), one can calculate B2ðÞ based on the knowledge of eðkÞ only. Hence, it is not a function of state norm.

By following the same procedure as in the proof of Lemma 3, and using Lemma 4, one can show that the non-negative function B2ðÞ is actually bounded above from zero for all eðkÞ 2 S; i.e., for all non-zero xðkÞ 2 Rn_:

Lemma 5

For the closed-loop system (1) and (17), there exists a constant a > 0 such that infeðkÞ2S½B2ðeðkÞÞ ¼ a > 0

One can now establish the exponential stability for the proposed new control law. Theorem 3

If the bilinear system (1) satisfies the Assumptions A1 and A2, the nonlinear control law (17) stabilizes the system exponentially.

Proof

Define a Lyapunov function candidate V ðkÞ ¼ xTPxðkÞ; where P is as in (2). The increment of

V ðkÞalong (1) and (17) is given by

V ðk þ1Þ  V ðkÞ

¼ xT_{ðk þ 1ÞPxðk þ 1Þ  x}T_ðkÞPxðkÞ

¼ 2xTðkÞATPNxðkÞuðkÞ þ ðNxðkÞÞTP ðNxðkÞÞuðkÞ2

¼  a2 eðkÞ r½ð2  rÞbeðkÞþ e ð2beðkÞþ eÞ2 jjxðkÞjj2 xT_ðkÞPxðkÞV ðkÞ

Since beðkÞis uniformly bounded, one has

r½ð2  rÞbeðkÞþ e

ð2beðkÞþ eÞ2

5c > 0

for some positive constant c: The last inequality in DV ðkÞ can then be deduced to

V ðk þ1Þ  V ðkÞ4  a2_eðkÞc lP

V ðkÞ40 ð23Þ

where lP is the maximum eigenvalue of the positive definite matrix P : Since V ðkÞ is

non-increasing, one has

V ðk þ iÞ5V ðk þ mÞ; i ¼0; 1;. . . ; m  1 ð24Þ

List the equations in (23) from k to k þ m  1; and utilize (24),

V ðk þ1Þ  V ðkÞ 4  a2eðkÞ c lP V ðkÞ4  a2eðkÞ c lP V ðk þ mÞ V ðk þ2Þ  V ðk þ 1Þ 4  a2_eðkþ1Þc lP V ðk þ1Þ4  a2_eðkþ1Þc lP V ðk þ mÞ
ðk þ mÞ  V ðk þ m  1Þ4  a2_eðkþm1Þc lP V ðk þ m 1Þ4  a2_eðkþm1Þc lP V ðk þ mÞ

Adding the above inequalities together, and quoting Lemma 5, one obtains

V ðk þ mÞ  V ðkÞ4  Pkþmi¼k 1 a2eðiÞ h _{i c} lP V ðk þ mÞ4 ac lP V ðk þ mÞ

Rearranging the last inequality gives

V ðk þ mÞ4 1 1 þ ac=lP V ðkÞ and hence, V ðkmÞ4skV ð0Þ where s ¼ 1 1 þ ac=lP 51 ð25Þ

One concludes from (25) that the Lyapunov function V ðkÞ converges exponentially to zero, and so does xðkÞ by the definition of V ðkÞ: &

Remark

Theorem 3 proves that the proposed control (17) produces an exponential deay rate. The exponential decay rate in Theorem 3 is superior to the power-law decay rate in Theorem 2 in two ways. (1) The exponential decay rate is much faster than the power-law decay rate as the time gets large. A clear demonstration of this point is shown in Figure 1 below. (2) When the closed-loop system is exponentially stable, it is robust against small unmodelled nonlinearities and small parametric perturbations. For a proof of this, please refer to Reference [10, Theorem 121].

5. MULTI-INPUT BILINEAR SYSTEMS

In the previous section, the exponentially stabilizing control is introduced for the single-input case. In this section, the multi-input case will be discussed. In particular, a two-input case will be

examined. Consider the following two-input bilinear system:

xðk þ1Þ ¼ AxðkÞ þ u1ðkÞN1xðkÞ þ u2ðkÞN2xðkÞ xð0Þ ¼ x0 ð26Þ where xðkÞ 2 Rn _{is the system state vector, u}

1ðkÞ and u2ðkÞ are two scalar control inputs, and A;

N1; N22 Rn
nare constant square matrices. The multi-input bilinear system (26) is assumed to satisfy the same neutral stability assumption A1 in (2) and the multi-input controllability assumption below.

Assumption A20

There exists an integer m > 0 such that

rank½A1N1x; A2N1Ax; . . . ; AmN1Am1x

A1N2x; A2N2Ax; . . . ; AmN2Am1x ¼ n for any non-zero x in Rn_:

An exponentially stabilizing control for the above system is as follows.

U ðkÞ ¼ u1ðkÞ u₂ðkÞ " # ¼ r NT eðkÞPNeðkÞ þ eI  1 N_eTðkÞPAeðkÞ; r 2 ð0; 2 ð27Þ

where r 2 ð0; 2 is a positive control gain, P is the positive definite matrix in (2), Ne¼ ½N1e
ðkÞ; N2eðkÞ; and eðkÞ the normalized state in (10).

The stability analysis for the multi-input control law follows exactly the single-input case, and is omitted here. Instead, a simulation example is used to verify the effectiveness of the proposed control design.

Example

Consider a three-dimensional bilinear system (26) with

A ¼ 0:7500 0:4330 0:5000 0:5000 0:8660 0 0:4330 0:2500 0:8660 2 6 6 4 3 7 7 5; N1¼ 0 0 0 0 0 0 0 0 1 2 6 6 4 3 7 7 5; N2¼ 0 0 0 0 0 1 0 0 0 2 6 6 4 3 7 7 5

and the initial condition is xTð0Þ ¼ ½0:5; 0:5; 1: For comparison, the system is simulated with two different control schemes. In the first simulation, the passivity control in Reference [6] is applied to the system, and the 2-norm of the system state versus time is shown by the dash line in Figure 1. In the second simulation, control (27) in this paper is applied to the system with design parameters P ¼ I; r ¼ 1 and e ¼ 0:1; and the result shown by the solid line in Figure 1. The figure clearly indicates that the decay rate from the new control (27) is exponential, and is much faster than the dash line resulting from the passivity control [6]. The system state under the new control virtually reaches the origin at around 12 s.

6. CONCLUSIONS

Conventional controls for discrete-time homogeneous bilinear systems with (neutrally) stable dynamics can stabilize the system asymptotically, but not exponentially. In this paper, a new nonlinear control is constructed to stabilize the bilinear system exponentially. Achieving exponential stability has two important implications: (1) the system decay rate is much better improved than the previous non-exponential decay rate; (2) the controlled system becomes robust with respect to small parametric perturbations, as is shown in Reference [10].

However, the achieved exponential decay rate still has a limited exponent no matter how the control design parameters are chosen. A more challenging problem then is to construct a control that can produce any specified exponential decay rate, and this will be a subject to be pursued in the future.

ACKNOWLEDGEMENTS

The work in this paper is supported in part by the National Science Council of the Republic of China under the grant number NSC89-2213-E-002-087.

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