Adaptive Control for a Discrete-time First-order Nonlinear System with Both Parametric and Non-parametric Uncertainties Hongbin Ma, Kai-Yew Lum and Shuzhi Sam Ge Abstract— A simple first-order discrete-time nonlinear system, which has both parametric uncertainty and nonparametric uncertainty, is studied in this paper. The uncertainty of non-parametric part is characterized by a Lipschitz constant L, and the nonlinearity of parametric part is characterized by an exponent index b. An adaptive controller is constructed for this model in both cases of b = 1 and b > 1, and its closed-loop stability is established under some conditions. When √ b = 1, the conditions given reveal the magic number 23 + 2 which appeared in previous study on capability and limitations of the feedback mechanism. Index Terms— Feedback mechanism, Capability and limitation, Parametric uncertainty, Non-parametric uncertainty, Adaptive control

I. I NTRODUCTION Feedback, a fundamental concept in automatic control, aims to reduce the effects of the plant uncertainty on the desired control performance. Because of the essence of the feedback control, there has been much effort devoted to the control of uncertain dynamical systems in the history of control, particularly in the areas of adaptive control and robust control (e.g. [1]–[6]). In these areas, considerable progress has been made in dealing with uncertainties of dynamical systems; however, systematic and quantitative characterization of the maximum capability and limitation of the whole feedback mechanism only appeared in the last decade. The term “feedback mechanism” refers to all possible feedback control laws and hence it is not restricted in a class of special control laws. A brief survey on this challenging topic can be found in the plenary lecture [7] by Guo in International Congress of Mathematicians, 2002. The first step towards this direction was made in [8], where Guo attempted to answer the following question for a nontrivial example of discrete-time nonlinear polynomial plant model with parametric uncertainty: What is the largest nonlinearity that can be dealt with by feedback? More specifically, in [8], for the following nonlinear uncertain system yt+1 = θφt + ut + wt+1 ,

φt = O(ytb ),

b>0

(I.1) where θ is the unknown parameter, b characterizes the nonlinear growth rate of the system, and {wt } is the GausHongbin Ma and Kai-Yew Lum are with the Temasek Laboratories, National University of Singapore, Singapore 117508. Email: {tslmh,tsllumky}@nus.edu.sg Shuzhi Sam Ge is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576; the Interactive Digital Media Institute, National University of Singapore, Singapore 117576. Email: [email protected]

sian noise sequence, a critical stability result is found — system (I.1) is not a.s. globally stabilizable if and only if b ≥ 4. This result indicates that there exist limitations of the feedback mechanism in controlling the discretetime nonlinear adaptive systems, which is not seen in the corresponding continuous-time nonlinear systems (see [8], [9]). The “impossibility” result has been extended to some classes of uncertain nonlinear systems with unknown vector parameters in [10], [11] and a similar result for system (I.1) with bounded noise is obtained in [12]. Stimulated by the pioneering work in [3], a series of efforts ([13]–[16]) have been made to explore the maximum capability and limitations of feedback mechanism. Among these work, a breakthrough for non-parametric uncertain systems was made by Xie and Guo in [13], where a class of first-order discrete-time dynamical control systems yt+1 = f (yt ) + ut + wt+1 ,

f (·) ∈ F(L)

(I.2)

is studied and another interesting critical stability phenomenon is proved by using new techniques which are totally different from those in [8]. More specifically, in [13], F(L) is a class of nonlinear functions satisfying Lipschitz condition, hence the Lipschitz constant L can characterize the size of the uncertainty set F(L). √ Xie and Guo obtained the following results: if L < 32 + 2, then there exists a feedback control law such that for any f ∈ F(L), the corresponding closed√ loop control system is globally stable; and if L ≥ 32 + 2, then for any feedback control law and any y0 ∈ R1 , there always exists some f ∈ F(L) such that the corresponding closed-loop system is √unstable. So for system (I.2), the “magic” number 23 + 2 characterizes the capability and limits of the whole feedback mechanism. The impossibility part of the above results has been generalized to similar highorder discrete-time nonlinear systems with single Lipschitz constant [14] and multiple Lipschitz constants [11]. From the work mentioned above, we can see two different threads: one is focused on parametric nonlinear systems and the other one is focused on non-parametric nonlinear systems. By examining the techniques in these threads, we find that different difficulties exist in the two threads, different controllers are designed to deal with the uncertainties and completely different methods are used to explore the capability and limitations of the feedback mechanism. Motivated by these work, we want to explore the following problems: When both parametric and non-parametric uncertainties are present in the system, what is the maximum capability of feedback mechanism in dealing with these uncertainties? And how to design feedback control laws to deal with both

kinds of internal uncertainties? These problems stimulate our research in this paper, and we shall study a discrete-time firstorder nonlinear system with both two kinds of uncertainties, which sheds light on the aforementioned problems. The remainder of this paper is organized as follows. Problem formulation together with some assumptions is given first in Section II, then an adaptive controller to be studied is designed in Section III, which aims at dealing with both non-parametric and parametric uncertainties. The closed-loop stability of the proposed controller is presented and rigorously proved √ in Section IV, which reveals again the magic number 32 + 2 in case where the uncertain parametric part is of linear growth rate. And then effectiveness of the designed adaptive controller is shown by some simulation examples in Section V. Finally Section VI provides a summary with some concluding remarks. II. P ROBLEM F ORMULATION Consider the following system yt+1 = f (yt ) + θφt + ut + wt+1

(II.1)

where yt , ut and wt are the output, input and noise, respectively; f (·) ∈ F(L) is an unknown function (the set F(L) will be defined later) and θ is an unknown parameter. In this system, two kinds of uncertainties exist: (i) Internal uncertainties are embodied in both non-parametric part f (yt ) and θφt ; (ii) External uncertainties are embodied in the noise part wt+1 . Previous exploration on capability and limitation of feedback mechanism motivates the following interesting question: How to quantify the uncertainties which can be dealt with the feedback mechanism when both nonparametric and parametric uncertainties are present in the system? To make further study, the following assumptions are used throughout this paper: Assumption 2.1: The unknown function f : R → R belongs to the following uncertainty set F(L) = {f : |f (x) − f (y)| ≤ L|x − y| + c}

(II.2)

where c is an arbitrary non-negative constant. Assumption 2.2: The noise sequence {wt } is bounded, i.e. |wt | ≤ w where w is an arbitrary positive constant. Assumption 2.3: The tracking signal {yt∗ } is bounded, i.e. |yt∗ | ≤ S, ∀t ≥ 0

(II.3)

where S is a positive constant. Assumption 2.4: In the parametric part θφt , we have no any a priori information of the unknown parameter θ, but φt = g(yt ) is measurable and satisfies 2) M 0 ≤ | g(xx1b)−g(x |≤M −xb 1

2

(II.4)

for any x1 6= x2 , where M 0 ≤ M are two positive constants and b ≥ 1 is a constant.

Remark 2.1: Assumption 2.4 implies that function g(·) has linear growth rate when b = 1. Especially when g(x) = x, we can take M = M 0 = 1. Condition (II.4) need only hold for sufficiently large x1 and x2 , however we require it holds for all x1 6= x2 to simplify the proof. Remark 2.2: Assumption 2.4 excludes the case where g(·) is a bounded function, which can be handled easily by 0 previous research. In fact, in that case wt+1 = θφt + wt+1 must be bounded, hence by the result √ of [13], system (II.1) is stabilizable if and only if L < 23 + 2. III. A DAPTIVE C ONTROLLER D ESIGN In this section, we construct a unified adaptive controller for both cases of b = 1 and b > 1. For convenience, we introduce some notations which are used in later parts. Let I = [a, b] be an interval, then ∆ m(I) = 21 (a + b) denotes the center point of interval I, ∆ and r(I) = 21 |b − a| denotes the radius of interval I. And correspondingly, we let I(x, δ) = [x − δ, x + δ] denote a closed interval centered at x ∈ R with radius δ ≥ 0. A. Estimate of Parametric Part At time t, we can use the following information: y0 , y1 , · · · , yt , u0 , u1 , · · · , ut−1 and φ0 , φ1 , · · · , φt . Define ∆ (III.1) zj = yj+1 − uj and ∆

z −z

T

It =

(i,j)∈Jt

I( φjj −φii ,

L|yj −yi | |φj −φi |

+

2w+c |φj −φi | )

(III.2)

where ∆

Jt = {(i, j) ∈ N : i < j < t, φi 6= φj }

(III.3)

then, we can take θˆt = m(It ),

δt = r(It )

(III.4)

as the estimate of parameter θ at time t and corresponding estimate error bound, respectively. With θˆt and δt defined above, θ¯t = θˆt + δt and θt = θˆt − δt are the estimates of the upper and lower bounds of the unknown parameter θ, respectively. According to Eq. (III.2), obviously we can see that {θ¯t } is a non-increasing sequence and {θt } is nondecreasing. B. Estimate of Non-parametric Part Since the non-parametric part f (yt ) may be unbounded and the parametric part is also unknown, generally speaking it is not easy to estimate the non-parametric part directly. To resolve this problem, we choose to estimate ∆

gt = θφt + f (yt ) as a whole part rather than to estimate f (yt ) directly. In this way, consequently, we can obtain the estimate of f (yt ) by removing the estimate of parametric part from the estimate of gt .

Define

it = arg min |yt − yi | i
(III.5)

then, we get = gt − zit + zit = [θφt + f (yt )] − [θφit + f (yit ) + wit +1 ] + zit = [θ(φt − φit ) + zit ] + [f (yt ) − f (yit ) − wit +1 ] (III.6) Thus, intuitively, we can take gt

∆ gˆt = θˆt (φt − φit ) + zit = θˆt (φt − φit ) + (yit +1 − uit ) (III.7) as the estimate of gt at time t.

C. Design of Control ut Let

∆ ¯bt = max yi = max(¯bt−1 , yt ) i≤t

bt = min yi = min(bt−1 , yt ).

(III.8)

i≤t

Under Assumptions 2.1-2.4, we can design the following control law  ∗ −ˆ gt + yt+1 if |yt − yit | ≤ D ut = (III.9) 1 ¯ −ˆ gt + 2 (bt + bt ) if |yt − yit | > D where D is an appropriately large constant, which will be addressed in the proof later. Remark 3.1: The controller designed above is different from most traditional adaptive controllers in its special form, information utilization and computational complexity. To reduce its computational complexity, the interval It given by Eq. (III.2) can be calculated recursively based on the idea in Eq. (III.8). More implementation details are omitted here to save space. IV. S TABILITY OF C LOSED - LOOP S YSTEM In this section, we shall investigate the closed-loop stability of system (II.1) using the adaptive controller given above. We only discuss the case that the parametric part is of linear growth rate, i.e. b = 1. For the case where the parametric part is of nonlinear growth rate, i.e. b > 1, though simulations show that the constructed adaptive controller can stabilize the system under some conditions, we have not rigorously established corresponding theoretical results; further investigation is needed in the future to yield deeper understanding. A. Main Results The adaptive controller constructed in last section has the following property: √ L 3 Theorem 4.1: When b = 1, M M 0 < 2 + 2, the controller defined by (III.1)— (III.9) can guarantee that the output {yt } of the closed-loop system is bounded. More precisely, we have lim sup |yt − yt∗ | ≤ 2w + c. (IV.1) t→∞

Based on Theorem 4.1, we can classify the capability and limitations of feedback mechanism for the system (II.1) in case of b = 1 as follows:

Corollary 4.1: For the system (II.1) with both parametric and non-parametric uncertainties, the following results can be obtained in case of b = 1: √ L (i) If M < 23 + 2, then there exists a feedback M0 control law guaranteeing that the closed-loop system is stabilized. (ii) When φt = yt (i.e. g(x) = x), the presence of uncertain parametric √ part θφt does not reduce the critical value 32 + 2 of the feedback mechanism which is determined by the uncertainties of non-parametric part. Proof of Corollary 4.1: (i) This result follows from Theorem 4.1 directly. (ii) When g(x) = x, we can take M = M 0 = 1. In this case, the sufficiency can be immediately obtained via Theorem 4.1; on the other hand, the necessity can be obtained by the “impossibility” part of Theorem 1 √ in [13]. In fact, if L ≥ 32 + 2, for any given control law {ut }, we need only take the parameter θ = 0, then by [13, Theorem 2.1], there exists a function f such that system (II.1) cannot be stabilized by the given control law. 2 Remark 4.1: As we have mentioned in the introduction part, system (I.2), a special case of system (II.1), has been studied in [13]. Comparing system (II.1) and system (I.2), we can see that system (II.1) has also parametric uncertainty besides non-parametric uncertainty and noise disturbance. Hence intuitively speaking, it will be more difficult for the feedback mechanism to deal with uncertainties in system (II.1) than those in system (I.2). Noting that M 0 ≤ M , we know this fact has been partially verified by Theorem 4.1. And Corollary 4.1 (ii) indicates that in the special case of φt = yt , since the structure of parametric part is completely determined, the uncertainty in non-parametric part becomes the main difficulty in designing controller, and the parametric uncertainty has no influence on the capability of the feedback mechanism, that is to say, the feedback mechanism can still deal with the non-parametric √ uncertainty characterized by the set F(L) with L < 32 + 2. Remark 4.2: Theorem 4.1 is also consistent with classic results on adaptive control for linear systems. In fact, when L = 0, the non-parametric part f (yt ) vanishes, consequently system (II.1) becomes a linear-in-parameter system yt+1 = θφt + ut + wt+1

(IV.2)

where θ is the unknown parameter, and φt = g(yt ) can have arbitrary linear growth rate because by Theorem 4.1, we can see that no restrictions are imposed on the values of M and M 0 when L = 0. Based on the knowledge from existing adaptive control theory [3], system (IV.2) can be always stabilized by algorithms such as minimum-variance adaptive controller no matter how large the θ is. Thus the special case of Theorem 4.1 reveals again the well-known result in a new way, where the adaptive controller is defined by Eq. (III.9) together with Eqs. (III.1)—(III.8). √ L 3 Corollary 4.2: If b = 1, M 2, c = w = 0, M0 < 2 + then the adaptive controller defined by (III.1)— (III.9) can asymptotically stabilize the corresponding noise-free system,

i.e.

where lim |yt − yt∗ | = 0.

t→∞

(IV.3)

B. Proof of Theorem 4.1 To prove Theorem 4.1, we need the following Lemmas: Lemma 4.1: Assume {xn } is a bounded sequence of real numbers, then we must have lim min |xn − xi | = 0. (IV.4) n→∞ i
lim

n→∞

∆

=

# yt+1

∆

=

Obviously Dij = Dji . In the latter case, i.e. when φt 6= φit , for any (i, j) ∈ Jt , noting that zj − zi

(yj+1 − uj ) − (yi+1 − ui ) θ(φj − φi ) + [f (yj ) − f (yi )] +[wj+1 − wi+1 ]

= =

we obtain that zj − zi wj+1 − wi+1 θ− = −Di,j − . φj − φi φj − φi Therefore θ˜t + Dt,i

(IV.12)

(IV.13)

z −z z −z = (θ − φjj −φii ) − (θˆt − φjj −φii ) + Dt,it w −wi+1 = Dt,it − Di,j − j+1 + ∆i,j (t) φj −φi (IV.14)

t

zj − zi − θˆt . (IV.15) φj − φi √ L 3 Step 2: Since M M 0 < 2 + √ 2, there exists a constant L 3 2. Let  > 0 such that M M0 +  < 2 + ∆i,j (t) =

hi < ∞.

(IV.6)

i=0

Proof: See [13, Lemma 3.3]. 2 Proof of Theorem 4.1: We divide the proof into four steps. In Step 1, we deduce the basic relation between yt+1 and θ˜t , and then a key inequality describing the upper bound of |yt − yit | is established in Step 2. Consequently, in Step 3, we prove that |yt −yit | → ∞ as t → ∞ if yt is not bounded, and hence the boundedness of output sequence {yt } can be guaranteed. Finally, in the last step, the bound of tracking error can be further estimated based on the stability result obtained in Step 3. Step 1: Let θ˜t

f (yi ) − f (yj ) . φi − φj

where

where x+ = max(x, 0), ∀x ∈ R, then we must have n X

∆

Di,j =

θ − θˆt θφt + f (yt ) + wt+1 − gˆt

(IV.7)

then, by definition of ut and Eq. (III.9), obviously we get  # ∗ yt+1 + yt+1 if |yt − yit | ≤ D yt+1 = (IV.8) # 1 ¯ yt+1 + 2 (bt + bt ) if |yt − yit | > D # Now we discuss yt+1 . By Eq. (III.7) and Eq. (II.1), we get # yt+1

= θφt + f (yt ) + wt+1 − gˆt = θφt + f (yt ) + wt+1 − θˆt (φt − φit ) −(θφit + f (yit ) + wit +1 ) = (θ − θˆt )(φt − φit ) +[f (yt ) − f (yit )] + (wt+1 − wit +1 ) = θ˜t (φt − φit ) + [f (yt ) − f (yit )] + (wt+1 − wit +1 ) (IV.9) In case of φt = φit , i.e. yt = yit , obviously we get # |yt+1 | = |wt+1 − wit +1 | ≤ 2w;

(IV.10)

otherwise, we get # yt+1 = (θ˜t + Dt,it )(φt − φit ) + (wt+1 − wit +1 ) (IV.11)

∆

Bt = [bt , ¯bt ], ∆Bt = Bt − Bt−1

(IV.16)

and consequently |Bt | = ¯bt − bt , |∆Bt | = |Bt | − |Bt−1 |

(IV.17)

By the definition of bt , ¯bt and Bt , we obtain that  |Bt | if yt+1 ∈ Bt |Bt+1 | = 1 1 ¯ |B | + |y − (b + b )| if yt+1 6∈ Bt t t+1 t 2 2 t (IV.18) By the definition of it , obviously we get  = |∆Bt | if yt 6∈ Bt−1 |yt − yit | (IV.19) ≤ |∆Bi | if yt ∈ ∆Bi ⊂ Bt−1 Step 3: Based on Assumption 2.4, for any fixed  > 0, we can take constants D and D0 such that |φi − φj | > D0 > 4M (2w+c) when |yi − yj | > D. Now we are ready to show  that for any s > 0, there always exists t > s such that |yt − yit | ≤ D . In fact, suppose that it is not true, then there must exist s > 0 such that |yt −yit | > D for any t > s, correspondingly |φt − φit | > D0 . Consequently, by the definition of D, for sufficiently large t and j < t, we obtain that wj+1 − wij +1 2w 1 | |≤| 0|< ; (IV.20) φ j − φ ij D 4M together with the definition of θˆt , we know that for any s < i < j < t, |∆i,j (t)| = |

zj − zi L 2w + c − θˆt | ≤ 0 + . φj − φi M |φj − φi |

(IV.21)

hence for s < j < t, i = ij , we get |∆j,ij (t)| = |∆ij ,j (t)| ≤

2w + c L 1 L + ≤ 0+ . 0 0 M D M 4M (IV.22)

Now we consider Dt,it − Dj,ij . Let dn = Dn,in , then, by the definition of Di,j , noting that |yj − yi | ≥ |yj − yij | > D for any j > s, we obtain that |Di,j | = |

yi − yj L f (yi ) − f (yj ) |·| | ≤ 0, yi − yj φi − φj M

(IV.23)

so we can conclude that {dn , n > s} is bounded. Then, by Lemma 4.1, we conclude that lim min |dt − dj | = 0.

t→∞ s
(IV.24)

Consequently there exists s0 > s such that for any t > s0 , we can always find a corresponding j = j(t) satisfying 1 |Dt,it − Dj,ij | = |dt − dj | < . (IV.25) 4M Summarizing the above, for any t > s0 , by taking j = j(t), we get wj+1 −wi+1 + ∆ij ,j (t)| φj −φij wj+1 −wi+1 |Dt,it − Dij ,j | + | φj −φi | + |∆ij ,j (t)| j 1 1 L 1 4M  + 4M  + ( M 0 + 4M ) L 3 M 0 + 4M 

|θ˜t + Dt,it | = |Dt,it − Dij ,j − ≤ ≤ = ∆

= L . (IV.26) Therefore

we can easily obtain that {|θ˜t |} is bounded, say |θ˜t | ≤ L0 . Considering that # = θ˜t (φt − φit ) + [f (yt ) − f (yit )] + (wt+1 − wit +1 ) yt+1 (IV.34) we can conclude that # ∗ ≤ |yt+1 + yt+1 | 0 ≤ L |φt − φit | + (L|yt − yit | + c) + 2w ≤ Y (IV.35) where Y = L0 M D + LD + c + 2w + S. The proof below is similar to that in [13]. Let

|yt+1 |

t0 = inf {t : |yt | ≤ Y }, tn = inf {t : |yt | ≤ Y }. t>tn−1

t>0

(IV.36) Because of the result obtained above, we conclude that for any n ≥ 1, tn is well-defined and tn < ∞. Let vn = ytn , then obviously {vn } is bounded. Then, by applying Lemma 4.1, we get min |vn − vi | → 0 i
(IV.37)

as n → ∞. Thus for any ε > 0, there exists an integer n0 such that for any n > n0 , min |vn − vi | < ε. i
(IV.38)

So φ −φ |(θ˜t + Dt,it ) ytt −yiit (yt − yit ) + (wt+1 − wit +1 )| t |ytn − yitn | = min |ytn − yi | ≤ min |ytn − yti | < ε. (IV.39) i D, we know that |ytn +1 | ≤ L0 M ε + Lε + c + 2w + S ≤ Y (IV.40) 1 # ¯ (IV.28) for any n > n . yt+1 = yt+1 + (bt + bt ). 0 2 Thus based on definition of tn , we conclude that tn+1 = From Eq. (IV.26) together with the result in Step 2, we obtain t + 1! Therefore for any t ≥ tn0 , n that |yt | ≤ Y (IV.41) |Bt | ≤ |Bt+1 | ≤ max{|Bt |, 12 |Bt | + |yt+1 − 12 (bt + ¯bt )|} # 1 = max{|Bt |, 2 |Bt | + |yt+1 |} which means that the sequence {yt } is bounded. (IV.29) Finally, by applying Lemma 4.1 again, for sufficiently Thus noting (IV.27), we obtain the the following key inequallarge t, |yt − yit | ≤ ε, consequently ity: # ∗ |yt+1 − yt+1 | = |yt+1 | ≤ L0 M ε + Lε + c + 2w. (IV.42) |∆Bt | ≤ (L M |yt − yit | + 2w − 12 |Bt |)+ (IV.30) Because of arbitrariness of ε, Theorem 4.1 is true. 2 where V. S IMULATION S TUDY √ 3 L 3 3 L M = ( ML0 + 4M )M = M 2. M0 + 4  < 2 + In this section, two simulation examples will be given (IV.31) to illustrate the effects of the adaptive controller designed 0 Considering the arbitrariness of t > s , together with above. In both simulations, the tracking signal is taken as Lemma 4.2, we obtain that t and the noise sequence is i.i.d. randomly yt∗ = 10 sin 10 P |∆Bj | < ∞, taken from uniform distribution U (0, 1). The simulation (IV.32) j>s0 results for two examples are depicted in Figure 1 and Figure and consequently {|Bt |} must be bounded. By applying 2, respectively. In each figure, the output sequence yt and the Lemma 4.1 again, we conclude that reference sequence yt∗ are plotted in the top-left subfigure; ∆ ∗ |yt − yit | ≤ min |yt − yi | → 0 (IV.33) the tracking error sequence et = yt − yt is plotted in the i 0, upper and lower estimated bounds is plotted in the bottomthere always exists t > s such that |yt − yit | ≤ D. Then, right subfigure. # |yt+1 | = ≤

Fig. 1. Simulation example 1: (g(x) = x, b = 1, M = M 0 = 1)

Fig. 2. Simulation example 2: (g(x) = x2 , b = 2, M = M 0 = 1)

Simulation Example 1: This example is for case of b = 1, and the unknown plant is

of the capability and limitations of the feedback mechanism. We have constructed a unified adaptive controller which can be used in both cases of b = 1 and b > 1. When the parametric part is of linear growth rate (b = 1), we have proved the closed-loop stability under some assumptions √ L 3 and a simple algebraic condition M 2, which M0 < 2 + reveals essential connections with the known magic number √ L = 23 + 2 discovered in recent work [13] on the study of feedback mechanism capability.

yt+1 = f (yt ) + θg(yt ) + wt+1 , f (·) ∈ F(L) (V.1) √ with L = 2.9 < 32 + 2, g(x) = x (i.e. b = 1, M = M 0 = 1), and f (x) = 1.4x sin log(|x| + 1). (V.2) For this example, we can verify that |f 0 (x)| = 1.4| sin log(|x| + 1) ±

x cos log(|x|+1) | |x|+1

≤ 2.8 < L (V.3) consequently |f (x) − f (y)| < L|x − y|, i.e. f (·) ∈ F(L). Simulation Example 2: This example is for case of b > 1, and the unknown plant is yt+1 = f (yt ) + θg(yt ) + wt+1 ,

f (·) ∈ F(L)

2

(V.4)

0

with L = 2.9, g(x) = x (i.e. b = 2, M = M = 1), and f (x) = 2x + sin x2 .

(V.5)

For this example, we can verify that |f (x) − f (y)| < L|x − y| + 2, i.e. f (·) ∈ F(L). From the simulation results, we can see that in both examples, the adaptive controller can track the reference signal successfully. The simulation study verified our theoretical result and indicate that under some conditions, the adaptive control law constructed in this paper can deal with both parametric and non-parametric uncertainties, even in some cases when the parametric part is of nonlinear growth rate. In case of b = 1, the stabilizability criteria have been completely characterized by a simple algebraic condition; however, in case of b > 1, it is very difficult to give complete theoretical characterization. Note that usually more accurate estimate of parameter can be obtained in case of b > 1 than in case of b = 1, however, worse transient performance may be encountered. VI. C ONCLUSION In this paper we investigate a simple first-order nonlinear system with both non-parametric uncertainties and parametric uncertainties, where these two kinds of uncertainties are both taken into consideration the first time in the exploration

R EFERENCES [1] G. Zames, “Feedback and optimal sensitivity: Model reference transformations, weighted seminorms and approximate inverses,” IEEE Transactions on Automatic Control, vol. 23, pp. 301–320, 1981. [2] K. J. Astr¨om and B. Wittenmark, Adaptive Control, 2nd ed. Reading, MA: Addison-Wesley, 1995. [3] H. F. Chen and L. Guo, Identification and Stochastic Adaptive Control. Boston, MA: Birkh¨auser, 1991. [4] K. Zhou, J. C. Doyle, and K.Glover, Robust and Optimal Control. Prentice-Hall, 1996. [5] P. A. Ioannou and J. Sun, Robust Adaptive Control. Engle-wood Cliffs, NJ: Prentice-Hall, 1996. [6] M. Krsti´c, I. Kanellakopoulos, and P. V. Kokotovi´c, Nonlinear and Adaptive Control Design. New York: John Wiley & Sons, 1995. [7] L. Guo, “Exploring the capability and limits of the feedback mechanism,” in Proceedings of ICM2002, Beijing, 2002, (invited lecture). [8] ——, “On critical stability of discrete-time adaptive nonlinear control,” IEEE Transactions on Automatic Control, vol. 42, no. 11, pp. 1488– 1499, 1997. [9] I. Kanellakopoulos, “A discrete-time adaptive nonlinear system,” IEEE Transactions on Automatic Control, vol. 39, Nov. 1994. [10] L. L. Xie and L. Guo, “Fundamental limitations of discrete-time adaptive nonlinear control,” IEEE Transactions on Automatic Control, vol. 44, no. 9, pp. 1777–1782, 1999. [11] H. B. Ma, “Further results on limitations to the capability of feedback,” International Journal of Control, 2007, (in press). http://www.tandf.co.uk/journals DOI:10.1080/00207170701218333. [12] C. Y. Li and L. L. Xie, “On robust stability of discrete-time adaptive nonlinear control,” Systems and Control Letters, vol. 55, no. 6, pp. 452–458, June 2006. [13] L. L. Xie and L. Guo, “How much uncertainty can be dealt with by feedback?” IEEE Transactions on Automatic Control, vol. 45, no. 12, pp. 2203–2217, 2000. [14] Y. X. Zhang and L. Guo, “A limit to the capability of feedback,” IEEE Transactions on Automatic Control, vol. 47, no. 4, pp. 687–692, 2002. [15] F. Xue and L. Guo, “Necessary and sufficient conditions for adaptive stabilizability of jump linear systems,” Communications in Information and Systems, vol. 1, no. 2, pp. 205–224, Apr. 2001. [16] H. B. Ma and L. Guo, “An “impossibility” theorem on second-order discrete-time nonlinear control systems,” in Proceedings of the 24th Chinese Control Conference. South China University of Technology Press: Guangzhou, July. 2005, pp. 57–61.