Universal control of nonlinear systems with unknown nonlinearity and growth rate by adaptive output feedback

Automatica 47 (2011) 2211–2217

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Brief paper

Universal control of nonlinear systems with unknown nonlinearity and growth rate by adaptive output feedback✩ Min-Sung Koo a , Ho-Lim Choi b,∗ , Jong-Tae Lim a a

Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea

b

Department of Electrical Engineering, Dong-A University, 840 Hadan2-Dong, Saha-gu, Busan, 604-714, Republic of Korea

article

abstract

info

Article history: Received 2 June 2010 Received in revised form 3 January 2011 Accepted 23 March 2011 Available online 25 August 2011

Over the past several years, triangular or feedforward nonlinear systems have received considerable attention and there are many independent results for stabilization or regulation problems. These results are often specifically targeted for only one of these systems and they are applicable when the system nonlinearity is known a priori. In this paper, we propose an adaptive output feedback controller coupled with switching logic to universally regulate both triangular and feedforward systems without recognizing the system nonlinearity. Moreover, with our control scheme, the linear growth rate is not required to be known. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Output feedback Unknown nonlinearity Adaptive switching

1. Introduction We consider the global adaptive output feedback control for a family of single-input single-output uncertain nonlinear systems x˙ = Ax + Bu + δ(t , x, u),

y = Cx

(1)

where x ∈ Rn , u ∈ R, and y ∈ R are the system state, input, and output, respectively. The system matrices (A, B) is the Brunovsky canonical pair and C = [1, 0, . . . , 0]. The system nonlinearity is δ(t , x, u) = [δ1 (t , x, u), . . . , δn (t , x, u)]T where δi (t , x, u): R × Rn × R → R, i = 1, . . . , n are C 1 with respect to all the variables. We first address two linear growth conditions for δ(t , x, u). (B1) (Triangular condition) For i = 1, . . . , n, there exists an unknown constant γ ≥ 0 such that

|δi (t , x, u)| ≤ γ (|x1 | + · · · + |xi |). (2) (B2) (Feedforward condition) For i = 1, . . . , n − 2, there exists an unknown constant γ ≥ 0 such that |δi (t , x, u)| ≤ γ (|xi+2 | + · · · + |xn | + |u|)

(3)

with |δn−1 (t , x, u)| ≤ γ |u| and δn (t , x, u) = 0. Here, we address some research flows related with (B1) and (B2) in the following. First, regarding (B1), when γ is known,

(B1) reduces to the condition assumed in Qian and Lin (2002). Then, under (B1), some output feedback controllers are proposed independently in Choi and Lim (2005b) and Lei and Lin (2006), respectively. Second, regarding (B2), when γ is known, the control method of Qian and Lin (1998) is available while the authors use the full state information. Then, under (B2), the control method of Ye and Unbehauen (2004) is applicable, but their controllers still use the full state feedback. The common feature of the results of Choi and Lim (2005b), Lei and Lin (2006), Praly (2003), Qian and Lin (1998, 2002) and Ye and Unbehauen (2004) is that they are specifically targeted for either triangular or feedforward systems. Then, in Choi and Lim (2004, 2005a), the authors propose unified state and output feedback controllers where the system is stabilized under both (B1) and (B2) when γ is known. Most recently, in Choi and Lim (2010b), the authors added adaptive feature such that γ is no longer required to be known. However, the results of Choi and Lim Choi and Lim (2004, 2005a, 2010b) still commonly require a priori knowledge on the system nonlinearity before their proposed controllers are designed. We now further extend the control problem by addressing the following assumption on the system nonlinearity δ(t , x, u): Assumption 1. Nonlinearity δ(t , x, u) as a whole belongs to either (B1) or (B2), but its structure is not known a priori.

✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Zhihua Qu under the direction of Editor Andrew R. Teel. ∗ Corresponding author. Tel.: +82 051 200 7734; fax: +82 051 200 7743. E-mail address: [email protected] (H.-L. Choi).

0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.07.002

Our control problem is to globally regulate the system (1) by output feedback under Assumption 1. Naturally, any of aforementioned existing results are not applicable because the system cannot be checked whether it satisfies their proposed conditions. The

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M.-S. Koo et al. / Automatica 47 (2011) 2211–2217

main difficulty with this control problem is that the so-called control directions are opposite for (B1) and (B2). More specifically speaking, under (B1), high-gain feedback control approaches are usually needed as observed in Choi and Lim (2004, 2005a,b), Lei and Lin (2006), Praly (2003) and Qian and Lin (2002). On the other hand, under (B2), some low-gain controllers are often suggested (Choi & Lim, 2005a; Krishnamurthy & Khorrami, 2003; Qian & Lin, 1998). Thus, we need to develop an adaptive controller which provides a bi-directional gain-tuning approach to tackle both unknown γ and nonlinear structure, i.e, a universal feature, under a limited resource (output feedback). Notably, there are some switching schemes in Jiang (1999), Liu and Ge (2005) and Ye (2005), which inspired us in solving our control problem along with the dynamic gain approach in Choi and Lim (2010a,b) and Lei and Lin (2006). Obviously, the methods of Jiang (1999), Liu and Ge (2005) and Ye (2005) are not applicable to our control problem because their system nonlinearities are expressed as functions of output. The main contribution of the paper is that we suggest a universal output feedback controller using adaptive switching scheme for global regulation of the output feedback system (1) in the presence of the unknown nonlinear structure and linear growth rate. 2. Main result The output feedback controller is given by u = K (ϵ(t ))z

(4)

z˙ = Az + Bu − L(ϵ(t ))(y − Cz )

(5)

where K (ϵ(t )) = [k1 /ϵ(t ) , . . . , kn /ϵ(t )] and L(ϵ(t )) = [l1 /ϵ(t ), . . . , ln /ϵ(t )n ]T , ϵ(t ) > 0. The design of dynamic gain ϵ(t ) will be introduced later on with a switching logic. For mathematical convenience, we define the following. Define Eϵ(t ) := diag[1, . . . , ϵ(t )n−1 ], K := K (ϵ(t ))|ϵ(t )=1 , L := L(ϵ(t ))|ϵ(t )=1 , AK := A + BK , AL = A + LC , AK (ϵ(t )) := A + BK (ϵ(t )), AL (ϵ(t )) := A + L(ϵ(t ))C . ‖ · ‖ denotes the Euclidean norm and ‖x‖1 denotes one norm as ‖x‖1 = |x1 | + · · · + |xn |. Define e = [e1 , . . . , en ]T where ei = xi − zi for i = 1, . . . , n. From (1) and (5), the error dynamics is n

e˙ = AL (ϵ(t ))e + δ(t , x, u)

(6)

and from (1), (4) and (5), we have z˙ = AK (ϵ(t ))z − L(ϵ(t ))Ce.

(7)

Lemma 1. Let Vo (e) = eT PL (ϵ(t ))e for the error dynamics (6) and Vc (z ) = z T PK (ϵ(t ))z for the system (7) where Pj (ϵ(t )) = Eϵ(t ) Pj Eϵ(t ) 2 such as ATj (ϵ(t ))Pj (ϵ(t )) + Pj (ϵ(t ))T Aj (ϵ(t )) = −ϵ(t )−1 Eϵ( t ) and ATj Pj + Pj Aj = −I for j = K , L where I is an n × n identity matrix. Then, we obtain V˙ o (e) ≤ −(ϵ(t )−1 − σ1 |˙ϵ (t )|ϵ(t )−1 )‖Eϵ(t ) e‖2

+ 2σ2 ‖Eϵ(t ) e‖‖Eϵ(t ) δ(t , x, u)‖1

(8)

and V˙ c (z ) ≤ −(ϵ(t )−1 − σ3 |˙ϵ (t )|ϵ(t )−1 )‖Eϵ(t ) z ‖2

+ 2σ4 ϵ(t )−1 ‖Eϵ(t ) e‖‖Eϵ(t ) z ‖

(9)



∫

Mφ (ϵ(s))‖Eϵ(s) z (s)‖ds ‖Eϵ(t ) e‖ ≤ b ‖Eϵ(0) e(0)‖ + 0  ∫ t  t −1 eb s (‖D‖|˙ϵ (τ )|ϵ(τ ) +Mφ (ϵ(τ )))dτ × 1+

×

  |˙ϵ (s)| ‖D‖ + Mφ (ϵ(s)) ds ϵ(s)

(10)

where φ ∈ {√ 1, 2∑ }, b is a finite constant independent of ϵ(t ), n M1 (ϵ(t )) = γ n j=1 ϵ(t )j−1 for the (B1) case and M2 (ϵ(t )) =

γ

√

n(1 + ‖K ‖)

∑n−1 j =1

ϵ(t )−(j+1) for the (B2) case.

Proof. Proof is in the Appendix.



Next, consider the following two gain dynamics as

|e1 | + ‖z ‖1 ϵ1 (t )n+1 , ϵ1 (t0 ) = 1 (11) |e1 | + ‖z ‖1 + 1 |e1 | + ‖z ‖1 ϵ˙2 (t ) = ϵ2 (t )−1 , ϵ2 (t0 ) = 1 (12) |e1 | + ‖z ‖1 + 1 with ϵ1 (t0 ) = 1 and ϵ2 (t0 ) = 1. From (11) to (12), we have closedform solutions of ϵ1 (t ) and ϵ2 (t ) as  − 1n ∫ t 1 |e1 (s)| + ‖z (s)‖1 ϵ1 (t ) = +n ds (13) ϵ1 (t0 )n t0 |e1 (s)| + ‖z (s)‖1 + 1  ∫ t |e1 (s)| + ‖z (s)‖1 ϵ2 (t ) = 2 ds + ϵ2 (t0 )2 . (14) | e 1 (s)| + ‖z (s)‖1 + 1 t0 ϵ˙1 (t ) = −

Note that there exists the static relation between ϵ1 (t ) and ϵ2 (t ) as n−1 ϵ1−n (t ) − 0.5ϵ22 (t ) = n−1 − 0.5. We clearly see that ϵ1 (t ) is nonincreasing and ϵ2 (t ) is nondecreasing. Lemma 3. Under Assumption 1, the output feedback controller (4) and (5) is engaged with ϵ(t ) = ϵ1 (t ) or ϵ(t ) = ϵ2 (t ). Then, t ‖z ‖1 → 0, |e1 | → 0, and 0 |e1 (s)| + ‖z (s)‖ds < ∞ as t → ∞ if limt →∞ ϵ1 (t ) = ϵ¯1 or limt →∞ ϵ2 (t ) = ϵ¯2 , 0 < ϵ¯1 ≤ 1, 1 ≤ ϵ¯2 < ∞. Proof. Proof is in the Appendix.



Next, we introduce a switching scheme such that the controller (4) and (5) can regulate all states of the closed-loop system (1) with two gain dynamics (11) and (12) and a switching logic shown in the following. Initialization: 1 |+‖z ‖1 • Set two gain dynamics: ϵ˙1 (t ) = − |e|1e|+‖ ϵ (t )n+1 , ϵ˙2 (t ) = z ‖1 +1 1

|e1 |+‖z ‖1 ϵ ( t ) −1 |e1 |+‖z ‖1 +1 2

with ϵ1 (t0 ) = ϵ2 (t0 ) = 1. • Set either φ0 = 1 or φ0 = 2. Note that φi means that the gain dynamics evolve according to ϵ(t ) = ϵφi (t ), φi ∈ H defined by H = {1, 2}, for ti ≤ t < ti+1 , i = 0, 1, 2, . . .. • Define the variables depending on ti , for ti ≤ t < ti+1 , θφi (ti ) =    3 ϵφi (ti ) N¯ φi (ti ) + ϵφi (ti )−φi + 2 ‖Eϵφi (ti ) z (ti )‖ , N¯ φi (ti ) = 1 +  (i+1)(1+M¯ (t ))t ∑ 1 φi i i with N ¯ φi (ti )m(ti ) ij− ¯ φ0 (t0 ) = M =0 (j + 1)θφj (tj ) e ∑n ∑ n −1 j −1 ¯ −(j+1) ¯ 1, M1 (ti ) = ϵ (t ) , M2 (ti ) = , and j=1 ϵ1 (ti )  j=1 21 i  1 −2 2 m(ti ) = max ϵ1 (ti ) , ϵ2 (ti ) . • Set i = 0 at t = t0 and t0 = 0. Switching logic: Step 1: Set ϵ(t ) = ϵφi (t ). Step 2: If

Proof. Proof is in the Appendix.

∫

Lemma 2. Suppose that Assumption 1 holds. Assume that AL is Hurwitz with distinct eigenvalues. Then, we obtain



0



where σ1 = 2‖D‖‖PL ‖, σ2 = ‖PL ‖, σ3 = 2‖D‖‖PK ‖, σ4 = ‖PK ‖‖L‖‖C ‖ with D = diag [0, 1, . . . , n − 1]. 

t

t

3

|e1 (s)| + ϵ(s)−φi + 2 ‖Eϵ(s) z (s)‖ds ≤ (i + 1)θφi (ti ) ti

go to Step 2. Otherwise, go to Step 3.

(15)

M.-S. Koo et al. / Automatica 47 (2011) 2211–2217

Step 3: Set ti+1 = t, φi+1 ∈ H \ {φi }. Let i = i + 1, go to Step 1. Here, we briefly give an intuition about how the switching logic is designed and how it works. We design two dynamic gains as ϵ1 (t ) for (B1) case and ϵ2 (t ) for (B2) case. From the result of Choi and Lim (2010b), it is deduced that the integral function (15) has a finite value if ϵ1 (t ) becomes sufficiently small for (B1) case and ϵ2 (t ) becomes sufficiently large for (B2) case. Since the nonlinearity structure is not known, two dynamic gains ϵ1 (t ) and ϵ2 (t ) are switched alternately by monitoring the integral function (15). Eventually, the integral function (15) has a finite value. In turn, the switching stops in a finite time, and ϵ(t ) converges to a finite value. In principle, our control approach is similar to ‘unfalsified control approach’ in Angeli and Mosca (2002), Cabral and Safonov (2004), Safonov and Tsao (1997) and Stefanovic and Safonov (2008). Theorem 1. Suppose that Assumption 1 holds. Select K and L such that AK and AL are Hurwitz with distinct eigenvalues, respectively. Then, the controller (4) and (5) with the switching logic globally regulates all states of the system (1) and limt →∞ ϵ(t ) = ϵ¯ , 0 < ϵ¯ < ∞. Proof. Note that there is no finite escape phenomenon with the proposed controller because of the linear growth condition for each (B1) and (B2). 

Thus, we only need to consider a case that ϵ1 (t ) < ϵ1∗ . Let i1 ∈ H1 be the smallest switching index such that ϵ1 (t ) < ϵ1∗ for t ≥ ti1 . (1−h)

Then, from (16) to (17), we have V˙ 1 (e, z ) ≤ − λ ϵ1 (t )−1 V1 (e, z ) 2 ¯ 1 defined by H¯ 1 = {i|i = i1 + for t ∈ [ti , ti+1 ) where i ∈ H 2(j − 1), j = 1, 2, . . .} ∈ H1 . This inequality yields V1 (e, z ) ≤ − (1λ−h)

V1 (e(ti ), z (ti ))e for t ∈ [ti , ti+1 ),



λ1 (‖Eϵ1 (t ) e‖2 + ϵ1 (t )‖Eϵ1 (t ) z ‖2 ) ≤ V1 (e, z ) ≤ λ2 (‖Eϵ1 (t ) e‖2 + ϵ1 (t )‖Eϵ1 (t ) z ‖2 )

(16)

where λ1 = min{λmin (PK ), λmin (PL )} and λ2 = max{λmax (PK ), λmax (PL )}, for t ∈ [tk , tk+1 ). Note that, V˙ 1 (e, z ) = V˙ o (e) + ϵ1 (t ) V˙ c (z ) + ϵ˙1 (t )Vc (z ) ≤ V˙ o (e) + ϵ1 (t )V˙ c (z ) and ‖Eϵ1 (t ) δ(t , x, u)‖1 ≤ M1 (ϵ1 (t ))‖Eϵ1 (t ) x‖ ≤ M1 (ϵ1 (t ))(‖Eϵ1 (t ) e‖ + ‖Eϵ1 (t ) z ‖) for t ∈ [tk , tk+1 ). Then, from the above inequalities and Lemma 1, we have

(1 − h) (‖Eϵ1 (t ) e‖2 + ϵ1 (t )‖Eϵ1 (t ) z ‖2 ) ϵ1 (t ) [ ]T [ ] ‖E ϵ 1 ( t ) e ‖ ‖Eϵ1 (t ) e‖ − T (ϵ1 (t )) (17) ‖E ϵ 1 ( t ) z ‖ ‖Eϵ1 (t ) z ‖   T T where 0 < h < 1, T (ϵ1 (t )) = T11 T12 , T11 = hϵ1 (t )−1 − σ1 21 22 V˙ 1 (e, z ) ≤ −

|ϵ˙1 (t )|ϵ1 (t )−1 − 2σ2 M1 (ϵ1 (t )), T22 = ϵ1 (t )(hϵ1 (t )−1 − σ3 |ϵ˙1 (t )| ϵ1 (t )−1 ), and T12 = T21 = −(σ2 M1 (ϵ1 (t )) + σ4 ). Recall that ϵ1 (t ) is nonincreasing. So, it is clear that there exists ϵ1∗ such as T (ϵ1 (t )) > 0 if ϵ1 (t ) < ϵ1∗ . By Lemma 3, if ϵ1 (t ) converges to a value larger than t or equal to ϵ1∗ , then we have 0 |e1 (s)| + ‖z (s)‖ds < ∞ as t → ∞. t Using 0 |e1 (s)|+‖z (s)‖ds < ∞ and ϵ1∗ ≤ ϵ1 (t ) ≤ 1, it is clear that t 1 there exists k∗1 ∈ H1 such that t ∗ |e1 (s)| + ϵ1 (s) 2 ‖Eϵ1 (s) z (s)‖ds ≤ k 1 t |e1 (s)| + ‖z (s)‖ds ≤ (k∗1 + 1)θ1 (tk∗1 ). Then, switching stops in t ∗ k 1

finite time.

2

t

ti

ϵ1 (s)−1 ds

for t ∈ [ti , ti+1 ). Then, we obtain,

‖Eϵ1 (t ) e‖2 + ϵ1 (t )‖Eϵ1 (t ) z ‖2  λ2  ≤ ‖Eϵ1 (ti ) e(ti )‖2 + ϵ1 (ti )‖Eϵ1 (ti ) z (ti )‖2 λ1 h) − (12− λ

t

ϵ1 (s)−1 ds

. (18) ¯ Let it ∈ H1 be the smallest integer satisfying   it > max ρ1 , ρ2 (19)      √ 2λ nλ where ρ1 = (1−2h) λ 2 1 + ‖Eϵ(0) e(0)‖ + γ n (b + 1) and 1 √ ρ2 = b(γ n + ‖D‖). t It is clear that t ϵ1 (s)−1 ds ≥ ϵ1 (tit )−1 (t − tit ). After the it th i ×e

2

ti

t

switching, from (18) and above inequality, we obtain

∫

Part 1: We show that the dynamic gain is switched only a finite number of times. Consider two cases: Case (I): the system nonlinearity δ(t , x, u) actually satisfies (B1), but is not known a priori; Case (II): the system nonlinearity δ(t , x, u) actually satisfies (B2), but is not known a priori. Case (I): We will show that only a finite number switchings occur by proving that ϵ1 (t ) can only be engaged finite times. Let l1 denote the switching index when ϵ1 (t ) is engaged for the first time. Thus, l1 is either 0 or 1. Define V1 (e, z ) = Vo (e)+ϵ1 (t )Vc (z ) as a Lyapunov function for t ∈ [tk , tk+1 ), k ∈ H1 defined by H1 = {k|k = l1 + 2(j − 1), j = 1, 2, . . .} where Vo (e) = eT PL (ϵ1 (t ))e and Vc (z ) = z T PK (ϵ1 (t ))z. We have

2213

t

 ϵ1 (s)‖Eϵ1 (s) z (s)‖ds

|e1 (s)| + tit

 ≤

nλ2 

‖Eϵ1 (tit ) e(tit )‖2 + ϵ1 (tit )‖Eϵ1 (tit ) z (tit )‖2

λ1 ∫ t ×

e

h) −1 − (12− λ ϵ1 (tit ) (s−tit ) 2

ds

tit



nλ2 ϵ1 (tit ) ( 1 − h) λ 1    (20) × ‖Eϵ1 (tit ) e(tit )‖ + ϵ1 (tit )‖Eϵ1 (tit ) z (tit )‖ .  tit ∑it −1 √ ¯ 1 (tit ), 0 ‖Eϵ(s) z (s)‖ds ≤ m(tit ) j=0 Note that M1 (t ) ≤ γ nM ϵ (t )| (j + 1)θφj (tj ) for t ∈ [0, tit ], and |˙ϵ( ≤ 1 from (11) to (12). Then, t)

≤

2λ2

by Lemma 2 and (19), we obtain

‖Eϵ1 (tit ) e(tit )‖   ∫ ti t ≤ b ‖Eϵ(0) e(0)‖ + M1 (ϵ(s))‖Eϵ(s) z (s)‖ds 0  ∫ ti  t  b stit ‖D‖+M1 (ϵ(τ ))dτ × 1+ ‖D‖ + M1 (ϵ(s)) e ds 0   i t −1 − √ ¯ 1 (tit )m(tit ) ≤ b ‖Eϵ(0) e(0)‖ + γ nM (j + 1)θφj (tj ) j=0

√    ¯ × 1 + eb(‖D‖+γ nM1 (tit ))tit − 1 b−1 √ ≤ b(‖Eϵ(0) e(0)‖ + γ n)(1 + b−1 )N¯ 1 (tit ).



(21)

Substituting (21) into (20) and using (19), we have

∫

t

|e1 (s)| +

 ϵ1 (s)‖Eϵ1 (s) z (s)‖ds

tit

   ≤ ρ1 ϵ1 (tit ) N¯ 1 (tit ) + ϵ1 (tit )‖Eϵ1 (tit ) z (tit )‖ ≤ (it + 1)θ1 (tit ).

(22)

From (22), no further switching occurs for t > tit according to the switching logic.

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M.-S. Koo et al. / Automatica 47 (2011) 2211–2217



Case(II): We will show that only a finite number switchings occur by proving that ϵ2 (t ) can only be engaged finitely many times. Let l2 denote the switching index when ϵ2 (t ) is engaged for the first time. Thus, l2 is either 0 or 1. Define V2 (e, z ) = Vo (e)+ϵ2 (t )−1 Vc (z ) as a Lyapunov function for t ∈ [tk , tk+1 ), k ∈ H2 defined by H2 = {k|k = l2 + 2(j − 1), j = 1, 2, . . .} where Vo (e) = eT PL (ϵ2 (t ))e and Vc (z ) = z T PK (ϵ2 (t ))z. Also, for t ∈ [tk , tk+1 ), we have

≤

(1 − h) (‖Eϵ2 (t ) e‖2 + ϵ2 (t )−1 ‖Eϵ2 (t ) z ‖2 ) V˙ 2 (e, z ) ≤ − ϵ2 (t ) [ ]T [ ] ‖Eϵ2 (t ) e‖ ‖Eϵ2 (t ) e‖ − F (ϵ2 (t )) (24) ‖Eϵ2 (t ) z ‖ ‖Eϵ2 (t ) z ‖   F F where F (ϵ2 (t )) = F11 F12 with F11 = ϵ2 (t )−2 (hϵ2 (t ) − σ1 21 22 |ϵ˙2 (t )|ϵ2 (t ) − 2σ2 ϵ2 (t ) M2 (ϵ2 (t ))), F22 = ϵ2 (t ) (h − σ3 |˙ϵ2 (t )|), and F12 = F21 = −ϵ2 (t )−2 (σ2 ϵ2 (t )2 M2 (ϵ2 (t )) + σ4 ). Recall that ϵ2 (t ) is nondecreasing. So, it is clear that there exists ϵ2∗ such as F (ϵ2 (t )) > 0 if ϵ2 (t ) > ϵ2∗ . By Lemma 3, if ϵ2 (t ) converges to a value t less than or equal to ϵ2∗ , then we have 0 |e1 (s)| + ‖z (s)‖ds < ∞ t as t → ∞. Using 0 |e1 (s)| + ‖z (s)‖ds < ∞ and 1 ≤ ϵ2 (t ) ≤ t ϵ2∗ , it is clear that there exists k∗2 ∈ H2 such that t ∗ |e1 (s)| + k 2  − 12 ∗ n −1 t ϵ2 (s) ‖Eϵ2 (s) z (s)‖ds ≤ ϵ2 |e1 (s)| + ‖z (s)‖ds ≤ (k∗2 + t ∗ k 2

1)θ2 (tk∗ ). Then, switching stops in finite time. 2 Thus, we only need to consider a case that ϵ2 (t ) > ϵ2∗ . Let i2 ∈ H2 be the smallest number of switchings such that ϵ2 (t ) > ϵ2∗ for t ≥ ti2 . Then, from (23) and (24), we have V˙ 2 (e, z ) ≤

− (1λ−2h) ϵ2 (t )−1 V2 (e, z )

¯ 2 defined by for t ∈ [ti , ti+1 ) where i ∈ H ¯ H2 = {i|i = i2 + 2(j − 1), j = 1, 2, . . .} ∈ H2 . This inequality yields  − (1λ−h)

V2 (e, z ) ≤ V2 (e(ti ), z (ti ))e we obtain, for t ∈ [ti , ti+1 ),



2

t ti

ϵ2 (s)−1 ds

for t ∈ [ti , ti+1 ). Then,

‖Eϵ2 (t ) e‖2 + ϵ2 (t )−1 ‖Eϵ2 (t ) z ‖2  λ2  ≤ ‖Eϵ2 (ti ) e(ti )‖2 + ϵ2 (ti )−1 ‖Eϵ2 (ti ) z (ti )‖2 λ1 ×

 − (1−h) t ϵ (s)−1 ds e 2λ2 ti 2



.

(25)

2λ2 (1−h)

(26) nλ2

2λ2 1−c

√

)(1 + (‖Eϵ(0) e(0)‖ + γ n(1 + λ1 √ ‖K ‖))(b + 1)) and ρ4 = b(γ n(1 + ‖K ‖) + ‖D‖). t |e (s)|+‖z (s)‖1 From (14) and t |e (1s)|+‖z (s)‖ + ds ≤ t − tif , we obtain if 1 1 1  t ϵ2 (t ) ≤ 2(t − tif ) + ϵ2 (tif )2 . This yields t ϵ2 (s)−1 ds ≥ if  2 2(t − tif ) + ϵ2 (tif ) − ϵ2 (tif ). After the if th switching, from (25) where ρ3 =

(1 +

and above inequalities, we obtain t

∫

|e1 (s)| + √ ti

f

1

ϵ2 (s)

‖Eϵ2 (s) z (s)‖ds



2(s−ti )+ϵ2 (ti )2 −ϵ2 (ti ) f

f

f

ds

f



2λ2

(1 − h)

nλ 1

 1+

λ2

2λ2



1−h

ϵ2 (tif )

√

¯ 2 (tif ) and n(1 + ‖K ‖)M

 tif 0

(27)

‖Eϵ(s) z (s)‖ds ≤

|˙ϵ (t )| ϵ(t )

∑if −1

m(tif ) j=0 (j + 1)θφj (tj ) for t ∈ [0, tif ], and ≤ 1 from (11) and (12). Then, by Lemma 2 and (26), similar to (21), we obtain

  √ ‖Eϵ2 (tif ) e(tif )‖ ≤ b ‖Eϵ(0) e(0)‖ + γ n(1 + ‖K ‖) × (1 + b−1 )N¯ 2 (tif ).

(28)

Substituting (28) into (27) and using (26), we have

∫

t

1

|e1 (s)| + √ ‖Eϵ (s) z (s)‖ds ϵ2 (s) 2 ti f   1 ≤ ρ3 ϵ2 (tif ) N¯ 2 (tif ) +  ‖Eϵ2 (tif ) z (tif )‖ ϵ2 (tif ) ≤ (if + 1)θ2 (tif ).

(29)

From (29), no further switching occurs for t > tif according to our switching logic. Part 2: We will show that if the switching is finite, then the closedloop system (1) is regulated. Let ti∗ be the final switching time. Then, by the switching logic (15), for t ∈ [ti∗ , ∞), we obtain

∫

t ti ∗

3

|e1 (s)| + ϵ(s)−φi∗ + 2 ‖Eϵ(s) z (s)‖ds ≤ (i∗ + 1)θφi∗ (ti∗ ).

For the case of φi∗ = 1, using the fact that

√ t

(30)

t

ϵ (s)n (|e1 (s)| + ti ∗ 1 √ √ |e1 (s)| + ϵ1 (s)‖Eϵ1 (s) z (s)‖ds ≤ n(i∗ +

‖z (s)‖1 )ds ≤ n t ∗ i 1)θ1 (ti∗ ), we have ∫ t ∫ t ϵ1 (t ) ϵ˙1 (s) |e1 (s)| + ‖z (s)‖1 ln ds = − ϵ1 (s)n ds = ϵ1 (ti∗ ) ti∗ ϵ1 (s) ti∗ |e1 (s)| + ‖z (s)‖1 + 1 ∫ t ≥− ϵ1 (s)n (|e1 (s)| + ‖z (s)‖1 )ds √ ≥ − n(i∗ + 1)θ1 (ti∗ ).

(31) √

− n(i∗ +1)θ1 (ti∗ )

 

‖Eϵ2 (tif ) z (tif )‖2

ti∗

¯ 2 be the smallest integer satisfying Let if ∈ H if ≥ max ρ3 , ρ4



2

Note that M2 (t ) ≤ γ

−2

2

h) − (12− λ

ϵ2 (tif )

  1 × ‖Eϵ2 (tif ) e(tif )‖ +  ‖Eϵ (t ) z (tif )‖ . ϵ2 (tif ) 2 if

(23)

Note that, for t ∈ [tk , tk+1 ), V˙ 2 (e, z ) = (V˙ o (e) + ϵ2 (t )−1 V˙ c (z ) − ϵ˙2 (t )ϵ2 (t )−2 Vc (z )) ≤ (V˙ o (e) + ϵ2 (t )−1 V˙ c (z )) and ‖Eϵ2 (t ) δ(t , x, u)‖1 ≤ M2 (ϵ2 (t ))(‖Eϵ2 (t ) z ‖+‖Eϵ2 (t ) e‖) with |u| ≤ ϵ2 (t )−n ‖K ‖ ‖Eϵ2 (t ) z ‖. Then, from the above inequalities and Lemma 1, we obtain

1

‖Eϵ2 (tif ) e(tif )‖2 + e

ti

λ1 (‖Eϵ2 (t ) e‖2 + ϵ2 (t )−1 ‖Eϵ2 (t ) z ‖2 )



λ2 ∫ t ×

≤

≤ V2 (e, z ) ≤ λ2 (‖Eϵ2 (t ) e‖2 + ϵ2 (t )−1 ‖Eϵ2 (t ) z ‖2 ).

nλ 1

. Thus, there From (31), we obtain ϵ1 (t ) ≥ ϵ1 (ti∗ )e exists 0 < ϵ¯1 ≤ ϵ1 (ti∗ ) such that limt →∞ ϵ1 (t) = ϵ¯1 . t For the case of φi∗ = 2, using that t ∗ ϵ2 (s)−1 (|e1 (s)| +

√ t ‖z (s)‖1 )ds ≤ n t ∗ |e1 (s)| + i 1)θ2 (ti∗ ), we have ∫ t ϵ2 (t ) − ϵ2 (ti∗ ) = ϵ˙2 (s)ds

i

√ 1 ‖Eϵ (s) z (s)‖ds 2 ϵ2 (s)

≤

√

n(i∗ +

ti∗ t

∫ =

ti∗

|e1 (s)| + ‖z (s)‖1 ϵ2 (s)−1 ds |e1 (s)| + ‖z (s)‖1 + 1

t

∫

ϵ2 (s)−1 (|e1 (s)| + ‖z (s)‖1 )ds

≤ ti∗

√ ≤

n(i∗ + 1)θ2 (ti∗ ).

(32)

M.-S. Koo et al. / Automatica 47 (2011) 2211–2217

2215

√

From (32), we obtain ϵ2 (t ) ≤ n(i∗ + 1)θ2 (ti∗ )+ϵ2 (ti∗ ). Thus, there exists ϵ2 (ti∗ ) ≤ ϵ¯2 < ∞ such that limt →∞ ϵ2 (t ) = ϵ¯2 . Then, the engaged dynamic gain after the final switching converges to a finite positive constant. By Lemma 3, we obtain ‖z ‖1 → 0, |e1 | → 0, t t ‖z (s)‖ds < ∞ and t ∗ |e1 (s)| < ∞ as t → ∞. Then, we show t∗ i

i

that all states of the system (1) are regulated for each case (B1) and (B2).

t

t |x1 (s)|ds ≤ t ∗ |z1 (s)| + i |e1 (s)|ds < ∞ and x1 → 0, we have δ1 (t , x, u) → 0 and x2 → 0. Generally, we have δi (t , x, u) → 0 and xi+1 → 0 from t x (s)ds < ∞ and xj → 0, i ∈ [1, n − 1], j ∈ [1, i]. Therefore, ti∗ i x → 0 as t → ∞. t t (ii) Consider the case (B2). Since t ∗ ‖u(s)‖ds < ∞ from t ∗ ‖z (s)‖ i i ds < ∞ and limt →∞ ϵ(t ) = ϵ¯ , we have xn → c1 where c1 is a ∑n−h+1 finite constant. Then, with ci , i ∈ [1, n − 1], xh → j=1  t 1 n−h−j+1 c t +α ( t ) where α ( t ) = δ ( s , x ( s j h h h n − h − 1 n − h−1 ), ti ∗ (n−h−j+1)  sn−h−1 u(sn−h−1 )) + t ∗ δh+1 (sn−h−2 , x(sn−h−2 ), u(sn−h−2 )) + · · · + i  s1 δ ( s , x ( s ), u ( s ))ds0 · · · dsn−h−2 dsn−h−1 , h ∈ [2, n − 1]. It n − 1 0 0 0 ti∗ is clear that the order of α2(t ) is O (t n−3 ) for n ≥ 3 and α2 (t ) is t t 0 for n ≤ 2. Since x1 = t ∗ x2 (s)ds + t ∗ δ1 (s, x(s), u(s))ds + i i     t x1 (0) and x1 → 0, we obtain c1 = 0 to satisfy  t ∗ x2 (s)ds ≤ i t t |δ1 (s, x(s), u(s))|ds + |x1 (0)| ≤ γ t ∗ (|x3 (s)| + · · · + |xn (s)| + ti∗ i |u(s)|)ds + |x1 (0)| as t → ∞. Similarly, we derive ci = 0, i ∈ [1, n − 1]. Therefore, x → 0 as t → ∞. (i) Consider the case (B1). Since

ti∗

3. Illustrative examples Consider the following system x˙ 1 = x2 + δ1 (t , x, u) x˙ 2 = u + δ2 (t , x, u) y = x1 .

(33)

Let δ(t , x, u) = [δ1 (t , x, u), δ2 (t , x, u)] satisfy Assumption 1. Obviously, the existing control methods are not applicable to the above system under Assumption 1 because we do not know the structure of δ(t , x, u). T

Fig. 1. State trajectories, ϵ(t ), and the number of switching for the case: (a) φ0 = 2 of the system (i) and (b) φ0 = 1 of the system (ii).

We set the parameters of controllers (4) and (5) as K = [−3, −2] and L = [2, 3]T with the initialization being set as proposed.  Then, PK =

1.25 0.25

0.25 0.25

and PL =

0.5 −0.5

−0.5 1

. For verification,

let the nonlinearity be (i) δ(t , x, u) = [0.2x1 sin x1 , x1 + 0.2x2 ]T for the case (B1) and (ii) δ(t , x, u) = [

1/3 10ux1

1+x21

, 0]T for the case

(B2). Then, we apply the same controller with the switching logic for the case (a) φ0 = 2 with nonlinearity (i) and (b) φ0 = 1 with nonlinearity (ii). It is shown in Fig. 1 that the systems are universally regulated by our adaptive controller without a priori knowledge on their nonlinearity structure and linear growth rate. Additionally, in Fig. 2(a) and (b), simulation comparisons with existing methods are made for the case that the nonlinearity structure is known a priori. Our controller is compared with Lei and Lin (2006) for (B1) case and with Choi and Lim (2010b) for (B2) case, respectively. The simulation results show that the proposed control matches the performance of controllers that rely on knowledge of the nonlinearity structure. 4. Conclusions In this paper, we have proposed a universal controller to a class of nonlinear systems whose nonlinearity structure is not

Fig. 2. State trajectories: (a) for the (B1) case and (b) for the (B2) case.

2216

M.-S. Koo et al. / Automatica 47 (2011) 2211–2217

known a priori. The proposed controller globally regulates the considered system by engaging adaptive gain with switching logic. More specifically, the proposed controller is applicable to both classes of triangular and feedforward systems with the exact same controller parameters, which is a universal feature. The validity of our result is shown through the analysis and simulations.

A and b = ‖Q ‖ ‖Q −1 ‖ such that e L



A By using e L



t

1 a dτ e 0 ϵ(τ ) ,

by b

t 0

Aj Eϵ(t ) and Pj (ϵ(t )) = Eϵ(t ) Pj Eϵ(t ) to ATj Pj + Pj Aj = −I, we obtain 2 ATj (ϵ)Pj (ϵ) + Pj (ϵ)Aj (ϵ) = −ϵ(t )−1 Eϵ( t ) for j = K , L. We set a

Note that eT P˙ L (ϵ(t ))e = eT (E˙ ϵ(t ) PL Eϵ(t ) + Eϵ(t ) PL E˙ ϵ(t ) )e and E˙ ϵ(t ) = Eϵ(t ) ∆ = ∆Eϵ(t ) where ∆ = ϵ˙ (t )ϵ(t )−1 D. Then, we obtain eT P˙ L (ϵ(t ))e ≤ 2‖D‖ ‖PL ‖ |˙ϵ (t )|ϵ(t )−1 ‖Eϵ(t ) e‖2 .

(35)

Then, we obtain the inequality (8). Next, we set a Lyapunov function Vc (z ) = z T PK (ϵ(t ))z. Then, we have V˙ c (z ) = z˙ T PK (ϵ(t ))z + z T PK (ϵ(t ))˙z + z T P˙ K (ϵ(t ))z 2

T

+ 2z T Eϵ(t ) PK Eϵ(t ) L(ϵ(t ))Ce.

(36)

From (35), z T P˙ K (ϵ(t ))z ≤ 2‖D‖ ‖PK ‖ |˙ϵ (t )|ϵ(t )−1 ‖Eϵ(t ) z ‖2 . Also, note that Eϵ(t ) L(ϵ(t ))Ce = ϵ(t )−1 LCEϵ(t ) e. Thus, we obtain the inequality (9).  Proof of Lemma 2. From (6), we obtain dEϵ(t ) e dt

=

1

ϵ(t )

AL Eϵ(t ) e +

ϵ˙ (t ) DEϵ(t ) e + Eϵ(t ) δ(t , x, u). ϵ(t )

(37)

Note that ‖Eϵ(t ) δ(t , x, u)‖1 ≤ Mφ (ϵ(t ))(‖Eϵ(t ) e‖ + ‖Eϵ(t ) z ‖) from Assumption 1. Using this inequality and (37), we have

 t 1   A dτ  ‖Eϵ(t ) e‖ ≤ ‖Eϵ(0) e(0)‖ e L 0 ϵ(τ )  ∫ t  t 1   A dτ  + Mφ (ϵ(s))‖Eϵ(s) z (s)‖ e L s ϵ(τ )  ds 0 ∫ t   + ‖D‖ |˙ϵ (s)|ϵ(s)−1 + Mφ (ϵ(s)) 0

 t  A × ‖Eϵ(s) e(s)‖ e L s

1

ϵ(τ ) dτ

   ds.

b‖Eϵ(0) e(0)‖ +

≤

 t 0

‖D‖ |˙ϵ (s)|ϵ(s)−1 +

By the Gronwall–Bellman in-

t

a

s

1

dτ

(39)

t 0

t

1 0 ϵ(τ ) dτ

and using that

s 0

Mφ (ϵ(k))‖Eϵ(k)

Mφ (ϵ(s))‖Eϵ(s) z (s)‖ds, we obtain (10).

Proof of Lemma 3. First, since δ(t , x, u) satisfies the global linear growth condition under Assumption 1, there is no finite-time escape phenomenon. From (13) and (14), ¯1 or t →∞ ϵ1 (t ) = ϵ  t lim |e (s)|+‖z (s)‖1 limt →∞ ϵ2 (t ) = ϵ¯2 implies that limt →∞ 0 |e (1s)|+‖z (s)‖ + ds exists 1 1

1

1 1 and is finite. Since |e |+‖ is uniformly continuous on [0, ∞), it z ‖1 +1 1 is obvious that ‖z ‖1 → 0 and |e1 | → 0 as t → ∞ by Barbalat’s Lemma (Khalil, 2002). Since ‖z ‖1 → 0 and |e1 | → 0 as t → ∞, it is clear that supt {|e1 | + ‖z ‖1 + 1} < ∞. This, together with  t |e (s)|+‖z (s)‖1 limt →∞ 0 |e (1s)|+‖z (s)‖ + ds < ∞, yields 1

|e |+‖z ‖

1

1

t

∫

|e1 (s)| + ‖z (s)‖1 ds   ∫ t |e1 (s)| + ‖z (s)‖1 = (|e1 (s)| + ‖z (s)‖1 + 1) ds |e1 (s)| + ‖z (s)‖1 + 1 0 ∫ t |e1 (s)| + ‖z (s)‖1 ≤ sup{|e1 | + ‖z ‖1 + 1} ds | e t 1 (s)| + ‖z (s)‖1 + 1 0 < ∞ as t → ∞. (40) t From (40) and ‖z ‖ ≤ ‖z ‖1 , we obtain 0 |e1 (s)| + ‖z (s)‖ds < ∞ as t → ∞. 0

≤ −ϵ(t ) ‖Eϵ(t ) z ‖ + z P˙K (ϵ(t ))z −1

1 a dτ e 0 ϵ(τ )

0

z (k)‖dk ≤ (34)

and multiplying (38)

t

+ b

1

ϵ(τ ) dτ .

t

−a

≤ −ϵ(t )−1 ‖Eϵ(t ) e‖2 + eT P˙L (ϵ(t ))e

1 s ϵ(τ ) dτ

t   ≤ be−a s

1 a 0 ϵ(τ ) dτ

Multiplying (39) by e

V˙ o (e) = e˙ PL (ϵ(t ))e + e PL (ϵ(t ))˙e + e P˙ L (ϵ(t ))e

1 s ϵ(τ ) dτ

Mφ (ϵ(s))‖Eϵ(s) z (s)‖e 0 ϵ(τ ) ds ≤ b‖Eϵ(0) e(0)‖ + b 0 ∫ t    t b ‖D‖ |˙ϵ (τ )|ϵ(τ )−1 +Mφ (ϵ(τ )) dτ e s +b 0   × ‖D‖ |˙ϵ (s)|ϵ(s)−1 + Mφ (ϵ(s)) ‖Eϵ(0) e(0)‖ ∫ s  k 1 a dτ Mφ (ϵ(k))‖Eϵ(k) z (k)‖e 0 ϵ(τ ) dk ds. +

−1 Proof of Lemma 1. Applying the relation Aj (ϵ(t )) = ϵ(t )−1 Eϵ( t)

+ 2‖PL ‖ ‖Eϵ(t ) e‖ ‖Eϵ(t ) δ(t , x, u)‖1 .

s

1 a dτ e 0 ϵ(τ ) ds.

∫

Appendix

T

s

t

t

1 dτ a e 0 ϵ(τ ) ds

Mφ (ϵ(s)) ‖Eϵ(s) e(s)‖ equality (Khalil, 2002),

This study was supported by research funds from Dong-A University.

T

  ≤ be−a

we have ‖Eϵ(t ) e‖



‖Eϵ(t ) e‖e

T

1 s ϵ(τ ) dτ

Mφ (ϵ(s))‖Eϵ(s) z (s)‖

Acknowledgment

Lyapunov function Vo (e) = eT PL (ϵ(t ))e. Then, we have

t

(38)

Since AL is Hurwitz and has distinct eigenvalues, we can express AL = Q Aˆ L Q −1 where Aˆ L is a diagonal matrix with eigenvalues of AL as its elements and Q is a similarity matrix. Then, it is clear that there exist positive constants a = |λmin (AL )|

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M.-S. Koo et al. / Automatica 47 (2011) 2211–2217 Jiang, Z. P. (1999). A combined backstepping and small-gain approach to adaptive output feedback control. Automatica, 35(7), 1131–1139. Khalil, H. K. (2002). Nonlinear systems (3rd ed.) Upper Saddle River, NJ 07458: Prentice Hall. Krishnamurthy, P., & Khorrami, F. (2003). Global robust control of feedforward systems: state-feedback and output feedback. In: Proc. of 37th IEEE Conf. Decision and Control, Maui, Hawaii USA, pp. 6145–6150. Lei, H., & Lin, W. (2006). Universal adaptive control of nonlinear systems with unknown growth rate by output feedback. Automatica, 42(10), 1783–1789. Liu, Y., & Ge, S. (2005). Output feedback adaptive stabilization for nonlinear systems with unknown direction control coefficients In: Proc. of American Control Conf., Portland, OR, USA, pp. 4696–4701, 2005. Praly, L. (2003). Asymptotic stabilization via ouput feedback for lower triangular systems with output dependent incremental rate. IEEE Transactions on Automatic Control, 48(6), 1103–1108. Qian, C., & Lin, W. (2002). Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm. IEEE Transactions on Automatic Control, 47(10), 1710–1715. Qian, C., & Lin, W. (1998). New results on global stabilization of feedforward systems via small feedback. In: Proc. of 37th IEEE Conf. Decision Control, Tampa, Fl, pp. 879–884. Safonov, M. G., & Tsao, T. C. (1997). The unfalsified control concept and learning. IEEE Transactions on Automatic Control, 42(6), 843–847. Stefanovic, M., & Safonov, M. G. (2008). Safe adaptive switching control: stability and convergence. IEEE Transactions on Automatic Control, 53(9), 2012–2021. Ye, X. (2005). Adaptive output feedback control of nonlinear systems with unknown nonlinearities. Automatica, 41(8), 1367–1374. Ye, X., & Unbehauen, H. (2004). Global adaptive stabilization for a class of feedforward nonlinear systems. IEEE Transactions on Automatic Control, 49(5), 786–792.

2217 Min-Sung Koo received the B.S.E. degree in 2004 and M.S. degree in 2006 and Ph.D degree in 2011 from the Department of Electrical Engineering, KAIST (Korea Advanced Institute of Science and Technology), Daejeon, Korea, respectively. She is a postdoctoral researcher at KAIST. Her research interests include nonlinear system, switching system, high-order system and time-delay systems. She is a member of ICROS.

Ho-Lim Choi received the B.S.E. degree from the Department of Electrical Engineering, The University of Iowa, USA in 1996, and M.S. degree in 1999 and Ph.D degree in 2004, from KAIST (Korea Advanced Institute of Science and Technology), respectively. Currently, he is an assistant professor at the Department of Electrical Engineering, Dong-A university, Busan. His research interests are in the nonlinear control problems with emphasis on feedback linearization, gain scheduling, singular perturbation, output feedback, time-delay systems. He is a member of IEEE, IEICE, KIEE, ICROS. Jong-Tae Lim received the B.S.E.E. degree from Yonsei University, Seoul, Korea, in 1975, the M.S.E.E. degree from the Illinois Institute of Technology, Chicago, in 1983, and the Ph.D. degree in Computer, Information and Control Engineering from the University of Michigan, Ann Arbor, in 1986. He is currently a professor in the Division of Electrical Engineering at the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology. His research interests are in the areas of system and control theory, communication networks, and discrete event systems. He is a member of IEEE, IEICE, KIEE, and KITE.