Stabilisation of non-linear systems with unknown ... - Semantic Scholar
International Journal of Systems Science Vol. 41, No. 6, June 2010, 673–678
Stabilisation of non-linear systems with unknown growth rate by adaptive output feedback Ho-Lim Choia and Jong-Tae Limb* a
b
Department of Electrical Engineering, Dong-A University, 840 Hadan2-Dong, Saha-gu, Busan, 604-714, Korea; Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, Korea
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(Received 10 October 2008; final version received 2 June 2009) In recent years, several results have been proposed on the global stabilisation of non-linear systems with unknown linear growth rate. However, these works are limited in the sense that they consider only one particular form of non-linear systems – mostly either triangular or feedforward form. We propose an adaptive output feedback control scheme which can deal with both triangular or feedforward non-linear systems with unknown linear growth rate in a unified framework. Thus, our result broadens the class of non-linear systems under consideration over the existing results. Keywords: non-linear systems; output feedback; unknown growth rate
1. Introduction We consider a class of single-input single-output nonlinear systems given by x_ ¼ Ax þ Bu þ ðt, x, uÞ y ¼ Cx
ð1Þ ð2Þ
where x 2 Rn, u 2 R and y 2 R are the system state, input and output, respectively. The system matrices are A, B and C, where (A, B) is the Brunovsky canonical pair and C ¼ [1, 0, . . . , 0]. The non-linear term is (t, x, u) ¼ [1(t, x, u), . . . , n(t, x, u)]T, where i(t, x, u): R Rn R ! R, i ¼ 1, . . . , n are C1 with respect to all the variables. Regarding the system non-linearity (t, x, u), we introduce the following two conditions. (A1) For i ¼ 1, . . . , n, there exists an unknown constant 0 such that ji ðt, x, uÞj ðjx1 j þ þ jxi jÞ
ð3Þ
(A2) For i ¼ 1, . . . , n 2, there exists an unknown constant 0 such that ji ðt, x, uÞj ðjxiþ2 j þ þ jxn j þ jujÞ
ð4Þ
with jn1(t, x, u)j juj and n(t, x, u) ¼ 0. We note that (A1) is the so-called ‘triangular condition’ as assumed in Lei and Lin (2006). A similar type of triangular form is considered in Krishnamurthy and Khorrami (2008), but i(t, x, u) is only assumed to be a function of (x1, . . . , xi) excluding time-variable t and states (xiþ1, . . . , xn). Since (A1) is given as a *Corresponding author. Email: [email protected] ISSN 0020–7721 print/ISSN 1464–5319 online 2010 Taylor & Francis DOI: 10.1080/00207720903144529 http://www.informaworld.com
norm-bound condition, the states (xiþ1, . . . , xn) can be included in i(t, x, u) unlike Krishnamurthy and Khorrami (2008) (Example 1). The other condition (A2) is the so-called ‘feedforward condition’, which is a linear growth version of the ones in Xudong (2003) and Ye and Unbehauen (2004) and is a restricted version of the one in Krishnamurthy and Khorrami (2007). The class of non-linear systems which fit the system (1)–(2) under either (A1) or (A2) are well addressed in Qian and Lin (2002), Xudong (2003) and references therein. Now, we address how the existing results have been progressed along with the two conditions mentioned above. First, when the growth rate is known, (A1) reduces the conditions assumed in Freeman and Kokotovic´ (1993) and Qian and Lin (2002), where the state and output feedback controllers are proposed, respectively, for the global stabilisation of the system (1)–(2). Later on, the authors of Choi and Lim (2004, 2005a) extend the results of Freeman and Kokotovic´ (1993) and Qian and Lin (2002) by including both (A1) and (A2) with being known and suggest a single form of globally stabilising state/output feedback controllers. However, all these results require a priori knowledge on in order to determine the gains of controllers. Then, when the growth rate is unknown, some form of additional gain-tuning mechanism is needed for stabilising controllers. In Choi and Lim (2005b) and later in Lei and Lin (2006), the authors, respectively, suggest the adaptive output feedback control methods which solve the global stabilisation problem of the system (1)–(2) under (A1). The results of
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Xudong (2003) and Ye and Unbehauen (2004) can handle (A2) case, but they use the full-state feedback controller. The result of Krishnamurthy and Khorrami (2007) can also handle (A2). Thus, to the best of our knowledge, there have been no output feedback control methods which can be applied to both (A1) and (A2) cases in a unified framework. So, here is our extended control problem: The system is globally asymptotically stabilised (1)–(2) by output feedback when only the structure of (t, x, u) is known such that it belongs to either (A1) or (A2). In order to solve this problem, we utilise the structure of controller in Choi and Lim (2005a) and combine it with an adaptive gain-tuning mechanism which is appropriately engaged depending on (A1) or (A2) case. Note that (A1) and (A2) are global linear growth conditions, which may look to be somewhat limited in the first hand. However, as investigated in Mazenc Praly, and Dayawansa (1994), some extra growth conditions on the unmeasurable states of the system are usually necessary for the global stabilisation of non-linear systems via output feedback. For example, a global Lipschitz condition is assumed in Jo and Seo (2000b), global linear growth conditions are assumed in Choi and Lim (2005a, b), Lei and Lin (2006) and Qian and Lin (2002) for the global stabilisation problem of non-linear systems. Under a local Lipschitz condition, only local stabilisation is achieved by output feedback (Jo and Seo 2000a). In this sense, for the global stabilisation problem of non-linear systems by output feedback, (A1) and (A2) are not much restricted conditions, but more relaxed than the conditions in related results.
2. Adaptive output feedback controller First, we introduce the following dynamic output feedback controller u ¼ KððtÞÞz z_ ¼ Az þ Bu LððtÞÞð y CzÞ
ð5Þ ð6Þ
where K((t)) ¼ [k1/(t)n, . . . , kn/(t)] and L((t)) ¼ _ ¼ [l1/(t), . . . , ln/(t)n]T, (t)40. The dynamic gain (t) (y, (t)) is to be introduced later on. Here, we introduce some notations for convenience. Notations: We define E(t) :¼ diag[1, (t), . . . , (t)n1], K :¼ K((t))j(t)¼1, L :¼ L((t))j(t)¼1, AK :¼ A þ BK, AL :¼ A þ LC, AK((t)) :¼ A þ BK((t)) and AL((t)) :¼ A þ L((t))C. Then, using these notations, the controller u in (5) can be expressed as u ¼ (t)n KE(t)z. Now, we state the main result.
Theorem 2.1: Suppose that (i) (A1) or (A2) holds for the system (1)–(2); (ii) K and L are selected such that AK and AL are Hurwitz, respectively. Under (A1), the controller (5)–(6) is engaged with _ ¼ j yjðtÞ, ð0Þ ¼ 1 ðtÞ
ð7Þ
Under (A2), the controller (5)–(6) is engaged with _ ¼ j yj1=2 =ðtÞ, ð0Þ ¼ 1 ðtÞ
ð8Þ
Then, the closed-loop system (1)–(2) becomes globally asymptotically stable and limt!1 ðtÞ ¼ 5 1 for each case. Proof: First, we introduce brief mathematical setups for the proof. Since AK and AL are Hurwitz, as it follows from Choi and Lim (2005a), we can derive two Lyapunov equations of with ATj ððtÞÞPj ððtÞÞ þ Pj ððtÞÞAj ððtÞÞ ¼ ðtÞ1 E2ðtÞ with Pj((t)) ¼ E(t)PjE(t) where ATj Pj þ Pj Aj ¼ I for j ¼ K, L. Here, I denotes an n n identity matrix. (A1) case: Define ei ¼ xi zi, 1 i n. By subtracting (6) from (1), the error dynamics is obtained as e_ ¼ AL ððtÞÞe þ ðt, x, uÞ
ð9Þ
With (5) and (7), the closed-loop system becomes x_ ¼ AK ððtÞÞx þ ðt, x, uÞ BKððtÞÞe _ ¼ j yjðtÞ, ð0Þ ¼ 1 ðtÞ
ð10Þ ð11Þ
We begin with showing the existence of a unique solution. The vector field of the augmented system (9) and (10) is C1 with respect to its arguments. Regarding (11), Rwe have a closed-form solution of ðtÞ ¼ t expð 0 j yjdÞ. Thus, the corresponding solution (e(t), x(t), (t)) of (9)–(11) exists and is unique on [0, Tf) for some Tf 2 (0, 1]. Moreover, since ¼ AL((t))e and x_ ¼ AK ððtÞÞx are stable (time-varying) linear systems and (t, x, u) satisfies the global linear growth condition, it can be easily checked by using Gronwall– Bellman inequality that there is no finite-time escape phenomenon (Rugh 1996). Thus, Tf can be arbitrarily large. Now, we will show that (e(t), x(t), (t)) is welldefined and bounded on [0, Tf). First, we set a Lyapunov function Vo(e) ¼ eTPL((t))e. Then, along the trajectory of (9), V_ o ðeÞ ¼ e_T PL ððtÞÞe þ eT PL ððtÞÞe_ þ eT P_ L ððtÞÞe ¼ ðtÞ1 kE ðtÞek2 þ eT P_ L ððtÞÞe þ 2eT EðtÞ PL EðtÞ ðt, x, uÞ
ð12Þ
International Journal of Systems Science Under (A1), note that kE(t)(t, x, u)k1 n3/2kE(t)xk. Also, we have that eT P_ L ððtÞÞe ¼ eT ðE_ ðtÞ PL EðtÞ þ EðtÞ PL E_ ðtÞ Þe and E_ (t) ¼ E(t)D ¼ DE(t) where 1 _ diag½0, 1, . . . , n 1 D ¼ ðtÞðtÞ
ð13Þ
This leads to the following inequality: _ kEðtÞ ek e P_ L ððtÞÞe 2sn kPL kjðtÞjðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where sn ¼ nðn 1Þð2n 1Þ=6. Then, we have T
1
2
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1 _ ÞkEðtÞ ek2 V_ o ðeÞ ððtÞ1 1 jðtÞjðtÞ
þ 22 kEðtÞ ekkEðtÞ xk
ð15Þ
where 1 ¼ 2snkPLk and 2 ¼ n3/2kPLk, which are finite constants independent of (t) and . Next, we set a Lyapunov function Vc(x) ¼ xT PK((t))x. Then, along the trajectory of (10), V_ c ðxÞ ¼ x_ T PK ððtÞÞx þ xT PK ððtÞÞx_ þ xT P_ K ððtÞÞx ¼ ðtÞ1 kEðtÞ xk2 þ 2xT EðtÞ PK EðtÞ ðt, x, uÞ þ xT P_ K ððtÞÞx 2xT EðtÞ PK EðtÞ BKððtÞÞe ð16Þ Note that E(t)BK((t))e ¼ (t)1 BKE(t)e. Also, from (14), we can deduce that 1 _ kEðtÞ xk2 xT P_ K ððtÞÞx 2sn kPK kjðtÞjðtÞ
ð17Þ
Thus, we have 1 _ V_ c ðxÞ ððtÞ1 3 4 jðtÞjðtÞ ÞkEðtÞ xk2
þ 2ðtÞ1 5 kEðtÞ ekkEðtÞ xk
ð18Þ
where 3 ¼ 2n3/2kPKk, 4 ¼ 2snkPKk and 5 ¼ kPKk kKk, which all are finite constants independent of (t) and . Now, we set a composite Lyapunov function V1(e, x) ¼ Vo(e) þ (t)Vc(x) for the augmented system (9) and (10). Then, we have V_ 1 ðe, xÞ ¼ V_ o ðeÞ þ _ _ _ ðtÞV_ c ðxÞ þ ðtÞV c ðxÞ Vo ðeÞ þ ðtÞVc ðxÞ due to which _ ¼ j yjðtÞ 0, (t)40 and Vc(x) 0. Thus, using ðtÞ the results of (15) and (18), V_ 1 ðe, xÞ
kEðtÞ ek kEðtÞ xk
T
1
2
kEðtÞ ek kEðtÞ xk
ð19Þ
1 _ where 1 ¼ ðtÞ1 1 jðtÞjðtÞ , 2 ¼ ðtÞððtÞ1 1 _ Þ and ¼ ( 2 þ 5). 3 4 jðtÞjðtÞ From (19), V_ 1 ðe, xÞ becomes negative definite if and only if (i) 140 and (ii) 12 240. Regarding (i), it can be rewritten as Z t ð20Þ exp j yjd 1 j yj 4 0 0
Thus, there must exist a finite time t1 such that 140 for t1 t 5 Tf. Regarding (ii), it can be rewritten as Z t j yj j yjd 3 4 j yj 1 1 exp Rt expð 0 j yjdÞ 0 ð2 þ 5 Þ2 4 0
ð14Þ
675
ð21Þ
Thus, there must exist a finite time t2 such that 12 ( 2 þ 5)240 for t2 t 5 Tf. Now, let us define t* ¼ max{t1, t2}. Then, we have V_ 1 ðe, xÞ ðkEðtÞ ek2 þ kEðtÞ xk2 Þ pffiffiffiffiffiffiffi ðkEðtÞ ek2 þ k ðtÞEðtÞ xk2 Þ, 4 0
ð22Þ
for all t 2 [t*, Tf). The last inequality comes from the fact that (t) 1 with (7). Note that pffiffiffiffiffiffiffi mðkEðtÞ ek2 þ k ðtÞEðtÞ xk2 Þ V1 ðe, xÞ pffiffiffiffiffiffiffi MðkEðtÞ ek2 þ k ðtÞEðtÞ xk2 Þ ð23Þ where m ¼ min{min(PL), min(PK)} and max{max(PL), max(PK)}. From (22) and (23), it is easy to obtain
M¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi kEðtÞ ek2 þ k ðtÞEðtÞ xk2 rffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi M t kEðt Þ eðt Þk2 þ k ðt ÞEðt Þ xðt Þk2 exp m 2M ð24Þ
for t 2 [t*, Tf). Now, we assure that kE(t)ek, kE(t)xk and (t) are well-defined and bounded on [0, Tf). So, (e(t), x(t), L(t)) is well-defined and bounded on [0, Tf). Thus, (e(t), x(t), L(t)) is contained in a compact set and we can take Tf ¼ 1 (Khalil 2002). Then, we conclude that limt!1 ke(t)k ¼ 0 and limt!1 kx(t)k ¼ 0 from (24) _ ¼ 0 from (7), which results in and in turn, limt!1 ðtÞ limt!1 ðtÞ ¼ 5 1. (A2) case: The analysis for (A2) case is very similar to (A1) case. Again, using (9) and (10) with (8), the augmented closed-loop system is e_ ¼ AL ððtÞÞe þ ðt, x, uÞ x_ ¼ AK ððtÞÞx þ ðt, x, uÞ BKððtÞÞe _ ¼ j yj1=2 =ðtÞ, ð0Þ ¼ 1 ðtÞ
ð25Þ ð26Þ ð27Þ
Note that (27) has a closed-form solution of ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rt 2 0 j yj1=2 d þ ð0Þ2 . Thus, we only need to show that there exists a Lyapunov function V2(e, x) such that V_ 2 ðe, xÞ along the trajectories of (25) and (26) is negative definite on t 2 [t*, Tf). Then, one can analogously show that (e(t), x(t), L(t)) is well-defined and bounded for t 2 [0, 1) and eventually limt!1ke(t)k ¼ 0, limt!1 respectively. kx(t)k ¼ 0 and limt!1 kðtÞk ¼ ,
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Using Vo(e) and Vc(x) defined previously, we set V2(e, x) ¼ Vo(e) þ (t)1Vc(x) for the augmented system (25) and (26). Then, V_ 2 ðe,xÞ ¼ V_ o ðeÞ þ ðtÞ1 V_ c ðxÞ 1 _ _ _ ¼ _ due to ðtÞ ðtÞ2 ðtÞV c ðxÞ Vo ðeÞ þ ðtÞ Vc ðxÞ 1=2 j yj =ðtÞ 0, (t) 1 and Vc(x) 0. First, we observe V_ o ðeÞ with respect to (25). Note that juj (t)nkKkE(t)ek þ kE(t)xk). Using this, under (A2), we can obtain that kE(t)(t, x, u)k1 pffiffiffi (t)2m(kE(t)ek þ kE(t)xk) where m ¼ ðn 2Þ n þ ðn 1ÞkKk. Then, using (12),
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1 _ V_ o ðeÞ ððtÞ1 1 jðtÞjðtÞ ðtÞ2 2 ÞkEðtÞ ek2
þ 2ðtÞ2 2 kEðtÞ ekkEðtÞ xk
ð28Þ
where 1 ¼ 2snkPLk and 2 ¼ mkPLk, which are finite constants independent of (t) and . Next, we observe V_ c ðxÞ with respect to (26). Using (16), we have the following inequality: 1 _ V_ c ðxÞ ððtÞ1 2ðtÞ2 3 4 jðtÞjðtÞ ÞkEðtÞ xk
þ 2ððtÞ2 3 þ ðtÞ1 5 ÞkEðtÞ ekkEðtÞ xk
ð29Þ
where 3 ¼ mkPKk, 4 ¼ 2snkPKk and 5 ¼ kPKk kKk, which all are finite constants independent of (t) and . Now, using (28) and (29), kEðtÞ ek T 1 _ V2 ðe, xÞ kEðtÞ xk
2
kEðtÞ ek kEðtÞ xk
3. Illustrative examples Example 1: Consider the following triangular system taken from Lei and Lin (2006): x_ 1 ¼ x2 þ
x1 ð1 c1 x2 Þ2 þ x22
x_ 2 ¼ u þ lnð1 þ ðx22 Þc2 Þ y ¼ x1
ð31Þ
where c1 and c2 1 are unknown constants. This system belongs to (A1) case. With K ¼ [4, 4] and L ¼ [4, 4]T, the controller (5), (6) with (7) is engaged. For simulation, we set c1 ¼ 1, c2 ¼ 2, (x1(0), x2(0)) ¼ (1, 5) and (z1(0), z2(0)) ¼ (10, 2) (same as Lei and Lin (2006)). The simulation result is shown in Figure 1. Note that, in terms of performance, our result yields faster state convergence with less transitional fluctuation than the one in Lei and Lin (2006). For readers’ convenience, we provide the simulation result of Lei and Lin (2006) in Figure 1(b). (a) 5
x1 z1
0 –5 –10
0
1
2
3
4
5
10
ð30Þ
_ where 1 ¼ ðtÞ2 ððtÞ 1 jðtÞjðtÞ 2 Þ, 2 ¼ ðtÞ2 1 _ ð1 2ðtÞ 3 4 jðtÞjÞ, and ¼ (t)2 1 (2 þq(t) 3 þ 5). Using the solution form of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R _ ¼ j yj1=2 , we can ðtÞ ¼ 2 0t j yj1=2 d þ ð0Þ2 and jðtÞjðtÞ easily check that there exists a time t* such that 140 and 12 240 for t 2 [t*, Tf) as done in (A1) case. Thus, V_ 2 ðe, xÞ is negative definite on t 2 [t*, Tf). The rest of the analysis is the repetition of (A1) case. œ Remark 2.2: The main reason why the existing results cannot handle (A1) and (A2) together is that (A1) often requires high-gain-type controllers while (A2) often requires low-gain-type controllers. So, the control directions are opposite for each case. Our controller is designed such that, once K and L are set, the high-gain adaptive gain (7) or the low-gain adaptive gain (8) is easily engaged depending on the structure of non-linearity.
6 x2 z2
0 –10 −20
0
1
2
3
4
5
1
6 ε
0.5 0
0
1
2
3 4 time [sec]
5
(b) 5
6 x1 x1hat
0 –5 –10
0
1
2
3
4
5
40
6 x2 x2hat
20 0 –20
0
1
2
3
4
5
4
6 L
3
Remark 2.3: The system (1)–(2) can be directly extended by including zero dynamics as follows: z_¼ q(z, x), x_ ¼ Ax þ Bu þ ðt, x, uÞ, y ¼ Cx. Then, it follows that if z_¼ q(z, x) is input-to-state stable, the whole system is globally asymptotically stabilisable (see Lemma 13.2 of Khalil (2002)).
2 1
0
1
2
3 4 time [sec]
5
6
Figure 1. System response of Example 1: (a) proposed method and (b) method by Lei and Lin (2006).
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International Journal of Systems Science Example 2: Consider the following feedforward system taken from Xudong (2003) and slightly modified by imposing the linear growth condition and adding unknown time-varying constants c1(t) and c2(t) x_ 1 ¼ x2 þ sin x3 þ c1 ðtÞx3 x_ 2 ¼ x3 þ c2 ðtÞu x_ 3 ¼ u
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y ¼ x1
ð32Þ
where c1(t) and c2(t) are bounded by some unknown constants. This system belongs to (A2) case. With K ¼ [27, 27, 9] and L ¼ [9, 27, 27]T, the controller (5), (6) with (8) is engaged. We set c1(t) ¼ sin 10t, c2(t) ¼ cos 10t, (x1(0), x2(0), x3(0)) ¼ (1, 1, 2) and, (z1(0), z2(0), z3(0)) ¼ (0, 0, 0) for simulation. The simulation result is shown in Figure 2. Our result is compared with the result of Krishnamurthy and Khorrami (2007). First, there is a tradeoff in the (a) 60 40 20 0 –20
x1 z1
0
100
200
300
400
500
10
600 x2 z2
5 0 –5
0
100
200
300
400
500
600 x3 z3
4 2 0 –2 (b) 60 40 20 0 –20
0
100
200
300 400 time [sec]
500
600 x1 x1hat
0
100
200
300
400
500
10
0 –5
0
100
200
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4
0 –2
0
100
200
300 400 time [sec]
500
Acknowledgement This work was supported by research funds from Dong-A University.
600 x2 x3hat
2
4. Conclusions We have presented an adaptive output feedback controller to globally stabilise non-linear systems whose non-linearity has the unknown linear growth rate. So, the so-called the universal feature is also retained in our work. Moreover, our method can deal not only with the triangular system but also with the feedforward system in a unified framework, whereas the existing results are limited to tackle only one specific form of non-linear systems. Thus, our result is generalised over the existing results. Notably, there can be several extensions. For example, in (3) and (4) can be generalised to be a function of y, i.e. (y). Or, some high-order terms can be imposed on (3) and (4). Or, a generalisation by allowing zero dynamics in (1), (3) and (4) can be studied. All these extensions can be pursued in a future work.
600 x2 x2hat
5
response of x1 between two results. Our result yields a slower response in x1, but there is a large overshoot in x1 by the method of Krishnamurthy and Khorrami (2007). Both trajectories are somewhat similar in x2 trajectories and our result yields a slightly faster response with less transitional fluctuation in x3 trajectories. For readers’ convenience, we provide the simulation result of Krishnamurthy and Khorrami (2007) in Figure 2(b). As a final note, the results of Choi and Lim (2005b) and Lei and Lin (2006) are applicable only to Example 1. The result of Krishnamurthy and Khorrami (2008) is not applicable to Example 1 because 1(t, x, u) clearly contains x2. The results of Krishnamurthy and Khorrami (2007), Xudong (2003) and Ye and Unbehauen (2004) are applicable only to Example 2. In contrast, our output feedback control method can deal with both cases in a unified way, thus it indeed broadens the class of non-linear systems under consideration over the existing results.
600
Figure 2. System response of Example 2: (a) proposed method and (b) method by Krishnamurthy and Khorrami (2007).
Notes on contributors Ho-Lim Choi received the BSE degree from the department of electrical engineering, The University of Iowa, USA in 1996, MS degree in 1999 and PhD degree in 2004, from KAIST (Korea Advanced Institute of Science and Technology). Currently, he is an Assistant Professor at the Department of Electrical Engineering,
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Dong-A university, Busan. His research interests are in the areas of non-linear control problems with emphasis on feedback linearisation, gain scheduling, singular perturbation, output feedback, time-delay systems. He is a member of IEEE, IEICE and ICROS. Jong-Tae Lim received the B.S.E.E. degree from Yonsei University, Seoul, Korea, in 1975, the MSEE degree from the Illinois Institute of Technology, Chicago, in 1983 and the PhD 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.
References Choi, H.-L., and Lim, J.-T. (2004), ‘On Robust Approximate Feedback Linearization’, IEICE Transactions on Fundamentals, E87-A(2), 502–504. Choi, H.-L., and Lim, J.-T. (2005), ‘Global Exponential Stabilization of a Class of Nonlinear Systems by Output Feedback’, IEEE Transactions on Automatic Control, 50(2), 255–257. Choi, H.-L., and Lim, J.-T. (2005), ‘Stabilization of a Class of Nonlinear Systems by Adaptive Output Feedback’, Automatica, 41(6), 1091–1097. Freeman, R.A., and Kokotovic´, P.V. (1993), ‘Design of softer Robust Nonlinear Control Laws’, Automatica, 29(6), 1425–1437.
Jo, N.H., and Seo, J.H. (2000a), ‘A State Observer for Nonlinear Systems and Its Application to Ball and Beam System’, IEEE Transactions on Automatic Control, 45(5), 968–973. Jo, N.H., and Seo, J.H. (2000b), ‘Input Output Linearization Approach to State Observer Design for Nonlinear System’, IEEE Transactions on Automatic Control, 45(12), 2388–2393. Khalil, H.K. (2002), Nonlinear Systems, Upper Saddle River, NJ: Prentice Hall. Krishnamurthy, P., and Khorrami, F., (2007) ‘Adaptive Output-feedback Control of Feedforward Systems with Uncertain Parameters Coupled with All States’ in Proceedings of the 2007 American Control ConferenceNew York, USA, pp. 480–485. Krishnamurthy, P., and Khorrami, F. (2008), ‘Dual Highgain-based Adaptive Output-feedback Control for a Class of Nonlinear Systems’, International Journal of Adaptive Control and Signal Processing, 22, 23–42. Lei, H., and Lin, W. (2006), ‘Universal Adaptive Control of Nonlinear Systems with Unknown Growth Rate by Output Feedback’, Automatica, 42(10), 1783–1789. Mazenc, F., Praly, L., and Dayawansa, W.P. (1994), ‘Global Stabilisation by Output Feedback: Examples and Counter Examples’, Systems and Control Letters, 23(2), 119–125. Qian, C., and 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. Rugh, W.J. (1996), Linear System Theory, Upper Saddle River, NJ: Prentice Hall. Xudong, Y. (2003), ‘Universal Stabilization of Feedforward Nonlinear Systems’, Automatica, 39(1), 141–147. Ye, X., and Unbehauen, H. (2004), ‘Global Adaptive Stabilization for a Class of Feedforward Nonlinear Systems’, IEEE Transactions on Automatic Control, 49(5), 786–792.
Stabilisation of non-linear systems with unknown growth rate by adaptive output feedback Ho-Lim Choia and Jong-Tae Limb* a
b
Department of Electrical Engineering, Dong-A University, 840 Hadan2-Dong, Saha-gu, Busan, 604-714, Korea; Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, Korea
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(Received 10 October 2008; final version received 2 June 2009) In recent years, several results have been proposed on the global stabilisation of non-linear systems with unknown linear growth rate. However, these works are limited in the sense that they consider only one particular form of non-linear systems – mostly either triangular or feedforward form. We propose an adaptive output feedback control scheme which can deal with both triangular or feedforward non-linear systems with unknown linear growth rate in a unified framework. Thus, our result broadens the class of non-linear systems under consideration over the existing results. Keywords: non-linear systems; output feedback; unknown growth rate
1. Introduction We consider a class of single-input single-output nonlinear systems given by x_ ¼ Ax þ Bu þ ðt, x, uÞ y ¼ Cx
ð1Þ ð2Þ
where x 2 Rn, u 2 R and y 2 R are the system state, input and output, respectively. The system matrices are A, B and C, where (A, B) is the Brunovsky canonical pair and C ¼ [1, 0, . . . , 0]. The non-linear term is (t, x, u) ¼ [1(t, x, u), . . . , n(t, x, u)]T, where i(t, x, u): R Rn R ! R, i ¼ 1, . . . , n are C1 with respect to all the variables. Regarding the system non-linearity (t, x, u), we introduce the following two conditions. (A1) For i ¼ 1, . . . , n, there exists an unknown constant 0 such that ji ðt, x, uÞj ðjx1 j þ þ jxi jÞ
ð3Þ
(A2) For i ¼ 1, . . . , n 2, there exists an unknown constant 0 such that ji ðt, x, uÞj ðjxiþ2 j þ þ jxn j þ jujÞ
ð4Þ
with jn1(t, x, u)j juj and n(t, x, u) ¼ 0. We note that (A1) is the so-called ‘triangular condition’ as assumed in Lei and Lin (2006). A similar type of triangular form is considered in Krishnamurthy and Khorrami (2008), but i(t, x, u) is only assumed to be a function of (x1, . . . , xi) excluding time-variable t and states (xiþ1, . . . , xn). Since (A1) is given as a *Corresponding author. Email: [email protected] ISSN 0020–7721 print/ISSN 1464–5319 online 2010 Taylor & Francis DOI: 10.1080/00207720903144529 http://www.informaworld.com
norm-bound condition, the states (xiþ1, . . . , xn) can be included in i(t, x, u) unlike Krishnamurthy and Khorrami (2008) (Example 1). The other condition (A2) is the so-called ‘feedforward condition’, which is a linear growth version of the ones in Xudong (2003) and Ye and Unbehauen (2004) and is a restricted version of the one in Krishnamurthy and Khorrami (2007). The class of non-linear systems which fit the system (1)–(2) under either (A1) or (A2) are well addressed in Qian and Lin (2002), Xudong (2003) and references therein. Now, we address how the existing results have been progressed along with the two conditions mentioned above. First, when the growth rate is known, (A1) reduces the conditions assumed in Freeman and Kokotovic´ (1993) and Qian and Lin (2002), where the state and output feedback controllers are proposed, respectively, for the global stabilisation of the system (1)–(2). Later on, the authors of Choi and Lim (2004, 2005a) extend the results of Freeman and Kokotovic´ (1993) and Qian and Lin (2002) by including both (A1) and (A2) with being known and suggest a single form of globally stabilising state/output feedback controllers. However, all these results require a priori knowledge on in order to determine the gains of controllers. Then, when the growth rate is unknown, some form of additional gain-tuning mechanism is needed for stabilising controllers. In Choi and Lim (2005b) and later in Lei and Lin (2006), the authors, respectively, suggest the adaptive output feedback control methods which solve the global stabilisation problem of the system (1)–(2) under (A1). The results of
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Xudong (2003) and Ye and Unbehauen (2004) can handle (A2) case, but they use the full-state feedback controller. The result of Krishnamurthy and Khorrami (2007) can also handle (A2). Thus, to the best of our knowledge, there have been no output feedback control methods which can be applied to both (A1) and (A2) cases in a unified framework. So, here is our extended control problem: The system is globally asymptotically stabilised (1)–(2) by output feedback when only the structure of (t, x, u) is known such that it belongs to either (A1) or (A2). In order to solve this problem, we utilise the structure of controller in Choi and Lim (2005a) and combine it with an adaptive gain-tuning mechanism which is appropriately engaged depending on (A1) or (A2) case. Note that (A1) and (A2) are global linear growth conditions, which may look to be somewhat limited in the first hand. However, as investigated in Mazenc Praly, and Dayawansa (1994), some extra growth conditions on the unmeasurable states of the system are usually necessary for the global stabilisation of non-linear systems via output feedback. For example, a global Lipschitz condition is assumed in Jo and Seo (2000b), global linear growth conditions are assumed in Choi and Lim (2005a, b), Lei and Lin (2006) and Qian and Lin (2002) for the global stabilisation problem of non-linear systems. Under a local Lipschitz condition, only local stabilisation is achieved by output feedback (Jo and Seo 2000a). In this sense, for the global stabilisation problem of non-linear systems by output feedback, (A1) and (A2) are not much restricted conditions, but more relaxed than the conditions in related results.
2. Adaptive output feedback controller First, we introduce the following dynamic output feedback controller u ¼ KððtÞÞz z_ ¼ Az þ Bu LððtÞÞð y CzÞ
ð5Þ ð6Þ
where K((t)) ¼ [k1/(t)n, . . . , kn/(t)] and L((t)) ¼ _ ¼ [l1/(t), . . . , ln/(t)n]T, (t)40. The dynamic gain (t) (y, (t)) is to be introduced later on. Here, we introduce some notations for convenience. Notations: We define E(t) :¼ diag[1, (t), . . . , (t)n1], K :¼ K((t))j(t)¼1, L :¼ L((t))j(t)¼1, AK :¼ A þ BK, AL :¼ A þ LC, AK((t)) :¼ A þ BK((t)) and AL((t)) :¼ A þ L((t))C. Then, using these notations, the controller u in (5) can be expressed as u ¼ (t)n KE(t)z. Now, we state the main result.
Theorem 2.1: Suppose that (i) (A1) or (A2) holds for the system (1)–(2); (ii) K and L are selected such that AK and AL are Hurwitz, respectively. Under (A1), the controller (5)–(6) is engaged with _ ¼ j yjðtÞ, ð0Þ ¼ 1 ðtÞ
ð7Þ
Under (A2), the controller (5)–(6) is engaged with _ ¼ j yj1=2 =ðtÞ, ð0Þ ¼ 1 ðtÞ
ð8Þ
Then, the closed-loop system (1)–(2) becomes globally asymptotically stable and limt!1 ðtÞ ¼ 5 1 for each case. Proof: First, we introduce brief mathematical setups for the proof. Since AK and AL are Hurwitz, as it follows from Choi and Lim (2005a), we can derive two Lyapunov equations of with ATj ððtÞÞPj ððtÞÞ þ Pj ððtÞÞAj ððtÞÞ ¼ ðtÞ1 E2ðtÞ with Pj((t)) ¼ E(t)PjE(t) where ATj Pj þ Pj Aj ¼ I for j ¼ K, L. Here, I denotes an n n identity matrix. (A1) case: Define ei ¼ xi zi, 1 i n. By subtracting (6) from (1), the error dynamics is obtained as e_ ¼ AL ððtÞÞe þ ðt, x, uÞ
ð9Þ
With (5) and (7), the closed-loop system becomes x_ ¼ AK ððtÞÞx þ ðt, x, uÞ BKððtÞÞe _ ¼ j yjðtÞ, ð0Þ ¼ 1 ðtÞ
ð10Þ ð11Þ
We begin with showing the existence of a unique solution. The vector field of the augmented system (9) and (10) is C1 with respect to its arguments. Regarding (11), Rwe have a closed-form solution of ðtÞ ¼ t expð 0 j yjdÞ. Thus, the corresponding solution (e(t), x(t), (t)) of (9)–(11) exists and is unique on [0, Tf) for some Tf 2 (0, 1]. Moreover, since ¼ AL((t))e and x_ ¼ AK ððtÞÞx are stable (time-varying) linear systems and (t, x, u) satisfies the global linear growth condition, it can be easily checked by using Gronwall– Bellman inequality that there is no finite-time escape phenomenon (Rugh 1996). Thus, Tf can be arbitrarily large. Now, we will show that (e(t), x(t), (t)) is welldefined and bounded on [0, Tf). First, we set a Lyapunov function Vo(e) ¼ eTPL((t))e. Then, along the trajectory of (9), V_ o ðeÞ ¼ e_T PL ððtÞÞe þ eT PL ððtÞÞe_ þ eT P_ L ððtÞÞe ¼ ðtÞ1 kE ðtÞek2 þ eT P_ L ððtÞÞe þ 2eT EðtÞ PL EðtÞ ðt, x, uÞ
ð12Þ
International Journal of Systems Science Under (A1), note that kE(t)(t, x, u)k1 n3/2kE(t)xk. Also, we have that eT P_ L ððtÞÞe ¼ eT ðE_ ðtÞ PL EðtÞ þ EðtÞ PL E_ ðtÞ Þe and E_ (t) ¼ E(t)D ¼ DE(t) where 1 _ diag½0, 1, . . . , n 1 D ¼ ðtÞðtÞ
ð13Þ
This leads to the following inequality: _ kEðtÞ ek e P_ L ððtÞÞe 2sn kPL kjðtÞjðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where sn ¼ nðn 1Þð2n 1Þ=6. Then, we have T
1
2
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1 _ ÞkEðtÞ ek2 V_ o ðeÞ ððtÞ1 1 jðtÞjðtÞ
þ 22 kEðtÞ ekkEðtÞ xk
ð15Þ
where 1 ¼ 2snkPLk and 2 ¼ n3/2kPLk, which are finite constants independent of (t) and . Next, we set a Lyapunov function Vc(x) ¼ xT PK((t))x. Then, along the trajectory of (10), V_ c ðxÞ ¼ x_ T PK ððtÞÞx þ xT PK ððtÞÞx_ þ xT P_ K ððtÞÞx ¼ ðtÞ1 kEðtÞ xk2 þ 2xT EðtÞ PK EðtÞ ðt, x, uÞ þ xT P_ K ððtÞÞx 2xT EðtÞ PK EðtÞ BKððtÞÞe ð16Þ Note that E(t)BK((t))e ¼ (t)1 BKE(t)e. Also, from (14), we can deduce that 1 _ kEðtÞ xk2 xT P_ K ððtÞÞx 2sn kPK kjðtÞjðtÞ
ð17Þ
Thus, we have 1 _ V_ c ðxÞ ððtÞ1 3 4 jðtÞjðtÞ ÞkEðtÞ xk2
þ 2ðtÞ1 5 kEðtÞ ekkEðtÞ xk
ð18Þ
where 3 ¼ 2n3/2kPKk, 4 ¼ 2snkPKk and 5 ¼ kPKk kKk, which all are finite constants independent of (t) and . Now, we set a composite Lyapunov function V1(e, x) ¼ Vo(e) þ (t)Vc(x) for the augmented system (9) and (10). Then, we have V_ 1 ðe, xÞ ¼ V_ o ðeÞ þ _ _ _ ðtÞV_ c ðxÞ þ ðtÞV c ðxÞ Vo ðeÞ þ ðtÞVc ðxÞ due to which _ ¼ j yjðtÞ 0, (t)40 and Vc(x) 0. Thus, using ðtÞ the results of (15) and (18), V_ 1 ðe, xÞ
kEðtÞ ek kEðtÞ xk
T
1
2
kEðtÞ ek kEðtÞ xk
ð19Þ
1 _ where 1 ¼ ðtÞ1 1 jðtÞjðtÞ , 2 ¼ ðtÞððtÞ1 1 _ Þ and ¼ ( 2 þ 5). 3 4 jðtÞjðtÞ From (19), V_ 1 ðe, xÞ becomes negative definite if and only if (i) 140 and (ii) 12 240. Regarding (i), it can be rewritten as Z t ð20Þ exp j yjd 1 j yj 4 0 0
Thus, there must exist a finite time t1 such that 140 for t1 t 5 Tf. Regarding (ii), it can be rewritten as Z t j yj j yjd 3 4 j yj 1 1 exp Rt expð 0 j yjdÞ 0 ð2 þ 5 Þ2 4 0
ð14Þ
675
ð21Þ
Thus, there must exist a finite time t2 such that 12 ( 2 þ 5)240 for t2 t 5 Tf. Now, let us define t* ¼ max{t1, t2}. Then, we have V_ 1 ðe, xÞ ðkEðtÞ ek2 þ kEðtÞ xk2 Þ pffiffiffiffiffiffiffi ðkEðtÞ ek2 þ k ðtÞEðtÞ xk2 Þ, 4 0
ð22Þ
for all t 2 [t*, Tf). The last inequality comes from the fact that (t) 1 with (7). Note that pffiffiffiffiffiffiffi mðkEðtÞ ek2 þ k ðtÞEðtÞ xk2 Þ V1 ðe, xÞ pffiffiffiffiffiffiffi MðkEðtÞ ek2 þ k ðtÞEðtÞ xk2 Þ ð23Þ where m ¼ min{min(PL), min(PK)} and max{max(PL), max(PK)}. From (22) and (23), it is easy to obtain
M¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi kEðtÞ ek2 þ k ðtÞEðtÞ xk2 rffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi M t kEðt Þ eðt Þk2 þ k ðt ÞEðt Þ xðt Þk2 exp m 2M ð24Þ
for t 2 [t*, Tf). Now, we assure that kE(t)ek, kE(t)xk and (t) are well-defined and bounded on [0, Tf). So, (e(t), x(t), L(t)) is well-defined and bounded on [0, Tf). Thus, (e(t), x(t), L(t)) is contained in a compact set and we can take Tf ¼ 1 (Khalil 2002). Then, we conclude that limt!1 ke(t)k ¼ 0 and limt!1 kx(t)k ¼ 0 from (24) _ ¼ 0 from (7), which results in and in turn, limt!1 ðtÞ limt!1 ðtÞ ¼ 5 1. (A2) case: The analysis for (A2) case is very similar to (A1) case. Again, using (9) and (10) with (8), the augmented closed-loop system is e_ ¼ AL ððtÞÞe þ ðt, x, uÞ x_ ¼ AK ððtÞÞx þ ðt, x, uÞ BKððtÞÞe _ ¼ j yj1=2 =ðtÞ, ð0Þ ¼ 1 ðtÞ
ð25Þ ð26Þ ð27Þ
Note that (27) has a closed-form solution of ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rt 2 0 j yj1=2 d þ ð0Þ2 . Thus, we only need to show that there exists a Lyapunov function V2(e, x) such that V_ 2 ðe, xÞ along the trajectories of (25) and (26) is negative definite on t 2 [t*, Tf). Then, one can analogously show that (e(t), x(t), L(t)) is well-defined and bounded for t 2 [0, 1) and eventually limt!1ke(t)k ¼ 0, limt!1 respectively. kx(t)k ¼ 0 and limt!1 kðtÞk ¼ ,
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Using Vo(e) and Vc(x) defined previously, we set V2(e, x) ¼ Vo(e) þ (t)1Vc(x) for the augmented system (25) and (26). Then, V_ 2 ðe,xÞ ¼ V_ o ðeÞ þ ðtÞ1 V_ c ðxÞ 1 _ _ _ ¼ _ due to ðtÞ ðtÞ2 ðtÞV c ðxÞ Vo ðeÞ þ ðtÞ Vc ðxÞ 1=2 j yj =ðtÞ 0, (t) 1 and Vc(x) 0. First, we observe V_ o ðeÞ with respect to (25). Note that juj (t)nkKkE(t)ek þ kE(t)xk). Using this, under (A2), we can obtain that kE(t)(t, x, u)k1 pffiffiffi (t)2m(kE(t)ek þ kE(t)xk) where m ¼ ðn 2Þ n þ ðn 1ÞkKk. Then, using (12),
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1 _ V_ o ðeÞ ððtÞ1 1 jðtÞjðtÞ ðtÞ2 2 ÞkEðtÞ ek2
þ 2ðtÞ2 2 kEðtÞ ekkEðtÞ xk
ð28Þ
where 1 ¼ 2snkPLk and 2 ¼ mkPLk, which are finite constants independent of (t) and . Next, we observe V_ c ðxÞ with respect to (26). Using (16), we have the following inequality: 1 _ V_ c ðxÞ ððtÞ1 2ðtÞ2 3 4 jðtÞjðtÞ ÞkEðtÞ xk
þ 2ððtÞ2 3 þ ðtÞ1 5 ÞkEðtÞ ekkEðtÞ xk
ð29Þ
where 3 ¼ mkPKk, 4 ¼ 2snkPKk and 5 ¼ kPKk kKk, which all are finite constants independent of (t) and . Now, using (28) and (29), kEðtÞ ek T 1 _ V2 ðe, xÞ kEðtÞ xk
2
kEðtÞ ek kEðtÞ xk
3. Illustrative examples Example 1: Consider the following triangular system taken from Lei and Lin (2006): x_ 1 ¼ x2 þ
x1 ð1 c1 x2 Þ2 þ x22
x_ 2 ¼ u þ lnð1 þ ðx22 Þc2 Þ y ¼ x1
ð31Þ
where c1 and c2 1 are unknown constants. This system belongs to (A1) case. With K ¼ [4, 4] and L ¼ [4, 4]T, the controller (5), (6) with (7) is engaged. For simulation, we set c1 ¼ 1, c2 ¼ 2, (x1(0), x2(0)) ¼ (1, 5) and (z1(0), z2(0)) ¼ (10, 2) (same as Lei and Lin (2006)). The simulation result is shown in Figure 1. Note that, in terms of performance, our result yields faster state convergence with less transitional fluctuation than the one in Lei and Lin (2006). For readers’ convenience, we provide the simulation result of Lei and Lin (2006) in Figure 1(b). (a) 5
x1 z1
0 –5 –10
0
1
2
3
4
5
10
ð30Þ
_ where 1 ¼ ðtÞ2 ððtÞ 1 jðtÞjðtÞ 2 Þ, 2 ¼ ðtÞ2 1 _ ð1 2ðtÞ 3 4 jðtÞjÞ, and ¼ (t)2 1 (2 þq(t) 3 þ 5). Using the solution form of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R _ ¼ j yj1=2 , we can ðtÞ ¼ 2 0t j yj1=2 d þ ð0Þ2 and jðtÞjðtÞ easily check that there exists a time t* such that 140 and 12 240 for t 2 [t*, Tf) as done in (A1) case. Thus, V_ 2 ðe, xÞ is negative definite on t 2 [t*, Tf). The rest of the analysis is the repetition of (A1) case. œ Remark 2.2: The main reason why the existing results cannot handle (A1) and (A2) together is that (A1) often requires high-gain-type controllers while (A2) often requires low-gain-type controllers. So, the control directions are opposite for each case. Our controller is designed such that, once K and L are set, the high-gain adaptive gain (7) or the low-gain adaptive gain (8) is easily engaged depending on the structure of non-linearity.
6 x2 z2
0 –10 −20
0
1
2
3
4
5
1
6 ε
0.5 0
0
1
2
3 4 time [sec]
5
(b) 5
6 x1 x1hat
0 –5 –10
0
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40
6 x2 x2hat
20 0 –20
0
1
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4
5
4
6 L
3
Remark 2.3: The system (1)–(2) can be directly extended by including zero dynamics as follows: z_¼ q(z, x), x_ ¼ Ax þ Bu þ ðt, x, uÞ, y ¼ Cx. Then, it follows that if z_¼ q(z, x) is input-to-state stable, the whole system is globally asymptotically stabilisable (see Lemma 13.2 of Khalil (2002)).
2 1
0
1
2
3 4 time [sec]
5
6
Figure 1. System response of Example 1: (a) proposed method and (b) method by Lei and Lin (2006).
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International Journal of Systems Science Example 2: Consider the following feedforward system taken from Xudong (2003) and slightly modified by imposing the linear growth condition and adding unknown time-varying constants c1(t) and c2(t) x_ 1 ¼ x2 þ sin x3 þ c1 ðtÞx3 x_ 2 ¼ x3 þ c2 ðtÞu x_ 3 ¼ u
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y ¼ x1
ð32Þ
where c1(t) and c2(t) are bounded by some unknown constants. This system belongs to (A2) case. With K ¼ [27, 27, 9] and L ¼ [9, 27, 27]T, the controller (5), (6) with (8) is engaged. We set c1(t) ¼ sin 10t, c2(t) ¼ cos 10t, (x1(0), x2(0), x3(0)) ¼ (1, 1, 2) and, (z1(0), z2(0), z3(0)) ¼ (0, 0, 0) for simulation. The simulation result is shown in Figure 2. Our result is compared with the result of Krishnamurthy and Khorrami (2007). First, there is a tradeoff in the (a) 60 40 20 0 –20
x1 z1
0
100
200
300
400
500
10
600 x2 z2
5 0 –5
0
100
200
300
400
500
600 x3 z3
4 2 0 –2 (b) 60 40 20 0 –20
0
100
200
300 400 time [sec]
500
600 x1 x1hat
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500
10
0 –5
0
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0 –2
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300 400 time [sec]
500
Acknowledgement This work was supported by research funds from Dong-A University.
600 x2 x3hat
2
4. Conclusions We have presented an adaptive output feedback controller to globally stabilise non-linear systems whose non-linearity has the unknown linear growth rate. So, the so-called the universal feature is also retained in our work. Moreover, our method can deal not only with the triangular system but also with the feedforward system in a unified framework, whereas the existing results are limited to tackle only one specific form of non-linear systems. Thus, our result is generalised over the existing results. Notably, there can be several extensions. For example, in (3) and (4) can be generalised to be a function of y, i.e. (y). Or, some high-order terms can be imposed on (3) and (4). Or, a generalisation by allowing zero dynamics in (1), (3) and (4) can be studied. All these extensions can be pursued in a future work.
600 x2 x2hat
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response of x1 between two results. Our result yields a slower response in x1, but there is a large overshoot in x1 by the method of Krishnamurthy and Khorrami (2007). Both trajectories are somewhat similar in x2 trajectories and our result yields a slightly faster response with less transitional fluctuation in x3 trajectories. For readers’ convenience, we provide the simulation result of Krishnamurthy and Khorrami (2007) in Figure 2(b). As a final note, the results of Choi and Lim (2005b) and Lei and Lin (2006) are applicable only to Example 1. The result of Krishnamurthy and Khorrami (2008) is not applicable to Example 1 because 1(t, x, u) clearly contains x2. The results of Krishnamurthy and Khorrami (2007), Xudong (2003) and Ye and Unbehauen (2004) are applicable only to Example 2. In contrast, our output feedback control method can deal with both cases in a unified way, thus it indeed broadens the class of non-linear systems under consideration over the existing results.
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Figure 2. System response of Example 2: (a) proposed method and (b) method by Krishnamurthy and Khorrami (2007).
Notes on contributors Ho-Lim Choi received the BSE degree from the department of electrical engineering, The University of Iowa, USA in 1996, MS degree in 1999 and PhD degree in 2004, from KAIST (Korea Advanced Institute of Science and Technology). Currently, he is an Assistant Professor at the Department of Electrical Engineering,
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Dong-A university, Busan. His research interests are in the areas of non-linear control problems with emphasis on feedback linearisation, gain scheduling, singular perturbation, output feedback, time-delay systems. He is a member of IEEE, IEICE and ICROS. Jong-Tae Lim received the B.S.E.E. degree from Yonsei University, Seoul, Korea, in 1975, the MSEE degree from the Illinois Institute of Technology, Chicago, in 1983 and the PhD 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.
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