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Stable adaptive controller design for uncertain phase shift
Annual Reviews in Control 29 (2005) 217–228 www.elsevier.com/locate/arcontrol
Stable adaptive controller design for uncertain phase shift Yoshihiko Miyasato * The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-Ku, Tokyo 106-8569, Japan Received 8 December 2004; accepted 1 June 2005
Abstract In order to construct stable adaptive control systems, several assumptions are needed, and those require some prior information of unknown plants, such as degrees and relative degrees. However, those assumptions sometimes become too restrictive, since it is difficult to obtain reasonable estimations of those indices a priori for unknown practical processes. This paper reviews the authors’ research works where some of those assumptions can be relaxed partially. The basic concepts behind those and the restrictions of the present version are summarized, and the future possibilities of the proposed methodologies are also discussed. # 2005 Elsevier Ltd. All rights reserved. Keywords: Adaptive control; Stability analysis; Model reference adaptive control
1. Introduction Every researcher who begins to study adaptive control, is confronted with the fact that several assumptions are needed to design stable adaptive control systems for unknown processes. Some of those assumptions require prior information of unknown plants. That seems a contradiction of adaptive control, since adaptive control methodologies have been developed for the case where system characteristics cannot be specified a priori. For stable adaptive controller designs, especially for stable model reference adaptive controller designs, the following assumptions are necessary. Assumption of adaptive control: (1) (2) (3) (4)
Degrees or upper bounds of degrees of plants are known. Relative degrees of plants are known. Signs of high-frequency gains of plants are known. Plants are minimum-phase processes.
The Assumption (3) (signs of high-frequent gains) is needed for the direct approach, but is not necessary for certain indirect approach where identification models are involved explicitly. The necessity of those assumptions is because the knowledge of those indices (especially, degrees and relative degrees) is utilized to determine the orders and * Tel.: +81 3 5421 8752; fax: +81 3 3446 1695. E-mail address: [email protected]. 1367-5788/$ – see front matter # 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.arcontrol.2005.06.001
structures of adaptive controllers with sufficient freedom of achieving control objective, such as model-following and stabilization. However, those assumptions sometimes become too restrictive, since it is difficult to obtain reasonable estimations of those indices a priori for unknown practical processes. Hence, the relaxation of those assumptions has been of great importance from both theoretical and practical point of views. The purpose of the present paper is to summarize the authors’ previous researches where the Assumptions (1)–(3) can be relaxed to some extent (Miyasato, 1997a, 1997b, 1998a, 1998b, 1998c, 1998d, 1999a, 1999b, 2000a, 2000b; Miyasato & Hanba, 1997). Especially, design methods of stable adaptive controllers for processes with uncertain relative degrees are provided. The proposed methodologies are composed of several kinds of control structures such as model reference control, stabilizing control and servo compensation, but the basic concept behind those is common to all of them, and that concept will be clarified without going into the detailed stability analysis. Additionally, the restriction of the present version and the future possibility are also discussed. 2. Assumption – prior information The present section summarize the reasons why those assumptions are required to construct stable adaptive control systems, and review the previous works to relax the assumptions.
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2.1. Assumption (1): Degree or upper bound of degree Model reference adaptive control achieves model-following control, and that control scheme is composed of state feedback (or equivalent procedure via output feedback) and feedforward compensations. The state feedback compensation allocates open-loop poles to desired closed-loop ones which cause polezero cancellation of plants. Therefore, the information on the degree or the upper bound of the degree of plants, is needed to specify the order of state feedback compensation, or the order of the equivalent dynamic compensation. One way to relax this assumption, is to introduce high-gain feedback scheme, where approximate model-following control is achieved under the several conditions. Related works are found in (Miller & Davison, 1991; Morse, 1987b). 2.2. Assumption (2): Relative degree The information on the relative degree is needed to specify the order of feedforward compensation, and also utilized to determine stable adaptation scheme which makes error processes strictly positive real. The relative degrees of strictly positive real transfer functions are restricted to 1, 0, and 1, and the order of the adaptation scheme achieving strictly positive realness, is concerned with the relative degree of the plant itself. Multi-controller assignments for several candidates of relative degrees and proper switching strategy of those adaptive controllers are useful way to solve the problem of uncertain relative degrees. Also, iterative identification and controller design procedures such as ‘‘wind-surfer algorithms’’ are practical solutions to uncertain relative degrees. Related works are found in (Anderson & Kosut, 1991; Morse, 1996; Morse, Mayne & Goodwin, 1992; Tao & Ioannou, 1989). On the contrary, stable adaptive controllers of single structures for uncertain relative degrees have been studied, where the property of the relative degree of strictly positive real transfer functions is utilized (Morse, 1985, 1987a). The present paper is considered as an extended version of those. 2.3. Assumption (3): Sign of high-frequency gain This assumption is necessary only for the case of the direct approach of adaptive control. In the indirect approach, where explicit identification models are involved, and the tuning of parameters are carried out based on identification errors, that assumption is not needed. On the contrary, the direct approach which does not utilize the knowledge on the sign of highfrequency gain, is also studied (Morse, 1985, 1987a)by introducing Nussbaum gain(Nussbaum, 1983). The present methods also deal with uncertain sign of high-frequency gains by utilizing the same strategy.
cancellation of plants. Hence, the zeros of plants should be stable in order that unstable pole-zero cancellation does not occur. The assumption of stable zeros is needed for model reference adaptive control and high-gain adaptive stabilization. However, adaptive pole-placement control, where there is no pole-zero cancellation, can be applied to plants with unstable zeros (Elliott, Cristi, & Das, 1985; Kreisselmeier, 1980, 1985). An alternative way is to utilize periodic feedback with multirate sampling. 3. Uncertain relative degree–main concept This paper considers the case where relative degrees are partially unknown. That is, relative degrees are known to be r; r þ 1, or r þ 2with known r 2 N, but they cannot be specified in more detail. The uncertainty of relative degrees r r þ 2 lead to the uncertainty of phase shifts 0 p of plants, and it is difficult to conquer uncertain phase shift 0 p even by the framework of the recent developed robust control theory. Contrary to it, in the proposed methodology, a single adaptive controller can deal with such uncertainty, where there is no switching of multiple controllers. In order to solve that problem, the property of strictly positive real transfer functions is utilized, that is, relative degrees of strictly positive real transfer functions are þ1, 0, and 1. This can be also certified in the next simple example. Consider the following three error systems with relative degrees þ1, 0, and 1, respectively d ˜ T vðtÞ ðCase 1Þ; þ l eðtÞ ¼ uðtÞ dt ˜ T vðtÞ ðCase 2Þ; eðtÞ ¼ uðtÞ d ˜ T vðtÞ ðCase 3Þ; þ l uðtÞ eðtÞ ¼ dt ˆ u; ˜ ¼ uðtÞ uðtÞ where eðtÞ is an error signal, u is an unknown parameter vector, ˆ is an estimate of u, and vðtÞ is a state vector. l is a positive uðtÞ constant (a design parameter). Each dimensions of those are determined properly. The problem here is to find out a single ˆ which is adequate for all of those three error adaptive law of uðtÞ processes. The answer for such question is given by ˙ˆ ¼ uðtÞ ˙˜ ¼ GvðtÞeðtÞ; uðtÞ
(1)
where G is a positive definite matrix. The fitness of that adaptive law can be certified from the following stability analysis. First consider Case 1. In this case, define V by ˜ T G1 uðtÞ ˜ V ¼ 12eðtÞ2 þ 12uðtÞ ð 0Þ;
2.4. Assumption (4): Minimum phase
and take the time derivative of Valong the trajectory of the error system (Case 1)
In order to achieve model-following control, the feedback compensation in the adaptive controller, allocates open-loop poles to desired closed-loop poles, which causes pole-zero
V˙ ¼ leðtÞ2 ð 0Þ: Hence, it follows that e 2 L1 \ L2 , and uˆ 2 L1 .
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
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ˆ ¼ uˆ 1 ðtÞ þ uˆ 2 ðtÞ; uðtÞ uˆ 1 ðtÞ ¼ GvðtÞeðtÞ;
For Case 2, V is defined by ˜ T G1 uðtÞ ˜ V ¼ 12uðtÞ ð 0Þ: Then, the time derivative of V along the error system (Case 2) is given by 2
V˙ ¼ eðtÞ ð 0Þ; and this yields e 2 L2 , and uˆ 2 L1 . For Case 3, V is defined with the filtered error signal e f ˜ T G1 uðtÞ ˜ ð 0Þ; V ¼ 12e f ðtÞ2 þ 12uðtÞ d þ l e f ðtÞ ¼ eðtÞ: dt The time derivative of V along the trajectory of the error system (Case 3) is given by
u˙ˆ 2 ðtÞ ¼ GflvðtÞ vðtÞgeðtÞ: ˙
(4)
In the main part of the proposed methodologies, the same relations between the error systems Case 10 , Case 20 , Case 30 (or equivalent forms Case 100 , Case 200 , Case 300 ) and the correspondent adaptive law (3)(or equivalently (4)), are utilized. It should be noted that the single adaptive law can deal with the three different error systems. Remark 1. The universal adaptive controllers proposed by Morse (1985, 1987a) utilize the relations between the error systems Case 10 , Case 20 (Case 100 , Case 200 ) and the correspondent adaptive law (3)(or equivalently (4)). The proposed methodologies can be considered as extended versions of those previous results.
V˙ ¼ e f ðtÞ2 ð 0Þ: Therefore, e f 2 L1 \ L2 , and uˆ 2 L1 are derived. The same procedure can be applied to the following three cases. 2 d ˜ T vðtÞ ðCase 10 Þ; þ l eðtÞ ¼ uðtÞ dt d ˜ T vðtÞ ðCase 20 Þ; þ l eðtÞ ¼ uðtÞ dt ˜ T vðtÞ eðtÞ ¼ uðtÞ
ðCase 30 Þ:
0
0
Then, Case 1 , Case 2 , and Case 3 are replaced by
d ˜ T vðtÞ þ l EðtÞ ¼ uðtÞ dt
Consider a single-input single-output linear system with a finite dimension not exceeding nas a controlled system d xðtÞ ¼ AxðtÞ þ buðtÞ; dt
(5)
yðtÞ ¼ cT xðtÞ:
(6)
00
ðCase 1 Þ;
For that system (5)(6), only the input uðtÞand the output yðtÞare assumed to be measurable, but the state xðtÞ and system parameters in A; b; c are unknown. Other assumptions about the system are almost the same as those in the usual model reference adaptive control.
T
˜ vðtÞ ðCase 200 Þ; EðtÞ ¼ uðtÞ d ˜ T vðtÞ ðCase 300 Þ; þ l uðtÞ EðtÞ ¼ dt respectively, and the proper adaptive law is given by ˙ˆ ¼ uðtÞ ˙˜ ¼ GvðtÞEðtÞ: uðtÞ
In this section, the proposed methodologies of stable model reference adaptive controller designs for uncertain relative degrees (phase shift), are given (Miyasato, 1997b, 1998a). In the first version, consider the case where the sign of highfrequency gains is known a priori. That assumption will be relaxed in the next section (Miyasato, 1998c, 2000b). 4.1. Problem statement
Those cases can be rewritten into the previous ones by introducing the filtered error signal EðtÞ. d þ l eðtÞ: (2) EðtÞ ¼ dt 0
4. Stable model reference adaptive controller design for uncertain phase shift
(3)
The fitness of (3)can be confirmed from the same stability analysis as Case 1–Case 3. When e˙ ðtÞ is not available for measurement, but vðtÞ is available, e˙ ðtÞ can be removed by ˙ utilizing the next equivalent adaptive law, which is easily deduced from (3)via integration by parts
AS-1. The controlled system is stabilizable and observable, and the zeros of it lie in C (minimum phase). AS-2. n, the upper bound on the dimension of the system, is known. AS-3. The sign of the high frequency gain b0 of cT ðsI AÞ1 b is known. In the following context, it is assumed that b0 > 0 without loss of generality. However, the assumption about the relative degree is quite different from that in the traditional case.
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Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
AS-4. The relative degree of cT ðsI AÞ1 b, denoted as n , is partly unknown. That is, it is known to be r; r þ 1, or r þ 2 with known r 2 N. A scalar function ðyM ðtÞÞ is introduced as a reference signal. For that ðyM ðtÞÞ, it is assumed that AS-5.
ðrþ2Þ yM ðtÞ; y˙ M ðtÞ; . . . ; yM ðtÞ
are uniformly bounded and
measurable.
Step r þ 2, but it deals with several error models in Step 1 (also, see Sections 4.4.2 and 4.4.3). Step1: Define state variables z1 ðtÞ; z2 ðtÞ by z1 ðtÞ yðtÞ yM ðtÞ;
(13)
z2 ðtÞ u fr ðtÞ a1 ðtÞ;
(14)
where a1 ðtÞ is a virtual input determined as follows: ˆ 1 ðtÞT v1 ðtÞ kˆ 1 ðtÞz1 ðtÞ; a1 ðtÞ ¼ F
Then the control problem of this section can be stated as follows: given an unknown system (5)(6)with known nð 2 NÞ and rð 2 NÞ, and with uncertain relative degree n ¼ r; r þ 1, or r þ 2, and given a known reference signal yM ðtÞ, determine a suitable controller such that the tracking error eðtÞ yðtÞ yM ðtÞ converges to zero asymptotically. 4.2. System representation
Lemma 1. On the assumptionsAS-1 andAS-2, the controlled system is represented as follows: n d þ l yðtÞ ¼ u1 yðtÞ þ uT2 vðtÞ þ b0 uðtÞ þ e0 ðtÞ; (7) dt vðtÞ ¼ ½v1 ðtÞT ; v2 ðtÞT T ; d v1 ðtÞ ¼ F0 v1 ðtÞ þ g0 yðtÞ; dt
(8) d v2 ðtÞ ¼ F0 v2 ðtÞ þ g0 uðtÞ; dt (9)
where vðtÞ; ðF0 ; g0 Þ; l are known scalars, vectors, and a matrix with proper dimensions, and e0 ðtÞ; u1 ; u2 ; b0 are unknown scalars and vectors u1 ; u2 ; b0 are system parameters of the controlled system. v1 ðtÞ; v2 ðtÞ, and vðtÞ are called state variable filters(Narendra & Annaswamy, 1989). ðF0 ; g0 Þis an n- or ðn 1Þ-dimensional controllable pair with a stable matrix F0 ; l is an arbitrary positive constant; e0 ðtÞ is an exponentially decaying term ðe0 ; e˙ 0 ðtÞ 2 L1 \ L2 Þ.
˙ˆ ðtÞ ¼ G flv ðtÞ v˙ ðtÞgz ðtÞ; F 12 11 1 1 1 kˆ 1 ðtÞ ¼ kˆ 11 ðtÞ þ kˆ 12 ðtÞ;
G11 ¼ GT11 > 0;
k˙ˆ 12 ðtÞ ¼ lg12 z1 ðtÞ2 ; g12 > 0;
v1 ðtÞ ¼ ½y fr ðtÞ; v fr ðtÞT ; yM ðtÞ; YM1 ðtÞ; YM2 ðtÞT ; d yM ðtÞ þ lyM ðtÞ; dt d YM2 ðtÞ ¼ YM1 ðtÞ þ lYM1 ðtÞ; dt ˆ ˆ F1 ðtÞ ¼ ½f1 ðtÞ; fˆ 2 ðtÞT ; pˆ 01 ðtÞ; pˆ 02 ðtÞ; pˆ 03 ðtÞT :
YM1 ðtÞ ¼
Next signals are also utilized in the construction of the adaptive control scheme ð1 i rÞ. ðy f 0 ðtÞ ¼ yðtÞÞ;
d v fi ðtÞ ¼ lv fi ðtÞ þ v fi1 ðtÞ dt
ðv f 0 ðtÞ ¼ vðtÞÞ;
d u fi ðtÞ ¼ lu fi ðtÞ þ u fi1 ðtÞ dt
ðu f 0 ðtÞ ¼ uðtÞÞ:
(10) (11) (12)
The proposed design method is based on the iterative backstepping procedure (Krstic´ et al., 1995)composed of Step 1 to
(16) (17)
(18)
(19) (20) (21)
ˆ 1 ðtÞ are tuning parameters, and F ˆ 1 ðtÞ is an current kˆ 1 ðtÞ and F estimate of unknown F1 defined by F1 ¼ ½f1 ; fT2 ; p01 ; p02 ; p03 T ; 1 f 1 ¼ p0 u 1 ; f 2 ¼ p0 u 2 ; p0 ¼ ; b0 p01 ¼ p0 ; p02 ¼ p03 ¼ 0; ðn ¼ rÞ; p02 ¼ p0 ; p03 ¼ p0 ;
p01 ¼ p03 ¼ 0; p01 ¼ p02 ¼ 0;
ðn ¼ r þ 1Þ; ðn ¼ r þ 2Þ:
(22)
Step 2: For z2 ðtÞ, determine a state variable z3 ðtÞ and a virtual input a2 ðtÞ by z3 ðtÞ u fr1 ðtÞ a2 ðtÞ;
4.3. Design procedure
d y fi ðtÞ ¼ ly fi ðtÞ þ y fi1 ðtÞ dt
ˆ 11 ðtÞ þ F ˆ 12 ðtÞ; ˆ 1 ðtÞ ¼ F F ˆ 11 ðtÞ ¼ G11 v1 ðtÞz1 ðtÞ; F
kˆ 11 ðtÞ ¼ 12g12 z1 ðtÞ2 ;
An input-output representation of the controlled system (5)(6)is derived. The following result is well-known (Narendra & Annaswamy, 1989).
(15)
(23)
a2 ðtÞ ¼ kˆ 20 ðtÞflu fr ðtÞ þ b1 ðtÞg kˆ 21 ðtÞz2 ðtÞ ˆ 2 ðtÞT v˜ 2 ðtÞ; k22 g 1 ðtÞ2 z2 ðtÞ F
(24)
k˙ˆ 20 ðtÞ ¼ g20 flu fr ðtÞ þ b1 ðtÞgz2 ðtÞ;
(25)
k˙ˆ 21 ðtÞ ¼ g21 z2 ðtÞ2 ;
(26)
˙ˆ ðtÞ ¼ G v˜ ðtÞz ðtÞ; F 2 22 2 2
(27)
g20 ; g21 > 0; G22 ¼ GT22 > 0; k22 > 0; @a1 @a1 v˙ 1 ðtÞ b1 ðtÞ ¼ lz1 ðtÞ; @v1 @z1
(28)
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
g 1 ðtÞ ¼
@a1 ˜ @a1 G1 v˜ 1 ðtÞ þ ; ˆ @z1 @K1 T
(29)
ˆ 1 ðtÞ ; kˆ 1 ðtÞ ; Kˆ 1 ðtÞ ¼ ½F T
v˜ 1 ðtÞ ¼ ½v1 ðtÞ ; z1 ðtÞ ; ˜ 1 ¼ block diagðG11 ; g12 Þ; G T
(30)
yM ðtÞT ;
(31)
tu3 ðtÞ ¼ G3 g 2 ðtÞv0 ðtÞz3 ðtÞ; tui ðtÞ ¼ t ui1 ðtÞ G3 g i1 ðtÞv0 ðtÞzi ðtÞði 4Þ;
(32) T
ˆ 2 ðtÞ ¼ ½fˆ 21 ðtÞ; fˆ 22 ðtÞ ; fˆ 23 ðtÞ ; F
(33)
v˜ 2 ðtÞ ¼ ½y fr1 ðtÞ; v fr1 ðtÞT ; YM1 ðtÞT ;
(34)
where k22 is an arbitrary positive constant. ˆ 2 ðtÞ are tuning parameters, and kˆ 20 ðtÞ and kˆ 20 ðtÞ; kˆ 21 ðtÞ; F ˆ 2 ðtÞ are also current estimates of k20 and F2 defined by F k20 ¼
ðn ¼ rÞ; ðn ¼ r þ 1; r þ 2Þ;
0 1
v0 ðtÞ ¼ ½y fr1 ðtÞ; v fr1 ðtÞT ; u fr1 ðtÞ; YM1 ðtÞT ;
(35)
F2 ¼ ½f21 ; fT22 ; f23 T ; f1 ðn ¼ rÞ; f21 ¼ 0 ðn ¼ r þ 1; r þ 2Þ; ðn ¼ rÞ; f2 f22 ¼ 0 ðn ¼ r þ 1; r þ 2Þ; p0 ðn ¼ rÞ; F23 ¼ 0 ðn ¼ r þ 1; r þ 2Þ:
ðiÞ
(45)
ðG3 ¼ GT3 > 0Þ;
ki1 and ki2 are arbitrary positive constants, and a˜ i ðtÞ are auxiliary signals defined in the final step. In (Step r þ 1), the actual control input uðtÞ is obtained in the following: uðtÞ ¼ arþ1 ðtÞ: (48) Step r þ 2: In the final step, the design procedure is ˆ completed by determining the adaptive law of QðtÞ and the auxiliary signals a˜ i ðtÞð4 i r þ 1Þ as follows:
(36)
˙ˆ QðtÞ ¼ turþ1 ðtÞ;
zi ðtÞ u friþ2 ðtÞ ai1 ðtÞ;
(37)
ziþ1 ðtÞ u friþ1 ðtÞ ai ðtÞ;
(38)
ˆ T v0 ðtÞ ai ðtÞ ¼ lu friþ2 ðtÞ þ bi1 ðtÞ zi1 ðtÞ þ g i1 ðtÞQðtÞ ki1 zi ðtÞ ki2 g i1 ðtÞ2 zi ðtÞ þ g ui1 ðtÞtui ðtÞ þ a˜ i ðtÞ; ki1 ; ki2 > 0; (39)
@z j
f˙u fr jþ2 ðtÞ b j1 ðtÞg;
i1 @ai1 ˜ @ai1 X @ai1 G1 v˜ 1 ðtÞ þ g ðtÞ; g i1 ðtÞ ¼ ˆ @z @z j j1 @K1 1 j¼2
g u1 ðtÞ ¼ g u2 ðtÞ ¼ 0; g ui1 ðtÞ ¼
g u3 ðtÞ ¼
ði 5Þ:
(50)
The next theorem is the first main result. Theorem 1. Consider a controlled system(5) and (6)with the assumptionsAS-1–AS-5. Then it follows that all the signals in the resulting adaptive control system (Step 1 to Step r þ 2) are uniformly bounded, and that the state variables zi ðtÞð1 i r þ 1Þ (where z1 ðtÞ ¼ eðtÞ) converge to zero asymptotically.
4.4. Outline of proof (41)
(42)
j¼4
ˆ 2 ðtÞT T ; Kˆ 2 ¼ ½kˆ 20 ðtÞ; kˆ 21 ðtÞ; F
g uk1 ðtÞG3 g i1 ðtÞv0 ðtÞzk ðtÞ
k¼4
(40)
@a3 ; ˆ @Q
i1 @ai1 X @ai1 g ðtÞ ði 5Þ; ˆ @z j u j1 @Q
i1 X
Remark 2. The similar idea was originally seen in Oya, Kobayashi and Yoshida (1995), where stable model reference adaptive controllers for uncertain relative degrees not exceeding 3, were provided.
@ai1 ˙ˆ @ai1 @ai1 bi1 ðtÞ ¼ v˙ i1 ðtÞ lz1 ðtÞ K2 ðtÞ þ ˆ @v @z1 @K2 i1
j¼2
(49)
a˜ 3 ðtÞ ¼ a˜ 4 ðtÞ ¼ 0;
Step i: ð3 i r þ 1Þ: For zi ðtÞ, determine a state variable ziþ1 ðtÞ and a virtual input ai ðtÞ as follows:
i1 X @ai1
(46)
ˆ where QðtÞ is a tuning vector and also a current estimate of Q defined by ½u1 ; uT2 ; b0 ; 1T ðn ¼ rÞ; Q¼ (47) 0 ðn ¼ r þ 1; r þ 2Þ:
a˜ i ðtÞ ¼
þ
(44)
vi1 ðtÞ ¼ ½vi2 ðtÞT ; y friþ2 ðtÞ; v friþ2 ðtÞT ; u friþ3 ðtÞ;
T
T
221
(43)
Several preparations are needed for proving Theorem 1. 4.4.1. Equivalent representations of adaptive laws The equivalent representations of the adaptive laws of ˆ 1 ðtÞand kˆ 1 ðtÞ are given in the forms of F ˙ˆ ðtÞ ¼ G v ðtÞZ ðtÞ; F 1 11 1 1
(51)
k˙ˆ 1 ðtÞ ¼ g12 z1 ðtÞZ1 ðtÞ;
(52)
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Z1 ðtÞ z˙1 ðtÞ þ lz1 ðtÞ:
(53)
Note that those cannot be realized in the schemes, since unknown z˙ 1 ðtÞ is included (16) and (17)are utilized as adaptive laws of The same relation is seen in (3) and (4).
actual adaptive in Z1 ðtÞ. Thus, ˆ 1 ðtÞ and kˆ 1 ðtÞ. F
4.4.2. Representations of state variable The following relations are obtained by taking the time derivatives of zi ðtÞ ð1 i r þ 1Þ, and Z1 ðtÞ: when n ¼ r þ m; m ¼ 0; 1; 2 m1 d þl Z1 ðtÞ dt ¼ b0 fFT1 v1 ðtÞ þ a1 ðtÞ þ z2 ðtÞ þ p0 e0 ðtÞg;
(54)
when n ¼ r Z1 ðtÞ ¼ b0 fFT2 v˜ 2 ðtÞ þ a2 ðtÞ þ z3 ðtÞ þ p0 e0 ðtÞg; Z1 ðtÞ ¼ QT v0 ðtÞ þ e0 ðtÞ; when n ¼ r þ 2; r þ 1; r
(55)
Fig. 1. Uncertain phase shift.
where s dtd . Category 1 is used to synthesize a virtual input a1 which stabilizes the first subsystem (z1 ). Category 2 is introduced to determine the adaptive law based on the same procedure as Section 3((3) and (4)). Thus, three degrees of uncertainties of relative degrees are involved in the first subsystem (z1 and Z1 ). The virtual control a1 and the adaptive laws of the first subsystem (Step 1) are determined so as to deal with three relative degrees 0, 1, 2 for control, and 1, 0, 1 for adaptive laws. From the second subsystem until the ðr þ 1Þth subsystem (Step 2 to Step r þ 1), the virtual inputs ai and the actual input u are derived so as to stabilize each subsystems of the forms
(56) zi ¼
1 fai þ ziþ1 þ Di ðz1 ; . . . ; zi Þg: sþl
z˙ i ðtÞ ¼ lu friþ2 ðtÞ þ ai ðtÞ þ ziþ1 ðtÞ bi1 ðtÞ ˙ˆ g i1 ðtÞZ1 ðtÞ g ui1 ðtÞQðtÞ
(57)
ð2 i rÞ; z˙ rþ1 ðtÞ ¼ lu f 1 ðtÞ þ uðtÞ br ðtÞ g r ðtÞZ1 ðtÞ ˙ˆ g ur ðtÞQðtÞ:
(58)
It is easily seen that each virtual input ai ðtÞ(or a1 ðtÞ) is determined so as to stabilize zi ðtÞ (or Z1 ðtÞ) (2 i r þ 1). Additional considerations are necessary for the regulation of Z1 ðtÞ, since the uncertainty of relative degree affects the representation of Z1 ðtÞ.
Also, the estimation schemes of unknown parameters are constructed in the iterative manner for the series of subsystems (z2 zrþ1 ). More precisely, the positive functions V and evaluation of those for each cases n ¼ r þ 2; r þ 1; r are given below. For more detailed stability analysis, see Miyasato (1997b, 1998a). Case 1 n ¼ r þ 2 : VðtÞis defined by VðtÞ
rþ1 k 1X Z1 ðtÞ2 þ zi ðtÞ2 2 2 i¼2
kb0 ˆ ˆ fF1 ðtÞ F1 gT G1 11 fF1 ðtÞ F1 g 2 1 kb0 fkˆ 1 ðtÞ k1 g2 1 X fkˆ 2 j ðtÞ k2 j g2 þ þ g2 j 2 j¼0 2 g12 g12
þ 4.4.3. Outline of stability analysis The basic design concept of the proposed adaptive controller is depicted in Figs. 1 and 2. In Step 1, consider the following error systems (Category 1), 8 a1 þ z2 þ D1 ðz1 Þ > > > < 1 fa þ z þ D ðz Þg 1 2 1 1 z1 ¼ s þ l > 1 > > : fa1 þ z2 þ D1 ðz1 Þg ðs þ lÞ2
ðn ¼ rÞ; ðn ¼ r þ 1Þ; ðn ¼ r þ 2Þ;
and the next error systems (Category 2), 8 fðs þ lÞða1 þ z2 þ D1 ðz1 ÞÞg > < a1 þ z2 þ D1 ðz1 Þ Z1 ¼ > : 1 fa1 þ z2 þ D1 ðz1 Þg sþl
ðn ¼ rÞ; ðn ¼ r þ 1Þ;
ðn ¼ r þ 2Þ;
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G1 þ fQðtÞ 3 fQðtÞ Qg; 2
(59)
where k is a positive constant to be determined in the proof, and k1 ; k21 are true values of kˆ 1 ðtÞ; kˆ 21 ðtÞ, respectively (k20 is defined by (35)). After several computation, the evaluation of Vis given below: Z t Z t rþ1 X VðtÞ Vð0Þ Ni zi ðtÞ2 dt Nrþ2 Z1 ðtÞ2 dt i¼1 2
Nrþ3 z1 ðtÞ þ N0 ;
0
0
(60)
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
223
After several computation, V is evaluated as follows: Z t rþ1 X zi ðtÞ2 dt Nrþ2 z1 ðtÞ2 þ N0 ; VðtÞ Vð0Þ Ni i¼1
0
(63) where N0 – Nrþ2 > 0. Thus, the same result is obtained.
5. Extension to unknown sign of high-frequency gain – Universal Model Reference Adaptive Controller The adaptive control systems in Section 4, can be modified into the case where the sign of the high-frequency gain is also unknown (Miyasato, 1998c, 2000b). To deal with such unknown element, Nussbaum gain is introduced into Step 1. 5.1. Problem statement Consider the same problem as ‘‘4.1 Problem Statement’’, where AS-1, AS-2, AS-4, AS-5 are assumed, but AS-3 is not necessary here. Fig. 2. Basic concept of design procedure.
5.2. Universal model reference adaptive controller where Ni > 0 ð0 i r þ 3Þ. Thus, it is shown that all signals are uniformly bounded, and that Z1 ðtÞ; z1 ðtÞ zrþ1 ðtÞ converge to zero asymptotically. Case 2 n ¼ r þ 1 :VðtÞ is defined by rþ1 1X kb0 ˆ ˆ fF1 ðtÞ F1 gT G1 VðtÞ zi ðtÞ2 þ 11 fF1 ðtÞ F1 g 2 i¼2 2 1 kb0 fkˆ 1 ðtÞ k1 g2 1 X fkˆ 2 j ðtÞ k2 j g2 þ þ 2 j¼0 2 g12 g2 j
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G1 þ fQðtÞ 3 fQðtÞ Qg; 2
Step 1: ˆ 1 ðtÞT v1 ðtÞ þ kˆ 1 ðtÞz1 ðtÞg; a1 ðtÞ ¼ mðtÞcos mðtÞfF
(64)
2
ˆ ˆ 1 ðtÞT G 1 F ˆ 1 ðtÞ þ k1 ðtÞ : mðtÞ ¼ F 11 g12
(65)
mðtÞcos mðtÞis a Nussbaum gain (Morse, 1985, 1987a; Nussbaum, 1983)to deal with the unknown sign of the highfrequency gain b0 . (61)
where k and k1 ; k21 are defined in the same way as (59). The evaluation of Vis given in the same form as n ¼ r þ 2 (60). Then, all signals are uniformly bounded and the state variables z1 ðtÞ zrþ1 ðtÞconverge to zero asymptotically. Case 3; n ¼ r : VðtÞis defined by 1 1 ˆ ˆ VðtÞ 2 z1 ðtÞ2 þ fF1 ðtÞ F1 gT G1 11 fF1 ðtÞ F1 g 2b0 2b0 1 1 fkˆ 1 ðtÞ k1 g2 1 X fkˆ 2 j ðtÞ k2 j g2 þ þ 2b0 2 j¼0 g12 g2 j rþ1 1 ˆ 1X T 1 ˆ þ fF zi ðtÞ2 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g þ 2 2 i¼3
1 ˆ ˆ QgT G1 þ fQðtÞ 3 fQðtÞ Qg: 2
The difference from the previous adaptive controller appears only in Step 1, and is given below:
(62)
Remark 3. The Nussbaum gain in this section is an example in many possible choices, which are discussed by Morse (1985, 1987a) and Nussbaum (1983). The additional condition for the Nussbaum gains in the present paper, is that those should be rtimes differentiable with respect to those arguments. That comes from the backstepping procedure. The next theorem is the second main theorem. Theorem 2. Consider a controlled system(5) and (6)with the assumptionsAS-1, AS-2, AS-4, AS-5. Then it follows that all the signals in the resulting adaptive control system (Step 10 , Step 2 to Step r þ 2) are uniformly bounded, and that the state variables zi ðtÞð1 i r þ 1Þðwhere z1 ðtÞ ¼ eðtÞÞ converge to zero asymptotically. 5.3. Outline of proof The stability analysis is carried out in the similar way to Section 4based on the same concept and procedures. However,
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Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
since Nussbaum gain term is included in Step 10 , the slightly different positive functions V are utilized for each cases n ¼ r þ 2; r þ 1; r. 5.3.1. Outline of stability analysis The positive functions V and evaluation of those for each cases n ¼ r þ 2; r þ 1; r are given below. For more detailed stability analysis, see Miyasato (1998c, 2000b). Case 1; n ¼ r þ 2 : VðtÞis defined by Z t rþ1 k 1X 2 VðtÞ ¼ Z1 ðtÞ þ kl Z1 ðtÞ2 dt þ zi ðtÞ2 2 2 0 i¼2 1 1X fkˆ 2 j ðtÞ k2 j g2 þ 2 j¼0 g2 j
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G3 1 fQðtÞ þ fQðtÞ Qg; 2
(66)
where k is a positive constant to be determined in the proof, and k20 ; k21 ; F2 ; Q are defined in the same way as the previous case. After several computation, the evaluation of VðtÞ is given by Z t Z t 2 VðtÞ Vð0Þ kl Z1 ðtÞ dt þ N1 Z1 ðtÞ2 dt 0
þ N2
Z
t
Z t rþ1 X z2 ðtÞ dt þ ki1 zi ðtÞ2 dt 2
0
0
i¼3
0
b0 k fmðtÞsin mðtÞ þ cos mðtÞg 2 pffiffiffiffiffiffiffiffiffi b0 k fmð0Þsin mð0Þ þ cos mð0Þg þ N3 k mðtÞ þ N4 k; 2 (67)
where N1 – N4 > 0. It is seen that the left-hand side of the inequality (67)is bounded from below. Thus, mðtÞ should be included in certain bounded region; mðtÞ 2 ½m ; mþ , where 0 m < mþ < 1. Hence, mðtÞ 2 L1 and then it is shown that VðtÞ 2 L1 . It follows that all signals are uniformly bounded, and that the state variables Z1 ðtÞ; z1 ðtÞ– zrþ1 ðtÞ converge to zero asymptotically.
Case 3; n ¼ r : VðtÞis defined by Z t rþ1 1 l 1X 2 2 VðtÞ ¼ z ðtÞ þ z ðtÞ dt þ zi ðtÞ2 1 1 2 i¼3 2b0 2 b0 2 0 þ
1 1X fkˆ 2 j ðtÞ k2 j g2 2 j¼0 g2 j
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G3 1 fQðtÞ þ fQðtÞ Qg; 2
(69)
where k20 ; k21 ; F2 ; Qare the same as the previous case. After several computation, VðtÞ is evaluated as follows: Z t Z t g 1 ðtÞ2 z2 ðtÞ2 dt 2 z2 ðtÞ dt þ k22 VðtÞ Vð0Þ þ N1 b20 0 0 Z rþ1 t X 1 þ ki1 zi ðtÞ2 dt fmðtÞsin mðtÞ þ cos mðtÞg 2b 0 0 i¼3 pffiffiffiffiffiffiffiffiffi 1 fmð0Þsin mð0Þ þ cos mð0Þg þ N2 mðtÞ 2b0 rþ1 X N4 þ N3 þ ; (70) 4k j2 j¼3 where N1 – N4 > 0. Note that the left-hand side of above inequality (70)is bounded from below. Hence, it follows that mðtÞ 2 ½m ; mþ , where 0 m < mþ < 1. Thus, mðtÞ 2 L1 and then it is shown that VðtÞ 2 L1 , and the same result as the previous two cases is obtained.
6. Adaptive stabilizing control for uncertain relative degree and unknown degree The methodologies in Sections 4 and 5can be applied to adaptive stabilization of plants with uncertain relative degrees and unknown degrees. First, consider the case where the sign of the high-frequency gain is known a priori (Miyasato, 1998d; Miyasato & Hanba, 1997). That assumption will be relaxed in the next section (Miyasato, 2000a). 6.1. Problem statement
Case 2; n ¼ r þ 1 : VðtÞ is defined by
VðtÞ ¼ k
Z 0
t
Z1 ðtÞ2 dt þ
rþ1 1 1X 1X fkˆ 2 j ðtÞ k2 j g2 zi ðtÞ2 þ 2 i¼2 2 j¼0 g2 j
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G3 1 fQðtÞ Qg; (68) þ fQðtÞ 2 where k and k20 ; k21 ; F2 ; Q are defined in the same way as the previous case. Then, the same evaluation form as n ¼ r þ 2 (67), is obtained.
A single-input single-output linear system with an additive nonlinear term, is considered as a controlled system: d xðtÞ ¼ AxðtÞ þ buðtÞ þ g f ðyðtÞ; tÞ; dt
(71)
yðtÞ ¼ cT xðtÞ:
(72)
AS-1, AS-3, AS-4 are assumed, but AS-2 is not needed (unknown n).
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
f ðyðtÞ; tÞ 2 Ris an unknown nonlinear term or disturbance. For f ðyðtÞ; tÞ, the following assumptions are introduced. AS-6. f ðy; tÞ is unknown, but known to be evaluated by f ðy; tÞ2 f1 y fðyÞ
ð f1 0;
f ð0; tÞ ¼ Fð0Þ ¼ 0Þ; (73)
where f1 is an unknown constant, and fðyÞ is a known function of y and r-times differentiable with respect to y. AS-7. The relative degree of cT ðsI AÞ1 g is greater than 1, when n ¼ r þ 2. Then, the control problem of this paper can be stated as follows: given a system (71)(72)with AS-1, AS-3, AS-4, AS-6, AS-7, determine a suitable controller such that the overall system is stabilized and the output yðtÞ goes to zero asymptotically.
225
6.3. Adaptive stabilizing control for uncertain relative degree and unknown relative degree The similar procedure is applied to the system representations (Lemma 2) with the known sign of the high-frequency gain b0 . Then, the proposed adaptive stabilizing controller is given below: Step1: Define state variables z1 ðtÞ; z2 ðtÞ by z1 ðtÞ eðtÞ ¼ yðtÞ rðtÞ;
(81)
z2 ðtÞ u fr ðtÞ a1 ðtÞ;
(82)
where a1 ðtÞ is a virtual input determined in the following way: a1 ðtÞ ¼ kˆ 11 ðtÞz1 ðtÞ kˆ 12 ðtÞu frþ1 ðtÞ kˆ 13 ðtÞfðz1 ðtÞÞ; (83)
6.2. System representation
kˆ 11 ðtÞ ¼ kˆ 111 ðtÞ þ kˆ 112 ðtÞ;
First, an input-output representation of the controlled system is derived by utilizing the zero dynamics of cT ðsI AÞ1 b. The following lemma is obtained.
1 kˆ 111 ðtÞ ¼ g11 z1 ðtÞ2 ðg11 > 0Þ; 2 ˙kˆ ðtÞ ¼ lg z ðtÞ2 ðkˆ ð0Þ 0Þ; 112 11 1 112
Lemma 2. On AS-1, AS-4, AS-6, AS-7, the controlled system(71), (72)is represented as follows:
kˆ 12 ðtÞ ¼ kˆ 121 ðtÞ þ kˆ 122 ðtÞ; kˆ 121 ðtÞ ¼ g12 u frþ1 ðtÞz1 ðtÞ
ðg12 > 0Þ;
(85)
k˙ˆ 122 ðtÞ ¼ g12 f2lu frþ1 ðtÞ u fr ðtÞgz1 ðtÞ;
when n ¼ r þ 2 YðtÞ ¼ u1 yðtÞ þ b0 u frþ1 ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ þ eðtÞ; (74) d YðtÞ þ lYðtÞ dt ¼ u2 yðtÞ þ u3 u frþ1 ðtÞ þ b0 u fr ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ
kˆ 13 ðtÞ ¼ kˆ 131 ðtÞ þ kˆ 132 ðtÞ; kˆ 131 ðtÞ ¼ g13 Fðz1 ðtÞÞ ðg13 > 0Þ; k˙ˆ 132 ðtÞ ¼ lg13 fðz1 ðtÞÞz1 ðtÞ ðkˆ 132 ð0Þ 0Þ; Z z FðzÞ ¼ fðyÞdy ð 0Þ:
(86)
0
þ eðtÞ; (75)
when n ¼ r þ 1 YðtÞ ¼ uyðtÞ þ b0 u fr ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ þ eðtÞ; (76) when n ¼ r YðtÞ ¼ uyðtÞ þ b0 u fr1 ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ þ eðtÞ; (77) d þ l fb0 u fr ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ þ eðtÞg; YðtÞ ¼ dt (78) where d YðtÞ yðtÞ þ lyðtÞ; dt
(84)
(79)
kˆ 11 ðtÞ; kˆ 12 ðtÞ; kˆ 13 ðtÞ are tuning parameters, and kˆ 12 ðtÞ is also a current estimate of k12 defined by 8 <0 ðn ¼ r; r þ 1Þ; u3 (87) k12 ¼ ðn ¼ r þ 2Þ: : b0 Step 2: For z2 ðtÞ, determine a state variable z3 ðtÞ and a virtual input a2 ðtÞ by z3 ðtÞ u fr1 ðtÞ a2 ðtÞ;
(88)
a2 ðtÞ ¼ kˆ 20 ðtÞflu fr ðtÞ þ b1 ðtÞg kˆ 21 ðtÞz2 ðtÞ k22 g 1 ðtÞ2 z2 ðtÞ;
(89)
k˙ˆ 20 ðtÞ ¼ g20 flu fr ðtÞ þ b1 ðtÞgz2 ðtÞ;
(90)
and u1 – u3 ; b0 ; u are unknown system parameters, l is an arbitrary positive constant, and eðtÞ is an exponentially decaying term. LðvðtÞÞðvðtÞ ¼ yðtÞ; f ðy; tÞÞ is defined by
k˙ˆ 21 ðtÞ ¼ g21 z2 ðtÞ2
(91)
LðeðtÞÞ G0 ðsÞeðtÞ;
b1 ðtÞ ¼
where G0 ðsÞ 2 RH1 .
(80)
ðkˆ 21 ð0Þ 0Þ;
@a1 @a1 u˙ frþ1 ðtÞ; lz1 ðtÞ þ @z1 @u frþ1
k22 ; g20 ; g21 > 0;
(92)
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Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
g 1 ðtÞ ¼
@a1 @a1 G1 v1 ðtÞ þ ; ˆ @z1 @K1
Kˆ 1 ðtÞ ¼ ½kˆ 11 ðtÞ; kˆ 12 ðtÞ; kˆ 13 ðtÞT ; T
(93)
In Step r þ 1, the actual control input uðtÞ is obtained in the following:
(94)
uðtÞ ¼ arþ1 ðtÞ:
v1 ðtÞ ¼ ½z1 ðtÞ; u frþ1 ðtÞ; Fðz1 ðtÞÞ ;
(95)
G1 ¼ diagðg11 ; g12 ; g13 Þ;
(96)
where kˆ 20 ðtÞ; kˆ 21 ðtÞ are tuning parameters, and k22 is an arbitrary positive constant. kˆ 20 ðtÞ is also a current estimate of k20 defined by 0 ðn ¼ rÞ; (97) k20 ¼ 1 ðn ¼ r þ 1; r þ 2Þ: Step i ð3 i r þ 1Þ: For zi ðtÞ, determine ziþ1 ðtÞ and ai ðtÞ in the following way: zi ðtÞ u friþ2 ðtÞ ai1 ðtÞ;
(98)
ziþ1 ðtÞ u friþ1 ðtÞ ai ðtÞ ð4 i rÞ;
(99)
ai ðtÞ ¼ lu friþ2 ðtÞ þ bi1 ðtÞ zi1 ðtÞ þ g i1 ðtÞb˜ˆ 0 ðtÞu fr1 ðtÞ ki1 zi ðtÞ ki2 g i1 ðtÞ2 zi ðtÞ þ g bi1 ðtÞtbi ðtÞ þ a˜ i ðtÞ ðki1 ; ki2 > 0Þ; (100) @ai1 ˙ˆ @ai1 ðriþ3Þ @ai1 bi1 ðtÞ ¼ ðtÞ lz1 ðtÞ K2 ðtÞ þ ðriþ3Þ u˙ f @z1 @Kˆ 2 @u f
þ
i1 X @ai1 j¼2
@z j
f˙u fr jþ2 ðtÞ b j1 ðtÞg; (101)
g i1 ðtÞ ¼
i1 @ai1 @ai1 X @ai1 G1 v1 ðtÞ þ g ðtÞ; @z1 @z j j1 @Kˆ 1 j¼2
(102) g b2 ðtÞ ¼ 0; g bi1 ðtÞ ¼
g b3 ðtÞ ¼
@a3 @bˆ˜ 0
ðg b1 ðtÞ ¼ 0Þ;
i1 @ai1 X @ai1 g ðtÞ ð5 i r þ 1Þ; ˆ @z j b j1 @b˜ 0 j¼4
(108)
Step r þ 2: In the final step, the design procedure is completed by determining the adaptive law of bˆ˜ 0 ðtÞ and the auxiliary signals a˜ i ðtÞ as follows: ˙ bˆ˜ 0 ðtÞ ¼ tbrþ1 ðtÞ; a˜ 4 ðtÞ ¼ 0; a˜ i ðtÞ ¼
(109)
i1 X g bk1 ðtÞg3 g i1 ðtÞu fr1 ðtÞzk ðtÞ k¼4
ð5 i r þ 1Þ:
(110)
The next theorem is the third main result. Theorem 3. Consider a controlled system(71) and (72)with the assumptionsAS-1, AS-3, AS-4, AS-6, AS-7. Then it follows that all the signals in the resulting adaptive control system (Step 1 to Step r þ 2) are uniformly bounded, and that the state variables zi ðtÞð1 i r þ 1Þðwhere z1 ðtÞ ¼ yðtÞÞ converge to zero asymptotically. Remark 4. Universal stabilizers for processes with uncertain degrees and relative degrees were studied in the previous works (Mareels, 1984; Ma˚rtensson, 1986; Miller & Davison, 1991; Hanba & Miyasato, 1998). Recently, Hoagg and Bernstein (2004) corrected and extended the result of Mareels (1984), and also proposed a novel universal stabilizer for processes with unknown-but-bounded relative degrees. All of those research works are based on high-gain stabilization, where dynamic compensations with tuning parameters are involved. However, those cannot be applied to model reference control (tracking control) problems, since the high-gain universal stabilizers have limited control structures. On the contrary, the present control schemes (Sections 4-7) utilize the property of strictly positive realness and have more flexible control structures, and can be applied to various control problems, such as model reference control and stabilization problems.
(103) tb3 ðtÞ ¼ g3 g 2 ðtÞu fr1 ðtÞz3 ðtÞ
ðg3 > 0Þ;
(104)
7. Extension to unknown sign of high-frequency gain – Universal Adaptive Stabilizer
tbi ðtÞ ¼ t bi1 ðtÞ g3 g i1 ðtÞu fr1 ðtÞzi ðtÞ ði 4Þ; Kˆ 2 ¼ ½kˆ 20 ðtÞ; kˆ 21 ðtÞT ; ðriþ3Þ
uf
ðtÞ ¼ ½u frþ1 ðtÞ; u fr ðtÞ; . . . ; u friþ3 ðtÞT ;
(105) (106)
where ki1 and ki2 are arbitrary positive constants, and a˜ i ðtÞ are auxiliary signals defined in Step r þ 2. bˆ˜ 0 ðtÞ is a tuning parameter and also a current estimate of b˜ 0 defined by ðn ¼ rÞ; b0 (107) b˜ 0 ¼ 0 ðn ¼ r þ 1; r þ 2Þ:
The adaptive control systems in Section 6, can be modified to the case where the sign of the high-frequency gain is also unknown (Miyasato, 2000a). To deal with such uncertainty, Nussbaum gain is introduced into Step 1. 7.1. Problem statement Consider the same problem as ‘‘6.1 Problem Statement’’, where AS-1, AS-4, AS-6, AS-7 are assumed, but AS-3 is not assumed here.
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
7.2. Universal adaptive stabilizer The difference from the previous adaptive controller is included in Step 1, and is given below: Step 10 : a1 ðtÞ ¼ mðtÞcos mðtÞ
fkˆ 11 ðtÞz1 ðtÞ þ kˆ 12 ðtÞu frþ1 ðtÞ þ kˆ 13 ðtÞfðz1 ðtÞÞg; (111) mðtÞ ¼
3 X
227
been realized until now. The present situation is ‘‘too much complicated to construct’’ just like the implementation of complex integrated electronic circuits. Although the present approach for uncertain relative degrees may not necessarily lead to practical usefulness because of its complicated structure compared with other ones such as switching or iterative design procedures, there still remains an interest for the question, that is, ‘‘what is the minimal prior information to construct stable adaptive control systems?’’ The present state is far from the final goal (solution), and it is still in the middle of the way now.
2
kˆ 1 j ðtÞ : g1 j j¼1
(112)
The next theorem is the last main theorem. Theorem 4. Consider a controlled system(71) and (72)with the assumptionsAS-1, AS-4, AS-6, AS-7. Then it follows that all the signals in the resulting adaptive control system (Step 10 , Step 2 to Step r þ 2) are uniformly bounded, and that the state variables zi ðtÞð1 i r þ 1Þðwherez1 ðtÞ ¼ yðtÞÞ converge to zero asymptotically. 8. Concluding remarks The methodologies of coping with n ¼ r þ 2 and n ¼ r þ 1, are essentially the same as the preceding researches (Morse, 1985, 1987a). The present version extends those to general relative degree cases via the backstepping procedure. On the contrary, when n ¼ r, strictly positive realness of ‘‘ ðs þ lÞ’’ for Z1 ðtÞis utilized, where an improper transfer function is contained. However, causal synthesis of an control input and adaptive laws are made possible via the proposed backstepping procedure. Thus, the structure n ¼ ris not included in the results (Morse, 1985, 1987a). The present control scheme combines those two composite structures (n ¼ r þ 2; r þ 1and n ¼ r), and extends to the general relative degree case via the backstepping techniques. The numbers of tuning parameters for each cases are given in Table 1, where a large difference is seen between relative degree uncertainty 3ðn ¼ r r þ 2Þand 2ðn ¼ r þ 1; r þ 2Þ. In the present methods, uncertainty of relative degrees 0–2 is assigned to the first subsystem, only. Then, uncertainty of relative degrees r r þ 2are accepted in the whole system structure. If such uncertainties could be assigned to multiple subsystems, then much more uncertainties of relative degrees would be dealt with, and finally, arbitrary uncertain relative degrees could be accepted. However, the extension of the proposed methodologies based on that principle, does not have
Table 1 Numbers of tuning parameters Relative degree uncertainty Model reference control Stabilizing control
3 6n þ 6 6
2 2n þ 4 4
1 2n þ 1 4
References Anderson, B. D. O., & Kosut, R. L. (1991). Adaptive robust control: on-line learning. In Proceedings of the 30th IEEE CDC (pp. 297–298). Elliott, H., Cristi, R., & Das, M. (1985). Global stability of adaptive pole placement algorithms. IEEE Transactions of Automatic Control, 30, 348–356. Hanba, S., & Miyasato, Y. (1998). High-gain output-feedback stabilization of linear and nonlinear systems with unknown degrees and relative degrees. In Proceedings of the 21st SICE Symposium on Dynamical System Theory (pp. 333–336) (in Japanese). Hoagg, J. B., & Bernstein, D. S. (2004). Direct adaptive dynamic compensation for minimum phase systems with unknown relative degrees. In Proceedings of the 43rd IEEE Conference on Decision and Control (pp. 183–188). Kreisselmeier, G. (1980). Adaptive control via adaptive observation and asymptotic feedback matrix synthesis. IEEE Transactions of Automatic Control, 25, 717–722. Kreisselmeier, G. (1985). An approach to stable indirect adaptive control. Automatica, 21, 425–431. Krstic´, M., Kanellakopoulos, I., & Kokotovic´, P. V. (1995). Nonlinear and adaptive control design. John Wiley and Sons. Mareels, I. (1984). A simple selftuning controller for stably invertible systems. Systems and Control Letters, 4, 5–16. Ma˚rtensson, B. (1986). The order of any stabilizing regulator is sufficient information for adaptive stabilization. Systems and Control Letters, 6, 299– 305. Miller, D. E., & Davison, E. J. (1991). An adaptive control which provides an arbitrarily good transient and steady-state responses. IEEE Transactions of Automatic Control, 36, 68–81. Miyasato, Y. (1997a). Adaptive servo controller for systems with unknown degrees and uncertain relative degrees. In Proceedings of the 1997 American Control Conference, 3 (pp. 1662–1666). Miyasato, Y. (1997b). Model reference adaptive control for systems with unknown relative degrees. In Proceedings of the European Control Conference 1997. Miyasato, Y. (1998a). A design method of model reference adaptive control systems with uncertain relative degrees. Transactions of the Society of Instrument and Control Engineers, 34, 87–95 (in Japanese). Miyasato, Y. (1998b). A design method of universal adaptive servo controller. In Proceedings of the 37th IEEE Conference on Decision and Control, 2 (pp. 2294–2299). Miyasato, Y. (1998c). A design method of universal model reference adaptive controller. In Proceedings of the 1998 American Control Conference, 1 (pp. 609–613). Miyasato, Y. (1998d). Nonlinear adaptive control for systems with unknown degrees and uncertain relative degrees. Transactions of the Institute of Systems, Control, and Informations, 11, 330–340 (in Japanese). Miyasato, Y. (1999a). A design method of adaptive servo control systems independent of the degrees and relative degrees. Transactions of the Society of Instrument and Control Engineers, 35, 450–452 (in Japanese). Miyasato, Y. (1999b). Model reference adaptive control for a class of nonlinear systems with unknown degrees and uncertain relative degrees. In Proceedings of the 1999 American Control Conference, 1 (pp. 571–575). Miyasato, Y. (2000a). A design method of universal adaptive stabilizer. IEEE Transaction on Automatic Control, 45, 2368–2373.
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Miyasato, Y. (2000b). A model reference adaptive controller for systems with uncertain relative degrees r; r þ 1 or r þ 2 and unknown signs of highfrequency gains. Automatica, 36, 889–896. Miyasato, Y., & Hanba, S. (1997). Adaptive control for nonlinear systems with unknown degrees and uncertain relative degrees.. In Y. Sawaragi, & S. Setsuo (Eds.), System Identification 1997 (SYSID’97) (pp. 903–908). England: Newblock Elsevier Science Ltd. Morse, A. S. (1985). A three-dimensional universal controller for the adaptive stabilization of any strictly proper minimum-phase system with relative degree not exceeding two. IEEE Transactions of Automatic Control, 30, 1188–1191. Morse, A. S. (1987a). A 4ðn þ 1Þ-dimensional model reference adaptive stabilizer for any relative degree one or two, minimum phase system of dimension n or less. Automatica, 23, 123–125. Morse, A. S. (1987b). High-gain adaptive stabilization. In Proceedings of the Carl Kranz Course. Morse, A. S. (1996). Supervisory control of families of linear set-point controllers. Part i. Exact matching. IEEE Transactions of Automatic Control, 41, 1413–1431. Morse, A. S., Mayne, D. Q., & Goodwin, G. C. (1992). Application of hysterisis switching in parameter adaptive control. IEEE Transactions Automatic Control, 37, 1343–1354. Narendra, K. S., & Annaswamy, A. M. (1989). Stable adaptive systems. Prentice-Hall.
Nussbaum, R. D. (1983). Some remarks on a conjecture in adaptive control. Systems and Control Letters, 3, 243–246. Oya, M., Kobayashi, T., & Yoshida, K. (1995). Model reference adaptive control of systems with unknown relative degree not exceeding three. Transactions of the Society of Instrument and Control Engineers, 31, 1432–1441 (in Japanese). Tao, G., & Ioannou, P. A. (1989). Model reference adaptive control for plants with unknown relative degrees. In Proceedings of 1989 ACC (pp. 2297– 2302). Yoshihiko Miyasato was born in Japan, in 1956. He received Bachelor of Engineering, Master of Engineering, and PhD degrees in Mathematical Engineering and Information Physics from the University of Tokyo, in 1979, 1981, and 1984, respectively. From 1984 until 1985, he was a research assistant at the University of Tokyo, and from 1985 until 1987, a research assistant at Chiba Institute of Technology. Since 1987, he has been an associate professor at The Institute of Statistical Mathematics. His research interests include system control theory, especially adaptive and nonlinear control, control of distributed parameter systems, motion control of robotic manipulators, adaptive control of nonlinear systems, modeling and control by neural networks, gain-scheduled control of LPV systems, and system identification. He got the best paper awards from the Society of Instrument and Control Engineers in Japan (SICE) in 1991 and 1996. He is a member of SICE, ISCIE (The Institute of Systems, Control and Information Engineers in Japan), and IEEE (senior member).
Stable adaptive controller design for uncertain phase shift Yoshihiko Miyasato * The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-Ku, Tokyo 106-8569, Japan Received 8 December 2004; accepted 1 June 2005
Abstract In order to construct stable adaptive control systems, several assumptions are needed, and those require some prior information of unknown plants, such as degrees and relative degrees. However, those assumptions sometimes become too restrictive, since it is difficult to obtain reasonable estimations of those indices a priori for unknown practical processes. This paper reviews the authors’ research works where some of those assumptions can be relaxed partially. The basic concepts behind those and the restrictions of the present version are summarized, and the future possibilities of the proposed methodologies are also discussed. # 2005 Elsevier Ltd. All rights reserved. Keywords: Adaptive control; Stability analysis; Model reference adaptive control
1. Introduction Every researcher who begins to study adaptive control, is confronted with the fact that several assumptions are needed to design stable adaptive control systems for unknown processes. Some of those assumptions require prior information of unknown plants. That seems a contradiction of adaptive control, since adaptive control methodologies have been developed for the case where system characteristics cannot be specified a priori. For stable adaptive controller designs, especially for stable model reference adaptive controller designs, the following assumptions are necessary. Assumption of adaptive control: (1) (2) (3) (4)
Degrees or upper bounds of degrees of plants are known. Relative degrees of plants are known. Signs of high-frequency gains of plants are known. Plants are minimum-phase processes.
The Assumption (3) (signs of high-frequent gains) is needed for the direct approach, but is not necessary for certain indirect approach where identification models are involved explicitly. The necessity of those assumptions is because the knowledge of those indices (especially, degrees and relative degrees) is utilized to determine the orders and * Tel.: +81 3 5421 8752; fax: +81 3 3446 1695. E-mail address: [email protected]. 1367-5788/$ – see front matter # 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.arcontrol.2005.06.001
structures of adaptive controllers with sufficient freedom of achieving control objective, such as model-following and stabilization. However, those assumptions sometimes become too restrictive, since it is difficult to obtain reasonable estimations of those indices a priori for unknown practical processes. Hence, the relaxation of those assumptions has been of great importance from both theoretical and practical point of views. The purpose of the present paper is to summarize the authors’ previous researches where the Assumptions (1)–(3) can be relaxed to some extent (Miyasato, 1997a, 1997b, 1998a, 1998b, 1998c, 1998d, 1999a, 1999b, 2000a, 2000b; Miyasato & Hanba, 1997). Especially, design methods of stable adaptive controllers for processes with uncertain relative degrees are provided. The proposed methodologies are composed of several kinds of control structures such as model reference control, stabilizing control and servo compensation, but the basic concept behind those is common to all of them, and that concept will be clarified without going into the detailed stability analysis. Additionally, the restriction of the present version and the future possibility are also discussed. 2. Assumption – prior information The present section summarize the reasons why those assumptions are required to construct stable adaptive control systems, and review the previous works to relax the assumptions.
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2.1. Assumption (1): Degree or upper bound of degree Model reference adaptive control achieves model-following control, and that control scheme is composed of state feedback (or equivalent procedure via output feedback) and feedforward compensations. The state feedback compensation allocates open-loop poles to desired closed-loop ones which cause polezero cancellation of plants. Therefore, the information on the degree or the upper bound of the degree of plants, is needed to specify the order of state feedback compensation, or the order of the equivalent dynamic compensation. One way to relax this assumption, is to introduce high-gain feedback scheme, where approximate model-following control is achieved under the several conditions. Related works are found in (Miller & Davison, 1991; Morse, 1987b). 2.2. Assumption (2): Relative degree The information on the relative degree is needed to specify the order of feedforward compensation, and also utilized to determine stable adaptation scheme which makes error processes strictly positive real. The relative degrees of strictly positive real transfer functions are restricted to 1, 0, and 1, and the order of the adaptation scheme achieving strictly positive realness, is concerned with the relative degree of the plant itself. Multi-controller assignments for several candidates of relative degrees and proper switching strategy of those adaptive controllers are useful way to solve the problem of uncertain relative degrees. Also, iterative identification and controller design procedures such as ‘‘wind-surfer algorithms’’ are practical solutions to uncertain relative degrees. Related works are found in (Anderson & Kosut, 1991; Morse, 1996; Morse, Mayne & Goodwin, 1992; Tao & Ioannou, 1989). On the contrary, stable adaptive controllers of single structures for uncertain relative degrees have been studied, where the property of the relative degree of strictly positive real transfer functions is utilized (Morse, 1985, 1987a). The present paper is considered as an extended version of those. 2.3. Assumption (3): Sign of high-frequency gain This assumption is necessary only for the case of the direct approach of adaptive control. In the indirect approach, where explicit identification models are involved, and the tuning of parameters are carried out based on identification errors, that assumption is not needed. On the contrary, the direct approach which does not utilize the knowledge on the sign of highfrequency gain, is also studied (Morse, 1985, 1987a)by introducing Nussbaum gain(Nussbaum, 1983). The present methods also deal with uncertain sign of high-frequency gains by utilizing the same strategy.
cancellation of plants. Hence, the zeros of plants should be stable in order that unstable pole-zero cancellation does not occur. The assumption of stable zeros is needed for model reference adaptive control and high-gain adaptive stabilization. However, adaptive pole-placement control, where there is no pole-zero cancellation, can be applied to plants with unstable zeros (Elliott, Cristi, & Das, 1985; Kreisselmeier, 1980, 1985). An alternative way is to utilize periodic feedback with multirate sampling. 3. Uncertain relative degree–main concept This paper considers the case where relative degrees are partially unknown. That is, relative degrees are known to be r; r þ 1, or r þ 2with known r 2 N, but they cannot be specified in more detail. The uncertainty of relative degrees r r þ 2 lead to the uncertainty of phase shifts 0 p of plants, and it is difficult to conquer uncertain phase shift 0 p even by the framework of the recent developed robust control theory. Contrary to it, in the proposed methodology, a single adaptive controller can deal with such uncertainty, where there is no switching of multiple controllers. In order to solve that problem, the property of strictly positive real transfer functions is utilized, that is, relative degrees of strictly positive real transfer functions are þ1, 0, and 1. This can be also certified in the next simple example. Consider the following three error systems with relative degrees þ1, 0, and 1, respectively d ˜ T vðtÞ ðCase 1Þ; þ l eðtÞ ¼ uðtÞ dt ˜ T vðtÞ ðCase 2Þ; eðtÞ ¼ uðtÞ d ˜ T vðtÞ ðCase 3Þ; þ l uðtÞ eðtÞ ¼ dt ˆ u; ˜ ¼ uðtÞ uðtÞ where eðtÞ is an error signal, u is an unknown parameter vector, ˆ is an estimate of u, and vðtÞ is a state vector. l is a positive uðtÞ constant (a design parameter). Each dimensions of those are determined properly. The problem here is to find out a single ˆ which is adequate for all of those three error adaptive law of uðtÞ processes. The answer for such question is given by ˙ˆ ¼ uðtÞ ˙˜ ¼ GvðtÞeðtÞ; uðtÞ
(1)
where G is a positive definite matrix. The fitness of that adaptive law can be certified from the following stability analysis. First consider Case 1. In this case, define V by ˜ T G1 uðtÞ ˜ V ¼ 12eðtÞ2 þ 12uðtÞ ð 0Þ;
2.4. Assumption (4): Minimum phase
and take the time derivative of Valong the trajectory of the error system (Case 1)
In order to achieve model-following control, the feedback compensation in the adaptive controller, allocates open-loop poles to desired closed-loop poles, which causes pole-zero
V˙ ¼ leðtÞ2 ð 0Þ: Hence, it follows that e 2 L1 \ L2 , and uˆ 2 L1 .
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
219
ˆ ¼ uˆ 1 ðtÞ þ uˆ 2 ðtÞ; uðtÞ uˆ 1 ðtÞ ¼ GvðtÞeðtÞ;
For Case 2, V is defined by ˜ T G1 uðtÞ ˜ V ¼ 12uðtÞ ð 0Þ: Then, the time derivative of V along the error system (Case 2) is given by 2
V˙ ¼ eðtÞ ð 0Þ; and this yields e 2 L2 , and uˆ 2 L1 . For Case 3, V is defined with the filtered error signal e f ˜ T G1 uðtÞ ˜ ð 0Þ; V ¼ 12e f ðtÞ2 þ 12uðtÞ d þ l e f ðtÞ ¼ eðtÞ: dt The time derivative of V along the trajectory of the error system (Case 3) is given by
u˙ˆ 2 ðtÞ ¼ GflvðtÞ vðtÞgeðtÞ: ˙
(4)
In the main part of the proposed methodologies, the same relations between the error systems Case 10 , Case 20 , Case 30 (or equivalent forms Case 100 , Case 200 , Case 300 ) and the correspondent adaptive law (3)(or equivalently (4)), are utilized. It should be noted that the single adaptive law can deal with the three different error systems. Remark 1. The universal adaptive controllers proposed by Morse (1985, 1987a) utilize the relations between the error systems Case 10 , Case 20 (Case 100 , Case 200 ) and the correspondent adaptive law (3)(or equivalently (4)). The proposed methodologies can be considered as extended versions of those previous results.
V˙ ¼ e f ðtÞ2 ð 0Þ: Therefore, e f 2 L1 \ L2 , and uˆ 2 L1 are derived. The same procedure can be applied to the following three cases. 2 d ˜ T vðtÞ ðCase 10 Þ; þ l eðtÞ ¼ uðtÞ dt d ˜ T vðtÞ ðCase 20 Þ; þ l eðtÞ ¼ uðtÞ dt ˜ T vðtÞ eðtÞ ¼ uðtÞ
ðCase 30 Þ:
0
0
Then, Case 1 , Case 2 , and Case 3 are replaced by
d ˜ T vðtÞ þ l EðtÞ ¼ uðtÞ dt
Consider a single-input single-output linear system with a finite dimension not exceeding nas a controlled system d xðtÞ ¼ AxðtÞ þ buðtÞ; dt
(5)
yðtÞ ¼ cT xðtÞ:
(6)
00
ðCase 1 Þ;
For that system (5)(6), only the input uðtÞand the output yðtÞare assumed to be measurable, but the state xðtÞ and system parameters in A; b; c are unknown. Other assumptions about the system are almost the same as those in the usual model reference adaptive control.
T
˜ vðtÞ ðCase 200 Þ; EðtÞ ¼ uðtÞ d ˜ T vðtÞ ðCase 300 Þ; þ l uðtÞ EðtÞ ¼ dt respectively, and the proper adaptive law is given by ˙ˆ ¼ uðtÞ ˙˜ ¼ GvðtÞEðtÞ: uðtÞ
In this section, the proposed methodologies of stable model reference adaptive controller designs for uncertain relative degrees (phase shift), are given (Miyasato, 1997b, 1998a). In the first version, consider the case where the sign of highfrequency gains is known a priori. That assumption will be relaxed in the next section (Miyasato, 1998c, 2000b). 4.1. Problem statement
Those cases can be rewritten into the previous ones by introducing the filtered error signal EðtÞ. d þ l eðtÞ: (2) EðtÞ ¼ dt 0
4. Stable model reference adaptive controller design for uncertain phase shift
(3)
The fitness of (3)can be confirmed from the same stability analysis as Case 1–Case 3. When e˙ ðtÞ is not available for measurement, but vðtÞ is available, e˙ ðtÞ can be removed by ˙ utilizing the next equivalent adaptive law, which is easily deduced from (3)via integration by parts
AS-1. The controlled system is stabilizable and observable, and the zeros of it lie in C (minimum phase). AS-2. n, the upper bound on the dimension of the system, is known. AS-3. The sign of the high frequency gain b0 of cT ðsI AÞ1 b is known. In the following context, it is assumed that b0 > 0 without loss of generality. However, the assumption about the relative degree is quite different from that in the traditional case.
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AS-4. The relative degree of cT ðsI AÞ1 b, denoted as n , is partly unknown. That is, it is known to be r; r þ 1, or r þ 2 with known r 2 N. A scalar function ðyM ðtÞÞ is introduced as a reference signal. For that ðyM ðtÞÞ, it is assumed that AS-5.
ðrþ2Þ yM ðtÞ; y˙ M ðtÞ; . . . ; yM ðtÞ
are uniformly bounded and
measurable.
Step r þ 2, but it deals with several error models in Step 1 (also, see Sections 4.4.2 and 4.4.3). Step1: Define state variables z1 ðtÞ; z2 ðtÞ by z1 ðtÞ yðtÞ yM ðtÞ;
(13)
z2 ðtÞ u fr ðtÞ a1 ðtÞ;
(14)
where a1 ðtÞ is a virtual input determined as follows: ˆ 1 ðtÞT v1 ðtÞ kˆ 1 ðtÞz1 ðtÞ; a1 ðtÞ ¼ F
Then the control problem of this section can be stated as follows: given an unknown system (5)(6)with known nð 2 NÞ and rð 2 NÞ, and with uncertain relative degree n ¼ r; r þ 1, or r þ 2, and given a known reference signal yM ðtÞ, determine a suitable controller such that the tracking error eðtÞ yðtÞ yM ðtÞ converges to zero asymptotically. 4.2. System representation
Lemma 1. On the assumptionsAS-1 andAS-2, the controlled system is represented as follows: n d þ l yðtÞ ¼ u1 yðtÞ þ uT2 vðtÞ þ b0 uðtÞ þ e0 ðtÞ; (7) dt vðtÞ ¼ ½v1 ðtÞT ; v2 ðtÞT T ; d v1 ðtÞ ¼ F0 v1 ðtÞ þ g0 yðtÞ; dt
(8) d v2 ðtÞ ¼ F0 v2 ðtÞ þ g0 uðtÞ; dt (9)
where vðtÞ; ðF0 ; g0 Þ; l are known scalars, vectors, and a matrix with proper dimensions, and e0 ðtÞ; u1 ; u2 ; b0 are unknown scalars and vectors u1 ; u2 ; b0 are system parameters of the controlled system. v1 ðtÞ; v2 ðtÞ, and vðtÞ are called state variable filters(Narendra & Annaswamy, 1989). ðF0 ; g0 Þis an n- or ðn 1Þ-dimensional controllable pair with a stable matrix F0 ; l is an arbitrary positive constant; e0 ðtÞ is an exponentially decaying term ðe0 ; e˙ 0 ðtÞ 2 L1 \ L2 Þ.
˙ˆ ðtÞ ¼ G flv ðtÞ v˙ ðtÞgz ðtÞ; F 12 11 1 1 1 kˆ 1 ðtÞ ¼ kˆ 11 ðtÞ þ kˆ 12 ðtÞ;
G11 ¼ GT11 > 0;
k˙ˆ 12 ðtÞ ¼ lg12 z1 ðtÞ2 ; g12 > 0;
v1 ðtÞ ¼ ½y fr ðtÞ; v fr ðtÞT ; yM ðtÞ; YM1 ðtÞ; YM2 ðtÞT ; d yM ðtÞ þ lyM ðtÞ; dt d YM2 ðtÞ ¼ YM1 ðtÞ þ lYM1 ðtÞ; dt ˆ ˆ F1 ðtÞ ¼ ½f1 ðtÞ; fˆ 2 ðtÞT ; pˆ 01 ðtÞ; pˆ 02 ðtÞ; pˆ 03 ðtÞT :
YM1 ðtÞ ¼
Next signals are also utilized in the construction of the adaptive control scheme ð1 i rÞ. ðy f 0 ðtÞ ¼ yðtÞÞ;
d v fi ðtÞ ¼ lv fi ðtÞ þ v fi1 ðtÞ dt
ðv f 0 ðtÞ ¼ vðtÞÞ;
d u fi ðtÞ ¼ lu fi ðtÞ þ u fi1 ðtÞ dt
ðu f 0 ðtÞ ¼ uðtÞÞ:
(10) (11) (12)
The proposed design method is based on the iterative backstepping procedure (Krstic´ et al., 1995)composed of Step 1 to
(16) (17)
(18)
(19) (20) (21)
ˆ 1 ðtÞ are tuning parameters, and F ˆ 1 ðtÞ is an current kˆ 1 ðtÞ and F estimate of unknown F1 defined by F1 ¼ ½f1 ; fT2 ; p01 ; p02 ; p03 T ; 1 f 1 ¼ p0 u 1 ; f 2 ¼ p0 u 2 ; p0 ¼ ; b0 p01 ¼ p0 ; p02 ¼ p03 ¼ 0; ðn ¼ rÞ; p02 ¼ p0 ; p03 ¼ p0 ;
p01 ¼ p03 ¼ 0; p01 ¼ p02 ¼ 0;
ðn ¼ r þ 1Þ; ðn ¼ r þ 2Þ:
(22)
Step 2: For z2 ðtÞ, determine a state variable z3 ðtÞ and a virtual input a2 ðtÞ by z3 ðtÞ u fr1 ðtÞ a2 ðtÞ;
4.3. Design procedure
d y fi ðtÞ ¼ ly fi ðtÞ þ y fi1 ðtÞ dt
ˆ 11 ðtÞ þ F ˆ 12 ðtÞ; ˆ 1 ðtÞ ¼ F F ˆ 11 ðtÞ ¼ G11 v1 ðtÞz1 ðtÞ; F
kˆ 11 ðtÞ ¼ 12g12 z1 ðtÞ2 ;
An input-output representation of the controlled system (5)(6)is derived. The following result is well-known (Narendra & Annaswamy, 1989).
(15)
(23)
a2 ðtÞ ¼ kˆ 20 ðtÞflu fr ðtÞ þ b1 ðtÞg kˆ 21 ðtÞz2 ðtÞ ˆ 2 ðtÞT v˜ 2 ðtÞ; k22 g 1 ðtÞ2 z2 ðtÞ F
(24)
k˙ˆ 20 ðtÞ ¼ g20 flu fr ðtÞ þ b1 ðtÞgz2 ðtÞ;
(25)
k˙ˆ 21 ðtÞ ¼ g21 z2 ðtÞ2 ;
(26)
˙ˆ ðtÞ ¼ G v˜ ðtÞz ðtÞ; F 2 22 2 2
(27)
g20 ; g21 > 0; G22 ¼ GT22 > 0; k22 > 0; @a1 @a1 v˙ 1 ðtÞ b1 ðtÞ ¼ lz1 ðtÞ; @v1 @z1
(28)
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
g 1 ðtÞ ¼
@a1 ˜ @a1 G1 v˜ 1 ðtÞ þ ; ˆ @z1 @K1 T
(29)
ˆ 1 ðtÞ ; kˆ 1 ðtÞ ; Kˆ 1 ðtÞ ¼ ½F T
v˜ 1 ðtÞ ¼ ½v1 ðtÞ ; z1 ðtÞ ; ˜ 1 ¼ block diagðG11 ; g12 Þ; G T
(30)
yM ðtÞT ;
(31)
tu3 ðtÞ ¼ G3 g 2 ðtÞv0 ðtÞz3 ðtÞ; tui ðtÞ ¼ t ui1 ðtÞ G3 g i1 ðtÞv0 ðtÞzi ðtÞði 4Þ;
(32) T
ˆ 2 ðtÞ ¼ ½fˆ 21 ðtÞ; fˆ 22 ðtÞ ; fˆ 23 ðtÞ ; F
(33)
v˜ 2 ðtÞ ¼ ½y fr1 ðtÞ; v fr1 ðtÞT ; YM1 ðtÞT ;
(34)
where k22 is an arbitrary positive constant. ˆ 2 ðtÞ are tuning parameters, and kˆ 20 ðtÞ and kˆ 20 ðtÞ; kˆ 21 ðtÞ; F ˆ 2 ðtÞ are also current estimates of k20 and F2 defined by F k20 ¼
ðn ¼ rÞ; ðn ¼ r þ 1; r þ 2Þ;
0 1
v0 ðtÞ ¼ ½y fr1 ðtÞ; v fr1 ðtÞT ; u fr1 ðtÞ; YM1 ðtÞT ;
(35)
F2 ¼ ½f21 ; fT22 ; f23 T ; f1 ðn ¼ rÞ; f21 ¼ 0 ðn ¼ r þ 1; r þ 2Þ; ðn ¼ rÞ; f2 f22 ¼ 0 ðn ¼ r þ 1; r þ 2Þ; p0 ðn ¼ rÞ; F23 ¼ 0 ðn ¼ r þ 1; r þ 2Þ:
ðiÞ
(45)
ðG3 ¼ GT3 > 0Þ;
ki1 and ki2 are arbitrary positive constants, and a˜ i ðtÞ are auxiliary signals defined in the final step. In (Step r þ 1), the actual control input uðtÞ is obtained in the following: uðtÞ ¼ arþ1 ðtÞ: (48) Step r þ 2: In the final step, the design procedure is ˆ completed by determining the adaptive law of QðtÞ and the auxiliary signals a˜ i ðtÞð4 i r þ 1Þ as follows:
(36)
˙ˆ QðtÞ ¼ turþ1 ðtÞ;
zi ðtÞ u friþ2 ðtÞ ai1 ðtÞ;
(37)
ziþ1 ðtÞ u friþ1 ðtÞ ai ðtÞ;
(38)
ˆ T v0 ðtÞ ai ðtÞ ¼ lu friþ2 ðtÞ þ bi1 ðtÞ zi1 ðtÞ þ g i1 ðtÞQðtÞ ki1 zi ðtÞ ki2 g i1 ðtÞ2 zi ðtÞ þ g ui1 ðtÞtui ðtÞ þ a˜ i ðtÞ; ki1 ; ki2 > 0; (39)
@z j
f˙u fr jþ2 ðtÞ b j1 ðtÞg;
i1 @ai1 ˜ @ai1 X @ai1 G1 v˜ 1 ðtÞ þ g ðtÞ; g i1 ðtÞ ¼ ˆ @z @z j j1 @K1 1 j¼2
g u1 ðtÞ ¼ g u2 ðtÞ ¼ 0; g ui1 ðtÞ ¼
g u3 ðtÞ ¼
ði 5Þ:
(50)
The next theorem is the first main result. Theorem 1. Consider a controlled system(5) and (6)with the assumptionsAS-1–AS-5. Then it follows that all the signals in the resulting adaptive control system (Step 1 to Step r þ 2) are uniformly bounded, and that the state variables zi ðtÞð1 i r þ 1Þ (where z1 ðtÞ ¼ eðtÞ) converge to zero asymptotically.
4.4. Outline of proof (41)
(42)
j¼4
ˆ 2 ðtÞT T ; Kˆ 2 ¼ ½kˆ 20 ðtÞ; kˆ 21 ðtÞ; F
g uk1 ðtÞG3 g i1 ðtÞv0 ðtÞzk ðtÞ
k¼4
(40)
@a3 ; ˆ @Q
i1 @ai1 X @ai1 g ðtÞ ði 5Þ; ˆ @z j u j1 @Q
i1 X
Remark 2. The similar idea was originally seen in Oya, Kobayashi and Yoshida (1995), where stable model reference adaptive controllers for uncertain relative degrees not exceeding 3, were provided.
@ai1 ˙ˆ @ai1 @ai1 bi1 ðtÞ ¼ v˙ i1 ðtÞ lz1 ðtÞ K2 ðtÞ þ ˆ @v @z1 @K2 i1
j¼2
(49)
a˜ 3 ðtÞ ¼ a˜ 4 ðtÞ ¼ 0;
Step i: ð3 i r þ 1Þ: For zi ðtÞ, determine a state variable ziþ1 ðtÞ and a virtual input ai ðtÞ as follows:
i1 X @ai1
(46)
ˆ where QðtÞ is a tuning vector and also a current estimate of Q defined by ½u1 ; uT2 ; b0 ; 1T ðn ¼ rÞ; Q¼ (47) 0 ðn ¼ r þ 1; r þ 2Þ:
a˜ i ðtÞ ¼
þ
(44)
vi1 ðtÞ ¼ ½vi2 ðtÞT ; y friþ2 ðtÞ; v friþ2 ðtÞT ; u friþ3 ðtÞ;
T
T
221
(43)
Several preparations are needed for proving Theorem 1. 4.4.1. Equivalent representations of adaptive laws The equivalent representations of the adaptive laws of ˆ 1 ðtÞand kˆ 1 ðtÞ are given in the forms of F ˙ˆ ðtÞ ¼ G v ðtÞZ ðtÞ; F 1 11 1 1
(51)
k˙ˆ 1 ðtÞ ¼ g12 z1 ðtÞZ1 ðtÞ;
(52)
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Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
Z1 ðtÞ z˙1 ðtÞ þ lz1 ðtÞ:
(53)
Note that those cannot be realized in the schemes, since unknown z˙ 1 ðtÞ is included (16) and (17)are utilized as adaptive laws of The same relation is seen in (3) and (4).
actual adaptive in Z1 ðtÞ. Thus, ˆ 1 ðtÞ and kˆ 1 ðtÞ. F
4.4.2. Representations of state variable The following relations are obtained by taking the time derivatives of zi ðtÞ ð1 i r þ 1Þ, and Z1 ðtÞ: when n ¼ r þ m; m ¼ 0; 1; 2 m1 d þl Z1 ðtÞ dt ¼ b0 fFT1 v1 ðtÞ þ a1 ðtÞ þ z2 ðtÞ þ p0 e0 ðtÞg;
(54)
when n ¼ r Z1 ðtÞ ¼ b0 fFT2 v˜ 2 ðtÞ þ a2 ðtÞ þ z3 ðtÞ þ p0 e0 ðtÞg; Z1 ðtÞ ¼ QT v0 ðtÞ þ e0 ðtÞ; when n ¼ r þ 2; r þ 1; r
(55)
Fig. 1. Uncertain phase shift.
where s dtd . Category 1 is used to synthesize a virtual input a1 which stabilizes the first subsystem (z1 ). Category 2 is introduced to determine the adaptive law based on the same procedure as Section 3((3) and (4)). Thus, three degrees of uncertainties of relative degrees are involved in the first subsystem (z1 and Z1 ). The virtual control a1 and the adaptive laws of the first subsystem (Step 1) are determined so as to deal with three relative degrees 0, 1, 2 for control, and 1, 0, 1 for adaptive laws. From the second subsystem until the ðr þ 1Þth subsystem (Step 2 to Step r þ 1), the virtual inputs ai and the actual input u are derived so as to stabilize each subsystems of the forms
(56) zi ¼
1 fai þ ziþ1 þ Di ðz1 ; . . . ; zi Þg: sþl
z˙ i ðtÞ ¼ lu friþ2 ðtÞ þ ai ðtÞ þ ziþ1 ðtÞ bi1 ðtÞ ˙ˆ g i1 ðtÞZ1 ðtÞ g ui1 ðtÞQðtÞ
(57)
ð2 i rÞ; z˙ rþ1 ðtÞ ¼ lu f 1 ðtÞ þ uðtÞ br ðtÞ g r ðtÞZ1 ðtÞ ˙ˆ g ur ðtÞQðtÞ:
(58)
It is easily seen that each virtual input ai ðtÞ(or a1 ðtÞ) is determined so as to stabilize zi ðtÞ (or Z1 ðtÞ) (2 i r þ 1). Additional considerations are necessary for the regulation of Z1 ðtÞ, since the uncertainty of relative degree affects the representation of Z1 ðtÞ.
Also, the estimation schemes of unknown parameters are constructed in the iterative manner for the series of subsystems (z2 zrþ1 ). More precisely, the positive functions V and evaluation of those for each cases n ¼ r þ 2; r þ 1; r are given below. For more detailed stability analysis, see Miyasato (1997b, 1998a). Case 1 n ¼ r þ 2 : VðtÞis defined by VðtÞ
rþ1 k 1X Z1 ðtÞ2 þ zi ðtÞ2 2 2 i¼2
kb0 ˆ ˆ fF1 ðtÞ F1 gT G1 11 fF1 ðtÞ F1 g 2 1 kb0 fkˆ 1 ðtÞ k1 g2 1 X fkˆ 2 j ðtÞ k2 j g2 þ þ g2 j 2 j¼0 2 g12 g12
þ 4.4.3. Outline of stability analysis The basic design concept of the proposed adaptive controller is depicted in Figs. 1 and 2. In Step 1, consider the following error systems (Category 1), 8 a1 þ z2 þ D1 ðz1 Þ > > > < 1 fa þ z þ D ðz Þg 1 2 1 1 z1 ¼ s þ l > 1 > > : fa1 þ z2 þ D1 ðz1 Þg ðs þ lÞ2
ðn ¼ rÞ; ðn ¼ r þ 1Þ; ðn ¼ r þ 2Þ;
and the next error systems (Category 2), 8 fðs þ lÞða1 þ z2 þ D1 ðz1 ÞÞg > < a1 þ z2 þ D1 ðz1 Þ Z1 ¼ > : 1 fa1 þ z2 þ D1 ðz1 Þg sþl
ðn ¼ rÞ; ðn ¼ r þ 1Þ;
ðn ¼ r þ 2Þ;
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G1 þ fQðtÞ 3 fQðtÞ Qg; 2
(59)
where k is a positive constant to be determined in the proof, and k1 ; k21 are true values of kˆ 1 ðtÞ; kˆ 21 ðtÞ, respectively (k20 is defined by (35)). After several computation, the evaluation of Vis given below: Z t Z t rþ1 X VðtÞ Vð0Þ Ni zi ðtÞ2 dt Nrþ2 Z1 ðtÞ2 dt i¼1 2
Nrþ3 z1 ðtÞ þ N0 ;
0
0
(60)
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
223
After several computation, V is evaluated as follows: Z t rþ1 X zi ðtÞ2 dt Nrþ2 z1 ðtÞ2 þ N0 ; VðtÞ Vð0Þ Ni i¼1
0
(63) where N0 – Nrþ2 > 0. Thus, the same result is obtained.
5. Extension to unknown sign of high-frequency gain – Universal Model Reference Adaptive Controller The adaptive control systems in Section 4, can be modified into the case where the sign of the high-frequency gain is also unknown (Miyasato, 1998c, 2000b). To deal with such unknown element, Nussbaum gain is introduced into Step 1. 5.1. Problem statement Consider the same problem as ‘‘4.1 Problem Statement’’, where AS-1, AS-2, AS-4, AS-5 are assumed, but AS-3 is not necessary here. Fig. 2. Basic concept of design procedure.
5.2. Universal model reference adaptive controller where Ni > 0 ð0 i r þ 3Þ. Thus, it is shown that all signals are uniformly bounded, and that Z1 ðtÞ; z1 ðtÞ zrþ1 ðtÞ converge to zero asymptotically. Case 2 n ¼ r þ 1 :VðtÞ is defined by rþ1 1X kb0 ˆ ˆ fF1 ðtÞ F1 gT G1 VðtÞ zi ðtÞ2 þ 11 fF1 ðtÞ F1 g 2 i¼2 2 1 kb0 fkˆ 1 ðtÞ k1 g2 1 X fkˆ 2 j ðtÞ k2 j g2 þ þ 2 j¼0 2 g12 g2 j
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G1 þ fQðtÞ 3 fQðtÞ Qg; 2
Step 1: ˆ 1 ðtÞT v1 ðtÞ þ kˆ 1 ðtÞz1 ðtÞg; a1 ðtÞ ¼ mðtÞcos mðtÞfF
(64)
2
ˆ ˆ 1 ðtÞT G 1 F ˆ 1 ðtÞ þ k1 ðtÞ : mðtÞ ¼ F 11 g12
(65)
mðtÞcos mðtÞis a Nussbaum gain (Morse, 1985, 1987a; Nussbaum, 1983)to deal with the unknown sign of the highfrequency gain b0 . (61)
where k and k1 ; k21 are defined in the same way as (59). The evaluation of Vis given in the same form as n ¼ r þ 2 (60). Then, all signals are uniformly bounded and the state variables z1 ðtÞ zrþ1 ðtÞconverge to zero asymptotically. Case 3; n ¼ r : VðtÞis defined by 1 1 ˆ ˆ VðtÞ 2 z1 ðtÞ2 þ fF1 ðtÞ F1 gT G1 11 fF1 ðtÞ F1 g 2b0 2b0 1 1 fkˆ 1 ðtÞ k1 g2 1 X fkˆ 2 j ðtÞ k2 j g2 þ þ 2b0 2 j¼0 g12 g2 j rþ1 1 ˆ 1X T 1 ˆ þ fF zi ðtÞ2 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g þ 2 2 i¼3
1 ˆ ˆ QgT G1 þ fQðtÞ 3 fQðtÞ Qg: 2
The difference from the previous adaptive controller appears only in Step 1, and is given below:
(62)
Remark 3. The Nussbaum gain in this section is an example in many possible choices, which are discussed by Morse (1985, 1987a) and Nussbaum (1983). The additional condition for the Nussbaum gains in the present paper, is that those should be rtimes differentiable with respect to those arguments. That comes from the backstepping procedure. The next theorem is the second main theorem. Theorem 2. Consider a controlled system(5) and (6)with the assumptionsAS-1, AS-2, AS-4, AS-5. Then it follows that all the signals in the resulting adaptive control system (Step 10 , Step 2 to Step r þ 2) are uniformly bounded, and that the state variables zi ðtÞð1 i r þ 1Þðwhere z1 ðtÞ ¼ eðtÞÞ converge to zero asymptotically. 5.3. Outline of proof The stability analysis is carried out in the similar way to Section 4based on the same concept and procedures. However,
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Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
since Nussbaum gain term is included in Step 10 , the slightly different positive functions V are utilized for each cases n ¼ r þ 2; r þ 1; r. 5.3.1. Outline of stability analysis The positive functions V and evaluation of those for each cases n ¼ r þ 2; r þ 1; r are given below. For more detailed stability analysis, see Miyasato (1998c, 2000b). Case 1; n ¼ r þ 2 : VðtÞis defined by Z t rþ1 k 1X 2 VðtÞ ¼ Z1 ðtÞ þ kl Z1 ðtÞ2 dt þ zi ðtÞ2 2 2 0 i¼2 1 1X fkˆ 2 j ðtÞ k2 j g2 þ 2 j¼0 g2 j
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G3 1 fQðtÞ þ fQðtÞ Qg; 2
(66)
where k is a positive constant to be determined in the proof, and k20 ; k21 ; F2 ; Q are defined in the same way as the previous case. After several computation, the evaluation of VðtÞ is given by Z t Z t 2 VðtÞ Vð0Þ kl Z1 ðtÞ dt þ N1 Z1 ðtÞ2 dt 0
þ N2
Z
t
Z t rþ1 X z2 ðtÞ dt þ ki1 zi ðtÞ2 dt 2
0
0
i¼3
0
b0 k fmðtÞsin mðtÞ þ cos mðtÞg 2 pffiffiffiffiffiffiffiffiffi b0 k fmð0Þsin mð0Þ þ cos mð0Þg þ N3 k mðtÞ þ N4 k; 2 (67)
where N1 – N4 > 0. It is seen that the left-hand side of the inequality (67)is bounded from below. Thus, mðtÞ should be included in certain bounded region; mðtÞ 2 ½m ; mþ , where 0 m < mþ < 1. Hence, mðtÞ 2 L1 and then it is shown that VðtÞ 2 L1 . It follows that all signals are uniformly bounded, and that the state variables Z1 ðtÞ; z1 ðtÞ– zrþ1 ðtÞ converge to zero asymptotically.
Case 3; n ¼ r : VðtÞis defined by Z t rþ1 1 l 1X 2 2 VðtÞ ¼ z ðtÞ þ z ðtÞ dt þ zi ðtÞ2 1 1 2 i¼3 2b0 2 b0 2 0 þ
1 1X fkˆ 2 j ðtÞ k2 j g2 2 j¼0 g2 j
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G3 1 fQðtÞ þ fQðtÞ Qg; 2
(69)
where k20 ; k21 ; F2 ; Qare the same as the previous case. After several computation, VðtÞ is evaluated as follows: Z t Z t g 1 ðtÞ2 z2 ðtÞ2 dt 2 z2 ðtÞ dt þ k22 VðtÞ Vð0Þ þ N1 b20 0 0 Z rþ1 t X 1 þ ki1 zi ðtÞ2 dt fmðtÞsin mðtÞ þ cos mðtÞg 2b 0 0 i¼3 pffiffiffiffiffiffiffiffiffi 1 fmð0Þsin mð0Þ þ cos mð0Þg þ N2 mðtÞ 2b0 rþ1 X N4 þ N3 þ ; (70) 4k j2 j¼3 where N1 – N4 > 0. Note that the left-hand side of above inequality (70)is bounded from below. Hence, it follows that mðtÞ 2 ½m ; mþ , where 0 m < mþ < 1. Thus, mðtÞ 2 L1 and then it is shown that VðtÞ 2 L1 , and the same result as the previous two cases is obtained.
6. Adaptive stabilizing control for uncertain relative degree and unknown degree The methodologies in Sections 4 and 5can be applied to adaptive stabilization of plants with uncertain relative degrees and unknown degrees. First, consider the case where the sign of the high-frequency gain is known a priori (Miyasato, 1998d; Miyasato & Hanba, 1997). That assumption will be relaxed in the next section (Miyasato, 2000a). 6.1. Problem statement
Case 2; n ¼ r þ 1 : VðtÞ is defined by
VðtÞ ¼ k
Z 0
t
Z1 ðtÞ2 dt þ
rþ1 1 1X 1X fkˆ 2 j ðtÞ k2 j g2 zi ðtÞ2 þ 2 i¼2 2 j¼0 g2 j
1 ˆ T 1 ˆ þ fF 2 ðtÞ F2 g G22 fF2 ðtÞ F2 g 2 1 ˆ ˆ QgT G3 1 fQðtÞ Qg; (68) þ fQðtÞ 2 where k and k20 ; k21 ; F2 ; Q are defined in the same way as the previous case. Then, the same evaluation form as n ¼ r þ 2 (67), is obtained.
A single-input single-output linear system with an additive nonlinear term, is considered as a controlled system: d xðtÞ ¼ AxðtÞ þ buðtÞ þ g f ðyðtÞ; tÞ; dt
(71)
yðtÞ ¼ cT xðtÞ:
(72)
AS-1, AS-3, AS-4 are assumed, but AS-2 is not needed (unknown n).
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
f ðyðtÞ; tÞ 2 Ris an unknown nonlinear term or disturbance. For f ðyðtÞ; tÞ, the following assumptions are introduced. AS-6. f ðy; tÞ is unknown, but known to be evaluated by f ðy; tÞ2 f1 y fðyÞ
ð f1 0;
f ð0; tÞ ¼ Fð0Þ ¼ 0Þ; (73)
where f1 is an unknown constant, and fðyÞ is a known function of y and r-times differentiable with respect to y. AS-7. The relative degree of cT ðsI AÞ1 g is greater than 1, when n ¼ r þ 2. Then, the control problem of this paper can be stated as follows: given a system (71)(72)with AS-1, AS-3, AS-4, AS-6, AS-7, determine a suitable controller such that the overall system is stabilized and the output yðtÞ goes to zero asymptotically.
225
6.3. Adaptive stabilizing control for uncertain relative degree and unknown relative degree The similar procedure is applied to the system representations (Lemma 2) with the known sign of the high-frequency gain b0 . Then, the proposed adaptive stabilizing controller is given below: Step1: Define state variables z1 ðtÞ; z2 ðtÞ by z1 ðtÞ eðtÞ ¼ yðtÞ rðtÞ;
(81)
z2 ðtÞ u fr ðtÞ a1 ðtÞ;
(82)
where a1 ðtÞ is a virtual input determined in the following way: a1 ðtÞ ¼ kˆ 11 ðtÞz1 ðtÞ kˆ 12 ðtÞu frþ1 ðtÞ kˆ 13 ðtÞfðz1 ðtÞÞ; (83)
6.2. System representation
kˆ 11 ðtÞ ¼ kˆ 111 ðtÞ þ kˆ 112 ðtÞ;
First, an input-output representation of the controlled system is derived by utilizing the zero dynamics of cT ðsI AÞ1 b. The following lemma is obtained.
1 kˆ 111 ðtÞ ¼ g11 z1 ðtÞ2 ðg11 > 0Þ; 2 ˙kˆ ðtÞ ¼ lg z ðtÞ2 ðkˆ ð0Þ 0Þ; 112 11 1 112
Lemma 2. On AS-1, AS-4, AS-6, AS-7, the controlled system(71), (72)is represented as follows:
kˆ 12 ðtÞ ¼ kˆ 121 ðtÞ þ kˆ 122 ðtÞ; kˆ 121 ðtÞ ¼ g12 u frþ1 ðtÞz1 ðtÞ
ðg12 > 0Þ;
(85)
k˙ˆ 122 ðtÞ ¼ g12 f2lu frþ1 ðtÞ u fr ðtÞgz1 ðtÞ;
when n ¼ r þ 2 YðtÞ ¼ u1 yðtÞ þ b0 u frþ1 ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ þ eðtÞ; (74) d YðtÞ þ lYðtÞ dt ¼ u2 yðtÞ þ u3 u frþ1 ðtÞ þ b0 u fr ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ
kˆ 13 ðtÞ ¼ kˆ 131 ðtÞ þ kˆ 132 ðtÞ; kˆ 131 ðtÞ ¼ g13 Fðz1 ðtÞÞ ðg13 > 0Þ; k˙ˆ 132 ðtÞ ¼ lg13 fðz1 ðtÞÞz1 ðtÞ ðkˆ 132 ð0Þ 0Þ; Z z FðzÞ ¼ fðyÞdy ð 0Þ:
(86)
0
þ eðtÞ; (75)
when n ¼ r þ 1 YðtÞ ¼ uyðtÞ þ b0 u fr ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ þ eðtÞ; (76) when n ¼ r YðtÞ ¼ uyðtÞ þ b0 u fr1 ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ þ eðtÞ; (77) d þ l fb0 u fr ðtÞ þ LðyðtÞÞ þ Lð f ðy; tÞÞ þ eðtÞg; YðtÞ ¼ dt (78) where d YðtÞ yðtÞ þ lyðtÞ; dt
(84)
(79)
kˆ 11 ðtÞ; kˆ 12 ðtÞ; kˆ 13 ðtÞ are tuning parameters, and kˆ 12 ðtÞ is also a current estimate of k12 defined by 8 <0 ðn ¼ r; r þ 1Þ; u3 (87) k12 ¼ ðn ¼ r þ 2Þ: : b0 Step 2: For z2 ðtÞ, determine a state variable z3 ðtÞ and a virtual input a2 ðtÞ by z3 ðtÞ u fr1 ðtÞ a2 ðtÞ;
(88)
a2 ðtÞ ¼ kˆ 20 ðtÞflu fr ðtÞ þ b1 ðtÞg kˆ 21 ðtÞz2 ðtÞ k22 g 1 ðtÞ2 z2 ðtÞ;
(89)
k˙ˆ 20 ðtÞ ¼ g20 flu fr ðtÞ þ b1 ðtÞgz2 ðtÞ;
(90)
and u1 – u3 ; b0 ; u are unknown system parameters, l is an arbitrary positive constant, and eðtÞ is an exponentially decaying term. LðvðtÞÞðvðtÞ ¼ yðtÞ; f ðy; tÞÞ is defined by
k˙ˆ 21 ðtÞ ¼ g21 z2 ðtÞ2
(91)
LðeðtÞÞ G0 ðsÞeðtÞ;
b1 ðtÞ ¼
where G0 ðsÞ 2 RH1 .
(80)
ðkˆ 21 ð0Þ 0Þ;
@a1 @a1 u˙ frþ1 ðtÞ; lz1 ðtÞ þ @z1 @u frþ1
k22 ; g20 ; g21 > 0;
(92)
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Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
g 1 ðtÞ ¼
@a1 @a1 G1 v1 ðtÞ þ ; ˆ @z1 @K1
Kˆ 1 ðtÞ ¼ ½kˆ 11 ðtÞ; kˆ 12 ðtÞ; kˆ 13 ðtÞT ; T
(93)
In Step r þ 1, the actual control input uðtÞ is obtained in the following:
(94)
uðtÞ ¼ arþ1 ðtÞ:
v1 ðtÞ ¼ ½z1 ðtÞ; u frþ1 ðtÞ; Fðz1 ðtÞÞ ;
(95)
G1 ¼ diagðg11 ; g12 ; g13 Þ;
(96)
where kˆ 20 ðtÞ; kˆ 21 ðtÞ are tuning parameters, and k22 is an arbitrary positive constant. kˆ 20 ðtÞ is also a current estimate of k20 defined by 0 ðn ¼ rÞ; (97) k20 ¼ 1 ðn ¼ r þ 1; r þ 2Þ: Step i ð3 i r þ 1Þ: For zi ðtÞ, determine ziþ1 ðtÞ and ai ðtÞ in the following way: zi ðtÞ u friþ2 ðtÞ ai1 ðtÞ;
(98)
ziþ1 ðtÞ u friþ1 ðtÞ ai ðtÞ ð4 i rÞ;
(99)
ai ðtÞ ¼ lu friþ2 ðtÞ þ bi1 ðtÞ zi1 ðtÞ þ g i1 ðtÞb˜ˆ 0 ðtÞu fr1 ðtÞ ki1 zi ðtÞ ki2 g i1 ðtÞ2 zi ðtÞ þ g bi1 ðtÞtbi ðtÞ þ a˜ i ðtÞ ðki1 ; ki2 > 0Þ; (100) @ai1 ˙ˆ @ai1 ðriþ3Þ @ai1 bi1 ðtÞ ¼ ðtÞ lz1 ðtÞ K2 ðtÞ þ ðriþ3Þ u˙ f @z1 @Kˆ 2 @u f
þ
i1 X @ai1 j¼2
@z j
f˙u fr jþ2 ðtÞ b j1 ðtÞg; (101)
g i1 ðtÞ ¼
i1 @ai1 @ai1 X @ai1 G1 v1 ðtÞ þ g ðtÞ; @z1 @z j j1 @Kˆ 1 j¼2
(102) g b2 ðtÞ ¼ 0; g bi1 ðtÞ ¼
g b3 ðtÞ ¼
@a3 @bˆ˜ 0
ðg b1 ðtÞ ¼ 0Þ;
i1 @ai1 X @ai1 g ðtÞ ð5 i r þ 1Þ; ˆ @z j b j1 @b˜ 0 j¼4
(108)
Step r þ 2: In the final step, the design procedure is completed by determining the adaptive law of bˆ˜ 0 ðtÞ and the auxiliary signals a˜ i ðtÞ as follows: ˙ bˆ˜ 0 ðtÞ ¼ tbrþ1 ðtÞ; a˜ 4 ðtÞ ¼ 0; a˜ i ðtÞ ¼
(109)
i1 X g bk1 ðtÞg3 g i1 ðtÞu fr1 ðtÞzk ðtÞ k¼4
ð5 i r þ 1Þ:
(110)
The next theorem is the third main result. Theorem 3. Consider a controlled system(71) and (72)with the assumptionsAS-1, AS-3, AS-4, AS-6, AS-7. Then it follows that all the signals in the resulting adaptive control system (Step 1 to Step r þ 2) are uniformly bounded, and that the state variables zi ðtÞð1 i r þ 1Þðwhere z1 ðtÞ ¼ yðtÞÞ converge to zero asymptotically. Remark 4. Universal stabilizers for processes with uncertain degrees and relative degrees were studied in the previous works (Mareels, 1984; Ma˚rtensson, 1986; Miller & Davison, 1991; Hanba & Miyasato, 1998). Recently, Hoagg and Bernstein (2004) corrected and extended the result of Mareels (1984), and also proposed a novel universal stabilizer for processes with unknown-but-bounded relative degrees. All of those research works are based on high-gain stabilization, where dynamic compensations with tuning parameters are involved. However, those cannot be applied to model reference control (tracking control) problems, since the high-gain universal stabilizers have limited control structures. On the contrary, the present control schemes (Sections 4-7) utilize the property of strictly positive realness and have more flexible control structures, and can be applied to various control problems, such as model reference control and stabilization problems.
(103) tb3 ðtÞ ¼ g3 g 2 ðtÞu fr1 ðtÞz3 ðtÞ
ðg3 > 0Þ;
(104)
7. Extension to unknown sign of high-frequency gain – Universal Adaptive Stabilizer
tbi ðtÞ ¼ t bi1 ðtÞ g3 g i1 ðtÞu fr1 ðtÞzi ðtÞ ði 4Þ; Kˆ 2 ¼ ½kˆ 20 ðtÞ; kˆ 21 ðtÞT ; ðriþ3Þ
uf
ðtÞ ¼ ½u frþ1 ðtÞ; u fr ðtÞ; . . . ; u friþ3 ðtÞT ;
(105) (106)
where ki1 and ki2 are arbitrary positive constants, and a˜ i ðtÞ are auxiliary signals defined in Step r þ 2. bˆ˜ 0 ðtÞ is a tuning parameter and also a current estimate of b˜ 0 defined by ðn ¼ rÞ; b0 (107) b˜ 0 ¼ 0 ðn ¼ r þ 1; r þ 2Þ:
The adaptive control systems in Section 6, can be modified to the case where the sign of the high-frequency gain is also unknown (Miyasato, 2000a). To deal with such uncertainty, Nussbaum gain is introduced into Step 1. 7.1. Problem statement Consider the same problem as ‘‘6.1 Problem Statement’’, where AS-1, AS-4, AS-6, AS-7 are assumed, but AS-3 is not assumed here.
Y. Miyasato / Annual Reviews in Control 29 (2005) 217–228
7.2. Universal adaptive stabilizer The difference from the previous adaptive controller is included in Step 1, and is given below: Step 10 : a1 ðtÞ ¼ mðtÞcos mðtÞ
fkˆ 11 ðtÞz1 ðtÞ þ kˆ 12 ðtÞu frþ1 ðtÞ þ kˆ 13 ðtÞfðz1 ðtÞÞg; (111) mðtÞ ¼
3 X
227
been realized until now. The present situation is ‘‘too much complicated to construct’’ just like the implementation of complex integrated electronic circuits. Although the present approach for uncertain relative degrees may not necessarily lead to practical usefulness because of its complicated structure compared with other ones such as switching or iterative design procedures, there still remains an interest for the question, that is, ‘‘what is the minimal prior information to construct stable adaptive control systems?’’ The present state is far from the final goal (solution), and it is still in the middle of the way now.
2
kˆ 1 j ðtÞ : g1 j j¼1
(112)
The next theorem is the last main theorem. Theorem 4. Consider a controlled system(71) and (72)with the assumptionsAS-1, AS-4, AS-6, AS-7. Then it follows that all the signals in the resulting adaptive control system (Step 10 , Step 2 to Step r þ 2) are uniformly bounded, and that the state variables zi ðtÞð1 i r þ 1Þðwherez1 ðtÞ ¼ yðtÞÞ converge to zero asymptotically. 8. Concluding remarks The methodologies of coping with n ¼ r þ 2 and n ¼ r þ 1, are essentially the same as the preceding researches (Morse, 1985, 1987a). The present version extends those to general relative degree cases via the backstepping procedure. On the contrary, when n ¼ r, strictly positive realness of ‘‘ ðs þ lÞ’’ for Z1 ðtÞis utilized, where an improper transfer function is contained. However, causal synthesis of an control input and adaptive laws are made possible via the proposed backstepping procedure. Thus, the structure n ¼ ris not included in the results (Morse, 1985, 1987a). The present control scheme combines those two composite structures (n ¼ r þ 2; r þ 1and n ¼ r), and extends to the general relative degree case via the backstepping techniques. The numbers of tuning parameters for each cases are given in Table 1, where a large difference is seen between relative degree uncertainty 3ðn ¼ r r þ 2Þand 2ðn ¼ r þ 1; r þ 2Þ. In the present methods, uncertainty of relative degrees 0–2 is assigned to the first subsystem, only. Then, uncertainty of relative degrees r r þ 2are accepted in the whole system structure. If such uncertainties could be assigned to multiple subsystems, then much more uncertainties of relative degrees would be dealt with, and finally, arbitrary uncertain relative degrees could be accepted. However, the extension of the proposed methodologies based on that principle, does not have
Table 1 Numbers of tuning parameters Relative degree uncertainty Model reference control Stabilizing control
3 6n þ 6 6
2 2n þ 4 4
1 2n þ 1 4
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Nussbaum, R. D. (1983). Some remarks on a conjecture in adaptive control. Systems and Control Letters, 3, 243–246. Oya, M., Kobayashi, T., & Yoshida, K. (1995). Model reference adaptive control of systems with unknown relative degree not exceeding three. Transactions of the Society of Instrument and Control Engineers, 31, 1432–1441 (in Japanese). Tao, G., & Ioannou, P. A. (1989). Model reference adaptive control for plants with unknown relative degrees. In Proceedings of 1989 ACC (pp. 2297– 2302). Yoshihiko Miyasato was born in Japan, in 1956. He received Bachelor of Engineering, Master of Engineering, and PhD degrees in Mathematical Engineering and Information Physics from the University of Tokyo, in 1979, 1981, and 1984, respectively. From 1984 until 1985, he was a research assistant at the University of Tokyo, and from 1985 until 1987, a research assistant at Chiba Institute of Technology. Since 1987, he has been an associate professor at The Institute of Statistical Mathematics. His research interests include system control theory, especially adaptive and nonlinear control, control of distributed parameter systems, motion control of robotic manipulators, adaptive control of nonlinear systems, modeling and control by neural networks, gain-scheduled control of LPV systems, and system identification. He got the best paper awards from the Society of Instrument and Control Engineers in Japan (SICE) in 1991 and 1996. He is a member of SICE, ISCIE (The Institute of Systems, Control and Information Engineers in Japan), and IEEE (senior member).