- Email: [email protected]
Robust high-order tuner of simplified structure
Automatica 35 (1999) 1409}1415
Brief Paper
Robust high-order tuner of simpli"ed structure夽 V.O. Nikiforov* Laboratory of Cybernetics and Control Systems, Institute of Fine Mechanics and Optics, Sablinskaya 14, Saint-Petersburg, 197101, Russia Received 14 May 1998; revised 20 November 1998; received in "nal form 2 February 1999
Abstract Recently, Morse has proposed a new adaptation scheme (so-called &&high-order tuner'') which generates as outputs not only tuned parameters, but also their time derivatives up to a certain order. In this paper a robusti"ed variant of the high-order tuner is introduced to adaptively stabilize a SISO linear plant subjected to external disturbances. In comparison with Morse's tuner, the proposed one has essentially simpli"ed structure. Namely, the total order of time-varying auxiliary "lters is reduced 2n times (where n is the order of the plant to be controlled). 1999 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive control; Robustness; High-order tuners; External disturbances
1. Introduction The problem of output-feedback adaptive control of linear systems has been attracting attention of the control community during the last decades. The earliest decisions were based on the certainty-equivalence principle and augmented error concept (Monopoli, 1974; Narendra and Valavani, 1978; Feuer and Morse, 1978). However, it was soon recognized that the proof of stability of the certainty equivalence schemes with adaptation laws forced by an augmented error is not trivial and can be accomplished after lengthy signal analysis (Feuer et al., 1978; Narendra et al., 1980; Morse, 1980). In this analysis, a crucial role is played by the rate of change of the adjustable parameters which is required to be square integrable. To satisfy this requirement, di!erent types of normalization are involved in the adaptation laws. In general, normalization slows down the adaptation and, as a result, leads to the poor transient performance (Zang and Bitmead, 1994). This motivates the search for new classes of adaptive controllers which can be tuned by unnormalized adaptation laws. Recently, such a class of adaptive controllers has been proposed by Krstic& et al. (1994) with the use of nonlinear design tools: integrator backstepping and nonlinear ***** * Tel.: #7(812)2191707; e-mail: [email protected]. 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor K.W. Lim under the direction of Editor C.C. Hang.
damping. Now it is well known that these new controllers possess several useful properties which were unattainable for the traditional adaptive systems (see Krstic& et al., 1995). Among those properties are strict passivity of the main adaptation loop, parametric robustness of the detuned system, a dramatic improvement of the transient performance without increasing the control e!ort. As a shortcoming of Krstic& 's controller one may consider its complicated structure (at least for the case of high relative degree systems demanding several design steps). An alternative approach to the design of adaptive controllers with unnormalized adaptation laws was proposed by Morse (1992) introducing the notion of a &&highorder tuner''. A tuner of order N was de"ned by Morse as an algorithm of adaptation that generates not only a set of adjusted parameters, but also the "rst N timederivatives of each parameter in the set. Ortega (1993) combined a high-order tuner with a dynamic certainty equivalence controller to solve the output-feedback adaptive control problem under ideal conditions (i.e. when there are no external disturbances or unmodelled dynamics). Ortega established also a priori computable bounds on the L and L norms of the tracking error. A high-order tuner can be viewed as an unnormalized gradient algorithm of adaptation plus time-varying "lters. In the author's opinion, the certain shortcomings of Morse's tuner consist in the following. First, as any adaptation law with pure integral action the high-order tuner is not robust in the sense that bounded external disturbances can cause unbounded parametric drift (this
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 5 1 - 5
1410
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
e!ect will be demonstrated in the present paper). The second shortcoming consists in the excess dynamic order of the tuner. Namely, we need to employ auxiliary timevarying "lters of the total order m;N (where m is the number of adjustable parameters and N is the tuner order dependent on the relative degree of the plant to be controlled). This can be impractical for multi-parameter systems with a high relative degree. Therefore, from the practical point of view, simpli"ed robust modi"cations of Morse's tuner are greatly needed. The main contributions of the paper are the following. First, the e!ect of external disturbances on stability of Morse's tuner is analyzed and possibility of unbounded parametric drift is demonstrated. Second, we propose a robusti"ed variant of high-order tuners expanding their applicability to the case of the presence of external disturbances. Third, we reduce the total dynamic order of the auxiliary "lters 2n times (where n is the order of the plant to be controlled) and, as a consequence, essentially simplify the structure of the resulting adaptive system. The paper is organized as follows. In Section 2 we pose the control problem and brie#y introduce a variant of the certainty equivalence controller which slightly di!ers from those considered by Ortega (1993). In Section 3, in self-contained form, Morse's high-order tuner is presented, its stability properties are analyzed and possibility of unbounded parametric drift caused by external disturbances is demonstrated. In Section 4 we present the main result introducing a robusti"ed high-order tuner and in Section 5 we demonstrate the simulation results.
2. Problem statement and controller structure We consider the problem of adaptive control of linear time-invariant SISO plants y"k
b(p) [u#d], a(p)
(1)
where y is the reference output, r is a piece-wise continu ous bounded reference input (command signal), a (p) is a monic Hurwitz polynomial of order o"n!m, b '0. The control objective is to force the output y to track a reference signal y with a certain accuracy while keeping all the closed-loop signals bounded. In order to design an appropriate control law, we "rst derive a suitable error model. Utilizing known results (see Feuer and Morse, 1978), we conclude that for any Hurwitz monic polynomial ¸(p) of order n!1, the tracking error e"y!y can be presented in the form k e" [u(t)2h#u#dM (t)]#, (3) a (p) where exponentially decays due to nonzero initial conditions, h3RL is the vector of unknown constant parameters, dM is a bounded function of time depending on d, u(t)3 RL is the standard regressor vector, i.e. 1 u(t)" [u(t), uR (t),2, uL\(t), y(t), ¸(p) yR (t),2, yL\(t), ¸(p)r(t)].
(4)
Furthermore, we de"ne a transfer function =(p) obeying the equality p#j =(p)" , a (p) where j is any positive constant. Obviously, =(p) is an asymptotically stable minimum phase transfer function with the relative degree o*"o!1 and 1 1 " =(p). a (p) p#j Therefore, the error model (3) can be written as
where y and u are the plant output and input, respectively, d is the bounded external disturbance, p"d/dt denotes the di!erential operator, a(p) and b(p) are monic coprime polynomials with unknown coe$cients. We made the following standard assumptions:
k e" [m(t)2h#=(p)u#d(t)]#, p#j
(1) the degrees n and m are known and o"n!m51; (2) the coe$cient k is nonzero and its sign is known (without loss of generality k is assumed to be positive); (3) the polynomial b(p) is Hurwitz.
Analysis of model (5) motivates us to choose the control law of the form
Let the desired behavior of the closed-loop system be speci"ed by the following reference model: b y " r, a (p)
(2)
(5)
where d(t)"=(p)dM (t) and m(t)"=(p)u(t).
u"!=(p)\m(t)2h) ,
(6)
(7)
where h) is the vector of adjustable parameters. Control (7) is given the name of dynamic certainty equivalence controller (Ortega, 1993). Substituting Eq. (7) into Eq. (5) we obtain k e" [m(t)2h#d(t)]#, p#j
(8)
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
where h"h!h) . Since the transfer function k/(p#j) is strictly positive real, then to tune the vector of adjustable parameters h) one can employ an adaptation law involving the accessible output e. When plant (1) has a relative degree equal to one (and therefore o*"0), the implementation of the control law (7) is trivial. If o"2, then o*"1 and implementation of Eq. (7) requires the "rst-order derivatives h) and m. This does not constitute a principal problem, because the derivatives of m(t) up to order o* are readily available without di!erentiation from Eq. (6), while h) can be obtained from a standard adaptation law of the integral type. However when o'2 (and therefore o*'1) the certainty equivalence controller (7) cannot be combined with standard adaptation laws, because they do not allow one to obtain the necessary derivatives of h) (t). To overcome this di$culty, one can employ a o*th-order tuner. Remark 2.1. In [11] Ortega considered a slightly di!erent certainty equivalence controller of the form (7) where =(p)"1/a (p). This controller results in a static error model and requires a oth-order tuner. The error model of form (8) (where d(t),0) was considered by Morse, who used an identi"cation-based adaptive controller.
3. Morse:s tuner and instability analysis When d(t),0, the certainty equivalence controller (7) can be combined with Morse's tuner of order o* whose structure and stability properties can be summarized in the form of the following theorem.
guarantees the uniform boundedness of all the closed-loop signals and asymptotic tracking lim e(t)"0. R Theorem 1 slightly reformulates results previously introduced by Morse (1992) and Ortega (1993). To provide a self-contained form of the paper and for the purposes of the subsequent analysis, we present below the proof of the theorem. Proof. De"ne auxiliary vectors z as G z "g #A\bt) , i"1, 2,2, 2n. G G G Then from Eqs. (9) and (10) we have
h) "c2z #t) , i"1, 2,2, 2n. G G G Introducing the parameter errors t "h !t) we can G G G derive the following error model to be analyzed: L L e "!je#k m t !k m c2z #d# , (14) G G G G G G zR "(1#km2m)Az #A\bm e, (15) G G G t "!m e, (16) G G where i"1, 2,2, 2n and exponentially decays. Let us select positive constants j , j and j obeying equality j #j #2nj "j and denote j*"j!j !j (i.e. j "j*/2n). Choose the following nonnegative time function
tKQ "m e, G G
1 1 L k L <(t)" e(t)# z (t)2Pz (t)# t (t) G G G 2 2 2 G G 1 # (q) dq. 4j R
g "(1#km2m)(Ag #bt) ), G G G
(10)
h) "c2g , G G
(11)
where i"1, 2,2, 2n and k is a design parameter. If n k' (kM "c"#"PA\b"), 2j*
(13)
zR "(1#km2m)Az #A\bm e, i"1, 2,2, 2n. G G G Since (c, A, b) realizes a (0)/a (p), it is clear that c2A\b"!1. Thus from Eqs. (11) and (13) we obtain
Theorem 1 (Morse's high-order tuner). Choose a monic asymptotically stable polynomial aN (p) of degree o* and let (c, A, b) be a minimal realization of aN (0)/aN (p). Consider a high-order tuner of the form (9)
1411
(17)
When d(t),0 (and therefore d(t),0), its time derivative along solutions of Eqs. (14)}(16) takes the form (12)
where kM 'k is the upper bound of k, j* is any positive constant satisfying the inequality 0(j*(j, and the positive-de,nite matrix P obeys the equality A2P#PA"!2I, then the certainty equivalence controller (7) with the high-order tuner (9)}(11), when applied to plant (1) with no external disturbances (i.e. d(t),0),
L L
1412
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
L 1 L 4!je! z! e #"e"" "!k m z G G G 4j G G L #"e" (kM "c"#"PA\b")"m ""z " G G G 1 L 4!j e! z! (j "e"! " " G 2(j G L ! [j e!(kM "c"#"PA\b")"e""m ""z "#kmz] G G G G G L L 4!j e! z! [j e!(kM "c"#"PA\b") G G G ;"e""m ""z "#km z]. G G G G If k obeys Eq. (12) then
L
L 4!j e! z40. (18) G G Since
1 a(p) 1 m" =(p) [1, p,2, pL\]y, ¸(p) k b(p)
[1, p,2, pL\]y, ¸(p)r
"H (p)y#=(p)e2 r, W L where e is the ith coordinate vector in RL and G 1 1 a(p) H (p)" =(p) [1, p,2, pL\], W ¸(p) k b(p)
(19)
[1, p,2, pL\], 0 . Since the 2n;1 transfer matrix H (p) is proper and W stable, the transfer function =(p) is strictly proper and stable, and the signals y(t) and r(t) are bounded, we conclude from Eq. (19) that m(t) is bounded, and from Eqs. (10) and (11) we obtain the boundedness of h) . Furthermore, rewriting Eq. (8) as yR "!jy#jy #yR #km2h#
we conclude that yR (t) is bounded. Then in view of Eq. (19) we obtain the boundedness of m (t), while Eqs. (10) and (11) guarantees the boundedness of h)G . Di!erentiating yR (t) with respect to time we show the boundedness of yK (t) and, as a consequence, of mG (t) and h) . In the same manner we can prove the boundedness of all mG(t) and h) G for 14i4o*. Then from Eq. (7) we conclude the boundedness of u. In turn, Eq. (4) means the boundedness of u. Thus, the boundedness of all the closed-loop signals is proved. Since
R lim
1413
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
to reduce the total dynamic order of the auxiliary "lters 2n times. This key idea is realized in the new robust highorder tuner introduced in the following section.
where the term d includes both the "ltered external disturbance d and exponentially decaying time function due to nonzero initial conditions. Choose the Lyapunov function (24). Its time derivative along solutions of (26)}(28) takes the form
4. Main result
The structure and stability properties of the new robust high-order tuner are established by the following theorem. Theorem 2 (robust high-order tuner of simpli"ed structure). Choose a monic asymptotically stable polynomial aN (p) of degree o* and let (c, A, b) be a minimal realization of aN (0)/aN (p). Consider the following high-order tuner gR "k(1#m2m)(Ag#be),
(20)
h) "c(mc2g!ph) ),
(21)
where k, c and p are design parameters. If k5k #k*, where
(22)
1 1 k "max j"PA\b", kM ("c"#"PA\b") , 4j 4p k* is any positive constant, kM is the upper bound of k (i.e. kM 'k), j and p are any positive constants satisfying inequalities 0(j (j and 0(p (p, and the positive de,nite matrix P obeys the equation A2P#PA"!2I, then adaptive controller (7) with the tuner (20)}(22), when applied to the plant (1), guarantees for any c'0 and p'0 the boundedness of all the closed-loop signals and exponential convergence of e(t), z(t) and hI (t) to the residual set D "+y, z, h : <(y, z, h)4i #d# #i "h",, (23) 0 where hI "h!hK , z"g#A\be, i and i are some positive constants, and 1 k 1 <(e, z, h)" e# z2Pz# h 2h . 2 2 2c
(24)
Proof. De"ne the following auxiliary vector: z"g#A\be.
(25)
It is worth noting that c2g"c2z!c2A\be"c2z#e. Then the closed-loop error model to be analyzed can be presented in the form e "!je#km2h#d,
(26)
zR "k(1#m2m)Az#A\b(!je#km2h#d),
(27)
h "!cme!cmc2z!cph#cph,
(28)
#kz2PA\bm2h#z2PA\bd!km2he!km2hc2z !kp"h "#kph 2h 4!je!k(1#m2m)"z"##d # "e"#j"PA\b""z""e" #k("c"#"PA\b")"z""m""h "#"PA\b"#d# "z" !kp"h "#kp"h ""h". Let us select the positive constants j , p (i"1, 2, 3) G G and k ( j"2, 3) such that H j #j #j "j, p #p #p "p, k #k "k*. If k obeys Eq. (22), then
!k
1
("c"#"PA\b")"m""z"!(p "h " 2(p
#d# #d# # ! (j "e"! 4j 2(j
"PA\b"#d# "PA\b" # ! (k "z"! #d# 4k 2(k
p kp !k (p "h "! h # "h" 4p 2(p #d# "PA\b" kp 4!i<# # #d# # "h", 4p 4j 4k where
(29)
2k i"min 2j , , 2ckp j . and j is the maximum eigenvalue of the matrix P. . Since
1 #d# "PA\b" kp # <' #d# # "h" 4p i 4j 4k we conclude that e(t), z(t) and h (t) are bounded. From Eq. (25) we obtain the boundedness of g(t). Using the same approach as in Theorem 1 we can show that m, u and u are also bounded. Thus the boundedness of all the closed-loop signals is proved.
1414
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
Furthermore, inequality (29) means that e(t), z(t) and h (t) exponentially converge to the residual set (23), where
1 1 1 1 kp i " # "PA\b" , i " . 4i j k 4i p The theorem is proved. ) It is clear from Eqs. (20) and (21) that the structure of the introduced simpli"ed tuner allows one to express the "rst o* time derivatives of h) (t) as known functions of the available signals. Therefore, this tuner can be successfully combined with the certainty equivalence controller (7). However, in contrast to Morse's tuner (9)}(11), the proposed one involves a single auxiliary "lter (20) of order o!1. Hence the resulting closed-loop system has the simpli"ed structure. Another peculiarity of the proposed tuner (20) and (21) is its applicability in the presence of external disturbances. The term !ph) proposed initially for the traditional adaptation laws by Ioannou and Kokotovic& (1984) guarantees the exponential convergence of the tracking error as well as parameter errors to a residual set.
second-order derivative hG) (t) in view of Eq. (21) can be expressed as hG) "c(mc2g#mc2g !ph) ), where m (t) can be obtained from Eq. (6), and g (t) and h) (t) are substituted by their analytical expressions from Eqs. (20) and (21). First, consider the problem of adaptive stabilization (i.e. when r(t),0 and y (t),0). The plant initial condi tions were set as y(0)"2, yR (0)"0, yK (0)"0, while the initial conditions of all auxiliary "lters are zero. The external disturbance is given by d(t)"2 sin t#sin 20t. As seen from Fig. 1, Morse's tuner (9)}(11) with k"100 demonstrates unbounded parametric drift, while the proposed robust tuner (20)}(21) with k"100, c"1 and p"0.5 provides the boundedness of all the closedloop signals and a su$ciently small stabilization error. Now let us consider the problem of adaptive tracking when r(t)"4 sin 0.8t, d(t)"2 sin 3t#0.1 sin 20t
5. Simulation results Consider the following unstable plant k y" [u#d] p(p#a p#a ) with unknown parameters a "2, a "!1 and k"2. We assume that the upper bound kM "4 is a priori known. The control objective is to track the output of the reference model 1 y" r. p#3p#3p#1 To construct the certainty equivalence controller let us choose 1 , a (p)"p#2p#1, =(p)" p#2p#1 and d(p)"p#p#0.125
A"
0 1 0 1 , b" , c" . !1 !2 1 0
Therefore, j"1 and
P"
3 1 . 1 1
In the case considered to implement the controller (7) we need to obtain h) (t) and hG) (t). The derivative h) (t) is readily available from the adaptation law (21), while the
Fig. 1. Adaptive stabilization: (a) adaptive system with Morse's tuner demonstrates unbounded parametric drift; (b) the system with the proposed robust tuner guarantees the boundedness of all the closedloop signals.
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
1415
possible in spite of the boundedness of the tracking error. To eliminate this undesirable a!ect, a robust variant of the high-order tuner is proposed. Embedding a decay term !ph) we were able not only to provide robustness of the closed-loop control system, but also essentially simplify its structure (in particular, the total dynamic order of the auxiliary time-varying "lters is reduced 2n times). Thus the main contribution of the paper consists in instability analysis of Morse's tuner and robust redesign of the high-order tuner with simultaneous dramatic simpli"cation of its structure.
References
Fig. 2. Adaptive tracking. Adaptive system with the proposed robust tuner provides boundedness of all the closed-loop signals and small residual tracking error.
and the initial conditions of the plant, reference model and all auxiliary "lters are set to zero. Fig. 2 shows transient processes in the closed-loop system with the proposed robust high-order tuner (20) and (21) (k"100, c"1 and p"0.5). As seen from the presented plots the certainty equivalence controller with the robust simpli"ed tuner demonstrates boundedness of all the closed-loop signals, perfect tracking with a small residual error (in spite of the presence of external disturbance) and an acceptable (not very large) amplitude of the control signal.
6. Conclusion This paper demonstrates that in the presence of external disturbances the new adaptation scheme proposed recently by Morse named high-order tuner has a drawback which is typical for adaptation laws with pure integral action. Namely, unbounded parametric drift is
Feuer, A., & Morse, A. S. (1978). Adaptive control of single-input, single-output linear systems. IEEE ¹ransactions on Automatic Control, 1-23(4), 557}569. Fuer, A., Barmish, B. R., & Morse, A. S. (1978). An unstable dynamical system associated with model reference adaptive control. IEEE ¹ransactions on Automatic Control, 23(3), 499}500. Ioannou, P. A., & KokotovicH , P. V. (1984). Instability analysis and improvement of robustness of adaptive control. Automatica, 20(5), 583}594. KrsticH , M., Kanellakopoulos, I., & KokotovicH , P. V. (1994). Nonlinear design of adaptive controllers for linear systems. IEEE ¹ransactions on Automatic Control, 39(4), 738}751. KrsticH , M., Kanellakopoulos, I., & KokotovicH , P. V. (1995). Nonlinear and adaptive control design. New York: Wiley. Monopoli, R. V. (1974). Model reference adaptive control with an augmented error signal. IEEE ¹ransactions on Automatic Control, 19(5), 474}484. Morse, A. S. (1980). Global stability of parameter-adaptive control systems. IEEE ¹ransactions on Automatic Control, 25(3), 433}438. Morse, A. S. (1992). High-order parameter tuners for the adaptive control of linear and nonlinear systems. In A. Isidori, & T. J. Tarn, eds., Systems, Models and Feedback: ¹heory and Applications (pp. 339}364) Basel: BirkhaK user. Narendra, K. S., & Valavani, L. S. (1978). Stable adaptive controller design * direct control. IEEE ¹ransactions on Automatic Control, 23(4), 570}583. Narendra, K. S., Lin, Y.-H., & Valavani, L. S. (1980). Stable adaptive controller design. Part II: Proof of stability, IEEE ¹ransactions on Automatic Control, 25(3), 440}448. Ortega, R. (1993). On Morse's new adaptive controller: Parameter convergence and transient performance. IEEE ¹ransactions on Automatic Control, 38(8), 1191}1202. Zang, Z., & Bitmead, R. R. (1994). Transient bounds for adaptive control systems. IEEE ¹ransactions on Automatic Control, 39, 171}175.
Vladimir O. Nikiforov was born in SaintPetersburg, Russia, on March 12, 1963. He received electrical engineer degree in 1986 and Ph.D. degree in 1991, both from Saint-Petersburg State Institute of Fine Mechanics and Optics. He is currently Associate Professor with the Department of Automatic and Remote Control in the same Institute. Research interests of Dr. Nikiforov include adaptive and nonlinear control, trajectory motion control systems and control applications to electro-mechanical systems.
Brief Paper
Robust high-order tuner of simpli"ed structure夽 V.O. Nikiforov* Laboratory of Cybernetics and Control Systems, Institute of Fine Mechanics and Optics, Sablinskaya 14, Saint-Petersburg, 197101, Russia Received 14 May 1998; revised 20 November 1998; received in "nal form 2 February 1999
Abstract Recently, Morse has proposed a new adaptation scheme (so-called &&high-order tuner'') which generates as outputs not only tuned parameters, but also their time derivatives up to a certain order. In this paper a robusti"ed variant of the high-order tuner is introduced to adaptively stabilize a SISO linear plant subjected to external disturbances. In comparison with Morse's tuner, the proposed one has essentially simpli"ed structure. Namely, the total order of time-varying auxiliary "lters is reduced 2n times (where n is the order of the plant to be controlled). 1999 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive control; Robustness; High-order tuners; External disturbances
1. Introduction The problem of output-feedback adaptive control of linear systems has been attracting attention of the control community during the last decades. The earliest decisions were based on the certainty-equivalence principle and augmented error concept (Monopoli, 1974; Narendra and Valavani, 1978; Feuer and Morse, 1978). However, it was soon recognized that the proof of stability of the certainty equivalence schemes with adaptation laws forced by an augmented error is not trivial and can be accomplished after lengthy signal analysis (Feuer et al., 1978; Narendra et al., 1980; Morse, 1980). In this analysis, a crucial role is played by the rate of change of the adjustable parameters which is required to be square integrable. To satisfy this requirement, di!erent types of normalization are involved in the adaptation laws. In general, normalization slows down the adaptation and, as a result, leads to the poor transient performance (Zang and Bitmead, 1994). This motivates the search for new classes of adaptive controllers which can be tuned by unnormalized adaptation laws. Recently, such a class of adaptive controllers has been proposed by Krstic& et al. (1994) with the use of nonlinear design tools: integrator backstepping and nonlinear ***** * Tel.: #7(812)2191707; e-mail: [email protected]. 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor K.W. Lim under the direction of Editor C.C. Hang.
damping. Now it is well known that these new controllers possess several useful properties which were unattainable for the traditional adaptive systems (see Krstic& et al., 1995). Among those properties are strict passivity of the main adaptation loop, parametric robustness of the detuned system, a dramatic improvement of the transient performance without increasing the control e!ort. As a shortcoming of Krstic& 's controller one may consider its complicated structure (at least for the case of high relative degree systems demanding several design steps). An alternative approach to the design of adaptive controllers with unnormalized adaptation laws was proposed by Morse (1992) introducing the notion of a &&highorder tuner''. A tuner of order N was de"ned by Morse as an algorithm of adaptation that generates not only a set of adjusted parameters, but also the "rst N timederivatives of each parameter in the set. Ortega (1993) combined a high-order tuner with a dynamic certainty equivalence controller to solve the output-feedback adaptive control problem under ideal conditions (i.e. when there are no external disturbances or unmodelled dynamics). Ortega established also a priori computable bounds on the L and L norms of the tracking error. A high-order tuner can be viewed as an unnormalized gradient algorithm of adaptation plus time-varying "lters. In the author's opinion, the certain shortcomings of Morse's tuner consist in the following. First, as any adaptation law with pure integral action the high-order tuner is not robust in the sense that bounded external disturbances can cause unbounded parametric drift (this
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 5 1 - 5
1410
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
e!ect will be demonstrated in the present paper). The second shortcoming consists in the excess dynamic order of the tuner. Namely, we need to employ auxiliary timevarying "lters of the total order m;N (where m is the number of adjustable parameters and N is the tuner order dependent on the relative degree of the plant to be controlled). This can be impractical for multi-parameter systems with a high relative degree. Therefore, from the practical point of view, simpli"ed robust modi"cations of Morse's tuner are greatly needed. The main contributions of the paper are the following. First, the e!ect of external disturbances on stability of Morse's tuner is analyzed and possibility of unbounded parametric drift is demonstrated. Second, we propose a robusti"ed variant of high-order tuners expanding their applicability to the case of the presence of external disturbances. Third, we reduce the total dynamic order of the auxiliary "lters 2n times (where n is the order of the plant to be controlled) and, as a consequence, essentially simplify the structure of the resulting adaptive system. The paper is organized as follows. In Section 2 we pose the control problem and brie#y introduce a variant of the certainty equivalence controller which slightly di!ers from those considered by Ortega (1993). In Section 3, in self-contained form, Morse's high-order tuner is presented, its stability properties are analyzed and possibility of unbounded parametric drift caused by external disturbances is demonstrated. In Section 4 we present the main result introducing a robusti"ed high-order tuner and in Section 5 we demonstrate the simulation results.
2. Problem statement and controller structure We consider the problem of adaptive control of linear time-invariant SISO plants y"k
b(p) [u#d], a(p)
(1)
where y is the reference output, r is a piece-wise continu ous bounded reference input (command signal), a (p) is a monic Hurwitz polynomial of order o"n!m, b '0. The control objective is to force the output y to track a reference signal y with a certain accuracy while keeping all the closed-loop signals bounded. In order to design an appropriate control law, we "rst derive a suitable error model. Utilizing known results (see Feuer and Morse, 1978), we conclude that for any Hurwitz monic polynomial ¸(p) of order n!1, the tracking error e"y!y can be presented in the form k e" [u(t)2h#u#dM (t)]#, (3) a (p) where exponentially decays due to nonzero initial conditions, h3RL is the vector of unknown constant parameters, dM is a bounded function of time depending on d, u(t)3 RL is the standard regressor vector, i.e. 1 u(t)" [u(t), uR (t),2, uL\(t), y(t), ¸(p) yR (t),2, yL\(t), ¸(p)r(t)].
(4)
Furthermore, we de"ne a transfer function =(p) obeying the equality p#j =(p)" , a (p) where j is any positive constant. Obviously, =(p) is an asymptotically stable minimum phase transfer function with the relative degree o*"o!1 and 1 1 " =(p). a (p) p#j Therefore, the error model (3) can be written as
where y and u are the plant output and input, respectively, d is the bounded external disturbance, p"d/dt denotes the di!erential operator, a(p) and b(p) are monic coprime polynomials with unknown coe$cients. We made the following standard assumptions:
k e" [m(t)2h#=(p)u#d(t)]#, p#j
(1) the degrees n and m are known and o"n!m51; (2) the coe$cient k is nonzero and its sign is known (without loss of generality k is assumed to be positive); (3) the polynomial b(p) is Hurwitz.
Analysis of model (5) motivates us to choose the control law of the form
Let the desired behavior of the closed-loop system be speci"ed by the following reference model: b y " r, a (p)
(2)
(5)
where d(t)"=(p)dM (t) and m(t)"=(p)u(t).
u"!=(p)\m(t)2h) ,
(6)
(7)
where h) is the vector of adjustable parameters. Control (7) is given the name of dynamic certainty equivalence controller (Ortega, 1993). Substituting Eq. (7) into Eq. (5) we obtain k e" [m(t)2h#d(t)]#, p#j
(8)
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
where h"h!h) . Since the transfer function k/(p#j) is strictly positive real, then to tune the vector of adjustable parameters h) one can employ an adaptation law involving the accessible output e. When plant (1) has a relative degree equal to one (and therefore o*"0), the implementation of the control law (7) is trivial. If o"2, then o*"1 and implementation of Eq. (7) requires the "rst-order derivatives h) and m. This does not constitute a principal problem, because the derivatives of m(t) up to order o* are readily available without di!erentiation from Eq. (6), while h) can be obtained from a standard adaptation law of the integral type. However when o'2 (and therefore o*'1) the certainty equivalence controller (7) cannot be combined with standard adaptation laws, because they do not allow one to obtain the necessary derivatives of h) (t). To overcome this di$culty, one can employ a o*th-order tuner. Remark 2.1. In [11] Ortega considered a slightly di!erent certainty equivalence controller of the form (7) where =(p)"1/a (p). This controller results in a static error model and requires a oth-order tuner. The error model of form (8) (where d(t),0) was considered by Morse, who used an identi"cation-based adaptive controller.
3. Morse:s tuner and instability analysis When d(t),0, the certainty equivalence controller (7) can be combined with Morse's tuner of order o* whose structure and stability properties can be summarized in the form of the following theorem.
guarantees the uniform boundedness of all the closed-loop signals and asymptotic tracking lim e(t)"0. R Theorem 1 slightly reformulates results previously introduced by Morse (1992) and Ortega (1993). To provide a self-contained form of the paper and for the purposes of the subsequent analysis, we present below the proof of the theorem. Proof. De"ne auxiliary vectors z as G z "g #A\bt) , i"1, 2,2, 2n. G G G Then from Eqs. (9) and (10) we have
h) "c2z #t) , i"1, 2,2, 2n. G G G Introducing the parameter errors t "h !t) we can G G G derive the following error model to be analyzed: L L e "!je#k m t !k m c2z #d# , (14) G G G G G G zR "(1#km2m)Az #A\bm e, (15) G G G t "!m e, (16) G G where i"1, 2,2, 2n and exponentially decays. Let us select positive constants j , j and j obeying equality j #j #2nj "j and denote j*"j!j !j (i.e. j "j*/2n). Choose the following nonnegative time function
tKQ "m e, G G
1 1 L k L <(t)" e(t)# z (t)2Pz (t)# t (t) G G G 2 2 2 G G 1 # (q) dq. 4j R
g "(1#km2m)(Ag #bt) ), G G G
(10)
h) "c2g , G G
(11)
where i"1, 2,2, 2n and k is a design parameter. If n k' (kM "c"#"PA\b"), 2j*
(13)
zR "(1#km2m)Az #A\bm e, i"1, 2,2, 2n. G G G Since (c, A, b) realizes a (0)/a (p), it is clear that c2A\b"!1. Thus from Eqs. (11) and (13) we obtain
Theorem 1 (Morse's high-order tuner). Choose a monic asymptotically stable polynomial aN (p) of degree o* and let (c, A, b) be a minimal realization of aN (0)/aN (p). Consider a high-order tuner of the form (9)
1411
(17)
When d(t),0 (and therefore d(t),0), its time derivative along solutions of Eqs. (14)}(16) takes the form (12)
where kM 'k is the upper bound of k, j* is any positive constant satisfying the inequality 0(j*(j, and the positive-de,nite matrix P obeys the equality A2P#PA"!2I, then the certainty equivalence controller (7) with the high-order tuner (9)}(11), when applied to plant (1) with no external disturbances (i.e. d(t),0),
L L
1412
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
L 1 L 4!je! z! e #"e"" "!k m z G G G 4j G G L #"e" (kM "c"#"PA\b")"m ""z " G G G 1 L 4!j e! z! (j "e"! " " G 2(j G L ! [j e!(kM "c"#"PA\b")"e""m ""z "#kmz] G G G G G L L 4!j e! z! [j e!(kM "c"#"PA\b") G G G ;"e""m ""z "#km z]. G G G G If k obeys Eq. (12) then
L
L 4!j e! z40. (18) G G Since
1 a(p) 1 m" =(p) [1, p,2, pL\]y, ¸(p) k b(p)
[1, p,2, pL\]y, ¸(p)r
"H (p)y#=(p)e2 r, W L where e is the ith coordinate vector in RL and G 1 1 a(p) H (p)" =(p) [1, p,2, pL\], W ¸(p) k b(p)
(19)
[1, p,2, pL\], 0 . Since the 2n;1 transfer matrix H (p) is proper and W stable, the transfer function =(p) is strictly proper and stable, and the signals y(t) and r(t) are bounded, we conclude from Eq. (19) that m(t) is bounded, and from Eqs. (10) and (11) we obtain the boundedness of h) . Furthermore, rewriting Eq. (8) as yR "!jy#jy #yR #km2h#
we conclude that yR (t) is bounded. Then in view of Eq. (19) we obtain the boundedness of m (t), while Eqs. (10) and (11) guarantees the boundedness of h)G . Di!erentiating yR (t) with respect to time we show the boundedness of yK (t) and, as a consequence, of mG (t) and h) . In the same manner we can prove the boundedness of all mG(t) and h) G for 14i4o*. Then from Eq. (7) we conclude the boundedness of u. In turn, Eq. (4) means the boundedness of u. Thus, the boundedness of all the closed-loop signals is proved. Since
R lim
1413
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
to reduce the total dynamic order of the auxiliary "lters 2n times. This key idea is realized in the new robust highorder tuner introduced in the following section.
where the term d includes both the "ltered external disturbance d and exponentially decaying time function due to nonzero initial conditions. Choose the Lyapunov function (24). Its time derivative along solutions of (26)}(28) takes the form
4. Main result
The structure and stability properties of the new robust high-order tuner are established by the following theorem. Theorem 2 (robust high-order tuner of simpli"ed structure). Choose a monic asymptotically stable polynomial aN (p) of degree o* and let (c, A, b) be a minimal realization of aN (0)/aN (p). Consider the following high-order tuner gR "k(1#m2m)(Ag#be),
(20)
h) "c(mc2g!ph) ),
(21)
where k, c and p are design parameters. If k5k #k*, where
(22)
1 1 k "max j"PA\b", kM ("c"#"PA\b") , 4j 4p k* is any positive constant, kM is the upper bound of k (i.e. kM 'k), j and p are any positive constants satisfying inequalities 0(j (j and 0(p (p, and the positive de,nite matrix P obeys the equation A2P#PA"!2I, then adaptive controller (7) with the tuner (20)}(22), when applied to the plant (1), guarantees for any c'0 and p'0 the boundedness of all the closed-loop signals and exponential convergence of e(t), z(t) and hI (t) to the residual set D "+y, z, h : <(y, z, h)4i #d# #i "h",, (23) 0 where hI "h!hK , z"g#A\be, i and i are some positive constants, and 1 k 1 <(e, z, h)" e# z2Pz# h 2h . 2 2 2c
(24)
Proof. De"ne the following auxiliary vector: z"g#A\be.
(25)
It is worth noting that c2g"c2z!c2A\be"c2z#e. Then the closed-loop error model to be analyzed can be presented in the form e "!je#km2h#d,
(26)
zR "k(1#m2m)Az#A\b(!je#km2h#d),
(27)
h "!cme!cmc2z!cph#cph,
(28)
#kz2PA\bm2h#z2PA\bd!km2he!km2hc2z !kp"h "#kph 2h 4!je!k(1#m2m)"z"##d # "e"#j"PA\b""z""e" #k("c"#"PA\b")"z""m""h "#"PA\b"#d# "z" !kp"h "#kp"h ""h". Let us select the positive constants j , p (i"1, 2, 3) G G and k ( j"2, 3) such that H j #j #j "j, p #p #p "p, k #k "k*. If k obeys Eq. (22), then
!k
1
("c"#"PA\b")"m""z"!(p "h " 2(p
#d# #d# # ! (j "e"! 4j 2(j
"PA\b"#d# "PA\b" # ! (k "z"! #d# 4k 2(k
p kp !k (p "h "! h # "h" 4p 2(p #d# "PA\b" kp 4!i<# # #d# # "h", 4p 4j 4k where
(29)
2k i"min 2j , , 2ckp j . and j is the maximum eigenvalue of the matrix P. . Since
1 #d# "PA\b" kp # <' #d# # "h" 4p i 4j 4k we conclude that e(t), z(t) and h (t) are bounded. From Eq. (25) we obtain the boundedness of g(t). Using the same approach as in Theorem 1 we can show that m, u and u are also bounded. Thus the boundedness of all the closed-loop signals is proved.
1414
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
Furthermore, inequality (29) means that e(t), z(t) and h (t) exponentially converge to the residual set (23), where
1 1 1 1 kp i " # "PA\b" , i " . 4i j k 4i p The theorem is proved. ) It is clear from Eqs. (20) and (21) that the structure of the introduced simpli"ed tuner allows one to express the "rst o* time derivatives of h) (t) as known functions of the available signals. Therefore, this tuner can be successfully combined with the certainty equivalence controller (7). However, in contrast to Morse's tuner (9)}(11), the proposed one involves a single auxiliary "lter (20) of order o!1. Hence the resulting closed-loop system has the simpli"ed structure. Another peculiarity of the proposed tuner (20) and (21) is its applicability in the presence of external disturbances. The term !ph) proposed initially for the traditional adaptation laws by Ioannou and Kokotovic& (1984) guarantees the exponential convergence of the tracking error as well as parameter errors to a residual set.
second-order derivative hG) (t) in view of Eq. (21) can be expressed as hG) "c(mc2g#mc2g !ph) ), where m (t) can be obtained from Eq. (6), and g (t) and h) (t) are substituted by their analytical expressions from Eqs. (20) and (21). First, consider the problem of adaptive stabilization (i.e. when r(t),0 and y (t),0). The plant initial condi tions were set as y(0)"2, yR (0)"0, yK (0)"0, while the initial conditions of all auxiliary "lters are zero. The external disturbance is given by d(t)"2 sin t#sin 20t. As seen from Fig. 1, Morse's tuner (9)}(11) with k"100 demonstrates unbounded parametric drift, while the proposed robust tuner (20)}(21) with k"100, c"1 and p"0.5 provides the boundedness of all the closedloop signals and a su$ciently small stabilization error. Now let us consider the problem of adaptive tracking when r(t)"4 sin 0.8t, d(t)"2 sin 3t#0.1 sin 20t
5. Simulation results Consider the following unstable plant k y" [u#d] p(p#a p#a ) with unknown parameters a "2, a "!1 and k"2. We assume that the upper bound kM "4 is a priori known. The control objective is to track the output of the reference model 1 y" r. p#3p#3p#1 To construct the certainty equivalence controller let us choose 1 , a (p)"p#2p#1, =(p)" p#2p#1 and d(p)"p#p#0.125
A"
0 1 0 1 , b" , c" . !1 !2 1 0
Therefore, j"1 and
P"
3 1 . 1 1
In the case considered to implement the controller (7) we need to obtain h) (t) and hG) (t). The derivative h) (t) is readily available from the adaptation law (21), while the
Fig. 1. Adaptive stabilization: (a) adaptive system with Morse's tuner demonstrates unbounded parametric drift; (b) the system with the proposed robust tuner guarantees the boundedness of all the closedloop signals.
V.O. Nikiforov/Automatica 35 (1999) 1409}1415
1415
possible in spite of the boundedness of the tracking error. To eliminate this undesirable a!ect, a robust variant of the high-order tuner is proposed. Embedding a decay term !ph) we were able not only to provide robustness of the closed-loop control system, but also essentially simplify its structure (in particular, the total dynamic order of the auxiliary time-varying "lters is reduced 2n times). Thus the main contribution of the paper consists in instability analysis of Morse's tuner and robust redesign of the high-order tuner with simultaneous dramatic simpli"cation of its structure.
References
Fig. 2. Adaptive tracking. Adaptive system with the proposed robust tuner provides boundedness of all the closed-loop signals and small residual tracking error.
and the initial conditions of the plant, reference model and all auxiliary "lters are set to zero. Fig. 2 shows transient processes in the closed-loop system with the proposed robust high-order tuner (20) and (21) (k"100, c"1 and p"0.5). As seen from the presented plots the certainty equivalence controller with the robust simpli"ed tuner demonstrates boundedness of all the closed-loop signals, perfect tracking with a small residual error (in spite of the presence of external disturbance) and an acceptable (not very large) amplitude of the control signal.
6. Conclusion This paper demonstrates that in the presence of external disturbances the new adaptation scheme proposed recently by Morse named high-order tuner has a drawback which is typical for adaptation laws with pure integral action. Namely, unbounded parametric drift is
Feuer, A., & Morse, A. S. (1978). Adaptive control of single-input, single-output linear systems. IEEE ¹ransactions on Automatic Control, 1-23(4), 557}569. Fuer, A., Barmish, B. R., & Morse, A. S. (1978). An unstable dynamical system associated with model reference adaptive control. IEEE ¹ransactions on Automatic Control, 23(3), 499}500. Ioannou, P. A., & KokotovicH , P. V. (1984). Instability analysis and improvement of robustness of adaptive control. Automatica, 20(5), 583}594. KrsticH , M., Kanellakopoulos, I., & KokotovicH , P. V. (1994). Nonlinear design of adaptive controllers for linear systems. IEEE ¹ransactions on Automatic Control, 39(4), 738}751. KrsticH , M., Kanellakopoulos, I., & KokotovicH , P. V. (1995). Nonlinear and adaptive control design. New York: Wiley. Monopoli, R. V. (1974). Model reference adaptive control with an augmented error signal. IEEE ¹ransactions on Automatic Control, 19(5), 474}484. Morse, A. S. (1980). Global stability of parameter-adaptive control systems. IEEE ¹ransactions on Automatic Control, 25(3), 433}438. Morse, A. S. (1992). High-order parameter tuners for the adaptive control of linear and nonlinear systems. In A. Isidori, & T. J. Tarn, eds., Systems, Models and Feedback: ¹heory and Applications (pp. 339}364) Basel: BirkhaK user. Narendra, K. S., & Valavani, L. S. (1978). Stable adaptive controller design * direct control. IEEE ¹ransactions on Automatic Control, 23(4), 570}583. Narendra, K. S., Lin, Y.-H., & Valavani, L. S. (1980). Stable adaptive controller design. Part II: Proof of stability, IEEE ¹ransactions on Automatic Control, 25(3), 440}448. Ortega, R. (1993). On Morse's new adaptive controller: Parameter convergence and transient performance. IEEE ¹ransactions on Automatic Control, 38(8), 1191}1202. Zang, Z., & Bitmead, R. R. (1994). Transient bounds for adaptive control systems. IEEE ¹ransactions on Automatic Control, 39, 171}175.
Vladimir O. Nikiforov was born in SaintPetersburg, Russia, on March 12, 1963. He received electrical engineer degree in 1986 and Ph.D. degree in 1991, both from Saint-Petersburg State Institute of Fine Mechanics and Optics. He is currently Associate Professor with the Department of Automatic and Remote Control in the same Institute. Research interests of Dr. Nikiforov include adaptive and nonlinear control, trajectory motion control systems and control applications to electro-mechanical systems.