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Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

FrA10.1

A Discrete-Time Periodic Adaptive Control Approach for Parametric-strict-feedback Systems Deqing Huang, Jian-Xin Xu, and Zhongsheng Hou Abstract— A periodic adaptive control approach is proposed for a class of discrete-time parametric-strict-feedback systems with unknown periodic coefficients which include the control gains. Using the information of the past few periods, the proposed adaptive controller updates the parameters periodically in a pointwise manner over one entire period. By means of backstepping design and through rigorous derivation, we show that the proposed controller guarantees the boundedness of all the closed-loop signals, and achieves the asymptotic tracking convergence. To the end, an illustrative example is presented.

I. INTRODUCTION In this work we extend the results of periodic adaptive control (PAC) of nonlinear systems in companion form [1] to more general classes of nonlinear systems in parametricstrict-feedback. The central idea of PAC is to classify parameters into periodic and nonperiodic cases, instead of slow time-varying and rapid time-varying cases. The periodic variations are encountered in many real systems such as system parameters [2] [3] or exogenous disturbances [4]-[6]. Considering the fact that the value of a periodic parameter will be invariant if the time is being shifted by one period, it is natural to extend the classic adaptive control, which is updated in two consecutive instances, to PAC which is achieved by updating parameters in the same instance of two consecutive periods. Analogously, the boundedness of the parametric estimate and convergence analysis are discussed by considering the difference between two consecutive periods, that is, convergence is asymptotical with respect to the number of periods, instead of the time instances. To deal with the unmatched structure and periodic uncertainties, we extend the backstepping technique [7]-[9] to the periodic adaptation for discrete-time systems. However, due to the nature of periodic coefficients, existing adaptive backstepping designs are not applicable. In this work, our main contribution is to explore and construct a new PAC law which employs multiple periods of previous states and tracking errors in the backstepping approach, hence warrant the asymptotic convergence of the output tracking for nonlinear plants with unmatched uncertainties in parametric-strictfeedback form. In discrete-time adaptive backstepping design, how to deal with the uncertain input gains is a challenging issue. As indicated in [10], the challenge here is that the elegantly devised coordinate mapping in many existing backstepping D. Huang and J.-X. Xu are with Department of Electrical and Computer Engineering National University of Singapore, Singapore 117576, elex-

[email protected] Z. Hou is with Advanced Control Systems Lab, Beijing Jiaotong University, Beijing, China, 100044

978-1-4244-3872-3/09/$25.00 ©2009 IEEE

designs for discrete-time nonlinear systems, such as [11][15], is not directly applicable when the control gains are unknown. In [10], the uncertain input gains are all constants. When periodic input gains are concerned, the adaptive backstepping design becomes much more difficult and complicated. Another main contribution of this work is to present a complete adaptive backstepping control design for this challenging problem, that is, nonlinear systems with unknown periodic input gains. The paper is organized as follows. In Section II, we present the problem formulation and address the difficulties encountered. In Section III, n steps of periodic adaptive control law are proposed. Section IV gives the convergence analysis. To the end, an illustrative example is provided in Section V. Throughout this paper, k · k denotes the Euclidean norm. For notational convenience, in mathematical expressions, fk represents f(k). II. P ROBLEM F ORMULATION Consider a class of strict-feedback discrete-time systems with time-varying uncertainties, described by T xi,k+1 = θ 0k ξ 0i (xi,k ) + bi,kxi+1,k , T

xn,k+1

=

θ 0k

yk

=

x1,k .

ξ 0n(xn,k ) + bn,k uk ,

T

(1)

where xi = [x1, x2, · · · , xi] , uk and yk represent the system input and output respectively, θ0k is the unknown periodic parametric vector in Rm . For each 1 ≤ i ≤ n, ξ0i (xi,k ) denotes the known nonlinearity which is continuous and satisfies ξ0i (0) = 0. For notational convenience, ξ0i (xi,k ) is denoted by ξ0i,k in the remaining parts of the paper. The unknown periodic function bi,k ∈ C[0, ∞) is system control gain. The prior information with regards to bi,k is that the control direction is known and invariant, that is, bi,k is either positive or negative and nonsingular for all k. Without loss of generality, assume that bi,k ≥ bmin where bmin > 0 is a known lower bound. The control objective here is to force the output of the system (1) to track the bounded reference signal ym . 0 Note that each unknown parameter, θj,k or bi,k , may have its own period Nj or Ni,b, j = 1, · · · , m, i = 1, · · ·, n. The periodic adaptive control will still be applicable if there exists a common period N , such that each Nj and Ni,b can divide N with an integer quotient. This is always true in discrete-time since Nj and Ni,b are integers, therefore N can be either the least common multiple of Nj and Ni,b ,

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FrA10.1 or simply the product of Nj and Ni,b when both are prime. Therefore, N can be used as the updating period. Now, we first recall the following assumption presented in [7], which implies the sector-bounded condition or the restrictive growth condition [1]. Then the Key Technical Lemma [8] can be applied for convergence analysis. Assumption 1: All the states x1,k , x2,k, · · · , xn,k are measured variables and kξ 0i (xi ) − ξ 0i (yi )k kξ0n(xn )k

≤ ≤

dikxi − yi k, ∀xi , yi ∈ Ri , dnkxnk (2)

where dj > 0, 1 ≤ j ≤ n. Based on Assumption 1 and by using backstepping technique and the projection algorithm, we extend the idea in [7] to more general case (1) and provide global results for such a nonlinearly parameterized problem. The results can be easily extended to the least-squares algorithm [8].

iT h 0 0 , · · · , θˆ1,m , ˆb1 is chosen as θˆ1,1 (z − z2,k−N )ξ1,k−N ˆ 1,k−N + 1,k−N +1 ˆ 1,k = L θ , (5) θ c + |ξ1,k−N |2 T ˆ1,k−N + where c > 0. Let a = aT1 , a2 denote the vector θ (z1,k−N+1 −z2,k−N )ξ 1,k−N , the semi-saturator is defined as c+|ξ 1,k−N |2 T aT1 , a2 , a2 ≥ bmin , (6) L[a] = T T a1 , bmin , a2 < bmin . ˆ1,k is updated by using the information We can see that θ ˆ 1,k−N , x2,k−N , x1,k−N +1, which are all available in the θ current step. Step i (2 ≤ i ≤ n − 1). Let i−j−1 0 ξ¯j,i(k)

ξ0j

=

T

xn,k+1

=

θ 0k

yk

=

x1,k ,

ξ 0n,k + ˆbn,k uk + ˜bn,kuk ,

(3)

ˆbl+1,k−N +i−j−l zi−j+2,k − S1 , · · ·

l=1 i−j−1 Y l=1

where Q0 ˆ 1 ≤ j ≤ i − 1, the product operator satisfies l=1 bl,k = 1, and Sm is equal to m m 0 T XY ˆb−1 ˆ p,k−N +i−j+m−p θ r,k−N +i−j+m−r

˜0 θ 1,k

T

˜T ξ , θ 1,k 1,k

T

˜0 θ 1,k

¯0 ξ r,i−j+m (k),

where m = 1, · · ·, j − 1. Furthermore, make the following transformation 0 T ˆ ξ0i,k zi+1,k = ˆbi,k xi+1,k + θ i,k +

1,k

z1,k+1 = z2,k +

! ˆbl+j−1,k−N +i−j−l zi,k − Sj−1 ,

r=1 p=r

Assume n ≤ N + 1 and construct the adaptation law as follows. Step 1. Let z1,k = x1,k and z2,k = ˆb1,k x2,k + 0 T ˆ θ ξ0 , then 1,k

ˆbl,k−N +i−j−l zi−j+1,k ,

l=1 i−j−1 Y

III. D ISCRETE -T IME P ERIODIC A DAPTIVE C ONTROL The presence of the uncertain system input gains makes the controller design in each step of backstepping more complex. To derive the periodic adaptive control law, define ˆbi,k to be the estimation of bi,k and ˜bi,k = bi,k − ˆbi,k , the system dynamics (1) can be rewritten as T xi,k+1 = θ 0k ξ 0i,k + ˆbi,k xi+1,k + ˜bi,k xi+1,k,

Y

(4)

i−1 i−1 Y X r=1 p=r

Then,

0 T 0 ˆb−1 ˆ θ ξ¯r,i (k). r,k−N +i−r p,k−N +i−p T

ˆ0 , θ 1,k

˜ 1,k = where θ , ˜b1,k , = − and T 0 T T ˆ ξ1,k = ξ01,k , ˆb−1 . Note that ξ01,k 1,k z2,k − θ 1,k θ 0k

˜ ξ zi,k+1 = ˆbi−1,k+1zi+1,k + θ i,k i,k+1 + χi,k+1 , T T ˜ i,k = θ ˜0 , ˜bi,k , where θ i,k

the computation of ξ1,k requires the inverse of the system input gain estimate ˆb1,k and may cause a singularity in the solution if the estimate of the gain is zero. To ensure this never occurs a semi-saturator must be applied on the gain estimator such that the estimate never goes below the ˆ1 = lower bound. For this purpose, the update law for θ

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h T ξ i,k+1 = ˆbi−1,k+1 ξ 0i,k , 0 T ˆbi−1,k+1ˆb−1 zi+1,k − θ ˆ ξ 0i,k i,k i,k −

i−1 i−1 X Y r=1 p=r

ˆb−1 p,k−N +i−p

T ˆ0 θ r,k−N +i−r

0 ξ¯r,i (k)

(7)

!#T

,

FrA10.1 χi,k+1 = −

T ˆbi−1,k+1 0 0 ˆ θ ξ¯i−1,i(k) i−1,k−N +1

iT h ˆn = θˆ0 , · · · , θˆ0 , ˆbn is chosen as The update law for θ n,1 n,m ξ n,k−N +1 ˆ n,k = L θ ˆ n,k−N + (zn,k−N +1 θ c + kξn,k−N +1k2

ˆbi−1,k−N +1 T ˆbi−1,k+1 ˆ0 + θ ξ0i−1,k+1 − i−1,k+1 ˆbi−1,k−N +1 i−2 i−2 0 T X Y 0 ˆb−1 ˆ θ ξ¯r,i (k) × r,k−N +i−r p,k−N +i−p

−ˆbn−1,k−N +1

+

ˆb−1 p,k−N +i−p

r=1 p=r

T ˆ0 θ r,k−N +i−r

−χn,k−N +1 )] .

0 ξ¯r,i−1(k

+ 1).

iT h ˆ i = θˆ0 , · · · , θˆ0 , ˆbi is chosen as The update law for θ i,1 i,m ˆi,k θ

ξi,k−N +1 (zi,k−N +1 c + |ξi,k−N +1 |2 i −ˆbi−1,k−N +1zi+1,k−N − χi,k−N +1 . (8)

ˆi,k−N + L θ

=

ˆ i,k is updated with measurable states xi+1,k−N , Note that θ ˆ w ,k−w , where w1 = xi,k−N +1 , and previous estimates θ 1 2 1, · · · , i, w2 = N, · · · , 2N − 1, which are all available in the current step. Step n. This is the last step of the design procedure. We select the control law n−1 Y

uk = ˆb−1 n,k

p=1

−

n−1 X n−1 Y

0 T ˆ ˆb−1 ξ0n,k p,k−N +n−p ym (k + n) − θ n,k

ˆb−1 p,k−N +n−p

r=1 p=r

0 ˆ θ

r,k−N +n−r

T

0 ξ¯r,n (k)

!

,

= ˆbn−1,k+1

zn,k+1

p=1 T ˜ ξ +θ n,k n,k+1

˜ n,k = where θ

˜0 θ n,k

T

, ˜bn,k

T

(10)

, χn,k+1 takes the form as

p=1

−

0 T ˆ ˆb−1 ξ0n,k p,k−N +n−p ym (k + n) − θ n,k

n−1 X n−1 Y r=1 p=r

ˆb−1 p,k−N +n−p

T ˆ0 θ r,k−N +n−r

0 ξ¯r,n (k)

=

z1,k+1 − z2,k ,

˜T ξ θ i,k i,k+1

=

zi,k+1 − ˆbi−1,k+1zi+1,k − χi,k+1,

˜T ξ θ n,k n,k+1

=

zn,k+1 − ˆbn−1,k+1

i = 2, · · · , n − 1, n−1 Y

ˆb−1 p,k−N +n−p

p=1

×ym (k + n) − χn,k+1.

χi,k+1 with i = n, h T ξn,k+1 = ˆbn−1,k+1 ξ0n,k , ˆbn−1,k+1ˆb−1 n,k n−1 Y

˜T ξ θ 1,k 1,k

(9)

ˆb−1 p,k−N +n−p ym (k + n) + χn,k+1,

(11)

ˆ n,k , the information xn,k−N +1, ym (k−N + For calculating θ ˆ n), θw1 ,k−w2 , where w1 = 1, · · ·, N, w2 = N, · · · , 2N − 1, are all adopted. Note that in each step the system is transformed from x-domain to z-domain. Actually, the amount zi+1,k can be regarded as an predictor of xi,k+1 in current Step k. Due to the existence of unknown control gains, it is not easy to find such corresponding predictors for all xi,k+1 because the higher the order of system the more estimation information or previous predictors will be engulfed in the following when we do backstepping design step by step. This can be seen from the above design procedure. On the other hand, in order to derive the parameter estimations in current step we usually choose the parametric estimation error in the previous period as the feedback information, which however is obviously not available because of the parametric uncertainties. Investigating the relationships (4), (7), and (10), the estimation errors for unknown parameters can be expressed in z-domain as

¯0 (k) takes the same form as ξ¯0 (k) with i = n. where ξ j,n j,i Then n−1 Y

ˆb−1 p,k−2N +n−p ym (k − N + n)

p=1

r=1 p=r

i−2 i−2 X Y

n−1 Y

Thus, in the actual projection algorithm we use the information in the above right hand sides to replace the unavailable parameter estimate errors. Moreover, we can prove the following properties of the projection algorithm, which are useful in our convergence analysis. Owing to space limitation, details of the proof are ignored here. Lemma 1: ˜ i,k k ≤ kθ ˜i,k−N k, i = 1, · · · , n, (P1) kθ (12) (P2)

!#T

.

(P3)

6622

(z1,k+1 − z2,k )2 = 0, k→∞ c + kξ 1,kk2 lim

(13)

(zi,k+1 − ˆbi−1,k+1zi+1,k − χi,k+1)2 = 0, (14) k→∞ c + kξi,k+1k2 lim

FrA10.1 "

where 2 ≤ i ≤ n − 1, 2 ηn,k = 0, (15) (P4) lim k→∞ c + kξ n,k+1k2 Qn−1 where ηn,k = zn,k+1 − ˆbn−1,k+1 p=1 ˆb−1 p,k−N +n−p ym (k + n) − χn,k+1,

(P5)

ˆi,k − θ ˆi,k−N k = 0, lim kθ

k→∞

i = 1, · · · , n.

(16)

The preceding control law and parametric updating law were built as n ≤ N + 1. As can be seen from (5), (8), (9) and (11), in the current Step k the future information to Step k − N + n − 1 will be used for control if n > N + 1, which is however not available. The following procedure can ˆ 0 in (5), be applied to solve this problem: for all ˆbp,q and θ p,q (8), (9) and (11), if the index q is greater than k, then they can be substituted by the latest corresponding signals which are available in a point-wise manner. For instance, assume N = 3 and consider the estimate ˆbp,q . By this rule ˆbp,k+3w+1 → ˆbp,k−2, ˆbp,k+3w+2 → ˆbp,k−1, ˆbp,k+3w+3 → ˆbp,k , where w = 0, 1, 2, · · · denotes the number of integral periods being shifted. After these substitutions, the control law and parametric updating law become implementable for the discrete system. Subsequently, the properties (P1)-(P5) can be similarly achieved. Moreover, note that the information at most in the previous ˆ i,k . 2N − 1 steps should be known beforehand to calculate θ ˆ ˆ For the sake of initialization, always assume θ i,k = θ ini, ˆini is a given vector, i = 1, · · · , n, k = 1, · · · , 2N − where θ 1.

bn = 0, 0, · · ·, 0, ˆbn−1,k+1

n−1 Y

ˆb−1 p,k−N +n−p

p=1

#T

.

Next, we move to the convergence analysis based on Lemma 1. Theorem 1: All the signals in the closed-loop system, consisting of (1), (5), (8), (9), (11), and (17), are uniformly bounded. In addition, y → ym as k → ∞. Proof. The property (12) induces that all the parameter estimates are bounded. Next, assume z1,k is bounded and prove the boundedness of other quantities in all steps. Step 1. If z1,k is bounded, ξ01,k is bounded by 0 T ˆ ξ01,k , we Assumption 1. Since z2,k = ˆb1,k x2,k + θ 1,k have 0 T 0 ˆ z x2,k = ˆb−1 ξ θ − 2,k 1,k . 1,k 1,k Then,

ξ 1,k =

h

ξ 01,k

T

, x2,k

iT

.

On the can derive x2,k = we further other hand,Tfrom (1) T 0 0 −1 −1 b1,k x1,k+1 − θk ξ1,k = b1,k z1,k+1 − θ 0k ξ01,k . Since the positive property and periodicity of b1,k implies 0 the boundedness of b−1 1,k , the periodicity of θ k implies the 0 boundedness of θ k , and the boundedness of z1,k implies the boundedness of ξ 01,k, x2,k and then ξ1,k are also bounded. Then, considering the property (13) we have lim (z1,k+1 − z2,k ) = 0,

k→∞

(18)

implying z2,k is also bounded. Thus, z1 , z2 and x1, x2 are all bounded. Since all the parameter estimates are 0 T ˆ b1,k+1 ˆ bounded, χ θ = − ξ01 (z2,k ) + 2,k+1 1,k−N +1 ˆ IV. C ONVERGENCE A NALYSIS b1,k−N+1 0 T ˆ θ ξ 01,k+1 is also bounded. Summarizing the n steps, the closed-loop system is ex1,k+1 pressed as Step i (2 ≤ i ≤ n−1). Recursively, assume that until Step T i − 1 the quantities zr,k , xr,k and χr,k+1 are all bounded, ˜ ξ θ where r ≤ i. Next to prove zi+1,k , xi+1,k and χi+1,k+1 are 1,k .1,k+1 0 T . zk+1 = An zk + bnym (k + n) + ˆ . ξ 0i,k + + Ψk+1 , (17)also bounded. Since zi+1,k = ˆbi,kxi+1,k + θ i,k T 0 T ˜ ξ Pi−1 Qi−1 ˆ−1 0 θ n,k n,k+1 ˆ ξ¯r,i (k). we have r=1 p=r bp,k−N +i−p θ r,k−N +i−r = [z1,k, z2,k , · · · , zn,k]T , Ψ = where zk 0 T −1 △ ˆ ˆ T x = b ξ0i,k i+1,k [χ1, χ2, · · · , χn] with χ1 = 0, i,k zi+1,k − θ i,k ! i−1 i−1 T 0 Y X 0 1 0 ··· 0 0 −1 ¯ ˆ ˆ − bp,k−N +i−p θ r,k−N +i−r ξ r,i (k) . 0 0 ˆb1,k+1 · · · 0 p=r r=1 .. .. .. An = ... ... , . . . iT h T 0 0 Then, ξi,k+1 = ˆbi−1,k+1 ξ 0i,k , ˆbi−1,k+1xi+1,k . On 0 · · · ˆbn−2,k+1 the other hand, from (1) we further derive xi+1,k = 0 0 0 ··· 0

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FrA10.1 0 T 0 x b−1 ξ − θ i,k+1 i,k . In Step i − 1, we already have k i,k that xi,k is bounded. Moreover, the boundedness of xj,k, j = 1, · · · , i and Assumption 1 guarantee the boundedness of ξ0i,k . Thus, xi+1,k and ξi,k+1 are bounded. Then, considering the property (14) we have lim (zi,k+1 − ˆbi−1,k+1zi+1,k − χi,k+1) = 0,

k→∞

(19)

Since the estimate zi,k , ˆbi−1,k and χi,k are bounded, (19) implies zi+1,k is also bounded. Summarily, zr,k , xr,k , r = 1, · · · , i + 1 are all bounded till Step i. From the expression of χi+1,k+1, it is still bounded. Step n. Before this step, we have proven that zr,k , xr,k and χr,k+1 are all bounded, where r = 1, · · · , n. Actually, the boundedness of all the parameter estimates and zr,k induces 0 the boundedness of ξ¯j,i(k), 1 ≤ j ≤ i − 1, 1 ≤ i ≤ n. Together with the boundedness of reference signal, we can get the boundedness ofh input signal uk by its expression. iT T Observing ξ , ˆbn−1,k+1uk , = ˆbn−1,k+1 ξ 0

(z1,k+1 − z2,k )2 (z1,k+1 − z2,k )2 = 0, = lim k→∞ c + kξ 0 k2 + ξ 2 k→∞ c + kξ1,k k2 1,k 1,m+1,k lim

there exist k0 and K2 > 0 such that z2,k = K2 z1,k for k ≥ k0 . Similarly, from (2), (14) and (15) we have zi,k = Ki |z1,k |, Ki > 0, 3 ≤ i ≤ n for k ≥ k0 since the parameter estimates are all bounded. Based on this observation, applying the Key Technical Lemma to (15) readily yields the boundedness of zn . This, in turn, implies the boundedness of zn−1 in view of (zn−1,k+1 − ˆbn−2,k+1zn,k − χn−1,k+1)2 = 0. k→∞ c + kξn−1,k+1k2 lim

n,k

n,k+1

such a nonlinear function should be bounded. Then, from property (15), lim ηn,k = 0.

(20)

k→∞

In view (16), it follows that of (18)-(20) and property 0 limk→∞ ξ¯j−1,j (k) − ξ0j−1,k+1 = 0, j = 2, · · ·, n, and 0 0 limk→∞ ξ¯i,j (k) − ξ¯i,j−1(k + 1) = 0, i = 1, · · · , j − 2, j = 3, · · ·, n, limk→∞ Ψk+1 = 0. As a result, as k→∞ zk+1 → An zk + bnym (k + n).

(21)

Thus, from the last limit property in (21) lim (ηn,k + χn,k+1) = 0.

k→∞

(22)

On the other hand, by using the first n − 1 limit properties in (21) lim zn,k+1 − ˆb−1 n−2,k+2zn−1,k+2 k→∞ ! n−2 Y −1 ˆ b z1,k+n = lim zn,k+1 − p,k+n−p

k→∞

=

Therefore, yk = x1,k = z1,k → ym (k) because of property (16) as k → ∞. Now, assume that z1 is unbounded. Since ξ01 satisfies (2) and

p=1

0.

Repeating the same argument gives the boundedness of z2 and hence z1 , which contradicts the assumption of z1 being unbounded. V. I LLUSTRATIVE E XAMPLE Consider the following system by using projection algorithm x1,k+1

=

0 0 θ1,k sin x1,k + θ2,k cos(x1,k + 0.5) + b1,k x2,k , (24)

x2,k+1

=

0 0 x1,k cos x2,k + b2,k(25) uk x2,k sin x1,k + 0.6θ2,k 0.5θ1,k

0 0 T 0 0 where θ 0k = [θ1,k θ2,k ] , θ1,k = 1.3+1.2 sin(πk/5), θ2,k = 3.0 + 0.5 cos(πk/2), b1,k = 1.0 − 0.6 cos(πk/5), b2,k = 2.2 − 0.5 sin(πk/10). The minimum common period is N = 20. Obviously, equations (24) and (25) have the form of equaT tion (1), where ξ 01(x1,k ) = [sin x1,k cos(x1,k + 0.5)] , T 0 ξ2 (x1,k , x2,k) = [0.5x2,k sin x1,k 0.6x1,k cos x2,k ] satisfying the Assumption (2). The control objective is to let state x1 track the reference signal ym (k) = 1.6 − 0.8 cos(πk/5). Assume

(23)

ˆ ini = [1 1 1]T . bmin = 0.2, c = 1, θ

Considering (22) and (23) together, lim

k→∞

−

ˆbn−1,k+1

n−1 Y

ˆb−1 p,k−N +n−p ym (k + n)

p=1

n−2 Y p=1

ˆb−1 p,k+n−p z1,k+n

!

= 0.

The results are shown in Fig. 1 and Fig. 2. In Fig. 1, we use |ei |sup to record the maximum absolute tracking error during the i-th period. From Fig. 1, boundedness of control signal and convergence of tracking error can be obtained. From Fig. 2, boundedness of adaptation signals can be further derived, 0 where only θˆ1,1 and ˆb1 are presented.

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FrA10.1 VI. C ONCLUSION In this paper we present a complete design of adaptive control for parametric-strict-feedback nonlinear systems with periodic parametric uncertainties. It can be seen from the adaptive laws derived in this paper, the controller becomes extremely complicated when there exist periodic uncertain inputs gains. The proposed backsteppng PAC is the extension of the classical adaptive backstepping designs for constant parametric uncertainties, as well as extension of the existing PAC which focuses on matched parametric uncertainties. Our next aim is to further extend PAC to systems non-affine-inparameters such as concave or convex functions, extend PAC to systems with unknown directions, or other scenarios that involve periodic perturbations.

(a)

R EFERENCES

(b) Fig. 1. Tracking error and control input profiles of the system with timevarying parameters by using periodic projection algorithm: (a) tracking error e = ym − x1 ; (b) control input uk .

(a)

(b) Fig. 2.

0 ; (b) ˆ Parametric adaptation signals: (a) θˆ1,1 b1 .

[1] K. Abidi and J.-X. Xu, “A discrete-time periodic adaptive control approach for time-varying parameters with known periodicity”, IEEE Trans. Automat. Control, Vol. 53, No. 2, pp.575-581, 2008. [2] J. X. Xu, S. K. Panda, Y. J. Pan, T. H. Lee, and B. H. Lam, “A modular control scheme for PMSM speed control with pulsating torque minimization”, IEEE Trans. Ind. Electron., Vol. 51, No. 3, pp.526-536, 2004. [3] P. R. Pagilla, B. Yu, and K. L. Pau, “Adaptive control of time-varying mechanical systems: Analysis and experiments”, IEEE/ASME Trans. Mechatron., Vol. 5, No. 4, pp.410-418, 2000. [4] J. M. Carrasco, E. Galvan, G. E. Valderrama, R. Ortega, and A. M. Stankovic, “Analysis and experimentation of nonlinear adaptive controllers for the series resonant converter”, IEEE Trans. Power Electron., Vol. 15, No. 3, pp.536-544, 2000. [5] K. Kaneko and R. Horowitz, “Repetitive and adaptive control of robot manipulators with velocity estimation” , IEEE Trans. Robot. Autom., Vol. 13, No. 2, pp.204-217, 1997. [6] Z. Wang, M. M. Polycarpou, J. G. Uber, and F. Shang, “Adaptive control of water quality in water distribution networks”, IEEE Trans. Control Syst. Technol., Vol. 14, No. 1, pp.149-156, 2006. [7] P.-C. Yeh and P. V. Kokotovi´c, “Adaptive control of a class of nonlinear discrete-time systems”, Int. J. Control, Vol. 62, pp.303-324, 1995. [8] G. C. Goodwin and K. S. Sin, Adaptive Filtering, Prediction, and Control, Englewood Cliffs, NJ: Prentice-Hall, 1984. [9] J. Zhao and I. Kanellakopoulos, “Active identification for discretetime nonlinear control-part I: output-feedback systems”, IEEE Trans. Automat. Control, Vol. 47, No. 2, pp.210-224, 2002. [10] S. S. Ge, C. Yang, and T. H. Lee, “Adaptive robust control of a class of nonlinear strict-feedback discrete-time systems with unknown control directions”, Systems & Control Letters, available online. [11] Y. Zhang, C. Wen, and Y. C. Soh, “Robust adaptive control of uncertain discrete-time systems ”, Automatica, Vol. 35, pp.321-329, 1999. [12] I. Kanellakopoulos, “A discrete-time adaptive nonlinear system”, IEEE Trans. Automat. Control, Vol. 39, No. 11, pp.2362-2365, 1994. [13] J. Zhao and I. Kanellakopoulos, “Active identification for discretetime nonlinear control-part II: strict-feedback systems”, IEEE Trans. Automat. Control, Vol. 47, No. 2, pp.225-240, 2002. [14] Y. Zhang, C. Wen, and Y. C. Soh, “Discrete-time robust backstepping adaptive control for nonlinear time-varying systems ”, IEEE Trans. Automat. Control, Vol. 45, No. 9, pp.1749-1755, 2000. [15] Y. Zhang, W. H. Chen, and Y. C. Soh, “Improved robust backstepping adaptive control for nonlinear discrete-time systems without overparameterization”, Automatica, Vol. 44, pp.864-867, 2008.

6625

Lihat lebih banyak...
FrA10.1

A Discrete-Time Periodic Adaptive Control Approach for Parametric-strict-feedback Systems Deqing Huang, Jian-Xin Xu, and Zhongsheng Hou Abstract— A periodic adaptive control approach is proposed for a class of discrete-time parametric-strict-feedback systems with unknown periodic coefficients which include the control gains. Using the information of the past few periods, the proposed adaptive controller updates the parameters periodically in a pointwise manner over one entire period. By means of backstepping design and through rigorous derivation, we show that the proposed controller guarantees the boundedness of all the closed-loop signals, and achieves the asymptotic tracking convergence. To the end, an illustrative example is presented.

I. INTRODUCTION In this work we extend the results of periodic adaptive control (PAC) of nonlinear systems in companion form [1] to more general classes of nonlinear systems in parametricstrict-feedback. The central idea of PAC is to classify parameters into periodic and nonperiodic cases, instead of slow time-varying and rapid time-varying cases. The periodic variations are encountered in many real systems such as system parameters [2] [3] or exogenous disturbances [4]-[6]. Considering the fact that the value of a periodic parameter will be invariant if the time is being shifted by one period, it is natural to extend the classic adaptive control, which is updated in two consecutive instances, to PAC which is achieved by updating parameters in the same instance of two consecutive periods. Analogously, the boundedness of the parametric estimate and convergence analysis are discussed by considering the difference between two consecutive periods, that is, convergence is asymptotical with respect to the number of periods, instead of the time instances. To deal with the unmatched structure and periodic uncertainties, we extend the backstepping technique [7]-[9] to the periodic adaptation for discrete-time systems. However, due to the nature of periodic coefficients, existing adaptive backstepping designs are not applicable. In this work, our main contribution is to explore and construct a new PAC law which employs multiple periods of previous states and tracking errors in the backstepping approach, hence warrant the asymptotic convergence of the output tracking for nonlinear plants with unmatched uncertainties in parametric-strictfeedback form. In discrete-time adaptive backstepping design, how to deal with the uncertain input gains is a challenging issue. As indicated in [10], the challenge here is that the elegantly devised coordinate mapping in many existing backstepping D. Huang and J.-X. Xu are with Department of Electrical and Computer Engineering National University of Singapore, Singapore 117576, elex-

[email protected] Z. Hou is with Advanced Control Systems Lab, Beijing Jiaotong University, Beijing, China, 100044

978-1-4244-3872-3/09/$25.00 ©2009 IEEE

designs for discrete-time nonlinear systems, such as [11][15], is not directly applicable when the control gains are unknown. In [10], the uncertain input gains are all constants. When periodic input gains are concerned, the adaptive backstepping design becomes much more difficult and complicated. Another main contribution of this work is to present a complete adaptive backstepping control design for this challenging problem, that is, nonlinear systems with unknown periodic input gains. The paper is organized as follows. In Section II, we present the problem formulation and address the difficulties encountered. In Section III, n steps of periodic adaptive control law are proposed. Section IV gives the convergence analysis. To the end, an illustrative example is provided in Section V. Throughout this paper, k · k denotes the Euclidean norm. For notational convenience, in mathematical expressions, fk represents f(k). II. P ROBLEM F ORMULATION Consider a class of strict-feedback discrete-time systems with time-varying uncertainties, described by T xi,k+1 = θ 0k ξ 0i (xi,k ) + bi,kxi+1,k , T

xn,k+1

=

θ 0k

yk

=

x1,k .

ξ 0n(xn,k ) + bn,k uk ,

T

(1)

where xi = [x1, x2, · · · , xi] , uk and yk represent the system input and output respectively, θ0k is the unknown periodic parametric vector in Rm . For each 1 ≤ i ≤ n, ξ0i (xi,k ) denotes the known nonlinearity which is continuous and satisfies ξ0i (0) = 0. For notational convenience, ξ0i (xi,k ) is denoted by ξ0i,k in the remaining parts of the paper. The unknown periodic function bi,k ∈ C[0, ∞) is system control gain. The prior information with regards to bi,k is that the control direction is known and invariant, that is, bi,k is either positive or negative and nonsingular for all k. Without loss of generality, assume that bi,k ≥ bmin where bmin > 0 is a known lower bound. The control objective here is to force the output of the system (1) to track the bounded reference signal ym . 0 Note that each unknown parameter, θj,k or bi,k , may have its own period Nj or Ni,b, j = 1, · · · , m, i = 1, · · ·, n. The periodic adaptive control will still be applicable if there exists a common period N , such that each Nj and Ni,b can divide N with an integer quotient. This is always true in discrete-time since Nj and Ni,b are integers, therefore N can be either the least common multiple of Nj and Ni,b ,

6620

FrA10.1 or simply the product of Nj and Ni,b when both are prime. Therefore, N can be used as the updating period. Now, we first recall the following assumption presented in [7], which implies the sector-bounded condition or the restrictive growth condition [1]. Then the Key Technical Lemma [8] can be applied for convergence analysis. Assumption 1: All the states x1,k , x2,k, · · · , xn,k are measured variables and kξ 0i (xi ) − ξ 0i (yi )k kξ0n(xn )k

≤ ≤

dikxi − yi k, ∀xi , yi ∈ Ri , dnkxnk (2)

where dj > 0, 1 ≤ j ≤ n. Based on Assumption 1 and by using backstepping technique and the projection algorithm, we extend the idea in [7] to more general case (1) and provide global results for such a nonlinearly parameterized problem. The results can be easily extended to the least-squares algorithm [8].

iT h 0 0 , · · · , θˆ1,m , ˆb1 is chosen as θˆ1,1 (z − z2,k−N )ξ1,k−N ˆ 1,k−N + 1,k−N +1 ˆ 1,k = L θ , (5) θ c + |ξ1,k−N |2 T ˆ1,k−N + where c > 0. Let a = aT1 , a2 denote the vector θ (z1,k−N+1 −z2,k−N )ξ 1,k−N , the semi-saturator is defined as c+|ξ 1,k−N |2 T aT1 , a2 , a2 ≥ bmin , (6) L[a] = T T a1 , bmin , a2 < bmin . ˆ1,k is updated by using the information We can see that θ ˆ 1,k−N , x2,k−N , x1,k−N +1, which are all available in the θ current step. Step i (2 ≤ i ≤ n − 1). Let i−j−1 0 ξ¯j,i(k)

ξ0j

=

T

xn,k+1

=

θ 0k

yk

=

x1,k ,

ξ 0n,k + ˆbn,k uk + ˜bn,kuk ,

(3)

ˆbl+1,k−N +i−j−l zi−j+2,k − S1 , · · ·

l=1 i−j−1 Y l=1

where Q0 ˆ 1 ≤ j ≤ i − 1, the product operator satisfies l=1 bl,k = 1, and Sm is equal to m m 0 T XY ˆb−1 ˆ p,k−N +i−j+m−p θ r,k−N +i−j+m−r

˜0 θ 1,k

T

˜T ξ , θ 1,k 1,k

T

˜0 θ 1,k

¯0 ξ r,i−j+m (k),

where m = 1, · · ·, j − 1. Furthermore, make the following transformation 0 T ˆ ξ0i,k zi+1,k = ˆbi,k xi+1,k + θ i,k +

1,k

z1,k+1 = z2,k +

! ˆbl+j−1,k−N +i−j−l zi,k − Sj−1 ,

r=1 p=r

Assume n ≤ N + 1 and construct the adaptation law as follows. Step 1. Let z1,k = x1,k and z2,k = ˆb1,k x2,k + 0 T ˆ θ ξ0 , then 1,k

ˆbl,k−N +i−j−l zi−j+1,k ,

l=1 i−j−1 Y

III. D ISCRETE -T IME P ERIODIC A DAPTIVE C ONTROL The presence of the uncertain system input gains makes the controller design in each step of backstepping more complex. To derive the periodic adaptive control law, define ˆbi,k to be the estimation of bi,k and ˜bi,k = bi,k − ˆbi,k , the system dynamics (1) can be rewritten as T xi,k+1 = θ 0k ξ 0i,k + ˆbi,k xi+1,k + ˜bi,k xi+1,k,

Y

(4)

i−1 i−1 Y X r=1 p=r

Then,

0 T 0 ˆb−1 ˆ θ ξ¯r,i (k). r,k−N +i−r p,k−N +i−p T

ˆ0 , θ 1,k

˜ 1,k = where θ , ˜b1,k , = − and T 0 T T ˆ ξ1,k = ξ01,k , ˆb−1 . Note that ξ01,k 1,k z2,k − θ 1,k θ 0k

˜ ξ zi,k+1 = ˆbi−1,k+1zi+1,k + θ i,k i,k+1 + χi,k+1 , T T ˜ i,k = θ ˜0 , ˜bi,k , where θ i,k

the computation of ξ1,k requires the inverse of the system input gain estimate ˆb1,k and may cause a singularity in the solution if the estimate of the gain is zero. To ensure this never occurs a semi-saturator must be applied on the gain estimator such that the estimate never goes below the ˆ1 = lower bound. For this purpose, the update law for θ

6621

h T ξ i,k+1 = ˆbi−1,k+1 ξ 0i,k , 0 T ˆbi−1,k+1ˆb−1 zi+1,k − θ ˆ ξ 0i,k i,k i,k −

i−1 i−1 X Y r=1 p=r

ˆb−1 p,k−N +i−p

T ˆ0 θ r,k−N +i−r

0 ξ¯r,i (k)

(7)

!#T

,

FrA10.1 χi,k+1 = −

T ˆbi−1,k+1 0 0 ˆ θ ξ¯i−1,i(k) i−1,k−N +1

iT h ˆn = θˆ0 , · · · , θˆ0 , ˆbn is chosen as The update law for θ n,1 n,m ξ n,k−N +1 ˆ n,k = L θ ˆ n,k−N + (zn,k−N +1 θ c + kξn,k−N +1k2

ˆbi−1,k−N +1 T ˆbi−1,k+1 ˆ0 + θ ξ0i−1,k+1 − i−1,k+1 ˆbi−1,k−N +1 i−2 i−2 0 T X Y 0 ˆb−1 ˆ θ ξ¯r,i (k) × r,k−N +i−r p,k−N +i−p

−ˆbn−1,k−N +1

+

ˆb−1 p,k−N +i−p

r=1 p=r

T ˆ0 θ r,k−N +i−r

−χn,k−N +1 )] .

0 ξ¯r,i−1(k

+ 1).

iT h ˆ i = θˆ0 , · · · , θˆ0 , ˆbi is chosen as The update law for θ i,1 i,m ˆi,k θ

ξi,k−N +1 (zi,k−N +1 c + |ξi,k−N +1 |2 i −ˆbi−1,k−N +1zi+1,k−N − χi,k−N +1 . (8)

ˆi,k−N + L θ

=

ˆ i,k is updated with measurable states xi+1,k−N , Note that θ ˆ w ,k−w , where w1 = xi,k−N +1 , and previous estimates θ 1 2 1, · · · , i, w2 = N, · · · , 2N − 1, which are all available in the current step. Step n. This is the last step of the design procedure. We select the control law n−1 Y

uk = ˆb−1 n,k

p=1

−

n−1 X n−1 Y

0 T ˆ ˆb−1 ξ0n,k p,k−N +n−p ym (k + n) − θ n,k

ˆb−1 p,k−N +n−p

r=1 p=r

0 ˆ θ

r,k−N +n−r

T

0 ξ¯r,n (k)

!

,

= ˆbn−1,k+1

zn,k+1

p=1 T ˜ ξ +θ n,k n,k+1

˜ n,k = where θ

˜0 θ n,k

T

, ˜bn,k

T

(10)

, χn,k+1 takes the form as

p=1

−

0 T ˆ ˆb−1 ξ0n,k p,k−N +n−p ym (k + n) − θ n,k

n−1 X n−1 Y r=1 p=r

ˆb−1 p,k−N +n−p

T ˆ0 θ r,k−N +n−r

0 ξ¯r,n (k)

=

z1,k+1 − z2,k ,

˜T ξ θ i,k i,k+1

=

zi,k+1 − ˆbi−1,k+1zi+1,k − χi,k+1,

˜T ξ θ n,k n,k+1

=

zn,k+1 − ˆbn−1,k+1

i = 2, · · · , n − 1, n−1 Y

ˆb−1 p,k−N +n−p

p=1

×ym (k + n) − χn,k+1.

χi,k+1 with i = n, h T ξn,k+1 = ˆbn−1,k+1 ξ0n,k , ˆbn−1,k+1ˆb−1 n,k n−1 Y

˜T ξ θ 1,k 1,k

(9)

ˆb−1 p,k−N +n−p ym (k + n) + χn,k+1,

(11)

ˆ n,k , the information xn,k−N +1, ym (k−N + For calculating θ ˆ n), θw1 ,k−w2 , where w1 = 1, · · ·, N, w2 = N, · · · , 2N − 1, are all adopted. Note that in each step the system is transformed from x-domain to z-domain. Actually, the amount zi+1,k can be regarded as an predictor of xi,k+1 in current Step k. Due to the existence of unknown control gains, it is not easy to find such corresponding predictors for all xi,k+1 because the higher the order of system the more estimation information or previous predictors will be engulfed in the following when we do backstepping design step by step. This can be seen from the above design procedure. On the other hand, in order to derive the parameter estimations in current step we usually choose the parametric estimation error in the previous period as the feedback information, which however is obviously not available because of the parametric uncertainties. Investigating the relationships (4), (7), and (10), the estimation errors for unknown parameters can be expressed in z-domain as

¯0 (k) takes the same form as ξ¯0 (k) with i = n. where ξ j,n j,i Then n−1 Y

ˆb−1 p,k−2N +n−p ym (k − N + n)

p=1

r=1 p=r

i−2 i−2 X Y

n−1 Y

Thus, in the actual projection algorithm we use the information in the above right hand sides to replace the unavailable parameter estimate errors. Moreover, we can prove the following properties of the projection algorithm, which are useful in our convergence analysis. Owing to space limitation, details of the proof are ignored here. Lemma 1: ˜ i,k k ≤ kθ ˜i,k−N k, i = 1, · · · , n, (P1) kθ (12) (P2)

!#T

.

(P3)

6622

(z1,k+1 − z2,k )2 = 0, k→∞ c + kξ 1,kk2 lim

(13)

(zi,k+1 − ˆbi−1,k+1zi+1,k − χi,k+1)2 = 0, (14) k→∞ c + kξi,k+1k2 lim

FrA10.1 "

where 2 ≤ i ≤ n − 1, 2 ηn,k = 0, (15) (P4) lim k→∞ c + kξ n,k+1k2 Qn−1 where ηn,k = zn,k+1 − ˆbn−1,k+1 p=1 ˆb−1 p,k−N +n−p ym (k + n) − χn,k+1,

(P5)

ˆi,k − θ ˆi,k−N k = 0, lim kθ

k→∞

i = 1, · · · , n.

(16)

The preceding control law and parametric updating law were built as n ≤ N + 1. As can be seen from (5), (8), (9) and (11), in the current Step k the future information to Step k − N + n − 1 will be used for control if n > N + 1, which is however not available. The following procedure can ˆ 0 in (5), be applied to solve this problem: for all ˆbp,q and θ p,q (8), (9) and (11), if the index q is greater than k, then they can be substituted by the latest corresponding signals which are available in a point-wise manner. For instance, assume N = 3 and consider the estimate ˆbp,q . By this rule ˆbp,k+3w+1 → ˆbp,k−2, ˆbp,k+3w+2 → ˆbp,k−1, ˆbp,k+3w+3 → ˆbp,k , where w = 0, 1, 2, · · · denotes the number of integral periods being shifted. After these substitutions, the control law and parametric updating law become implementable for the discrete system. Subsequently, the properties (P1)-(P5) can be similarly achieved. Moreover, note that the information at most in the previous ˆ i,k . 2N − 1 steps should be known beforehand to calculate θ ˆ ˆ For the sake of initialization, always assume θ i,k = θ ini, ˆini is a given vector, i = 1, · · · , n, k = 1, · · · , 2N − where θ 1.

bn = 0, 0, · · ·, 0, ˆbn−1,k+1

n−1 Y

ˆb−1 p,k−N +n−p

p=1

#T

.

Next, we move to the convergence analysis based on Lemma 1. Theorem 1: All the signals in the closed-loop system, consisting of (1), (5), (8), (9), (11), and (17), are uniformly bounded. In addition, y → ym as k → ∞. Proof. The property (12) induces that all the parameter estimates are bounded. Next, assume z1,k is bounded and prove the boundedness of other quantities in all steps. Step 1. If z1,k is bounded, ξ01,k is bounded by 0 T ˆ ξ01,k , we Assumption 1. Since z2,k = ˆb1,k x2,k + θ 1,k have 0 T 0 ˆ z x2,k = ˆb−1 ξ θ − 2,k 1,k . 1,k 1,k Then,

ξ 1,k =

h

ξ 01,k

T

, x2,k

iT

.

On the can derive x2,k = we further other hand,Tfrom (1) T 0 0 −1 −1 b1,k x1,k+1 − θk ξ1,k = b1,k z1,k+1 − θ 0k ξ01,k . Since the positive property and periodicity of b1,k implies 0 the boundedness of b−1 1,k , the periodicity of θ k implies the 0 boundedness of θ k , and the boundedness of z1,k implies the boundedness of ξ 01,k, x2,k and then ξ1,k are also bounded. Then, considering the property (13) we have lim (z1,k+1 − z2,k ) = 0,

k→∞

(18)

implying z2,k is also bounded. Thus, z1 , z2 and x1, x2 are all bounded. Since all the parameter estimates are 0 T ˆ b1,k+1 ˆ bounded, χ θ = − ξ01 (z2,k ) + 2,k+1 1,k−N +1 ˆ IV. C ONVERGENCE A NALYSIS b1,k−N+1 0 T ˆ θ ξ 01,k+1 is also bounded. Summarizing the n steps, the closed-loop system is ex1,k+1 pressed as Step i (2 ≤ i ≤ n−1). Recursively, assume that until Step T i − 1 the quantities zr,k , xr,k and χr,k+1 are all bounded, ˜ ξ θ where r ≤ i. Next to prove zi+1,k , xi+1,k and χi+1,k+1 are 1,k .1,k+1 0 T . zk+1 = An zk + bnym (k + n) + ˆ . ξ 0i,k + + Ψk+1 , (17)also bounded. Since zi+1,k = ˆbi,kxi+1,k + θ i,k T 0 T ˜ ξ Pi−1 Qi−1 ˆ−1 0 θ n,k n,k+1 ˆ ξ¯r,i (k). we have r=1 p=r bp,k−N +i−p θ r,k−N +i−r = [z1,k, z2,k , · · · , zn,k]T , Ψ = where zk 0 T −1 △ ˆ ˆ T x = b ξ0i,k i+1,k [χ1, χ2, · · · , χn] with χ1 = 0, i,k zi+1,k − θ i,k ! i−1 i−1 T 0 Y X 0 1 0 ··· 0 0 −1 ¯ ˆ ˆ − bp,k−N +i−p θ r,k−N +i−r ξ r,i (k) . 0 0 ˆb1,k+1 · · · 0 p=r r=1 .. .. .. An = ... ... , . . . iT h T 0 0 Then, ξi,k+1 = ˆbi−1,k+1 ξ 0i,k , ˆbi−1,k+1xi+1,k . On 0 · · · ˆbn−2,k+1 the other hand, from (1) we further derive xi+1,k = 0 0 0 ··· 0

6623

FrA10.1 0 T 0 x b−1 ξ − θ i,k+1 i,k . In Step i − 1, we already have k i,k that xi,k is bounded. Moreover, the boundedness of xj,k, j = 1, · · · , i and Assumption 1 guarantee the boundedness of ξ0i,k . Thus, xi+1,k and ξi,k+1 are bounded. Then, considering the property (14) we have lim (zi,k+1 − ˆbi−1,k+1zi+1,k − χi,k+1) = 0,

k→∞

(19)

Since the estimate zi,k , ˆbi−1,k and χi,k are bounded, (19) implies zi+1,k is also bounded. Summarily, zr,k , xr,k , r = 1, · · · , i + 1 are all bounded till Step i. From the expression of χi+1,k+1, it is still bounded. Step n. Before this step, we have proven that zr,k , xr,k and χr,k+1 are all bounded, where r = 1, · · · , n. Actually, the boundedness of all the parameter estimates and zr,k induces 0 the boundedness of ξ¯j,i(k), 1 ≤ j ≤ i − 1, 1 ≤ i ≤ n. Together with the boundedness of reference signal, we can get the boundedness ofh input signal uk by its expression. iT T Observing ξ , ˆbn−1,k+1uk , = ˆbn−1,k+1 ξ 0

(z1,k+1 − z2,k )2 (z1,k+1 − z2,k )2 = 0, = lim k→∞ c + kξ 0 k2 + ξ 2 k→∞ c + kξ1,k k2 1,k 1,m+1,k lim

there exist k0 and K2 > 0 such that z2,k = K2 z1,k for k ≥ k0 . Similarly, from (2), (14) and (15) we have zi,k = Ki |z1,k |, Ki > 0, 3 ≤ i ≤ n for k ≥ k0 since the parameter estimates are all bounded. Based on this observation, applying the Key Technical Lemma to (15) readily yields the boundedness of zn . This, in turn, implies the boundedness of zn−1 in view of (zn−1,k+1 − ˆbn−2,k+1zn,k − χn−1,k+1)2 = 0. k→∞ c + kξn−1,k+1k2 lim

n,k

n,k+1

such a nonlinear function should be bounded. Then, from property (15), lim ηn,k = 0.

(20)

k→∞

In view (16), it follows that of (18)-(20) and property 0 limk→∞ ξ¯j−1,j (k) − ξ0j−1,k+1 = 0, j = 2, · · ·, n, and 0 0 limk→∞ ξ¯i,j (k) − ξ¯i,j−1(k + 1) = 0, i = 1, · · · , j − 2, j = 3, · · ·, n, limk→∞ Ψk+1 = 0. As a result, as k→∞ zk+1 → An zk + bnym (k + n).

(21)

Thus, from the last limit property in (21) lim (ηn,k + χn,k+1) = 0.

k→∞

(22)

On the other hand, by using the first n − 1 limit properties in (21) lim zn,k+1 − ˆb−1 n−2,k+2zn−1,k+2 k→∞ ! n−2 Y −1 ˆ b z1,k+n = lim zn,k+1 − p,k+n−p

k→∞

=

Therefore, yk = x1,k = z1,k → ym (k) because of property (16) as k → ∞. Now, assume that z1 is unbounded. Since ξ01 satisfies (2) and

p=1

0.

Repeating the same argument gives the boundedness of z2 and hence z1 , which contradicts the assumption of z1 being unbounded. V. I LLUSTRATIVE E XAMPLE Consider the following system by using projection algorithm x1,k+1

=

0 0 θ1,k sin x1,k + θ2,k cos(x1,k + 0.5) + b1,k x2,k , (24)

x2,k+1

=

0 0 x1,k cos x2,k + b2,k(25) uk x2,k sin x1,k + 0.6θ2,k 0.5θ1,k

0 0 T 0 0 where θ 0k = [θ1,k θ2,k ] , θ1,k = 1.3+1.2 sin(πk/5), θ2,k = 3.0 + 0.5 cos(πk/2), b1,k = 1.0 − 0.6 cos(πk/5), b2,k = 2.2 − 0.5 sin(πk/10). The minimum common period is N = 20. Obviously, equations (24) and (25) have the form of equaT tion (1), where ξ 01(x1,k ) = [sin x1,k cos(x1,k + 0.5)] , T 0 ξ2 (x1,k , x2,k) = [0.5x2,k sin x1,k 0.6x1,k cos x2,k ] satisfying the Assumption (2). The control objective is to let state x1 track the reference signal ym (k) = 1.6 − 0.8 cos(πk/5). Assume

(23)

ˆ ini = [1 1 1]T . bmin = 0.2, c = 1, θ

Considering (22) and (23) together, lim

k→∞

−

ˆbn−1,k+1

n−1 Y

ˆb−1 p,k−N +n−p ym (k + n)

p=1

n−2 Y p=1

ˆb−1 p,k+n−p z1,k+n

!

= 0.

The results are shown in Fig. 1 and Fig. 2. In Fig. 1, we use |ei |sup to record the maximum absolute tracking error during the i-th period. From Fig. 1, boundedness of control signal and convergence of tracking error can be obtained. From Fig. 2, boundedness of adaptation signals can be further derived, 0 where only θˆ1,1 and ˆb1 are presented.

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FrA10.1 VI. C ONCLUSION In this paper we present a complete design of adaptive control for parametric-strict-feedback nonlinear systems with periodic parametric uncertainties. It can be seen from the adaptive laws derived in this paper, the controller becomes extremely complicated when there exist periodic uncertain inputs gains. The proposed backsteppng PAC is the extension of the classical adaptive backstepping designs for constant parametric uncertainties, as well as extension of the existing PAC which focuses on matched parametric uncertainties. Our next aim is to further extend PAC to systems non-affine-inparameters such as concave or convex functions, extend PAC to systems with unknown directions, or other scenarios that involve periodic perturbations.

(a)

R EFERENCES

(b) Fig. 1. Tracking error and control input profiles of the system with timevarying parameters by using periodic projection algorithm: (a) tracking error e = ym − x1 ; (b) control input uk .

(a)

(b) Fig. 2.

0 ; (b) ˆ Parametric adaptation signals: (a) θˆ1,1 b1 .

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