Robust adaptive control of discrete nominally stabilizable plants
Applied Mathematics and Computation 150 (2004) 555–583 www.elsevier.com/locate/amc
Robust adaptive control of discrete nominally stabilizable plants S. Alonso-Quesada *, M. De la Sen Dpto. de Electricidad y Electr onica, Facultad de Ciencias, Universidad del Paıs Vasco, UPV-EHU, 48940 Leioa (Bizkaia), Spain
Abstract This paper presents an indirect pole-placement based adaptive control scheme for discrete linear time-invariant non-necessarily inversely stable and stabilizable systems. The scheme is a pole-placement type without requiring the plant inverse stability assumption. The control objective is that the plant output should asymptotically track in the absence of unmodeled dynamics and bounded disturbances a reference signal, given by an arbitrary stable filter, under a bounded tracking error while ensuring robust closed-loop stability. The adaptive stabilizability and the robustness of this system under the presence of unmodeled dynamics and, possibly, bounded disturbances, are proved without assuming the controllability of the modeled plant. A normalized least-squares algorithm with a relative adaptation dead-zone and Ôa posterioriÕ modification of the parameter estimates is used to ensure the controllability of the modified estimation plant model at all sampling instants and at the limit. Such a property is crucial to solve the stable poleplacement problem. On the other hand, the relative adaptation dead-zone is included for robustness purposes under unmodeled dynamics and, possibly, bounded disturbances. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Robust control; Unmodeled dynamics; Adaptive stabilizability; Inversely unstable plant; Tracking; Adaptation dead-zone
1. Introduction The robust adaptive stability of discrete linear time-invariant systems has been successfully developed in the last two decades. In the beginning, as in [1] *
Corresponding author. E-mail address: [email protected] (S. Alonso-Quesada).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00291-1
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and [2], most of the proofs of stability of the adaptive control algorithms proposed were based on a set of standard assumptions. Such assumptions were: (i) the knowledge of an upper bound of the system order, (ii) the knowledge of the system relative degree, (iii) the plant being inversely stable and (iv) the knowledge of the sign of the high frequency gain of the plant as well as an upper bound for its absolute value. Later, works whose goal was to relax some of these aforementioned assumptions were presented. In this way, the studies in [3] and [4] relaxed the assumptions relative to the high frequency gain by including Nussbaum gain in the adaptation laws. Therefore, in the paper [5] the assumption of inverse stability as well as that of the knowledge of the high frequency plant gain were relaxed. That work focused on the robust adaptative stabilization of discrete linear time-invariant plants. The controllability of the nominal plant model and an overbounding function for the contribution of the unmodeled dynamics and external disturbances were supposed known for purposes of achieving robust adaptive stability. Such a stability result was established without either requiring the injection of persistent excitation probing signals into the system or assuming any prior knowledge of the plant parameters. The necessary and sufficient condition for the system stabilization via an adaptive pole-placement scheme was established in [6]. Such a condition is simply that the asymptotically reached estimated model of the stabilizable plant be controllable, which is weaker than the controllability of the true plant required in [5]. If the system is not inversely stable then the controllability of the estimated model of the plant has to be ensured when the time tends to infinity to achieve the adaptive stabilization via pole-placement. Basically, two different approaches have been used to circumvent the regions in the parameter space corresponding to uncontrollable models. One of them relies on the use of excitation probing signals, [7–11], while the other one is based on the use of a suitable modification of the plant parameter estimates as in [5,6,12–15]. The estimation modification algorithm used in [5,13,15] includes a hysteresis switching function what it bears a very heavy computation burden before the convergence of the estimates. The modification algorithm presented in [6] involves the use of a hysteresis-free switching function. Then, the convergence of the estimates is usually reached with a light computation burden, but with a high number of switches. This paper relaxes the controllability condition of the plant nominal model required in [5] for robust adaptive stabilizability of the closed-loop system under the presence of unmodeled dynamics. The control objective is to ensure the stability of the closed-loop system with a bounded tracking error. The proposed scheme is of pole-placement type while it ensures asymptotic reference tracking properties when the reference sequence becomes constant. The scheme is valid to operate in both non-adaptive and adaptive environments.
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The only assumption about the true plant is that it is stabilizable. This paper is an extension of the problem solved in [12] to the case of discrete-time plants. A normalized least-squares algorithm with a relative adaptation dead-zone and a parameter modification is used to update the plant parameters at each sampling instant. The above combined technique is the basis to prove the convergence of the estimated parameters and the boundedness of all the signals in the closed-loop system. The knowledge of an upper-bounding function for the contribution of the unmodeled dynamics and disturbances to the output is required to design the relative adaptation dead-zone, [16]. In that way, the adaptation is frozen when an aumented identification error lies below a known upper bound of the absolute value of the contribution of the unmodeled dynamics and bounded disturbances to the output. Two designs for the parameter modification are presented to fulfil the control objective. Both of them use a hysteresis-free switching function which does not change its value unless it is crucial to ensure the controllability of the estimation model of the plant. In this way, the number of switches and/or the computational cost is reduced with regard to the modification algorithms used in the previous works [5,6,13,15]. It is proved for both modification algorithms that the normalized modified identification error belongs to a residual set defined by an absolute normalized upper bound for the contribution of the unmodeled dynamics and bounded noise to the output. The paper is organized as follows. Section 2 presents the problem statement with the control objective and the structure of the control law. Section 3 is devoted to formulate the unmodified and modified parameter estimation algorithms. Two alternative algorithms are studied for the parameter modification, both of them providing a controllable plant estimated model for all sampling instants and at the limit. Section 4 establishes the main result of the stability analysis. Section 5 presents a numerical simulation to show the performance of the proposed control scheme. Finally, conclusions end the paper.
2. Problem statement Consider the following linear discrete-time system Aðq1 ÞyðkÞ ¼ Bðq1 ÞuðkÞ þ gðkÞ
ð1Þ
which is not necessarily controllable and can include stable uncontrollable modes as the roots of Aðq1 Þ ¼ 0, where Aðq1 Þ ¼ A0 ðq1 ÞA0 ðq1 Þ ¼ 1 þ a1 q1 þ þ an qn Bðq1 Þ ¼ B0 ðq1 ÞA0 ðq1 Þ ¼ bnm qmn þ bnmþ1 qmn1 þ þ bn qn
ð2Þ
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with q1 yðkÞ yðk 1Þ, m 6 n, gðkÞ being the contribution of unmodeled dynamics and bounded noise of any order to the output and A0 ðq1 Þ denoting the possible n0 P 0 stable common factors of Aðq1 Þ and Bðq1 Þ. If n0 ¼ 0 then A0 ðq1 Þ ¼ 1. Eq. (1) may be rewritten as yðkÞ ¼ hT /ðkÞ þ gðkÞ
ð3Þ
where h ¼ ½ bnm
bn
/ðkÞ ¼ ½ uðk n þ mÞ
a1
an T ; uðk nÞ
yðk 1Þ
T
yðk nÞ
ð4Þ are the true plant parameters and measurable regression vectors, respectively. 2.1. Case of known plant and absence of unmodeled dynamics (g() 0) In this case, the control objective is the tracking between the system output sequence and the reference sequence ym ðkÞ with a prescribed pole-placement, being zero the tracking error when the reference sequence becomes constant. The control law designed to meet this objective is: uðkÞ ¼ Kym ðkÞ Rðq1 ÞuðkÞ Sðq1 ÞyðkÞ ¼
K Sðq1 Þ y yðkÞ ðkÞ m 1 þ Rðq1 Þ 1 þ Rðq1 Þ
ð5Þ
where ym ðkÞ is given by a stable filter Wm ðq1 Þ ¼ Bm ðq1 Þ=Am ðq1 Þ ¼ ðb0nm mm qmm nm þ þ b0nm qnm Þ=ð1 þ a01 q1 þ þ a0nm qnm Þ; Rðq1 Þ ¼ rnm qðnmÞ þ þ rnn0 1 qðnn0 1Þ ; Sðq1 Þ ¼ s0 þ s1 q1 þ þ snn0 1 qðnn0 1Þ
ð6Þ
with ri ; sj 2 R, for i 2 fn m; . . . ; n n0 1g and j 2 f0; . . . ; n n0 1g, being calculated from the following diophantine equation ð1 þ Rðq1 ÞÞAðq1 Þ þ Sðq1 ÞBðq1 Þ ¼ Cðq1 Þ
ð7Þ
where Cðq1 Þ ¼ C 0 ðq1 ÞA0 ðq1 Þ ¼ 1 þ c1 q1 þ þ c2n1 qð2n1Þ is a Hurwitz polynomial. Eq. (7) is solvable since the n0 stable common factors of Bðq1 Þ and Aðq1 Þ are included in Cðq1 Þ. If the plant is controllable then all of the factors of the Hurwitz polynomial Cðq1 Þ can be freely chosen. Note that Eq. (7) is equivalent to that obtained by replacing A ! A0 , B ! B0 and C ! C 0
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after cancelling A0 and that Eq. (1) is equivalent neglecting initial conditions to the equation obtained by replacing A ! A0 , B ! B0 and g ! g0 ¼ ð1=A0 Þg. On the other hand, the control parameter K will be calculated such that the tracking of the reference sequence be perfect when it becomes constant. By introducing the control law (5) in the transfer function of the plant W0 ðq1 Þ ¼ Bðq1 Þ=Aðq1 Þ the transfer function of the closed-loop system is yðq1 Þ KBm ðq1 ÞB0 ðq1 ÞA0 ðq1 Þ ¼ r ðq1 Þ Am ðq1 ÞC 0 ðq1 ÞA0 ðq1 Þ
ð8Þ
which can include stable cancellations if the degree of A0 ðq1 Þ is non-zero and where the relationship (7) has been used. Its dynamics includes the poles of the stable filter Wm ðq1 Þ. The remaining poles can be fixed according to the arbitrary Hurwitz monic polynomial C 0 ðq1 Þ by the control polynomials Rðq1 Þ and Sðq1 Þ. Then, the dynamics of the closed-loop system is able to practically equate that of the stable filter Wm ðq1 Þ if the zeros of the chosen Hurwitz polynomial C 0 ðq1 Þ are sufficiently more stable than the poles of Wm ðq1 Þ. Note from (8) that the closed-loop characteristic equation is Am ðq1 ÞC 0 ðq1 Þ ¼ 0 and it has nm þ 2ðn n0 Þ 1 roots. However, note that the computational cost in solving the diophantine equation (7) is fixed irrespective of the degree of Am ðq1 Þ. Therefore, the number of closed-loop poles can be increased, if desired, without increasing the computational cost requested to generate the control input. The transfer function from the external input sequence r ðkÞ to the tracking error eðkÞ ¼ yðkÞ ym ðkÞ is Ge ðq1 Þ ¼ ¼
eðq1 Þ r ðq1 Þ ½KB0 ðq1 Þ ð1 þ Rðq1 ÞÞA0 ðq1 Þ Sðq1 ÞB0 ðq1 ÞBm ðq1 Þ Am ðq1 ÞC 0 ðq1 Þ
ð9Þ
where the relationship (7) has been used. The control parameter K can be calculated such that the absolute gain of this transfer function be zero when q1 ejT x ¼ 1 þ j0, with T being the sampling period and j the complex imaginary unity. In this way, the tracking error can be zero if the reference sequence becomes constant. The value of K to meet this objective is from (9): K¼
ð1 þ Rð1ÞÞA0 ð1Þ þ Sð1ÞB0 ð1Þ B0 ð1Þ
ð10Þ
If n ¼ m, nm ¼ mm and n0 ¼ 0, Eqs. (7) and (10) can be written more compactly as ½ 1 q1
q2nþ2
q2nþ1 ½MðhÞvc p ¼ 0
ð11Þ
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where vc ¼ ½ 1 þ r 0 r 1 p ¼ ½ 1 c1
rn1 s0 T c2n1 0
s1
sn1
T
K ;
ð12Þ
and MðhÞ ¼ 2
1 0 6 a1 1 6 6 .. .. 6 6 . . 6 6 an1 an2 6 6 6 an an1 6 6 0 an 6 6 6 .. .. 6 . . 6 6 0 0 6 6 n n 4 P P 1 þ ai 1 þ ai i¼1
i¼1
.. . .. .
0 0 .. . 1 a1 a2 .. . a nn P 1 þ ai i¼1
b0 b1 .. . bn1 bn 0 .. . 0 n P bi
0 b0 .. . bn2 bn1 bn .. . 0 n P bi
i¼0
i¼0
.. . .. .
0 0 .. . b0 b1 b2 .. . bn n P bi i¼0
3 0 0 7 7 7 .. 7 . 7 7 0 7 7 7 0 7 7 7 0 7 7 .. 7 . 7 7 7 0 7 7 n 5 P bi i¼0
ð13Þ is the Sylvester matrix of the true plant parameters. Eq. (11) is uniquely solvable if the determinant of MðhÞ is strictly non-zero, i.e., jDetðMðhÞÞj P d0 > 0
ð14Þ
for some design real small positive constant d0 . The relationship (14) will be referred to as the controllability condition of the true plant. Then, if the plant is controllable, i.e., Aðq1 Þ and Bðq1 Þ are relatively prime polynomials, then, the control parameters can be uniquely obtained from the algebraic system (11). Remark 1. Since the relative degree of the plant and its parameters are unknown, one can choose that the relative degree of the nominal model of the plant be zero (n ¼ m) and A0 ðq1 Þ ¼ 1 (n0 ¼ 0) to establish the formalism in a simpler way. In the case when the relative degree of the plant is non-zero the estimated parameters, issued from an estimation algorithm, corresponding to those added parameters will tend to zero. Remark 2. In the case when the polynomials Aðq1 Þ and Bðq1 Þ have stable common factors, the control parameters are obtained from a similar equation to (11) with the Sylvester determinant of A0 ðq1 Þ and B0 ðq1 Þ being strictly positive.
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2.2. Case of unknown plant The transfer function of the system (1) can be written as yðq1 Þ=uðq1 Þ ¼ W ðq1 Þ ¼ W0 ðq1 Þð1 þ mD1 ðq1 ÞÞ þ mD2 ðq1 Þ, where W0 ðq1 Þ ¼ Bðq1 Þ=Aðq1 Þ is the modeled part of the transfer function of the true plant, mD1 ðq1 Þ and mD2 ðq1 Þ are the transfer functions of the multiplicative and additive unmodeled dynamics, respectively, and m is a positive constant. D1 ðq1 Þ and D2 ðq1 Þ must be both exponentially stable and strictly proper so that Assumption 1 below be feasible.
Assumption 1. There exist real constants r 2 ð0; 1Þ, a0 P 0 and a P 0, and a constant vector v, which are known, such that jgðkÞj 6 gðkÞ ¼ aqðkÞ þ a0
ð15Þ
for all integer k P 0
where 0
qðkÞ ¼ sup fjvT xðk 0 Þjrkk g;
xðkÞ
0 6 k0 6 k
¼ ½ eðk 1Þ
eðk nÞ
uðk 1Þ
T
uðk nÞ
ð16Þ
for all integer k P 0, where eðkÞ ¼ yðkÞ ym ðkÞ is the tracking error with ym ðkÞ ¼ Wm ðq1 Þr ðkÞ and r ðkÞ being any arbitrary bounded external input sequence.
Remark 3. Assumption 1 is invoked in the design of a relative adaptation dead-zone for robustness purposes in the parameter estimation shown in Section 3. The width of the dead-zone is governed by the overbounding normalized function gn ðkÞ ¼ gðkÞ=ð1 þ k/ðkÞkÞ. This assumption is fulfilled by any system in which the signal gðkÞ is the sum of a bounded term plus a term related to uðkÞ by a strictly proper exponentially stable transfer function, [17]. The structure of the control law is maintained as in (5) while replacing the control parameters K, ri and si , for i 2 f0; . . . ; n 1g, by KðkÞ, ri ðkÞ and si ðkÞ, respectively, for all integer k P 0. The equations to obtain the control parameters are those of (7) and (10) substituting the plant parameters of the vector h by their corresponding estimates of hðkÞ ¼ T ½ b0 ðkÞ bn ðkÞa1 ðkÞ an ðkÞ for all integer k P 0. Such estimates are obtained from an estimation and modification algorithm which ensures the controllability of the estimated and modified model at all sampling instants, i.e., jDetðMðhðkÞÞÞj P d0 > 0 for all integer k P 0, as it will be shown in Section 3.
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3. Adaptive control If the plant parameters are unknown, an estimation algorithm has to calculate a controllable estimation plant model. Two alternative algorithms are presented which include a hysteresis-free switching function.
3.1. Estimation and modification algorithms A least-square estimation algorithm with normalization and a relative adaptation dead-zone is used to obtain an Ôa prioriÕ estimation of the plant parameters in Step 1. Then, a suitable modification is applied to such Ôa prioriÕ estimates to obtain a controllable Ôa posterioriÕ estimated model of the plant in Step 2. Step 1 (‘A priori’ estimation) The Ôa prioriÕ plant parameter estimates are obtained from the recursive equations ^ þ 1Þ ¼ hðkÞ ^ þ hðk
sðkÞP ðkÞ/n ðkÞen ðkÞ ; cðkÞ þ sðkÞ/Tn ðkÞP ðkÞ/n ðkÞ
P ðk þ 1Þ ¼ P ðkÞ
sðkÞP ðkÞ/n ðkÞ/Tn ðkÞP ðkÞ cðkÞ þ sðkÞ/Tn ðkÞP ðkÞ/n ðkÞ
ð17aÞ
where en ðkÞ ¼
eðkÞ ; 1 þ k/ðkÞk
^ eðkÞ ¼ yðkÞ /T ðkÞhðkÞ
ð17bÞ
are the normalized identification error and the identification error, respectively, ^ ¼ ½ b^ ðkÞ b^ ðkÞ a^ ðkÞ a^ ðkÞ T and hðkÞ ~ ¼ hðkÞ ^ h are the hðkÞ 0 n 1 n a priori estimation vector of the plant parameters and the parametrical error, respectively, /n ðkÞ ¼ /ðkÞ=ð1 þ k/ðkÞkÞ, P ð0Þ ¼ P T ð0Þ > 0 and cðkÞ is a real value sequence chosen such that 0 < c1 6 cðkÞ 6 c2 < 1, for all integer k P 0, with c1 and c2 being some design real constants. The sequence sðkÞ is the relative adaptation dead-zone defined as ( sðkÞ ¼
1=2 T ðkÞ/n ðkÞ 0 if wðkÞ 6 l 1 þ /n ðkÞPcðkÞ gn ðkÞ f ðkÞ=wðkÞ otherwise
where wðkÞ ¼ ðe2n ðkÞ þ /Tn ðkÞP 2 ðkÞ/n ðkÞÞ1=2 , and
ð17cÞ
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f ðkÞ 8 1=2 > /Tn ðkÞP ðkÞ/n ðkÞ > > > if wðkÞ 6 l 1 þ gn ðkÞ > /T ðkÞP ðkÞ/n ðkÞ > > gn ðkÞ otherwise : wðkÞ l 1 þ n cðkÞ
ð17dÞ with l > 1. ~ It will be then proved in Lemma 3.1 below that kP 1 ðkÞhðkÞk is bounded for all integer k P 0 and at the limit as k ! 1. Thus, a bounded vector sequence b ðkÞ 2 R2nþ1 exists such that the true plant parameter vector can be written ^ þ P ðkÞb ðkÞ. This fact provides the motivation to obtain the Ôa as h ¼ hðkÞ posterioriÕ modified estimates of the plant parameters in Step 2 below. Step 2 (‘A posteriori’ modified estimation) Two alternative algorithms are designed: (1) First modification algorithm (i) The modified estimates are calculated from the modification rule ^ þ P ðkÞbðkÞ hðkÞ ¼ hðkÞ
ð18Þ
where hðkÞ ¼ ½ b0 ðkÞ bn ðkÞ a1 ðkÞ an ðkÞ T is the Ôa posterioriÕ estimation vector of the plant parameters and bðkÞ 2 R2nþ1 is a switching sequence which takes values being equal to one of the constant vectors belonging to a finite set as which is computed in (ii) below: ð1Þ (ii) Let e0 be a zero ð2n þ 1Þ-dimensional vector. Let fei g be ð2n þ 1Þ-dimensional vectors with only one non-zero element which equalizes either +1 or ðjÞ to )1, for i 2 f1; . . . ; 2ð2n þ 1Þg. Similarly, let fei g be ð2n þ 1Þ-dimensional vectors with j non-zero elements, each of which equalizes either to +1 or to )1, for i 2 f1; . . . ; ½ð2n þ 1Þ!=ðð2n þ 1 jÞ!j!Þ2j g. The total number of such vectors is L¼
2X nþ1 j¼0
ð2n þ 1Þ! 2j ð2n þ 1 jÞ!j!
Denote these vectors by fej g, with j 2 ZL f0; 1; . . . ; L 1g, in the increasing order of the number of non-zero elements in ej . We now precisely define the sequence bðkÞ. Consider a real number 0 < d 1 and let the initial condition of bð0Þ be de0 . At each sampling instant, bðkÞ is defined by ^ þ P ðkÞbðk 1ÞÞÞj P d0 bðk 1Þ if jDetðMðhðkÞ bðkÞ ¼ ð19Þ dej0 otherwise
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for all integer k P 1, where d0 is a small real positive constant and ej0 is obtained by searching from the vectors fej g the smaller j0 2 ZL such that ^ þ P ðkÞdej ÞÞj P d0 . jDetðMðhðkÞ 0 (2) Second modification algorithm (i) The modified estimates are calculated from the modification rule (18), where bðkÞ ¼ pðkÞb0 ðkÞ, with pðkÞ 2 R being a scalar switching sequence and b0 ðkÞ 2 R2nþ1 a vector sequence. (ii) At each sampling instant, the components b0i ðkÞ for i 2 f1; 2; . . . ; 2n þ 1g, of the vector sequence b0 ðkÞ, are calculated from the following equations b0i ðkÞ ¼
DetðPi0 ðkÞÞ DetðP ðkÞÞ
ð20Þ
where Pi0 ðkÞ has 2n columns identied to those of P ðkÞ while the ith column is replaced with T nþ1 z}|{ v¼ : 0 0 0 0 0 1 (iii) The switching sequence pðkÞ is zero for k ¼ 0 and for integers k > 0 it is defined as follows, pðkÞ ¼
pðk 1Þ p0 ðkÞ
^ þ pðk 1ÞP ðkÞb0 ðkÞÞÞj P d0 if jDetðMðhðkÞ otherwise
ð21Þ
for some small real positive constant d0 . i.e., expression (21) implies that pðkÞ maintains the value that it had at the previous sampling instant if the estimated and modified plant model obtained with such an aforementioned value is controllable. Otherwise, the function pðkÞ switches from the value pðk 1Þ to a new value p0 ðkÞ, which depends on the conmutation sampling instant, to avoid non-controllable plant estimated models. Besides, it is convenient that the absolute value of p0 ðÞ be as small as possible so that parameters modification be smooth. Because of this, p0 ðÞ is obtained from the following algorithm: Algorithm to compute p0 ðÞ at the conmutation sampling instants This algorithm is splitted into a set of elementary steps: ^ þ p0 P ðkÞb0 ðkÞÞÞj and go to Step 2. Step 1: Set p0 ¼ 0, compute jDetðMðhðkÞ 0 ^ Step 2: If jDetðMðhðkÞ þ p0 P ðkÞb ðkÞÞÞj P d0 then go out, otherwise go to Step 3. Step 3: Increase the value of p0 by the operation p0 ¼ p0 þ Dp0 , with ^ þ p0 P ðkÞb0 ðkÞÞÞj and go to 0 < Dp0 1, compute jDetðMðhðkÞ Step 4.
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^ þ p0 P ðkÞb0 ðkÞÞÞj P d0 then set p1 ¼ p0 and go to Step 5, Step 4: If jDetðMðhðkÞ otherwise go to Step 3. Step 5: Set p0 ¼ 0 and go to Step 6. ^ þ Step 6: Decrease the value of p0 via p0 ¼ p0 Dp0 , compute jDetðMðhðkÞ 0 p0 P ðkÞb ðkÞÞÞj and go to Step 7. ^ þ p0 P ðkÞb0 ðkÞÞÞj P d0 then go to Step 8, otherwise go to Step 7: If jDetðMðhðkÞ Step 6. Step 8: If jp1 j 6 jp0 j then set p0 ¼ p1 , and go out.
Remark 4. From (17b) and (18), yðkÞ ¼ /T ðkÞhðkÞ þ eðkÞ bT ðkÞP ðkÞ/ðkÞ ¼ /T ðkÞhðkÞ þ ea ðkÞ
ð22Þ
is obtained where ea ðkÞ ¼ eðkÞ bT ðkÞP ðkÞ/ðkÞ is referred to as ‘a posteriori’ identification error. Such a model has uðkÞ and yðkÞ as input and output, respectively, and ea ðkÞ as an external disturbance. The controllability condition of the plant estimated model is jDetðMðhðkÞÞÞj > 0 where MðhðkÞÞ has the same structure as MðhÞ, in Eq. (13) of Section 2, by replacing the components of the true parameters vector h with the corresponding one of their modified estimates hðkÞ through (18). 3.2. Adaptive control law The control law is based in (5) and becomes: uðkÞ ¼ KðkÞym ðkÞ Rðq1 ; kÞuðkÞ Sðq1 ; kÞyðkÞ ¼
KðkÞ Sðq1 ; kÞ ym ðkÞ yðkÞ 1 1 þ Rðq ; kÞ 1 þ Rðq1 ; kÞ
ð23Þ
where the control parameters KðkÞ, ri ðkÞ and si ðkÞ (coefficients of the timevarying polynomials Rðq1 ; kÞ and Sðq1 ; kÞ), for i 2 f0; . . . ; n 1g, are calculated by similar equations to (7) and (10). In those equations, the parameters of the polynomials Aðq1 Þ and Bðq1 Þ must be substituted by their estimated and modified ones of Aðq1 Þ and Bðq1 Þ, respectively. The adaptive controller parameters can be obtained through a similar equation to (11). 3.3. Properties of the adaptive control system The convergence and stability schemeÕs properties are given in the following results. Step 1 of the estimation algorithm posses the properties given in the following lemma:
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Lemma 3.1 (i) P ðkÞ is symmetrical for all integer k P 0, uniformly bounded and it asymptotically converges to a finite (at least semidefinite positive) limit as k tends to infinity. ^ (ii) khðkÞk < 1, gn ðkÞ < 1, jgn ðkÞj < 1, jen ðkÞj < 1 and wðkÞ < 1 for all integer k P 0. Pk Pk Pk (iii) f ðkÞ < 1, sðkÞw2 ðkÞ < 1, i¼0 f 2 ðiÞ < 1, i¼0 f ðiÞ < 1 and i¼0 sðiÞ w2 ðiÞ < 1 for all integer k P 0, limk!1 f 2 ðkÞ ! 0, limk!1 f ðkÞ ! 0 and limk!1 sðkÞw2 ðkÞ ! 0: ^ hðk ^ 1Þk < 1 for all integer k P 1 and limk!1 khðkÞ ^ hðk ^ (iv) khðkÞ 1Þk ! 0. (v) 0 6 sðkÞ < 1, kP ðkÞk > 0 and sðkÞe2n ðkÞ < 1 for all integer k P 0, limk!1 sðkÞ ! 0 and limk!1 sðkÞe2n ðkÞ ! 0 ~ (vi) kP 1 ðkÞhðkÞk < 1 for all integer k P 0. The proof is given in Appendix A. Note that Property (v) ensures the nonsingularity of the matrix P ðkÞ, for all integer k P 0. In this way, the plant parameters are estimated from (17a)–(17d) until wðkÞ be smaller than lð1 þ ð/Tn ðkÞP ðkÞ/n ðkÞÞ=cðkÞÞ1=2 gn ðkÞ, provided that /ðkÞ does not become or it is closed to a constant vector prior to the convergence of the plant estimates to their limits. Step 2 applies, independent of the computation method of bðkÞ: Lemma 3.2 (i) There exists a finite number of switches in bðkÞ, it is bounded, for all integer k P 0, and asymptotically converges. (ii) hðkÞ is a bounded vector sequence, for all integer k P 0, and asymptotically converges to a finite limit. (iii) The control parameters KðkÞ, ri ðkÞ and si ðkÞ, for all integer k P 0 and i 2 f0; 1; . . . ; n 1g, are bounded sequences and asymptotically converge to finite limits. (iv) ean ðkÞ ¼ ea ðkÞ=ð1 þ k/ðkÞkÞ verifies lim
k!1
e2an ðkÞ
/Tn ðkÞP ðkÞ/n ðkÞ 2 2bmax ðkÞl 1 þ gn ðkÞ 6 0 cðkÞ 2
2
where bmax ðkÞ ¼ maxf1; kbðkÞk g, for all integer k P 0. The proof is given in Appendix B. Remark 5. In view of Property (iv) of Lemma 3.2, the normalized Ôa posterioriÕ identification error ean ðkÞ belongs to the residual set defined below
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
567
9 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi = < T / ðkÞP ðkÞ/ ðkÞ n l lim fgn ðkÞg De ¼ ean lim fean ðkÞg 6 2 lim bmax ðkÞ 1 þ n k!1 k!1 ; : k!1 cðkÞ
If gfn ðkÞ converges to zero, which can occur with a0 ¼ 0 in (15) if qðkÞ converges to zero, then, the residual set De converges to the zero equilibrium.
4. Stability analysis Eqs. (22) and (23) lead to the following time-varying 2n-auxiliary system xðkÞ ¼ Ac ðk 1Þxðk 1Þ þ B1 #1 ðkÞ þ B2 #2 ðkÞ
ð24aÞ
with xðk 1Þ ¼ ½ eðk 1Þ
eðk nÞ uðk 1Þ
uðk nÞ
T
ð24bÞ
#1 ðkÞ ¼ ½b0 ðk 1ÞðKðk 1Þ s0 ðk 1ÞÞ 1 r0 ðk 1Þym ðkÞ
n1 X
ðb0 ðk 1Þsi ðk 1Þ þ ð1 þ r0 ðk 1ÞÞai ðk 1ÞÞym ðk iÞ
i¼1
an ðk 1Þð1 þ r0 ðk 1ÞÞym ðk nÞ þ ð1 þ r0 ðk 1ÞÞea ðkÞ #2 ðkÞ ¼ Kðk 1Þym ðkÞ þ
n1 X
ðs0 ðk 1Þai ðk 1Þ si ðk 1ÞÞym ðk iÞ
i¼1
s0 ðk 1Þan ðk 1Þym ðk nÞ þ s0 ðk 1Þea ðkÞ ð24cÞ Ac ðk 1Þ ¼ 2 b0 s1 ð1 þ r0 Þa1 6 6 1 6 6 0 6 6 6 6 6 6 6 6 6 6 0 6 6 6 s0 a1 s1 6 6 0 6 6 6 0 6 6 6 6 6 6 6 4 0
b0 s2 ð1 þ r0 Þa2
ð1 þ r0 Þan
ð1 þ r0 Þb1 b0 r1
ð1 þ r0 Þbn
0
0
0
0
1
0
0
0
0
1
s0 a2 s2
0
0
0
s0 an
ðs0 b1 þ r1 Þ
s0 bn
0
0
1
0
0
0
0
1
0
0
0
0
1
0
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
ð24dÞ
568
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
BT1
¼ ½1
0
0 ;
BT2
¼
nþ1
0
0
z}|{ 1
0
ð24eÞ
First, note that Ac ðk 1Þ is uniformly bounded from Lemma 3.2 (Properties (ii) and (iii)). On the other hand, the number of switches of the sequence bðÞ is finite. This fact implies that the time interval between two consecutive switches is finite. Besides, the control parameters are bounded at all sampling instants since the controllability of the modified estimation plant is ensured and the closed-loop modified estimation model posses a stable dynamics. Then, there is no finite escape between the initial time instant and the sampling instant k1 T at which the last switch in bðÞ occurs. Thus, one can redefine the time origin as k1 T and study the system stability for k P k1 . The following theorem establishes the main result of the convergence analysis. Theorem 4.1 (Main result). The adaptive control law stabilizes the plant (1) in the sense that uðkÞ, eðkÞ and yðkÞ are bounded for all finite initial states and any bounded reference sequence r ðkÞ, subject to Assumption 1, provided that a in (15) is sufficiently small so that 1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0Þg ca supk1 6 k0 6 k 2bmax ðk 0 Þ 1 þ kmax fP lakvk c1 40
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569
for all integer k P 0 where T ¼ 0:01s, are considered. The following parameters are chosen in (15) for the computation of the overbounding function for the unmodeled dynamics: a ¼ 105 , a0 ¼ 0 and r ¼ 0:5. Also, v ¼ ½ 0:1 0:2 0:3 0:1 T in (16), the sequence cðkÞ ¼ 4 for all integer k P 0 in (17a) and l ¼ 1:0001 in (17c) and (17d), and 3 2 1 0:4 0:4 0:4 0:4 6 0:4 1 0:4 0:4 0:4 7 7 6 7 P ð0Þ ¼ 55 6 6 0:4 0:4 1 0:4 0:4 7 4 0:4 0:4 0:4 1 0:4 5 0:4 0:4 0:4 0:4 1 ^ ¼ ½ 1:3 0:7 0:5 1:2 0:9 are the initial conditions of the and hð0Þ estimation algorithm with d0 ¼ 0:001. The parameter d ¼ 105 for the first modification algorithm and Dp0 ¼ 104 for the second modification algorithm are chosen. The control objective is defined by the Hurwitz polynomial Cðq1 Þ ¼ 1 þ 0:4q1 0:25q2 0:1q3 . The simulation results are shown in Figs. 1–8. T
(i) Results with the first modification algorithm
80
60
40
ε
20
0
-20
-40 0
5
10
15
20
25
30
35
40
t
Fig. 1. Tracking error signal in the interval [0,50].
45
50
570
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583 100
80
60
40
u 20
0
-20
-40 0
5
10
15
20
25
30
35
40
45
50
t
Fig. 2. Control signal in the interval [0,50].
Fig. 3. Parameters of the plant estimated and modified model in the interval [0,10].
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
Fig. 4. Components of the vector ej0 in the interval [0,0.5].
(ii) Results with the second modification algorithm
20
15
10
5
ε 0
-5
-10
-15 0
5
10
15
20
25
30
35
40
t
Fig. 5. Tracking error signal in the interval [0,50].
45
50
571
572
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
30
20
10
u
0
-10
-20
-30 0
5
10
15
20
25 t
30
35
40
45
50
Fig. 6. Control signal in the interval [0,50].
Fig. 7. Parameters of the plant estimated and modified model in the interval [0,10].
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
573
0.15
0.1
0.05
0 π -0.05
-0.1
-0.15
-0.2
0
0.2
0.4
0.6
0.8
1 t
1.2
1.4
1.6
1.8
2
Fig. 8. Evolution of pðtÞ in the interval [0,2].
Remark 6. Note that Figs. 1, 2, 5 and 6 exhibit bounded jumps in the tracking error and control signals. These jumps occur at the sampling instants at which the sequence bðkÞ changes its value. The modified estimated parameters present also bounded jumps at the same sampling instants at which the sequence bðkÞ changes its value. The presence of such jumps is crucial to ensure the controllability of the estimated and modified model of the plant when the unmodified one losses (or it is close to lose) its controllability. Note also that the numerator and denominator polynomials of the modeled plant present the common factor ð1 0:3q1 Þ. I.e., the controllability condition for the nominal plant is not fulfilled. However, the adaptive system becomes stabilizable since the estimated model of the plant is controllable thanks to the modification algorithm. Note that bðkÞ switches a finite number of times before converging for both modification algorithms.
6. Conclusions An adaptive pole-placement based control algorithm has been presented that stabilizes an, in general, inversely unstable discrete-time system in the presence of unmodeled dynamics and, eventually, bounded noise. The algorithm includes the use of a relative adaptation dead-zone for the closed-loop
574
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robust adaptive stabilization. The controllability of the estimated model is ensured by incorporating an appropriate modification in the standard leastsquares estimation algorithm which is implemented as the first estimation level. The boundedness of the tracking error and all the remaining signals in the closed-loop system is ensured without providing the controllability of the nominal plant.
Acknowledgements The authors are very grateful to the Basque Country University UPV/EHU and MCYT by their support through projects 1/UPV/EHU 00I06.I06-EB8235/2000 and DPI 2000-0244, respectively.
Appendix A. Proof of Lemma 3.1 (i) The matrix P ð0Þ is chosen symmetrical and P ðk þ 1Þ P ðkÞ is also symmetrical from (17a) for all integer k P 0. Then, P ðkÞ is a monotonic nonincreasing symmetrical matrix sequence since P ðk þ 1Þ P ðkÞ P 0 for all integer k P 0 from Eq. (17a). Thus, if P ðk1 Þ ¼ 0 for any integer k1 P 0, then, P ðk1 þ 1Þ P ðk1 Þ ¼ 0 from (17a) and then, P ðkÞ ¼ 0 for all integer k P k1 . Consequently, P ðkÞ is upper bounded by P ð0Þ and lower bounded by zero, i.e., 0 6 P ðkÞ 6 P ð0Þ, and it also converges asymptotically to a finite limit as k tends to infinity because of it is monotonic non-increasing matrix sequence. ~ þ trðP ðkÞÞ (ii) Consider the non-negative sequence V ðkÞ ¼ h~T ðkÞP 1 ðkÞhðkÞ for all integer k P 0. The matrix inversion lemma, [18], applied to the second equation of (17a) yields P 1 ðk þ 1Þ ¼ P 1 ðkÞ þ
sðkÞ / ðkÞ/Tn ðkÞ cðkÞ n
ðA:1Þ
From Eqs. (17a)–(17d), (15) and (A.1), it follows that sðkÞw2 ðkÞ sðkÞ 2 gn ðkÞ þ T cðkÞ þ sðkÞ/n ðkÞP ðkÞ/n ðkÞ cðkÞ 2 l 1 f 2 ðkÞ 6 2 l cðkÞ þ sðkÞ/Tn ðkÞP ðkÞ/n ðkÞ
V ðk þ 1Þ V ðkÞ ¼
ðA:2Þ
where sðkÞ 2 ½0; 1Þ and sðkÞw2 ðkÞ ¼ f ðkÞwðkÞ P f 2 ðkÞ have been used. From (A.1) it follows that kmin ðP 1 ðkÞÞ P kmin ðP 1 ð0ÞÞ
ðA:3Þ
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575
for all integer k P 0. Eq. (A.2) leads to ~ þ tr P ðkÞ 6 h~T ð0ÞP 1 ð0Þhð0Þ ~ þ tr P ð0Þ V ðkÞ 6 V ð0Þ () h~T ðkÞP 1 ðkÞhðkÞ ðA:4Þ for all integer k P 0, and then 2 ~ kmax ðP 1 ð0ÞÞkhð0Þk þ tr P ð0Þ 2 2 ~ ~ P kmin ðP 1 ð0ÞÞkhðkÞk þ tr P ðkÞ () khðkÞk
6
kmax ðP ð0ÞÞ ~ tr P ð0Þ tr P ðkÞ khð0Þk2 þ l2 ð1 þ ð/Tn ðkÞP ðkÞ/n ðkÞÞ=cðkÞÞg2n ðkÞ is fulfilled. Then, (A.11) implies since sðiÞ 2 ½0; 1Þ: h i /T ðiÞP ðiÞ/ ðiÞ k X X sðiÞ w2 ðiÞ 1 þ n cðiÞ n g2n ðiÞ M2 ðkÞ ¼ M1 ðkÞ < 1 sðiÞ 6 cðiÞ þ sðiÞ/Tn ðiÞP ðiÞ/n ðiÞ i¼0 i2I2 i6k
ðA:12Þ with M2 ðkÞ inf i2I2 i6k
h i 8 9 T ðiÞ/n ðiÞ < w2 ðiÞ 1 þ /n ðiÞPcðiÞ g2n ðiÞ = :
cðiÞ þ sðiÞ/Tn ðiÞP ðiÞ/n ðiÞ
;
>0
for all integer k P 0. Then, limk!1 sðkÞ ! 0. Thus, it follows from (A.1) that P 1 ðk þ 1Þ ¼ P 1 ð0Þ þ
k X sðiÞ /n ðiÞ/Tn ðiÞ cðiÞ i¼0
ðA:13Þ
From (A.13) and the facts that kP ð0Þk > 0, /n ðkÞ is bounded, cðkÞ > 0 for all integer k P 0 and limk!1 sðkÞ ! 0, it follows that kP 1 ðkÞk < 1, kP ðkÞk > 0
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577
for all finite k and as k ! 1. Finally, it follows, from the definition of wðkÞ in Eqs. (17), that k X
sðiÞe2n ðiÞ 6
i¼0
k X
sðiÞw2 ðiÞ
i¼0
inf
f/Tn ðiÞP 2 ðiÞ/n ðiÞg
i2f0;1;...;kg
k X
sðiÞ < 1
i¼0
ðA:14Þ Pk
2
Pk
where the boundedness of i¼0 sðiÞw ðiÞ, i¼0 sðiÞ, /n ðkÞ and P ðkÞ for all integer k P 0 have been used. Thus, 0 6 sðkÞe2n ðkÞ < 1 for all integer k P 0 and limk!1 sðkÞe2n ðkÞ ! 0. (vi) From (17a) and (A.1), one gets ~ kP 1 ðkÞhðkÞk ~ 1Þk þ sðk 1Þ k/ ðk 1Þkj/T ðk 1Þhðk ~ 1Þj 6 kP 1 ðk 1Þhðk n cðk 1Þ n 1 þ cðk 1Þ sðk 1Þk/n ðk 1Þkjen ðk 1Þj þ cðk 1Þ ðA:15Þ ~ þ g ðiÞ for all integer i P 0, it follows that From en ðiÞ ¼ /Tn ðiÞhðiÞ n T ~ j/n ðiÞhðiÞj 6 jen ðiÞj þ jgn ðiÞj. By introducing this expression in (A.15), one obtains ~ kP 1 ðkÞhðkÞk ~ 1Þk þ sðk 1Þ k/ ðk 1Þkðjg ðk 1Þj 6 kP 1 ðk 1Þhðk n cðk 1Þ n þ ½2 þ cðk 1Þjen ðk 1ÞjÞ 1 ~ k/n ðiÞkðjgn ðiÞj þ ½2 þ cðiÞjen ðiÞjÞ þ sup 6 kP 1 ð0Þhð0Þk cðiÞ 0 6 i 6 k1 k1 X sðiÞ < 1 ðA:16Þ i¼0
for all integer k P 0 since jen ðkÞj < 1 and 0 < cðkÞ < 1.
Pk1 i¼0
sðiÞ < 1, k/n ðkÞk < 1, jgn ðkÞj < 1,
Appendix B. Proof of Lemma 3.2 (i) These properties have to be proved for both the first modification algorithm and the second modification algorithm. For the first modification algorithm, it has been proved in [6] that there exits ^ þ P ðkÞdei ÞÞj P a positive sufficient small real constant d0 such that jDetðMðhðkÞ d0 for at least one of the vectors ei at each sampling instant, for
578
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^ i 2 f0; . . . ; L 1g. Thus, from the fact that P ðkÞ and hðkÞ are bounded and converge (see Properties (i), (ii) and (iv) of Lemma 3.1) and from (19), it follows that the number of switches in bðkÞ is finite and then bðkÞ converges asymptotically. Besides, bðkÞ is uniformly bounded since it equates to the product of d with one of the aforementioned bounded constant vectors ei at each sampling instant. For the second modification algorithm, one has to prove that there exits a bounded sequence pðkÞ such that jDetðMðhðkÞÞÞj P d0 > 0, where the matrix defined as MðhðkÞÞ is: MðhðkÞÞ ¼ 2 1 0 6 a1 1 6 6 6 .. .. 6 . . 6 6 an1 an2 6 6 6 an an1 6 6 6 0 an 6 6 .. .. 6 . . 6 6 6 0 0 6 n n 4 P P 1 þ ai 1 þ ai i¼1
.. . .. .
0 0 .. . 1 a1 a2 .. .
b0 b1 .. . bn1 bn 0 .. .
0 b0 .. . bn2 bn1 bn .. .
.. . .. .
0 0 .. . b0 b1 b2 .. .
a nn P 1 þ ai
0 n P bi
0 n P bi
bn n P bi
i¼0
i¼0
i¼1
i¼1
i¼0
0 0 .. . 0 0 0 .. .
3
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 0 7 7 n 5 P bi i¼0
ðB:1Þ 0
From (18) and the expression bðkÞ ¼ pðkÞb ðkÞ, the vector hðkÞ can be written as: ^ þ pðkÞP ðkÞb0 ðkÞ hðkÞ ¼ hðkÞ
ðB:2Þ 0
where the components of the vector sequence b ðkÞ are calculated by means of (20). i.e., b0 ðkÞ is the solution of the matrix equation: P ðkÞb0 ðkÞ ¼ v
ðB:3Þ
where v¼
T
nþ1
0
0
0
z}|{ 1
0
0
0
2 R2nþ1 :
That is, the components of b0 ðkÞ are such that, at each sampling instant: piT ðkÞb0i ðkÞ ¼ 0 T pnþ1 ðkÞb0nþ1 ðkÞ
for i 2 f1; . . . ; ng \ fn þ 2; . . . ; 2n þ 1g; ¼1
ðB:4Þ
where pi ðkÞ denotes the i-column of the matrix P ðkÞ. By introducing (B.2) and (B.4) into (B.1), it follows that:
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583 MðhðkÞÞ ¼ 2 1 6 a^1 6 6 6 .. 6 . 6 6 a^n1 6 6 6 a^n 6 6 0 6 6 6 .. 6 . 6 6 6 0 6 n 4 P 1 þ a^i
0
0
1 .. .
.. .
0 .. . 1 a^1
a^n2 a^n1 a^n ... 0 n P 1 þ a^i
i¼1
.. .
b^0 b^1 .. . b^n1 b^n þ p 0 .. .
a^2 ... a^n 0 n nP 1 P 1 þ a^i b^n p b^i
i¼1
i¼1
0 b^0 .. . b^n2
0
.. .
0 .. . b^0
b^n1 b^n þ p .. .
.. .
b^1 b^2 .. .
0 b^n p
i¼0
nP 1
b^i
b^n þ p
b^n p
i¼0
579 3
0
7 7 7 7 7 7 7 7 7 7 0 7 7 0 7 7 7 .. 7 . 7 7 7 0 7 nP 1 5 b^n þ p þ b^i 0 .. . 0
nP 1
b^i
i¼0
i¼0
ðB:5Þ of determinant ^ DetðMðhðkÞÞÞ ¼ ðb^n ðkÞ þ pðkÞÞnþ1 þ gðhðkÞ; pðkÞÞ ¼ pðkÞ
nþ1
^ þ g0 ðhðkÞ; pðkÞÞ
ðB:6Þ
^ ^ where both gðhðkÞ; pðkÞÞ and g0 ðhðkÞ; pðkÞÞ are n-order polynomials in pðkÞ. From (B.6), the controllability condition of modified estimation plant model is: jpðkÞ
nþ1
^ þ g0 ðhðkÞ; pðkÞÞj P d0
ðB:7Þ
^ Note that function g0 ðhðkÞ; pðkÞÞ is a sum of a finite number of terms of the c0 ^ ci c ^ form pðkÞ hi ðkÞ . . . hj ðkÞ j for some non-negative integers c0 ; ci ; . . . ; cj , and ^ where h^i ðkÞ and h^j ðkÞ denote some components of the vector hðkÞ. Then, 2nþ1 0 ^ ^ g ðhðkÞ; pðkÞÞ is analytic for all pðkÞ 2 R and for all hðkÞ 2 R . Then, the expression DetðMðhðkÞÞÞ is a ðn þ 1Þ-order polynomial in p, with all coefficients well defined at all sampling instants. Thus, there exists some absolute bounded value for pðkÞ which verifies the controllability condition (B.7). Besides, the vector sequence b0 ðkÞ is uniformly bounded and it asymptotically converges from (20) and the boundedness and asymptotic convergence of P ðkÞ [Lemma 3.1, (i) ]. Then, pðkÞ asymptotically converges to a bounded constant from (21) ^ [Lemma 3.1, and the uniform boundedness and asymptotic convergence of hðkÞ (ii) and (iv)]. Thus, bðkÞ is uniformly bounded, has a finite number of switches and asymptotically converges. ^ are bounded. (ii) It follows directly from (i) since P ðkÞ and hðkÞ (iii) It follows directly since khðkÞk < 1 for all integer k P 0, hðkÞ converges to a limit as k ! 1 and jDetðMðhðkÞÞÞj P d0 > 0 for all integer k P 0. (iv) Since ean ðkÞ ¼ en ðkÞ bT ðkÞP ðkÞ/n ðkÞ, one obtains from the relation 2 ða þ bÞ 6 2ða2 þ b2 Þ for any a; b 2 R: e2an ðkÞ 6 2ðe2n ðkÞ þ kbðkÞk2 /Tn ðkÞP 2 ðkÞ/n ðkÞÞ /T ðkÞP ðkÞ/n ðkÞ 2 ) lim 2bmax ðkÞ w2 ðkÞ l2 1 þ n gn ðkÞ 6 0 t!1 cðkÞ ðB:8Þ
580
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
since limk!1 sðkÞ ! 0 so that w2 ðkÞ 6 l2 ð1 þ ð/Tn ðkÞP ðkÞ/n ðkÞÞ=cðkÞÞg2n ðkÞ as k ! 1.
Appendix C. Proof of Theorem 4.1 Ac ðkÞ is bounded because of hðkÞ, and the controller parameters KðkÞ and the coefficients of Rðq1 ; kÞ and Sðq1 ; kÞ are bounded, from Lemma 3.2(ii)–(iii). The eigenvalues of Ac ðkÞ are in jzj < 1 for all integer k P 0. Besides, k X
2
kAc ðk 0 Þ Ac ðk 0 1Þk 6 b0 þ b1 ðk k0 Þ
ðC:1Þ
k 0 ¼k0 þ1
for all integers k and k0 such that k > k0 P 0, some positive constants b0 and b1 with b1 being sufficiently small. Note that (C.1) is fulfilled with a suitable constant b0 , which takes into account the changes of value of the entries of Ac ðÞ when a switch in the sequence bðÞ occurs, and a slow enough estimation rate via suitable P ð0Þ and cðkÞ in (17a) so that b1 is sufficiently small. Thus, the time-varying homogeneous system xðk þ 1Þ ¼ Ac ðkÞxðkÞ is exponentially stable Qk1 0 and its transition matrix wðk; k 0 Þ ¼ j¼k0 Ac ðjÞ satisfies kwðk; k 0 Þk 6 c1 rkk for 0 all k P k 0 where r0 2 ð0; 1Þ and c1 is a norm-dependent constant, [19,20]. We now redefine the time origin k1 T as the sampling instant such that for kT > k1 T there are not changes of value in bðkÞ. From (24a), if follows that
xðkÞ ¼ wðk; k1 Þxðk1 Þ þ
k X
wðk; k 0 Þ½B1 #1 ðk 0 Þ þ B2 #2 ðk 0 Þ
ðC:2Þ
k 0 ¼k1
for all integer k P k1 P 0. From (C.2), one obtains 1 kxðk1 Þk þ kxðkÞk 6 c1 rkk 0
k X
0
c1 rkk ðc2 þ c3 jea ðk 0 ÞjÞ 0
k 0 ¼k1
6 c4 þ
k X
0
c5 rkk jea ðk 0 Þj 0
ðC:3Þ
k 0 ¼k1
P2 since k i¼1 Bi #i ðkÞk 6 c2 þ c3 jea ðkÞj, for some positive real constants c2 , c3 , c4 and c5 , from (24c) and (24e), Schwarz’s inequality, r0 2 ð0; 1Þ, and the boundedness of KðÞ, b0 ðÞ, ai ðÞ, for i 2 f1; . . . ; ng, the sequence ym ðÞ and kxðk1 Þk. From (B.8), e2an ðkÞ 6 2bmax ðkÞw2 ðkÞ so that
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kxðkÞk 6 c4 þ
k X
0 c5 rkk 0
k 0 ¼k1
581
1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /Tn ðk 0 ÞP ðk 0 Þ/n ðk 0 Þ 0 2bmax ðk Þl 1 þ cðk 0 Þ
gn ðk 0 Þð1 þ k/ðk 0 ÞkÞ þ
k X
c5 rkk 0
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2bmax ðk 0 Þf ðk Þð1 þ k/ðk 0 ÞkÞ
k 0 ¼k1
ðC:4Þ where wðkÞ has been split into the two additive terms f ðkÞ and 1=2 /Tn ðkÞP ðkÞ/n ðkÞ wðkÞ f ðkÞ 6 l 1 þ gn ðkÞ cðkÞ from (17d). It follows from (15), (16) and (C.4) that ( kxðkÞk 6 c6 þ c5 lakvk sup
k1 6 k 0 6 k
sup fkxðk 0 Þkg k1 6 k 0 6 k
þ
k X
c5 rkk 0
0
1=2 ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /Tn ðk 0 ÞP ðk 0 Þ/n ðk 0 Þ 0 2bmax ðk Þ 1 þ cðk 0 Þ
1 r0kk1 þ1 1 r0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2bmax ðk 0 Þf ðk Þð1 þ k/ðk 0 ÞkÞ
ðC:5Þ
k 0 ¼k1
for some positive constant c6 , where kP ðkÞk < 1 and kvk supk1 6 k0 6 k fkxðk 0 Þkg P 0 supk1 6 k0 6 k fjvT xðk 0 Þjrkk g have been used. Besides, the right-hand side of (C.5) is monotonic non-decreasing in k. Then, for k k1 and provided that a
Robust adaptive control of discrete nominally stabilizable plants S. Alonso-Quesada *, M. De la Sen Dpto. de Electricidad y Electr onica, Facultad de Ciencias, Universidad del Paıs Vasco, UPV-EHU, 48940 Leioa (Bizkaia), Spain
Abstract This paper presents an indirect pole-placement based adaptive control scheme for discrete linear time-invariant non-necessarily inversely stable and stabilizable systems. The scheme is a pole-placement type without requiring the plant inverse stability assumption. The control objective is that the plant output should asymptotically track in the absence of unmodeled dynamics and bounded disturbances a reference signal, given by an arbitrary stable filter, under a bounded tracking error while ensuring robust closed-loop stability. The adaptive stabilizability and the robustness of this system under the presence of unmodeled dynamics and, possibly, bounded disturbances, are proved without assuming the controllability of the modeled plant. A normalized least-squares algorithm with a relative adaptation dead-zone and Ôa posterioriÕ modification of the parameter estimates is used to ensure the controllability of the modified estimation plant model at all sampling instants and at the limit. Such a property is crucial to solve the stable poleplacement problem. On the other hand, the relative adaptation dead-zone is included for robustness purposes under unmodeled dynamics and, possibly, bounded disturbances. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Robust control; Unmodeled dynamics; Adaptive stabilizability; Inversely unstable plant; Tracking; Adaptation dead-zone
1. Introduction The robust adaptive stability of discrete linear time-invariant systems has been successfully developed in the last two decades. In the beginning, as in [1] *
Corresponding author. E-mail address: [email protected] (S. Alonso-Quesada).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00291-1
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and [2], most of the proofs of stability of the adaptive control algorithms proposed were based on a set of standard assumptions. Such assumptions were: (i) the knowledge of an upper bound of the system order, (ii) the knowledge of the system relative degree, (iii) the plant being inversely stable and (iv) the knowledge of the sign of the high frequency gain of the plant as well as an upper bound for its absolute value. Later, works whose goal was to relax some of these aforementioned assumptions were presented. In this way, the studies in [3] and [4] relaxed the assumptions relative to the high frequency gain by including Nussbaum gain in the adaptation laws. Therefore, in the paper [5] the assumption of inverse stability as well as that of the knowledge of the high frequency plant gain were relaxed. That work focused on the robust adaptative stabilization of discrete linear time-invariant plants. The controllability of the nominal plant model and an overbounding function for the contribution of the unmodeled dynamics and external disturbances were supposed known for purposes of achieving robust adaptive stability. Such a stability result was established without either requiring the injection of persistent excitation probing signals into the system or assuming any prior knowledge of the plant parameters. The necessary and sufficient condition for the system stabilization via an adaptive pole-placement scheme was established in [6]. Such a condition is simply that the asymptotically reached estimated model of the stabilizable plant be controllable, which is weaker than the controllability of the true plant required in [5]. If the system is not inversely stable then the controllability of the estimated model of the plant has to be ensured when the time tends to infinity to achieve the adaptive stabilization via pole-placement. Basically, two different approaches have been used to circumvent the regions in the parameter space corresponding to uncontrollable models. One of them relies on the use of excitation probing signals, [7–11], while the other one is based on the use of a suitable modification of the plant parameter estimates as in [5,6,12–15]. The estimation modification algorithm used in [5,13,15] includes a hysteresis switching function what it bears a very heavy computation burden before the convergence of the estimates. The modification algorithm presented in [6] involves the use of a hysteresis-free switching function. Then, the convergence of the estimates is usually reached with a light computation burden, but with a high number of switches. This paper relaxes the controllability condition of the plant nominal model required in [5] for robust adaptive stabilizability of the closed-loop system under the presence of unmodeled dynamics. The control objective is to ensure the stability of the closed-loop system with a bounded tracking error. The proposed scheme is of pole-placement type while it ensures asymptotic reference tracking properties when the reference sequence becomes constant. The scheme is valid to operate in both non-adaptive and adaptive environments.
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557
The only assumption about the true plant is that it is stabilizable. This paper is an extension of the problem solved in [12] to the case of discrete-time plants. A normalized least-squares algorithm with a relative adaptation dead-zone and a parameter modification is used to update the plant parameters at each sampling instant. The above combined technique is the basis to prove the convergence of the estimated parameters and the boundedness of all the signals in the closed-loop system. The knowledge of an upper-bounding function for the contribution of the unmodeled dynamics and disturbances to the output is required to design the relative adaptation dead-zone, [16]. In that way, the adaptation is frozen when an aumented identification error lies below a known upper bound of the absolute value of the contribution of the unmodeled dynamics and bounded disturbances to the output. Two designs for the parameter modification are presented to fulfil the control objective. Both of them use a hysteresis-free switching function which does not change its value unless it is crucial to ensure the controllability of the estimation model of the plant. In this way, the number of switches and/or the computational cost is reduced with regard to the modification algorithms used in the previous works [5,6,13,15]. It is proved for both modification algorithms that the normalized modified identification error belongs to a residual set defined by an absolute normalized upper bound for the contribution of the unmodeled dynamics and bounded noise to the output. The paper is organized as follows. Section 2 presents the problem statement with the control objective and the structure of the control law. Section 3 is devoted to formulate the unmodified and modified parameter estimation algorithms. Two alternative algorithms are studied for the parameter modification, both of them providing a controllable plant estimated model for all sampling instants and at the limit. Section 4 establishes the main result of the stability analysis. Section 5 presents a numerical simulation to show the performance of the proposed control scheme. Finally, conclusions end the paper.
2. Problem statement Consider the following linear discrete-time system Aðq1 ÞyðkÞ ¼ Bðq1 ÞuðkÞ þ gðkÞ
ð1Þ
which is not necessarily controllable and can include stable uncontrollable modes as the roots of Aðq1 Þ ¼ 0, where Aðq1 Þ ¼ A0 ðq1 ÞA0 ðq1 Þ ¼ 1 þ a1 q1 þ þ an qn Bðq1 Þ ¼ B0 ðq1 ÞA0 ðq1 Þ ¼ bnm qmn þ bnmþ1 qmn1 þ þ bn qn
ð2Þ
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with q1 yðkÞ yðk 1Þ, m 6 n, gðkÞ being the contribution of unmodeled dynamics and bounded noise of any order to the output and A0 ðq1 Þ denoting the possible n0 P 0 stable common factors of Aðq1 Þ and Bðq1 Þ. If n0 ¼ 0 then A0 ðq1 Þ ¼ 1. Eq. (1) may be rewritten as yðkÞ ¼ hT /ðkÞ þ gðkÞ
ð3Þ
where h ¼ ½ bnm
bn
/ðkÞ ¼ ½ uðk n þ mÞ
a1
an T ; uðk nÞ
yðk 1Þ
T
yðk nÞ
ð4Þ are the true plant parameters and measurable regression vectors, respectively. 2.1. Case of known plant and absence of unmodeled dynamics (g() 0) In this case, the control objective is the tracking between the system output sequence and the reference sequence ym ðkÞ with a prescribed pole-placement, being zero the tracking error when the reference sequence becomes constant. The control law designed to meet this objective is: uðkÞ ¼ Kym ðkÞ Rðq1 ÞuðkÞ Sðq1 ÞyðkÞ ¼
K Sðq1 Þ y yðkÞ ðkÞ m 1 þ Rðq1 Þ 1 þ Rðq1 Þ
ð5Þ
where ym ðkÞ is given by a stable filter Wm ðq1 Þ ¼ Bm ðq1 Þ=Am ðq1 Þ ¼ ðb0nm mm qmm nm þ þ b0nm qnm Þ=ð1 þ a01 q1 þ þ a0nm qnm Þ; Rðq1 Þ ¼ rnm qðnmÞ þ þ rnn0 1 qðnn0 1Þ ; Sðq1 Þ ¼ s0 þ s1 q1 þ þ snn0 1 qðnn0 1Þ
ð6Þ
with ri ; sj 2 R, for i 2 fn m; . . . ; n n0 1g and j 2 f0; . . . ; n n0 1g, being calculated from the following diophantine equation ð1 þ Rðq1 ÞÞAðq1 Þ þ Sðq1 ÞBðq1 Þ ¼ Cðq1 Þ
ð7Þ
where Cðq1 Þ ¼ C 0 ðq1 ÞA0 ðq1 Þ ¼ 1 þ c1 q1 þ þ c2n1 qð2n1Þ is a Hurwitz polynomial. Eq. (7) is solvable since the n0 stable common factors of Bðq1 Þ and Aðq1 Þ are included in Cðq1 Þ. If the plant is controllable then all of the factors of the Hurwitz polynomial Cðq1 Þ can be freely chosen. Note that Eq. (7) is equivalent to that obtained by replacing A ! A0 , B ! B0 and C ! C 0
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559
after cancelling A0 and that Eq. (1) is equivalent neglecting initial conditions to the equation obtained by replacing A ! A0 , B ! B0 and g ! g0 ¼ ð1=A0 Þg. On the other hand, the control parameter K will be calculated such that the tracking of the reference sequence be perfect when it becomes constant. By introducing the control law (5) in the transfer function of the plant W0 ðq1 Þ ¼ Bðq1 Þ=Aðq1 Þ the transfer function of the closed-loop system is yðq1 Þ KBm ðq1 ÞB0 ðq1 ÞA0 ðq1 Þ ¼ r ðq1 Þ Am ðq1 ÞC 0 ðq1 ÞA0 ðq1 Þ
ð8Þ
which can include stable cancellations if the degree of A0 ðq1 Þ is non-zero and where the relationship (7) has been used. Its dynamics includes the poles of the stable filter Wm ðq1 Þ. The remaining poles can be fixed according to the arbitrary Hurwitz monic polynomial C 0 ðq1 Þ by the control polynomials Rðq1 Þ and Sðq1 Þ. Then, the dynamics of the closed-loop system is able to practically equate that of the stable filter Wm ðq1 Þ if the zeros of the chosen Hurwitz polynomial C 0 ðq1 Þ are sufficiently more stable than the poles of Wm ðq1 Þ. Note from (8) that the closed-loop characteristic equation is Am ðq1 ÞC 0 ðq1 Þ ¼ 0 and it has nm þ 2ðn n0 Þ 1 roots. However, note that the computational cost in solving the diophantine equation (7) is fixed irrespective of the degree of Am ðq1 Þ. Therefore, the number of closed-loop poles can be increased, if desired, without increasing the computational cost requested to generate the control input. The transfer function from the external input sequence r ðkÞ to the tracking error eðkÞ ¼ yðkÞ ym ðkÞ is Ge ðq1 Þ ¼ ¼
eðq1 Þ r ðq1 Þ ½KB0 ðq1 Þ ð1 þ Rðq1 ÞÞA0 ðq1 Þ Sðq1 ÞB0 ðq1 ÞBm ðq1 Þ Am ðq1 ÞC 0 ðq1 Þ
ð9Þ
where the relationship (7) has been used. The control parameter K can be calculated such that the absolute gain of this transfer function be zero when q1 ejT x ¼ 1 þ j0, with T being the sampling period and j the complex imaginary unity. In this way, the tracking error can be zero if the reference sequence becomes constant. The value of K to meet this objective is from (9): K¼
ð1 þ Rð1ÞÞA0 ð1Þ þ Sð1ÞB0 ð1Þ B0 ð1Þ
ð10Þ
If n ¼ m, nm ¼ mm and n0 ¼ 0, Eqs. (7) and (10) can be written more compactly as ½ 1 q1
q2nþ2
q2nþ1 ½MðhÞvc p ¼ 0
ð11Þ
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where vc ¼ ½ 1 þ r 0 r 1 p ¼ ½ 1 c1
rn1 s0 T c2n1 0
s1
sn1
T
K ;
ð12Þ
and MðhÞ ¼ 2
1 0 6 a1 1 6 6 .. .. 6 6 . . 6 6 an1 an2 6 6 6 an an1 6 6 0 an 6 6 6 .. .. 6 . . 6 6 0 0 6 6 n n 4 P P 1 þ ai 1 þ ai i¼1
i¼1
.. . .. .
0 0 .. . 1 a1 a2 .. . a nn P 1 þ ai i¼1
b0 b1 .. . bn1 bn 0 .. . 0 n P bi
0 b0 .. . bn2 bn1 bn .. . 0 n P bi
i¼0
i¼0
.. . .. .
0 0 .. . b0 b1 b2 .. . bn n P bi i¼0
3 0 0 7 7 7 .. 7 . 7 7 0 7 7 7 0 7 7 7 0 7 7 .. 7 . 7 7 7 0 7 7 n 5 P bi i¼0
ð13Þ is the Sylvester matrix of the true plant parameters. Eq. (11) is uniquely solvable if the determinant of MðhÞ is strictly non-zero, i.e., jDetðMðhÞÞj P d0 > 0
ð14Þ
for some design real small positive constant d0 . The relationship (14) will be referred to as the controllability condition of the true plant. Then, if the plant is controllable, i.e., Aðq1 Þ and Bðq1 Þ are relatively prime polynomials, then, the control parameters can be uniquely obtained from the algebraic system (11). Remark 1. Since the relative degree of the plant and its parameters are unknown, one can choose that the relative degree of the nominal model of the plant be zero (n ¼ m) and A0 ðq1 Þ ¼ 1 (n0 ¼ 0) to establish the formalism in a simpler way. In the case when the relative degree of the plant is non-zero the estimated parameters, issued from an estimation algorithm, corresponding to those added parameters will tend to zero. Remark 2. In the case when the polynomials Aðq1 Þ and Bðq1 Þ have stable common factors, the control parameters are obtained from a similar equation to (11) with the Sylvester determinant of A0 ðq1 Þ and B0 ðq1 Þ being strictly positive.
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561
2.2. Case of unknown plant The transfer function of the system (1) can be written as yðq1 Þ=uðq1 Þ ¼ W ðq1 Þ ¼ W0 ðq1 Þð1 þ mD1 ðq1 ÞÞ þ mD2 ðq1 Þ, where W0 ðq1 Þ ¼ Bðq1 Þ=Aðq1 Þ is the modeled part of the transfer function of the true plant, mD1 ðq1 Þ and mD2 ðq1 Þ are the transfer functions of the multiplicative and additive unmodeled dynamics, respectively, and m is a positive constant. D1 ðq1 Þ and D2 ðq1 Þ must be both exponentially stable and strictly proper so that Assumption 1 below be feasible.
Assumption 1. There exist real constants r 2 ð0; 1Þ, a0 P 0 and a P 0, and a constant vector v, which are known, such that jgðkÞj 6 gðkÞ ¼ aqðkÞ þ a0
ð15Þ
for all integer k P 0
where 0
qðkÞ ¼ sup fjvT xðk 0 Þjrkk g;
xðkÞ
0 6 k0 6 k
¼ ½ eðk 1Þ
eðk nÞ
uðk 1Þ
T
uðk nÞ
ð16Þ
for all integer k P 0, where eðkÞ ¼ yðkÞ ym ðkÞ is the tracking error with ym ðkÞ ¼ Wm ðq1 Þr ðkÞ and r ðkÞ being any arbitrary bounded external input sequence.
Remark 3. Assumption 1 is invoked in the design of a relative adaptation dead-zone for robustness purposes in the parameter estimation shown in Section 3. The width of the dead-zone is governed by the overbounding normalized function gn ðkÞ ¼ gðkÞ=ð1 þ k/ðkÞkÞ. This assumption is fulfilled by any system in which the signal gðkÞ is the sum of a bounded term plus a term related to uðkÞ by a strictly proper exponentially stable transfer function, [17]. The structure of the control law is maintained as in (5) while replacing the control parameters K, ri and si , for i 2 f0; . . . ; n 1g, by KðkÞ, ri ðkÞ and si ðkÞ, respectively, for all integer k P 0. The equations to obtain the control parameters are those of (7) and (10) substituting the plant parameters of the vector h by their corresponding estimates of hðkÞ ¼ T ½ b0 ðkÞ bn ðkÞa1 ðkÞ an ðkÞ for all integer k P 0. Such estimates are obtained from an estimation and modification algorithm which ensures the controllability of the estimated and modified model at all sampling instants, i.e., jDetðMðhðkÞÞÞj P d0 > 0 for all integer k P 0, as it will be shown in Section 3.
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3. Adaptive control If the plant parameters are unknown, an estimation algorithm has to calculate a controllable estimation plant model. Two alternative algorithms are presented which include a hysteresis-free switching function.
3.1. Estimation and modification algorithms A least-square estimation algorithm with normalization and a relative adaptation dead-zone is used to obtain an Ôa prioriÕ estimation of the plant parameters in Step 1. Then, a suitable modification is applied to such Ôa prioriÕ estimates to obtain a controllable Ôa posterioriÕ estimated model of the plant in Step 2. Step 1 (‘A priori’ estimation) The Ôa prioriÕ plant parameter estimates are obtained from the recursive equations ^ þ 1Þ ¼ hðkÞ ^ þ hðk
sðkÞP ðkÞ/n ðkÞen ðkÞ ; cðkÞ þ sðkÞ/Tn ðkÞP ðkÞ/n ðkÞ
P ðk þ 1Þ ¼ P ðkÞ
sðkÞP ðkÞ/n ðkÞ/Tn ðkÞP ðkÞ cðkÞ þ sðkÞ/Tn ðkÞP ðkÞ/n ðkÞ
ð17aÞ
where en ðkÞ ¼
eðkÞ ; 1 þ k/ðkÞk
^ eðkÞ ¼ yðkÞ /T ðkÞhðkÞ
ð17bÞ
are the normalized identification error and the identification error, respectively, ^ ¼ ½ b^ ðkÞ b^ ðkÞ a^ ðkÞ a^ ðkÞ T and hðkÞ ~ ¼ hðkÞ ^ h are the hðkÞ 0 n 1 n a priori estimation vector of the plant parameters and the parametrical error, respectively, /n ðkÞ ¼ /ðkÞ=ð1 þ k/ðkÞkÞ, P ð0Þ ¼ P T ð0Þ > 0 and cðkÞ is a real value sequence chosen such that 0 < c1 6 cðkÞ 6 c2 < 1, for all integer k P 0, with c1 and c2 being some design real constants. The sequence sðkÞ is the relative adaptation dead-zone defined as ( sðkÞ ¼
1=2 T ðkÞ/n ðkÞ 0 if wðkÞ 6 l 1 þ /n ðkÞPcðkÞ gn ðkÞ f ðkÞ=wðkÞ otherwise
where wðkÞ ¼ ðe2n ðkÞ þ /Tn ðkÞP 2 ðkÞ/n ðkÞÞ1=2 , and
ð17cÞ
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563
f ðkÞ 8 1=2 > /Tn ðkÞP ðkÞ/n ðkÞ > > > if wðkÞ 6 l 1 þ gn ðkÞ > /T ðkÞP ðkÞ/n ðkÞ > > gn ðkÞ otherwise : wðkÞ l 1 þ n cðkÞ
ð17dÞ with l > 1. ~ It will be then proved in Lemma 3.1 below that kP 1 ðkÞhðkÞk is bounded for all integer k P 0 and at the limit as k ! 1. Thus, a bounded vector sequence b ðkÞ 2 R2nþ1 exists such that the true plant parameter vector can be written ^ þ P ðkÞb ðkÞ. This fact provides the motivation to obtain the Ôa as h ¼ hðkÞ posterioriÕ modified estimates of the plant parameters in Step 2 below. Step 2 (‘A posteriori’ modified estimation) Two alternative algorithms are designed: (1) First modification algorithm (i) The modified estimates are calculated from the modification rule ^ þ P ðkÞbðkÞ hðkÞ ¼ hðkÞ
ð18Þ
where hðkÞ ¼ ½ b0 ðkÞ bn ðkÞ a1 ðkÞ an ðkÞ T is the Ôa posterioriÕ estimation vector of the plant parameters and bðkÞ 2 R2nþ1 is a switching sequence which takes values being equal to one of the constant vectors belonging to a finite set as which is computed in (ii) below: ð1Þ (ii) Let e0 be a zero ð2n þ 1Þ-dimensional vector. Let fei g be ð2n þ 1Þ-dimensional vectors with only one non-zero element which equalizes either +1 or ðjÞ to )1, for i 2 f1; . . . ; 2ð2n þ 1Þg. Similarly, let fei g be ð2n þ 1Þ-dimensional vectors with j non-zero elements, each of which equalizes either to +1 or to )1, for i 2 f1; . . . ; ½ð2n þ 1Þ!=ðð2n þ 1 jÞ!j!Þ2j g. The total number of such vectors is L¼
2X nþ1 j¼0
ð2n þ 1Þ! 2j ð2n þ 1 jÞ!j!
Denote these vectors by fej g, with j 2 ZL f0; 1; . . . ; L 1g, in the increasing order of the number of non-zero elements in ej . We now precisely define the sequence bðkÞ. Consider a real number 0 < d 1 and let the initial condition of bð0Þ be de0 . At each sampling instant, bðkÞ is defined by ^ þ P ðkÞbðk 1ÞÞÞj P d0 bðk 1Þ if jDetðMðhðkÞ bðkÞ ¼ ð19Þ dej0 otherwise
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for all integer k P 1, where d0 is a small real positive constant and ej0 is obtained by searching from the vectors fej g the smaller j0 2 ZL such that ^ þ P ðkÞdej ÞÞj P d0 . jDetðMðhðkÞ 0 (2) Second modification algorithm (i) The modified estimates are calculated from the modification rule (18), where bðkÞ ¼ pðkÞb0 ðkÞ, with pðkÞ 2 R being a scalar switching sequence and b0 ðkÞ 2 R2nþ1 a vector sequence. (ii) At each sampling instant, the components b0i ðkÞ for i 2 f1; 2; . . . ; 2n þ 1g, of the vector sequence b0 ðkÞ, are calculated from the following equations b0i ðkÞ ¼
DetðPi0 ðkÞÞ DetðP ðkÞÞ
ð20Þ
where Pi0 ðkÞ has 2n columns identied to those of P ðkÞ while the ith column is replaced with T nþ1 z}|{ v¼ : 0 0 0 0 0 1 (iii) The switching sequence pðkÞ is zero for k ¼ 0 and for integers k > 0 it is defined as follows, pðkÞ ¼
pðk 1Þ p0 ðkÞ
^ þ pðk 1ÞP ðkÞb0 ðkÞÞÞj P d0 if jDetðMðhðkÞ otherwise
ð21Þ
for some small real positive constant d0 . i.e., expression (21) implies that pðkÞ maintains the value that it had at the previous sampling instant if the estimated and modified plant model obtained with such an aforementioned value is controllable. Otherwise, the function pðkÞ switches from the value pðk 1Þ to a new value p0 ðkÞ, which depends on the conmutation sampling instant, to avoid non-controllable plant estimated models. Besides, it is convenient that the absolute value of p0 ðÞ be as small as possible so that parameters modification be smooth. Because of this, p0 ðÞ is obtained from the following algorithm: Algorithm to compute p0 ðÞ at the conmutation sampling instants This algorithm is splitted into a set of elementary steps: ^ þ p0 P ðkÞb0 ðkÞÞÞj and go to Step 2. Step 1: Set p0 ¼ 0, compute jDetðMðhðkÞ 0 ^ Step 2: If jDetðMðhðkÞ þ p0 P ðkÞb ðkÞÞÞj P d0 then go out, otherwise go to Step 3. Step 3: Increase the value of p0 by the operation p0 ¼ p0 þ Dp0 , with ^ þ p0 P ðkÞb0 ðkÞÞÞj and go to 0 < Dp0 1, compute jDetðMðhðkÞ Step 4.
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565
^ þ p0 P ðkÞb0 ðkÞÞÞj P d0 then set p1 ¼ p0 and go to Step 5, Step 4: If jDetðMðhðkÞ otherwise go to Step 3. Step 5: Set p0 ¼ 0 and go to Step 6. ^ þ Step 6: Decrease the value of p0 via p0 ¼ p0 Dp0 , compute jDetðMðhðkÞ 0 p0 P ðkÞb ðkÞÞÞj and go to Step 7. ^ þ p0 P ðkÞb0 ðkÞÞÞj P d0 then go to Step 8, otherwise go to Step 7: If jDetðMðhðkÞ Step 6. Step 8: If jp1 j 6 jp0 j then set p0 ¼ p1 , and go out.
Remark 4. From (17b) and (18), yðkÞ ¼ /T ðkÞhðkÞ þ eðkÞ bT ðkÞP ðkÞ/ðkÞ ¼ /T ðkÞhðkÞ þ ea ðkÞ
ð22Þ
is obtained where ea ðkÞ ¼ eðkÞ bT ðkÞP ðkÞ/ðkÞ is referred to as ‘a posteriori’ identification error. Such a model has uðkÞ and yðkÞ as input and output, respectively, and ea ðkÞ as an external disturbance. The controllability condition of the plant estimated model is jDetðMðhðkÞÞÞj > 0 where MðhðkÞÞ has the same structure as MðhÞ, in Eq. (13) of Section 2, by replacing the components of the true parameters vector h with the corresponding one of their modified estimates hðkÞ through (18). 3.2. Adaptive control law The control law is based in (5) and becomes: uðkÞ ¼ KðkÞym ðkÞ Rðq1 ; kÞuðkÞ Sðq1 ; kÞyðkÞ ¼
KðkÞ Sðq1 ; kÞ ym ðkÞ yðkÞ 1 1 þ Rðq ; kÞ 1 þ Rðq1 ; kÞ
ð23Þ
where the control parameters KðkÞ, ri ðkÞ and si ðkÞ (coefficients of the timevarying polynomials Rðq1 ; kÞ and Sðq1 ; kÞ), for i 2 f0; . . . ; n 1g, are calculated by similar equations to (7) and (10). In those equations, the parameters of the polynomials Aðq1 Þ and Bðq1 Þ must be substituted by their estimated and modified ones of Aðq1 Þ and Bðq1 Þ, respectively. The adaptive controller parameters can be obtained through a similar equation to (11). 3.3. Properties of the adaptive control system The convergence and stability schemeÕs properties are given in the following results. Step 1 of the estimation algorithm posses the properties given in the following lemma:
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Lemma 3.1 (i) P ðkÞ is symmetrical for all integer k P 0, uniformly bounded and it asymptotically converges to a finite (at least semidefinite positive) limit as k tends to infinity. ^ (ii) khðkÞk < 1, gn ðkÞ < 1, jgn ðkÞj < 1, jen ðkÞj < 1 and wðkÞ < 1 for all integer k P 0. Pk Pk Pk (iii) f ðkÞ < 1, sðkÞw2 ðkÞ < 1, i¼0 f 2 ðiÞ < 1, i¼0 f ðiÞ < 1 and i¼0 sðiÞ w2 ðiÞ < 1 for all integer k P 0, limk!1 f 2 ðkÞ ! 0, limk!1 f ðkÞ ! 0 and limk!1 sðkÞw2 ðkÞ ! 0: ^ hðk ^ 1Þk < 1 for all integer k P 1 and limk!1 khðkÞ ^ hðk ^ (iv) khðkÞ 1Þk ! 0. (v) 0 6 sðkÞ < 1, kP ðkÞk > 0 and sðkÞe2n ðkÞ < 1 for all integer k P 0, limk!1 sðkÞ ! 0 and limk!1 sðkÞe2n ðkÞ ! 0 ~ (vi) kP 1 ðkÞhðkÞk < 1 for all integer k P 0. The proof is given in Appendix A. Note that Property (v) ensures the nonsingularity of the matrix P ðkÞ, for all integer k P 0. In this way, the plant parameters are estimated from (17a)–(17d) until wðkÞ be smaller than lð1 þ ð/Tn ðkÞP ðkÞ/n ðkÞÞ=cðkÞÞ1=2 gn ðkÞ, provided that /ðkÞ does not become or it is closed to a constant vector prior to the convergence of the plant estimates to their limits. Step 2 applies, independent of the computation method of bðkÞ: Lemma 3.2 (i) There exists a finite number of switches in bðkÞ, it is bounded, for all integer k P 0, and asymptotically converges. (ii) hðkÞ is a bounded vector sequence, for all integer k P 0, and asymptotically converges to a finite limit. (iii) The control parameters KðkÞ, ri ðkÞ and si ðkÞ, for all integer k P 0 and i 2 f0; 1; . . . ; n 1g, are bounded sequences and asymptotically converge to finite limits. (iv) ean ðkÞ ¼ ea ðkÞ=ð1 þ k/ðkÞkÞ verifies lim
k!1
e2an ðkÞ
/Tn ðkÞP ðkÞ/n ðkÞ 2 2bmax ðkÞl 1 þ gn ðkÞ 6 0 cðkÞ 2
2
where bmax ðkÞ ¼ maxf1; kbðkÞk g, for all integer k P 0. The proof is given in Appendix B. Remark 5. In view of Property (iv) of Lemma 3.2, the normalized Ôa posterioriÕ identification error ean ðkÞ belongs to the residual set defined below
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
567
9 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi = < T / ðkÞP ðkÞ/ ðkÞ n l lim fgn ðkÞg De ¼ ean lim fean ðkÞg 6 2 lim bmax ðkÞ 1 þ n k!1 k!1 ; : k!1 cðkÞ
If gfn ðkÞ converges to zero, which can occur with a0 ¼ 0 in (15) if qðkÞ converges to zero, then, the residual set De converges to the zero equilibrium.
4. Stability analysis Eqs. (22) and (23) lead to the following time-varying 2n-auxiliary system xðkÞ ¼ Ac ðk 1Þxðk 1Þ þ B1 #1 ðkÞ þ B2 #2 ðkÞ
ð24aÞ
with xðk 1Þ ¼ ½ eðk 1Þ
eðk nÞ uðk 1Þ
uðk nÞ
T
ð24bÞ
#1 ðkÞ ¼ ½b0 ðk 1ÞðKðk 1Þ s0 ðk 1ÞÞ 1 r0 ðk 1Þym ðkÞ
n1 X
ðb0 ðk 1Þsi ðk 1Þ þ ð1 þ r0 ðk 1ÞÞai ðk 1ÞÞym ðk iÞ
i¼1
an ðk 1Þð1 þ r0 ðk 1ÞÞym ðk nÞ þ ð1 þ r0 ðk 1ÞÞea ðkÞ #2 ðkÞ ¼ Kðk 1Þym ðkÞ þ
n1 X
ðs0 ðk 1Þai ðk 1Þ si ðk 1ÞÞym ðk iÞ
i¼1
s0 ðk 1Þan ðk 1Þym ðk nÞ þ s0 ðk 1Þea ðkÞ ð24cÞ Ac ðk 1Þ ¼ 2 b0 s1 ð1 þ r0 Þa1 6 6 1 6 6 0 6 6 6 6 6 6 6 6 6 6 0 6 6 6 s0 a1 s1 6 6 0 6 6 6 0 6 6 6 6 6 6 6 4 0
b0 s2 ð1 þ r0 Þa2
ð1 þ r0 Þan
ð1 þ r0 Þb1 b0 r1
ð1 þ r0 Þbn
0
0
0
0
1
0
0
0
0
1
s0 a2 s2
0
0
0
s0 an
ðs0 b1 þ r1 Þ
s0 bn
0
0
1
0
0
0
0
1
0
0
0
0
1
0
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
ð24dÞ
568
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
BT1
¼ ½1
0
0 ;
BT2
¼
nþ1
0
0
z}|{ 1
0
ð24eÞ
First, note that Ac ðk 1Þ is uniformly bounded from Lemma 3.2 (Properties (ii) and (iii)). On the other hand, the number of switches of the sequence bðÞ is finite. This fact implies that the time interval between two consecutive switches is finite. Besides, the control parameters are bounded at all sampling instants since the controllability of the modified estimation plant is ensured and the closed-loop modified estimation model posses a stable dynamics. Then, there is no finite escape between the initial time instant and the sampling instant k1 T at which the last switch in bðÞ occurs. Thus, one can redefine the time origin as k1 T and study the system stability for k P k1 . The following theorem establishes the main result of the convergence analysis. Theorem 4.1 (Main result). The adaptive control law stabilizes the plant (1) in the sense that uðkÞ, eðkÞ and yðkÞ are bounded for all finite initial states and any bounded reference sequence r ðkÞ, subject to Assumption 1, provided that a in (15) is sufficiently small so that 1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0Þg ca supk1 6 k0 6 k 2bmax ðk 0 Þ 1 þ kmax fP lakvk c1 40
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569
for all integer k P 0 where T ¼ 0:01s, are considered. The following parameters are chosen in (15) for the computation of the overbounding function for the unmodeled dynamics: a ¼ 105 , a0 ¼ 0 and r ¼ 0:5. Also, v ¼ ½ 0:1 0:2 0:3 0:1 T in (16), the sequence cðkÞ ¼ 4 for all integer k P 0 in (17a) and l ¼ 1:0001 in (17c) and (17d), and 3 2 1 0:4 0:4 0:4 0:4 6 0:4 1 0:4 0:4 0:4 7 7 6 7 P ð0Þ ¼ 55 6 6 0:4 0:4 1 0:4 0:4 7 4 0:4 0:4 0:4 1 0:4 5 0:4 0:4 0:4 0:4 1 ^ ¼ ½ 1:3 0:7 0:5 1:2 0:9 are the initial conditions of the and hð0Þ estimation algorithm with d0 ¼ 0:001. The parameter d ¼ 105 for the first modification algorithm and Dp0 ¼ 104 for the second modification algorithm are chosen. The control objective is defined by the Hurwitz polynomial Cðq1 Þ ¼ 1 þ 0:4q1 0:25q2 0:1q3 . The simulation results are shown in Figs. 1–8. T
(i) Results with the first modification algorithm
80
60
40
ε
20
0
-20
-40 0
5
10
15
20
25
30
35
40
t
Fig. 1. Tracking error signal in the interval [0,50].
45
50
570
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583 100
80
60
40
u 20
0
-20
-40 0
5
10
15
20
25
30
35
40
45
50
t
Fig. 2. Control signal in the interval [0,50].
Fig. 3. Parameters of the plant estimated and modified model in the interval [0,10].
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
Fig. 4. Components of the vector ej0 in the interval [0,0.5].
(ii) Results with the second modification algorithm
20
15
10
5
ε 0
-5
-10
-15 0
5
10
15
20
25
30
35
40
t
Fig. 5. Tracking error signal in the interval [0,50].
45
50
571
572
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
30
20
10
u
0
-10
-20
-30 0
5
10
15
20
25 t
30
35
40
45
50
Fig. 6. Control signal in the interval [0,50].
Fig. 7. Parameters of the plant estimated and modified model in the interval [0,10].
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
573
0.15
0.1
0.05
0 π -0.05
-0.1
-0.15
-0.2
0
0.2
0.4
0.6
0.8
1 t
1.2
1.4
1.6
1.8
2
Fig. 8. Evolution of pðtÞ in the interval [0,2].
Remark 6. Note that Figs. 1, 2, 5 and 6 exhibit bounded jumps in the tracking error and control signals. These jumps occur at the sampling instants at which the sequence bðkÞ changes its value. The modified estimated parameters present also bounded jumps at the same sampling instants at which the sequence bðkÞ changes its value. The presence of such jumps is crucial to ensure the controllability of the estimated and modified model of the plant when the unmodified one losses (or it is close to lose) its controllability. Note also that the numerator and denominator polynomials of the modeled plant present the common factor ð1 0:3q1 Þ. I.e., the controllability condition for the nominal plant is not fulfilled. However, the adaptive system becomes stabilizable since the estimated model of the plant is controllable thanks to the modification algorithm. Note that bðkÞ switches a finite number of times before converging for both modification algorithms.
6. Conclusions An adaptive pole-placement based control algorithm has been presented that stabilizes an, in general, inversely unstable discrete-time system in the presence of unmodeled dynamics and, eventually, bounded noise. The algorithm includes the use of a relative adaptation dead-zone for the closed-loop
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robust adaptive stabilization. The controllability of the estimated model is ensured by incorporating an appropriate modification in the standard leastsquares estimation algorithm which is implemented as the first estimation level. The boundedness of the tracking error and all the remaining signals in the closed-loop system is ensured without providing the controllability of the nominal plant.
Acknowledgements The authors are very grateful to the Basque Country University UPV/EHU and MCYT by their support through projects 1/UPV/EHU 00I06.I06-EB8235/2000 and DPI 2000-0244, respectively.
Appendix A. Proof of Lemma 3.1 (i) The matrix P ð0Þ is chosen symmetrical and P ðk þ 1Þ P ðkÞ is also symmetrical from (17a) for all integer k P 0. Then, P ðkÞ is a monotonic nonincreasing symmetrical matrix sequence since P ðk þ 1Þ P ðkÞ P 0 for all integer k P 0 from Eq. (17a). Thus, if P ðk1 Þ ¼ 0 for any integer k1 P 0, then, P ðk1 þ 1Þ P ðk1 Þ ¼ 0 from (17a) and then, P ðkÞ ¼ 0 for all integer k P k1 . Consequently, P ðkÞ is upper bounded by P ð0Þ and lower bounded by zero, i.e., 0 6 P ðkÞ 6 P ð0Þ, and it also converges asymptotically to a finite limit as k tends to infinity because of it is monotonic non-increasing matrix sequence. ~ þ trðP ðkÞÞ (ii) Consider the non-negative sequence V ðkÞ ¼ h~T ðkÞP 1 ðkÞhðkÞ for all integer k P 0. The matrix inversion lemma, [18], applied to the second equation of (17a) yields P 1 ðk þ 1Þ ¼ P 1 ðkÞ þ
sðkÞ / ðkÞ/Tn ðkÞ cðkÞ n
ðA:1Þ
From Eqs. (17a)–(17d), (15) and (A.1), it follows that sðkÞw2 ðkÞ sðkÞ 2 gn ðkÞ þ T cðkÞ þ sðkÞ/n ðkÞP ðkÞ/n ðkÞ cðkÞ 2 l 1 f 2 ðkÞ 6 2 l cðkÞ þ sðkÞ/Tn ðkÞP ðkÞ/n ðkÞ
V ðk þ 1Þ V ðkÞ ¼
ðA:2Þ
where sðkÞ 2 ½0; 1Þ and sðkÞw2 ðkÞ ¼ f ðkÞwðkÞ P f 2 ðkÞ have been used. From (A.1) it follows that kmin ðP 1 ðkÞÞ P kmin ðP 1 ð0ÞÞ
ðA:3Þ
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575
for all integer k P 0. Eq. (A.2) leads to ~ þ tr P ðkÞ 6 h~T ð0ÞP 1 ð0Þhð0Þ ~ þ tr P ð0Þ V ðkÞ 6 V ð0Þ () h~T ðkÞP 1 ðkÞhðkÞ ðA:4Þ for all integer k P 0, and then 2 ~ kmax ðP 1 ð0ÞÞkhð0Þk þ tr P ð0Þ 2 2 ~ ~ P kmin ðP 1 ð0ÞÞkhðkÞk þ tr P ðkÞ () khðkÞk
6
kmax ðP ð0ÞÞ ~ tr P ð0Þ tr P ðkÞ khð0Þk2 þ l2 ð1 þ ð/Tn ðkÞP ðkÞ/n ðkÞÞ=cðkÞÞg2n ðkÞ is fulfilled. Then, (A.11) implies since sðiÞ 2 ½0; 1Þ: h i /T ðiÞP ðiÞ/ ðiÞ k X X sðiÞ w2 ðiÞ 1 þ n cðiÞ n g2n ðiÞ M2 ðkÞ ¼ M1 ðkÞ < 1 sðiÞ 6 cðiÞ þ sðiÞ/Tn ðiÞP ðiÞ/n ðiÞ i¼0 i2I2 i6k
ðA:12Þ with M2 ðkÞ inf i2I2 i6k
h i 8 9 T ðiÞ/n ðiÞ < w2 ðiÞ 1 þ /n ðiÞPcðiÞ g2n ðiÞ = :
cðiÞ þ sðiÞ/Tn ðiÞP ðiÞ/n ðiÞ
;
>0
for all integer k P 0. Then, limk!1 sðkÞ ! 0. Thus, it follows from (A.1) that P 1 ðk þ 1Þ ¼ P 1 ð0Þ þ
k X sðiÞ /n ðiÞ/Tn ðiÞ cðiÞ i¼0
ðA:13Þ
From (A.13) and the facts that kP ð0Þk > 0, /n ðkÞ is bounded, cðkÞ > 0 for all integer k P 0 and limk!1 sðkÞ ! 0, it follows that kP 1 ðkÞk < 1, kP ðkÞk > 0
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577
for all finite k and as k ! 1. Finally, it follows, from the definition of wðkÞ in Eqs. (17), that k X
sðiÞe2n ðiÞ 6
i¼0
k X
sðiÞw2 ðiÞ
i¼0
inf
f/Tn ðiÞP 2 ðiÞ/n ðiÞg
i2f0;1;...;kg
k X
sðiÞ < 1
i¼0
ðA:14Þ Pk
2
Pk
where the boundedness of i¼0 sðiÞw ðiÞ, i¼0 sðiÞ, /n ðkÞ and P ðkÞ for all integer k P 0 have been used. Thus, 0 6 sðkÞe2n ðkÞ < 1 for all integer k P 0 and limk!1 sðkÞe2n ðkÞ ! 0. (vi) From (17a) and (A.1), one gets ~ kP 1 ðkÞhðkÞk ~ 1Þk þ sðk 1Þ k/ ðk 1Þkj/T ðk 1Þhðk ~ 1Þj 6 kP 1 ðk 1Þhðk n cðk 1Þ n 1 þ cðk 1Þ sðk 1Þk/n ðk 1Þkjen ðk 1Þj þ cðk 1Þ ðA:15Þ ~ þ g ðiÞ for all integer i P 0, it follows that From en ðiÞ ¼ /Tn ðiÞhðiÞ n T ~ j/n ðiÞhðiÞj 6 jen ðiÞj þ jgn ðiÞj. By introducing this expression in (A.15), one obtains ~ kP 1 ðkÞhðkÞk ~ 1Þk þ sðk 1Þ k/ ðk 1Þkðjg ðk 1Þj 6 kP 1 ðk 1Þhðk n cðk 1Þ n þ ½2 þ cðk 1Þjen ðk 1ÞjÞ 1 ~ k/n ðiÞkðjgn ðiÞj þ ½2 þ cðiÞjen ðiÞjÞ þ sup 6 kP 1 ð0Þhð0Þk cðiÞ 0 6 i 6 k1 k1 X sðiÞ < 1 ðA:16Þ i¼0
for all integer k P 0 since jen ðkÞj < 1 and 0 < cðkÞ < 1.
Pk1 i¼0
sðiÞ < 1, k/n ðkÞk < 1, jgn ðkÞj < 1,
Appendix B. Proof of Lemma 3.2 (i) These properties have to be proved for both the first modification algorithm and the second modification algorithm. For the first modification algorithm, it has been proved in [6] that there exits ^ þ P ðkÞdei ÞÞj P a positive sufficient small real constant d0 such that jDetðMðhðkÞ d0 for at least one of the vectors ei at each sampling instant, for
578
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
^ i 2 f0; . . . ; L 1g. Thus, from the fact that P ðkÞ and hðkÞ are bounded and converge (see Properties (i), (ii) and (iv) of Lemma 3.1) and from (19), it follows that the number of switches in bðkÞ is finite and then bðkÞ converges asymptotically. Besides, bðkÞ is uniformly bounded since it equates to the product of d with one of the aforementioned bounded constant vectors ei at each sampling instant. For the second modification algorithm, one has to prove that there exits a bounded sequence pðkÞ such that jDetðMðhðkÞÞÞj P d0 > 0, where the matrix defined as MðhðkÞÞ is: MðhðkÞÞ ¼ 2 1 0 6 a1 1 6 6 6 .. .. 6 . . 6 6 an1 an2 6 6 6 an an1 6 6 6 0 an 6 6 .. .. 6 . . 6 6 6 0 0 6 n n 4 P P 1 þ ai 1 þ ai i¼1
.. . .. .
0 0 .. . 1 a1 a2 .. .
b0 b1 .. . bn1 bn 0 .. .
0 b0 .. . bn2 bn1 bn .. .
.. . .. .
0 0 .. . b0 b1 b2 .. .
a nn P 1 þ ai
0 n P bi
0 n P bi
bn n P bi
i¼0
i¼0
i¼1
i¼1
i¼0
0 0 .. . 0 0 0 .. .
3
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 0 7 7 n 5 P bi i¼0
ðB:1Þ 0
From (18) and the expression bðkÞ ¼ pðkÞb ðkÞ, the vector hðkÞ can be written as: ^ þ pðkÞP ðkÞb0 ðkÞ hðkÞ ¼ hðkÞ
ðB:2Þ 0
where the components of the vector sequence b ðkÞ are calculated by means of (20). i.e., b0 ðkÞ is the solution of the matrix equation: P ðkÞb0 ðkÞ ¼ v
ðB:3Þ
where v¼
T
nþ1
0
0
0
z}|{ 1
0
0
0
2 R2nþ1 :
That is, the components of b0 ðkÞ are such that, at each sampling instant: piT ðkÞb0i ðkÞ ¼ 0 T pnþ1 ðkÞb0nþ1 ðkÞ
for i 2 f1; . . . ; ng \ fn þ 2; . . . ; 2n þ 1g; ¼1
ðB:4Þ
where pi ðkÞ denotes the i-column of the matrix P ðkÞ. By introducing (B.2) and (B.4) into (B.1), it follows that:
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583 MðhðkÞÞ ¼ 2 1 6 a^1 6 6 6 .. 6 . 6 6 a^n1 6 6 6 a^n 6 6 0 6 6 6 .. 6 . 6 6 6 0 6 n 4 P 1 þ a^i
0
0
1 .. .
.. .
0 .. . 1 a^1
a^n2 a^n1 a^n ... 0 n P 1 þ a^i
i¼1
.. .
b^0 b^1 .. . b^n1 b^n þ p 0 .. .
a^2 ... a^n 0 n nP 1 P 1 þ a^i b^n p b^i
i¼1
i¼1
0 b^0 .. . b^n2
0
.. .
0 .. . b^0
b^n1 b^n þ p .. .
.. .
b^1 b^2 .. .
0 b^n p
i¼0
nP 1
b^i
b^n þ p
b^n p
i¼0
579 3
0
7 7 7 7 7 7 7 7 7 7 0 7 7 0 7 7 7 .. 7 . 7 7 7 0 7 nP 1 5 b^n þ p þ b^i 0 .. . 0
nP 1
b^i
i¼0
i¼0
ðB:5Þ of determinant ^ DetðMðhðkÞÞÞ ¼ ðb^n ðkÞ þ pðkÞÞnþ1 þ gðhðkÞ; pðkÞÞ ¼ pðkÞ
nþ1
^ þ g0 ðhðkÞ; pðkÞÞ
ðB:6Þ
^ ^ where both gðhðkÞ; pðkÞÞ and g0 ðhðkÞ; pðkÞÞ are n-order polynomials in pðkÞ. From (B.6), the controllability condition of modified estimation plant model is: jpðkÞ
nþ1
^ þ g0 ðhðkÞ; pðkÞÞj P d0
ðB:7Þ
^ Note that function g0 ðhðkÞ; pðkÞÞ is a sum of a finite number of terms of the c0 ^ ci c ^ form pðkÞ hi ðkÞ . . . hj ðkÞ j for some non-negative integers c0 ; ci ; . . . ; cj , and ^ where h^i ðkÞ and h^j ðkÞ denote some components of the vector hðkÞ. Then, 2nþ1 0 ^ ^ g ðhðkÞ; pðkÞÞ is analytic for all pðkÞ 2 R and for all hðkÞ 2 R . Then, the expression DetðMðhðkÞÞÞ is a ðn þ 1Þ-order polynomial in p, with all coefficients well defined at all sampling instants. Thus, there exists some absolute bounded value for pðkÞ which verifies the controllability condition (B.7). Besides, the vector sequence b0 ðkÞ is uniformly bounded and it asymptotically converges from (20) and the boundedness and asymptotic convergence of P ðkÞ [Lemma 3.1, (i) ]. Then, pðkÞ asymptotically converges to a bounded constant from (21) ^ [Lemma 3.1, and the uniform boundedness and asymptotic convergence of hðkÞ (ii) and (iv)]. Thus, bðkÞ is uniformly bounded, has a finite number of switches and asymptotically converges. ^ are bounded. (ii) It follows directly from (i) since P ðkÞ and hðkÞ (iii) It follows directly since khðkÞk < 1 for all integer k P 0, hðkÞ converges to a limit as k ! 1 and jDetðMðhðkÞÞÞj P d0 > 0 for all integer k P 0. (iv) Since ean ðkÞ ¼ en ðkÞ bT ðkÞP ðkÞ/n ðkÞ, one obtains from the relation 2 ða þ bÞ 6 2ða2 þ b2 Þ for any a; b 2 R: e2an ðkÞ 6 2ðe2n ðkÞ þ kbðkÞk2 /Tn ðkÞP 2 ðkÞ/n ðkÞÞ /T ðkÞP ðkÞ/n ðkÞ 2 ) lim 2bmax ðkÞ w2 ðkÞ l2 1 þ n gn ðkÞ 6 0 t!1 cðkÞ ðB:8Þ
580
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
since limk!1 sðkÞ ! 0 so that w2 ðkÞ 6 l2 ð1 þ ð/Tn ðkÞP ðkÞ/n ðkÞÞ=cðkÞÞg2n ðkÞ as k ! 1.
Appendix C. Proof of Theorem 4.1 Ac ðkÞ is bounded because of hðkÞ, and the controller parameters KðkÞ and the coefficients of Rðq1 ; kÞ and Sðq1 ; kÞ are bounded, from Lemma 3.2(ii)–(iii). The eigenvalues of Ac ðkÞ are in jzj < 1 for all integer k P 0. Besides, k X
2
kAc ðk 0 Þ Ac ðk 0 1Þk 6 b0 þ b1 ðk k0 Þ
ðC:1Þ
k 0 ¼k0 þ1
for all integers k and k0 such that k > k0 P 0, some positive constants b0 and b1 with b1 being sufficiently small. Note that (C.1) is fulfilled with a suitable constant b0 , which takes into account the changes of value of the entries of Ac ðÞ when a switch in the sequence bðÞ occurs, and a slow enough estimation rate via suitable P ð0Þ and cðkÞ in (17a) so that b1 is sufficiently small. Thus, the time-varying homogeneous system xðk þ 1Þ ¼ Ac ðkÞxðkÞ is exponentially stable Qk1 0 and its transition matrix wðk; k 0 Þ ¼ j¼k0 Ac ðjÞ satisfies kwðk; k 0 Þk 6 c1 rkk for 0 all k P k 0 where r0 2 ð0; 1Þ and c1 is a norm-dependent constant, [19,20]. We now redefine the time origin k1 T as the sampling instant such that for kT > k1 T there are not changes of value in bðkÞ. From (24a), if follows that
xðkÞ ¼ wðk; k1 Þxðk1 Þ þ
k X
wðk; k 0 Þ½B1 #1 ðk 0 Þ þ B2 #2 ðk 0 Þ
ðC:2Þ
k 0 ¼k1
for all integer k P k1 P 0. From (C.2), one obtains 1 kxðk1 Þk þ kxðkÞk 6 c1 rkk 0
k X
0
c1 rkk ðc2 þ c3 jea ðk 0 ÞjÞ 0
k 0 ¼k1
6 c4 þ
k X
0
c5 rkk jea ðk 0 Þj 0
ðC:3Þ
k 0 ¼k1
P2 since k i¼1 Bi #i ðkÞk 6 c2 þ c3 jea ðkÞj, for some positive real constants c2 , c3 , c4 and c5 , from (24c) and (24e), Schwarz’s inequality, r0 2 ð0; 1Þ, and the boundedness of KðÞ, b0 ðÞ, ai ðÞ, for i 2 f1; . . . ; ng, the sequence ym ðÞ and kxðk1 Þk. From (B.8), e2an ðkÞ 6 2bmax ðkÞw2 ðkÞ so that
S. Alonso-Quesada, M. De la Sen / Appl. Math. Comput. 150 (2004) 555–583
kxðkÞk 6 c4 þ
k X
0 c5 rkk 0
k 0 ¼k1
581
1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /Tn ðk 0 ÞP ðk 0 Þ/n ðk 0 Þ 0 2bmax ðk Þl 1 þ cðk 0 Þ
gn ðk 0 Þð1 þ k/ðk 0 ÞkÞ þ
k X
c5 rkk 0
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2bmax ðk 0 Þf ðk Þð1 þ k/ðk 0 ÞkÞ
k 0 ¼k1
ðC:4Þ where wðkÞ has been split into the two additive terms f ðkÞ and 1=2 /Tn ðkÞP ðkÞ/n ðkÞ wðkÞ f ðkÞ 6 l 1 þ gn ðkÞ cðkÞ from (17d). It follows from (15), (16) and (C.4) that ( kxðkÞk 6 c6 þ c5 lakvk sup
k1 6 k 0 6 k
sup fkxðk 0 Þkg k1 6 k 0 6 k
þ
k X
c5 rkk 0
0
1=2 ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /Tn ðk 0 ÞP ðk 0 Þ/n ðk 0 Þ 0 2bmax ðk Þ 1 þ cðk 0 Þ
1 r0kk1 þ1 1 r0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2bmax ðk 0 Þf ðk Þð1 þ k/ðk 0 ÞkÞ
ðC:5Þ
k 0 ¼k1
for some positive constant c6 , where kP ðkÞk < 1 and kvk supk1 6 k0 6 k fkxðk 0 Þkg P 0 supk1 6 k0 6 k fjvT xðk 0 Þjrkk g have been used. Besides, the right-hand side of (C.5) is monotonic non-decreasing in k. Then, for k k1 and provided that a
