Improving Adaptive Feedforward Vibration ... - IEEE Xplore
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Improving adaptive feedforward vibration compensation by using integral + proportional adaptation Ioan Dor´e Landau, Tudor-Bogdan Airimit¸oaie Abstract— Analysis of various adaptive feedforward vibration compensation schemes has shown that a strictly positive real (SPR) condition has to be satisfied in order to guarantee the stability of the whole system [1], [2], [3], [4]. Filters have to be implemented in order to satisfy this condition. The problem becomes even more crucial in the presence of the internal mechanical coupling between the compensator system and the reference source (a correlated measurement with the disturbance) since some information is not available when adaptation starts (see [3]). It is therefore very important to relax the SPR condition at least in the initial phase and in the same time to improve the adaptation transients. It is shown in this paper that adding a proportional adaptation to the standard integral type parametric adaptation, the SPR condition can be relaxed and the adaptation transients are improved. Theoretical developments are enhanced by real time experimental results obtained on an active vibration control (AVC) system. Index Terms— active vibration control, adaptive feedforward compensation, adaptive control, stability, parameter estimation
it has not been used in the context of adaptive feedforward compensation. Furthermore, it was observed in adaptive control, that adding positive proportional adaptation will speed up the adaption transients ([9], [8]). ”Integral + Proportional” (IP) adaptation has been discussed from a stability point of view in [9], [8] for the case of constant integral adaptation gain. A stable IP adaptation with time varying integral adaptation gain has been introduced in [10]. The objective of this paper is to explore the advantages of adding proportional adaptation in the context of the adaptive feedforward compensation of vibrations both from the theoretical and applications points of view. The main contributions of the present paper are: •
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I. I NTRODUCTION Analysis of various adaptive feedforward vibration compensation schemes has shown that a strictly positive real (SPR) condition has to be satisfied in order to guarantee the stability of the whole system [1], [2], [3], [4]. Therefore an important issue in adaptive feedforward compensation is the design of filters either on the observed variables of the feedforward compensator ([3]) or on the residual acceleration ([5]) in order to satisfy positive realness conditions on some transfer functions required by the stability analysis. The problem becomes even more crucial in the presence of the internal mechanical coupling between the compensator system and the reference source (a correlated measurement with the disturbance) since some information is not available when adaptation starts (see [3]). In [3], based on work done in [6], it was shown that for small adaptation gains (slow adaptation) violation of the SPR condition in some frequency regions is acceptable provided that in the average the input-output product associated with this transfer function is positive. However, the performances are degraded with respect to the case when the SPR condition is satisfied. It is in fact a signal dependent condition. The problem of removing or relaxing the positive real condition can be also approached by adding a proportional adaptation to the widely used integral adaptation. While this approach is known in adaptive control ([7], [8]) apparently I.D. Landau and T.B. Airimit¸oaie are with the Control System Department of Gipsa-lab, St. Martin d’H´eres, 38402 France (e-mail: [ioan-dore.landau, tudor-bogdan.airimitoaie]@gipsa-lab.grenoble-inp.fr).
978-1-4673-2066-5/12/$31.00 978-1-4673-2064-1/12/$31.00 ©2012 ©2012 IEEE IEEE
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Development and stability analysis of ”Integral + Proportional” adaptation algorithms for adaptive feedforward compensation in the presence (or not) of an internal positive feedback coupling. Extension of the results of [7] for relaxation of the SPR condition in the context of adaptive feedforward compensation using ”Integral + Proportional” adaptation compensation. Application of the ”Integral + Proportional” adaptation algorithms to an active vibration control system featuring internal positive mechanical coupling. Comparisons between the new algorithms and the existing algorithms.
While the paper is developed in the context of AVC systems, the results are certainly applicable to active noise control (ANC) systems. The paper is organized as follows. The basic equations describing the system are given in section II. The ”Integral + Proportional” adaptive feedforward compensation algorithms will be developed and analyzed in Section III. The problem of SPR relaxation will be discussed in Section IV. The AVC system on which the algorithms will be tested, is presented in Section V. Section VI will present experimental results obtained on the active vibration control system with the various algorithms. II. BASIC E QUATIONS AND N OTATIONS The block diagrams associated with an AVC system are shown in Fig.1 in open loop (1(a)) and with the adaptive feedforward compensator in the presence of an internal positive feedback (1(b)). For a physical insight see also Section V. s(t) is the disturbance and w(t) is the correlated measurement with the disturbance. The primary (D), secondary (G) and reverse (positive coupling) (M) paths
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Global primary path
while y(t ˆ + 1), y(t), ˆ . . . are the measurements provided by the primary transducer1 . The unmeasurable a priori output of the secondary path will be denoted
Primary path Residual acceleration measurement
Measurement of the image of the disturbance
B∗ (q−1 ) zˆ0 (t + 1) = zˆ(t + 1|θˆ (t)) = G −1 u(t). ˆ AG (q )
(a) Global primary path
The a posteriori unmeasurable value of the output of the secondary path is denoted by
Residual acceleration measurement Primary path
(7)
+
zˆ(t + 1) = zˆ(t + 1|θˆ (t + 1)).
Positive feedback coupling
(8)
+
-1 Feedforward compensator
+
The measured primary signal (called also reference) satisfies the equation
Secondary path
+ Measurement of the image of the disturbance
y(t ˆ + 1) = w(t + 1) +
PAA Parameter adaptation algorithm
(b)
B∗M (q−1 ) u(t). ˆ AM (q−1 )
(9)
The measured residual error satisfies
Fig. 1. Feedforward AVC: (a) in open loop, (b) with adaptive feedforward compensator.
e0 (t + 1) = e(t + 1|θˆ (t)) = x(t + 1) + zˆ0 (t + 1).
(10)
The a priori adaptation error is represented in Fig. 1(b) are respectively characterized by the asymptotically stable transfer operators bX1 q−1 + ... + bXnB q−nBX BX (q−1 ) X , X(q ) = = AX (q−1 ) 1 + aX1 q−1 + ... + aXnA q−nAX −1
(1)
ν 0 (t + 1) = ν(t + 1|θˆ (t)) = −e0 (t + 1).
The a posteriori adaptation (residual) error (which is computed) will be given by ν(t + 1) = ν(t + 1|θˆ (t + 1)) = −x(t + 1) − zˆ(t + 1).
X
BX = q−1 B∗X ,
ˆ Mˆ and Dˆ denote the with ∀ X ∈ {D, G, M}. G, identified (estimated) models of G, M and D. The optimal feedforward filter (unknown) which will minimize the residual acceleration in the sense of a certain criterion is defined by N(q−1 ) =
R(q−1 ) r0 + r1 q−1 + . . . + rnR q−nR = . S(q−1 ) 1 + s1 q−1 + . . . + snS q−nS
(2)
ˆ −1 ) or N( ˆ θˆ , q−1 ) when The estimated filter is denoted by N(q ˆ q−1 ) it is a linear filter with constant coefficients or N(t, during estimation (adaptation) of its parameters. The input of the feedforward filter is denoted by y(t) ˆ and it corresponds to the measurement provided by the primary transducer (force or acceleration transducer in AVC or a microphone in ANC). In the absence of the compensation loop (open loop operation) y(t) ˆ = w(t). The a priori output of the estimated feedforward filter is given by uˆ0 (t + 1) = u(t ˆ + 1|θˆ (t)) = θˆ T (t)φ (t)
(3)
III. D EVELOPMENT AND A NALYSIS OF THE A LGORITHMS The block diagram of the adaptive feedforward compensator in the presence of an internal positive feedback associated with an AVC system is shown in Fig. (1(b)). The description, equations and notations of the various blocks and transfer functions have been presented in detail in [3] eqs. (1) - (22). Using Lemma 4.1 and eqs. (26) - (30) from [3], the a posteriori adaptation error, ν(t + 1), which is computed, is given by ν(t + 1) =
AM (q−1 )G(q−1 ) [θ − θˆ (t + 1)]T φ f (t), P(q−1 )L(q−1 )
T
θ =[s1 , . . . snS , r0 , . . . rnR ] = (4)
φ (t) =[−u(t), ˆ . . . − u(t ˆ − nS + 1), y(t ˆ + 1), . . . y(t ˆ − nR + 1)] =[φuTˆ (t), φyˆT (t)]
u(t ˆ + 1) = u(t ˆ + 1|θˆ (t + 1)) = θˆ T (t + 1)φ (t),
[θST ,
θRT ],
(14) (15)
θˆ (t) and φ (t) given in (4) and (5) respectively and φ f (t) = L(q−1 )φ (t),
(5)
and u(t), ˆ u(t ˆ − 1), . . . are the a posteriori outputs of the feedforward filter generated by
(13)
with P(q−1 ) =AM (q−1 )S(q−1 ) − BM (q−1 )R(q−1 ),
T
(12)
When using an estimated filter Nˆ with constant parameters: uˆ0 (t) = u(t), ˆ zˆ0 (t) = zˆ(t), and ν 0 (t) = ν(t).
where θˆ T (t) =[sˆ1 (t), . . . sˆnS (t), rˆ0 (t), . . . rˆnR (t)] =[θˆST (t), θˆRT (t)],
(11)
where L(q−1 ) =
(6) 4413
1 y(t ˆ + 1)
BL (q−1 ) AL (q−1 )
(16)
will be detailed later.
is available before adaptation of parameters starts at t + 1.
Eq. (13) has the standard form for an a posteriori adaptation error ([8]). The following ”Integral + Proportional” parameter adaptation algorithm (IP-PAA) is proposed: θˆI (t + 1) = θˆI (t) + ξ (t)FI (t)Φ(t)ν(t + 1), θˆP (t + 1) = FP (t)Φ(t)ν(t + 1), ν(t + 1) =
(17a) (17b)
ν 0 (t + 1)
, (17c) 1 + ΦT (t)(ξ (t)FI (t) + FP (t))Φ(t) FI (t)Φ(t)ΦT (t)FI (t) 1 FI (t) − λ (t) FI (t + 1) = , (17d) 1 λ1 (t) + ΦT (t)FI (t)Φ(t)
last case the SPR condition is satisfied provided that one has good estimations of AM , G, and P (for more details see [3]). Neglecting the non-commutativity of time varying operators, one has the following result: Theorem 3.1: Assuming that eq. (20) represents the evolution of the a posteriori adaptation error and that the IP-PAA (17a) - (17j) is used, one has lim ν(t + 1) = 0,
FP (t) = α(t)FI (t); α(t) > −0.5,
(17e)
[ν 0 (t + 1)2 ] = 0, t→∞ 1 + Φ(t)T F(t)Φ(t) ||Φ(t)|| is bounded,
F(t) = ξ (t)FI (t) + FP (t), λ2 (t) T ξ (t) = 1 + Φ (t)FP (t)Φ(t), λ1 (t) θˆ (t + 1) = θˆI (t + 1) + θˆP (t + 1),
(17f)
lim ν 0 (t + 1) = 0,
Φ(t) = φ f (t),
where ν(t + 1) is the adaptation error2 , λ1 (t) and λ2 (t) allow to obtain various profiles for the matrix adaptation gain FI (t) (see [8] for more details). By taking λ2 (t) ≡ 0 one obtains a constant adaptation gain matrix (and choosing FI = γI, γ > 0 one gets a scalar adaptation gain). For α(t) ≡ 0, one obtains the algorithm with integral adaptation gain introduced in [3]. Three choices for the filter L will be considered, leading to three different algorithms: Algorithm II
L=G L = Gˆ
Algorithm III
L=
Algorithm I
Aˆ M ˆ G Pˆ
(18)
Pˆ = Aˆ M Sˆ − Bˆ M Rˆ
(19)
where is an estimation of the characteristic polynomial of the internal feedback loop computed on the basis of available ˆ To use Algorithm estimates for the parameters of the filter N. III one has to start with Algorithm II where the SPR condition is in general not satisfied. Therefore, relaxing the SPR condition for Algorithm II is very important. A. Analysis of the Algorithms For Algorithms I, II, and III the equation for the a posteriori adaptation error has the form (eqs. (29) and (30) from [3]) ν(t + 1) = H(q−1 )[θ − θˆ (t + 1)]T Φ(t), AM (q−1 )G(q−1 ) , Φ = φ f = L(q−1 )φ . H(q ) = P(q−1 )L(q−1 ) −1
Thus for Algorithm II one has H(q−1 ) = Algorithm III one obtains 2 ν 0 (t + 1)
H(q−1 ) =
AM GPˆ . PAˆ M Gˆ
is the measured a priori adaptation error
AM G PGˆ
H 0 (q−1 ) = H(q−1 ) −
(21) and for
Note that in this
λ2 , max λ2 (t) ≤ λ2 < 2. t 2
(26)
is a SPR transfer function. The proof of (22) is given in Appendix I. For (23), (24), and (25), the proof follows [11], [3] and it is omitted. The proof in [10] for IP adaptation with time varying integral adaptation gain is given for ξ (t) = λ 1(t) + λ2 (t) T λ1 (t) Φ (t)FP (t)Φ(t).
1
To the knowledge of the authors, the proof for ξ (t) given in eq. (17g) is presented here for the first time. IV. R ELAXING THE P OSITIVE R EAL CONDITION An equivalent feedback system can be associated to the IP-PAA where the feedforward path is characterized by the transfer function H 0 (z−1 ). There is additional ”excess” of passivity in the feedback path (which depends upon the magnitude of the adaptation gains and the magnitude of Φ(t)) which can be transferred to the linear feedforward block in order to relax the SPR condition. This idea was prompted out in the context of recursive identification by Tomizuka and results have been given for the case of integral adaptation with constant gains and when the equivalent linear feedforward path is characterized by an all poles (no zeros) transfer function (see [7]). These results have been extended in [8] for IP adaptation with constant adaptation gain. Taking into account the poles-zeros structure of H(z−1 ), the results of [7], [8] should be extended for the situation described in this paper. One has the following result: Lemma 4.1: Given the discrete transfer function H(z−1 ) =
(20)
where
(25)
for any initial conditions θˆ (0), ν 0 (0), F(0), provided that
(17i) (17j)
(24)
t→∞
(17h)
0 < λ1 (t) ≤ 1, 0 ≤ λ2 (t) < 2, FI (0) > 0,
(23)
lim
λ2 (t)
(17g)
(22)
t→∞
B(z−1 ) b0 + b1 z−1 + . . . + bnB z−nB = , A(z−1 ) 1 + a1 z−1 + . . . + anA z−nA
(27)
under the hypothesis: H1) H(z−1 ) has all its zeros inside the unit circle, H2) b0 6= 0, H there exists a positive scalar gain K such that 1+KH is SPR. The proof of this lemma is presented in II. Using the above property, for the IP-PAA given by the eqs. (17a) - (17j) and (20) for λ2 (t) ≡ 0, λ1 (t) ≡ 1 (constant
4414
adaptation gain) and choosing K such such that one gets the equivalent feedback system
H 1+KH
is SPR,
H(z−1 ) y2 (t), (28) 1 + KH(z−1 ) θ˜I (t) = θˆI (t) − θ , (29) θ˜I (t + 1) = θ˜I (t) + ξ (t)FI Φ(t)ν(t + 1), (30) T T y2 (t) = Φ (t)θ˜I (t) + (Φ (t)F(t)Φ(t) − K)ν(t + 1), (31) ν(t + 1) = −
To prove the stability it remains to show that the new feedback path given by (31) is passive, i.e., it satisfies the Popov inequality t1
∑ y2 (t)u2 (t) ≥ −γ02 .
(32)
t=0
bellow plate M3 and is used for disturbance rejection. Two accelerometers positioned as in figure 2 measure the image of the disturbance and the residual acceleration e0 (t). The corresponding block diagrams in open loop operation and with the compensator system are shown in Figs. 1(a) and 1(b). VI. E XPERIMENTAL RESULTS A. System identification The procedure for identifying the models of the various paths has been described in [3]. Their frequency characteristics are shown in Fig. 3. The model orders for the secondary path (solid line) and the reverse path (dotted line) have been estimated to be: nBG = 14, nAG = 14, and nBM = 14, nAM = 14 respectively. The primary path model has been used only for simulations. The models show the presence of several low
Amplitude [dB]
The following theorem provides the necessary result. Theorem 4.1: The adaptive system described by eq. (13) 40 Secondary path and eqs. (17a) - (17j) for λ2 (t) ≡ 0 and λ1 (t) ≡ 1 is Reverse path 20 asymptotically stable provided that: Primary path H 0 T1) It exists a gain K such that 1+KH is SPR, T2) The adaptation gains FI and FP (t) and the observation −20 vector Φ(t) satisfy −40 t1 1 T 2 −60 ∑ Φ (t − 1) 2 FI + FP (t − 1) Φ(t − 1) − K ν (t) ≥ 0 t=0 −80 (33) 0 50 100 150 200 250 300 350 400 Frequency [Hz] for all t1 ≥ 0 or 1 Fig. 3. Frequency characteristics of the primary, secondary and reverse T Φ (t) FI + FP (t) Φ(t) > K > 0, (34) paths. 2 damped pairs of complex poles and complex zeros. Note for all t ≥ 0. that the primary path features a strong resonance at 108 Hz The proof of this theorem is given in Appendix III. exactly where the secondary path has a pair of low damped complex zeros (almost no gain). Therefore one can not expect V. A N ACTIVE VIBRATION CONTROL SYSTEM USING AN good attenuation around this frequency. INERTIAL ACTUATOR
B. Broadband disturbance rejection
Fig. 2.
An AVC system using a feedforward compensation - scheme.
Fig. 2 represents an AVC system using a measurement of the image of the disturbance and an inertial actuator for reducing the residual acceleration which has been used for real time experiments. The system is composed of three metal plates interconnected by springs. The one on top (M1) is equipped with an inertial actuator which generates the disturbance s(t) (Fig. 1). Another inertial actuator is located
The adaptive feedforward filter structure for all of the experiments has been nR = 3, nS = 4 (total of 8 parameters). This complexity does not allow to verify the ”perfect matching condition” (not enough parameters). A PRBS excitation on the global primary path will be considered as the disturbance. For the adaptive operation the Algorithm II has been used with scalar adaptation gain (λ1 (t) = 1, λ2 (t) = 0)3 . The experiments have been carried out by first applying the disturbance in open loop during 50 s and after that closing the loop with the adaptive feedforward algorithms. Time domain results obtained on the AVC system are shown in Fig. 4. The advantage of using an ”Integral + Proportional” PAA is an overall improvement of the transient behavior despite that the SPR condition on H(q−1 ) = APMGˆG is not satisfied as shown in Fig. 5 (the SPR condition is not satisfied around 83 Hz and around 116 Hz). Note that Fig. 5 corresponds to an estimation of this transfer function
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3 Note
that Algorithm II uses the same filtering as FuLMS algorithm.
development shows that the SPR condition can be relaxed and an improvement of the adaptation transients is obtained.
Residual acceleration [V]
Residual acceleration [V]
Plant output using broadband disturbance adaptive compensation after 50 seconds 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −0.5 −0.5 Integral + Proportional Integral −0.6 −0.6 40 50 60 70 80 40 50 60 70 80 Time [sec] Time [sec]
A PPENDIX I P ROOF OF RESULT (22)-T HEOREM 3.1
Fig. 4. Real time results obtained with Algorithm II using ”Integral” scalar adaptation gain (left) and ”Integral + Proportional” scalar adaptation gain (right).
One has to show that for the proposed algorithm there exist sequences of time varying positive semidefinite matrices P(t), Q(t), R(t), and a matrix sequence S(t) such that the equivalent feedback path associated with the algorithm satisfies Lemma 2 and the condition of Theorem 1 of [10]. Such a sequence is given below: P(t) = FI−1 (t), Q(t) =
150 120 60
def
def
0
T
−60 −90 −120 −150 0
Fig. 5.
50
100
150 200 250 Frequency [Hz]
300
350
400
(38)
λ22 (t)
fF (t) fF2P (t) λ1 (t) I λ2 (t) + λ2 (t) fF2P (t) + 2 fF (t) fFP (t) + 2 fFP (t). λ1 (t) I
R(t) = [2 − λ1 (t)] fFI (t) +
−30
(39)
A PPENDIX II P ROOF OF L EMMA 4.1
Phase of estimated H(z−1 ) for Algorithm II.
assuming Gˆ = G, Mˆ = M and P being estimated using equation (19) in which the parameters of Rˆ and Sˆ have been obtained by running the adaptation algorithm for 1500s. A variable α(t) in the PAA has been chosen, starting with an initial value of 200 and linearly decreasing to 100 (over a horizon of 25s). Fig. 6 shows the comparison between ”Integral” and ”Integral + Proportional” adaptation over an horizon of 1500s (Fig. 4 is a zoom of Fig. 6 covering only the first 30s after the introduction of the adaptive feedforward compensator). It is clear that ”Integral + Proportional” adaptation gives better results even on a long run. VII. C ONCLUSIONS The paper has shown that the ”Integral + Proportional” adaptation algorithms presented are useful in the context of adaptive feedforward vibration compensation. Theoretical
Proof: To analyse the strict positive realness of this transfer function, one has to check first that it’s real part is strictly positive. We then have: Re{
H(z−1 ) K · Re{H}2 + Re{H} + K · Im{H}2 } = . 1 + K · H(z−1 ) (1 + K · Re{H})2 + (K · Im{H})2 (40)
In eq. (40), the denominator is always strictly positive. Thus, the strict positive realness is satisfied if K is chosen such that the numerator of eq. (40) is also strictly positive. This is always true if K satisfies the relation K >−
Re{H(e− jω )} Re{H(e− jω )}2 + Im{H(e− jω )}2
, 0 ≤ ω ≤ π · fS ,
fS being the sampling frequency. Next, the stability of H/(1 + KH) is analyzed. Under hypothesis H2, the poles of H/(1 + KH) are given by the roots of the polynomial n
P(q−1 ) = 1 +
n
A B a p q−p + K ∑m=1 bm q−m ∑ p=1 1 + Kb0
(41)
and assuming K large enough such that Kbm a p , ∀m ∈ {1, . . . , nB }, p ∈ {1, . . . , nA }, P(q−1 ) ∼ = ( bm −m B 1 + ∑nm=1 if nB ≥ nA , b0 q ∼ = ap nA nB bm −m −p 1 + ∑m=1 b0 q + ∑ p=nB +1 1+Kb0 q if nB < nA .
Plant output using broadband disturbance adaptive compensation after 50 seconds 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −0.5 Integral −0.5 Integral + Proportional −0.6 −0.6 0 500 1000 1500 0 500 1000 1500 Time [sec] Time [sec] Residual acceleration [V]
Residual acceleration [V]
(37)
T
fFI (t) = Φ (t)FI (t)Φ(t), fFP (t) = Φ (t)FP (t)Φ(t),
30
−180
(36)
S(t) = [1 − λ1 (t)]Φ(t),
90
Phase [deg]
(35)
[1 − λ1 (t)]FI−1 (t),
Fig. 6. Real time results obtained with Algorithm II using ”Integral” scalar adaptation gain (left) and ”Integral + Proportional” scalar adaptation gain (right) over 1500s.
Thus for nB ≥ nA , the poles and the zeros of H/(1 + KH) become identical when K → ∞. For nB < nA , in addition to the poles identical to the zeros of B(q−1 ), nA − nB poles appear that go to zero as K → ∞. The hypothesis H1 has been introduced to assure the stability of the direct path when H2 is satisfied. Hypothesis H2 is necessary since if b0 = 0, H/(1 + KH) becomes unstable for large K.
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and (42), results
A PPENDIX III P ROOF OF T HEOREM 4.1 Proof: The proof is similar to that of Theorem 3.3 (p. +
-
-
1 θ˜IT (t + 1)Φ(t)ν(t + 1) = θ˜IT (t + 1)FI−1 θ˜I (t + 1)− 2 1 T 1 ˜T −1 ˜ − θI (t)FI θI (t) + Φ (t)FI Φ(t)ν 2 (t + 1), (49) 2 2 and summing up from t = 0 to t1 , one gets
+
+
+
1 θ˜IT (t + 1)FI−1 θ˜I (t) = θ˜IT (t + 1)FI−1 θ˜I (t + 1)+ 2 1 ˜T 1 −1 ˜ + θI (t)FI θI (t) − ΦT (t)FI Φ(t)ν 2 (t + 1). 2 2 Substituting the last equation back into (46) and using (48)
+ +
t1
Fig. 7. Equivalent feedback representation of the PAA with ”Integral + Proportional” adaptation with constant integral adaptation gain.
109) in [8] where Lemma 3.3 (p.110) is replaced by Lemma 4.1 of this paper. However the details of the proof of Theorem 3.3 in [8] are not given. For the sake of completeness, the details of the proof of Theorem 4.1 are given next. The proof is done by using Theorem 1 of [10]. The adaptive system can be rearranged into the one in Fig. 7, where θ˜I (t + 1) = θ˜I (t) + ξ (t)FI (t)Φ(t)ν(t + 1), y¯e2 (t) = ΦT (t)θ˜I (t) + ΦT (t)F(t)Φ(t)ν(t + 1),
t1
t1
t1
t=0
t=0
t=0
(42) (43)
(44)
Taking into consideration eqs. (42) and (43) y¯e2 (t)ue2 (t) = y¯e2 (t)ν(t + 1) = θ˜IT (t + 1)Φ(t)ν(t + 1)+ + ΦT (t)FP (t)Φ(t)ν 2 (t + 1).
(45)
The first term in the right hand side can be further expressed as (see also Lemma 3.2 of [8]) θ˜IT (t + 1)Φ(t)ν(t + 1) = θ˜IT (t + 1)FI−1 θ˜I (t + 1)− (46) − θ˜IT (t + 1)FI−1 θ˜I (t). On the other hand [θ˜I (t + 1) − θ˜I (t)]T FI−1 [θ˜I (t + 1) − θ˜I (t)] = = θ˜IT (t + 1)F −1 θ˜I (t + 1) + θ˜IT (t)F −1 θ˜I (t)− I
I
− 2θ˜IT (t + 1)FI−1 θ˜I (t) ≥ 0,
t=0
t1
t1 1 − K ∑ ν 2 (t + 1) − θ˜IT (0)FI−1 θ˜I (0). 2 t=0
(50)
From eq. (50) and the fact that FI is positive definite one concludes that t1
1
∑ ye2 (t)ue2 (t) ≥ − 2 θ˜IT (0)FI−1 θ˜I (0)
(51)
(47)
as long as K satisfies condition T 2 of the theorem, thus Popov’s inequality is satisfied and the adaptive system is asymptotically stable. R EFERENCES [1] C. Jacobson, C. Johnson, D. M. Cormick, and W. Sethares, “Stability of active noise control algorithms,” IEEE Signal Processing letters, vol. 8, no. 3, pp. 74–76, 2001. [2] A. Wang and W. Ren, “Convergence analysis of the filtered-u algorithm for active noise control,” Signal Processing, vol. 73, pp. 255– 266, 1999. [3] I. Landau, M. Alma, and T. Airimit¸oaie, “Adaptive feedforward compensation algorithms for active vibration control with mechanical coupling,” Automatica, vol. 47, no. 10, pp. 2185 – 2196, 2011. [4] I. Landau, T. Airimit¸oaie, and M. Alma, “An IIR Youla-Kucera parametrized adaptive feedforward compensator for active vibration control with mechanical coupling,” in Proceedings of the 50th IEEE Conference on Decision and Control, Orlando, USA, 2011. [5] A. Montazeri and J. Poshtan, “A computationally efficient adaptive IIR solution to active noise and vibration control systems,” IEEE Trans. on Automatic Control, vol. AC-55, pp. 2671 – 2676, 2010. [6] B. Anderson, R. Bitmead, C. Johnson, P. Kokotovic, R. Kosut, I. Mareels, L. Praly, and B. Riedle, Stability of adaptive systems. Cambridge Massachusetts , London, England: The M.I.T Press, 1986. [7] M. Tomizuka, “Parallel MRAS without compensation block,” Automatic Control, IEEE Transactions on, vol. 27, pp. 505 – 506, apr 1982. [8] I. D. Landau, R. Lozano, M. M’Saad, and A. Karimi, Adaptive control. London: Springer, 2nd ed., 2011. [9] I. D. Landau, Adaptive control, The model reference approach. New York and Basel: Marcel Dekker, Inc., 1st ed., 1979. [10] I. Landau and H. Silveira, “A stability theorem with applications to adaptive control,” Automatic Control, IEEE Transactions on, vol. 24, pp. 305 – 312, apr 1979. [11] I. Landau, “An extension of a stability theorem applicable to adaptive control,” Automatic Control, IEEE Transactions on, vol. 25, pp. 814 – 817, aug 1980.
from which, using FI−1 (t + 1) = λ1 (t)FI−1 (t) + λ2 (t)Φ(t)ΦT (t),
1 + ∑ Φ (t) FI + FP (t) Φ(t)ν 2 (t + 1)− 2 t=0 T
t=0
Under condition T 1, the linear feedforward block from u1 (t) to ν(t) belongs to the class L(0). Given the choice in adaptation gain (λ2 (t) ≡ 0, λ1 (t) ≡ 1), the necessary condition for asymptotic stability is only that the time-varying feedback block belongs to the class N(0) and, therefore, its input-output product verifies Popov’s inequality (32),
∑ ye2 (t)ue2 (t) = ∑ y¯e2 (t)ue2 (t) − K ∑ u2e2 (t) ≥ −γ02 .
1
∑ ye2 (t)ν(t + 1) = 2 θ˜IT (t1 + 1)FI−1 θ˜I (t1 + 1)+
(48) 4417
Improving adaptive feedforward vibration compensation by using integral + proportional adaptation Ioan Dor´e Landau, Tudor-Bogdan Airimit¸oaie Abstract— Analysis of various adaptive feedforward vibration compensation schemes has shown that a strictly positive real (SPR) condition has to be satisfied in order to guarantee the stability of the whole system [1], [2], [3], [4]. Filters have to be implemented in order to satisfy this condition. The problem becomes even more crucial in the presence of the internal mechanical coupling between the compensator system and the reference source (a correlated measurement with the disturbance) since some information is not available when adaptation starts (see [3]). It is therefore very important to relax the SPR condition at least in the initial phase and in the same time to improve the adaptation transients. It is shown in this paper that adding a proportional adaptation to the standard integral type parametric adaptation, the SPR condition can be relaxed and the adaptation transients are improved. Theoretical developments are enhanced by real time experimental results obtained on an active vibration control (AVC) system. Index Terms— active vibration control, adaptive feedforward compensation, adaptive control, stability, parameter estimation
it has not been used in the context of adaptive feedforward compensation. Furthermore, it was observed in adaptive control, that adding positive proportional adaptation will speed up the adaption transients ([9], [8]). ”Integral + Proportional” (IP) adaptation has been discussed from a stability point of view in [9], [8] for the case of constant integral adaptation gain. A stable IP adaptation with time varying integral adaptation gain has been introduced in [10]. The objective of this paper is to explore the advantages of adding proportional adaptation in the context of the adaptive feedforward compensation of vibrations both from the theoretical and applications points of view. The main contributions of the present paper are: •
•
I. I NTRODUCTION Analysis of various adaptive feedforward vibration compensation schemes has shown that a strictly positive real (SPR) condition has to be satisfied in order to guarantee the stability of the whole system [1], [2], [3], [4]. Therefore an important issue in adaptive feedforward compensation is the design of filters either on the observed variables of the feedforward compensator ([3]) or on the residual acceleration ([5]) in order to satisfy positive realness conditions on some transfer functions required by the stability analysis. The problem becomes even more crucial in the presence of the internal mechanical coupling between the compensator system and the reference source (a correlated measurement with the disturbance) since some information is not available when adaptation starts (see [3]). In [3], based on work done in [6], it was shown that for small adaptation gains (slow adaptation) violation of the SPR condition in some frequency regions is acceptable provided that in the average the input-output product associated with this transfer function is positive. However, the performances are degraded with respect to the case when the SPR condition is satisfied. It is in fact a signal dependent condition. The problem of removing or relaxing the positive real condition can be also approached by adding a proportional adaptation to the widely used integral adaptation. While this approach is known in adaptive control ([7], [8]) apparently I.D. Landau and T.B. Airimit¸oaie are with the Control System Department of Gipsa-lab, St. Martin d’H´eres, 38402 France (e-mail: [ioan-dore.landau, tudor-bogdan.airimitoaie]@gipsa-lab.grenoble-inp.fr).
978-1-4673-2066-5/12/$31.00 978-1-4673-2064-1/12/$31.00 ©2012 ©2012 IEEE IEEE
•
•
Development and stability analysis of ”Integral + Proportional” adaptation algorithms for adaptive feedforward compensation in the presence (or not) of an internal positive feedback coupling. Extension of the results of [7] for relaxation of the SPR condition in the context of adaptive feedforward compensation using ”Integral + Proportional” adaptation compensation. Application of the ”Integral + Proportional” adaptation algorithms to an active vibration control system featuring internal positive mechanical coupling. Comparisons between the new algorithms and the existing algorithms.
While the paper is developed in the context of AVC systems, the results are certainly applicable to active noise control (ANC) systems. The paper is organized as follows. The basic equations describing the system are given in section II. The ”Integral + Proportional” adaptive feedforward compensation algorithms will be developed and analyzed in Section III. The problem of SPR relaxation will be discussed in Section IV. The AVC system on which the algorithms will be tested, is presented in Section V. Section VI will present experimental results obtained on the active vibration control system with the various algorithms. II. BASIC E QUATIONS AND N OTATIONS The block diagrams associated with an AVC system are shown in Fig.1 in open loop (1(a)) and with the adaptive feedforward compensator in the presence of an internal positive feedback (1(b)). For a physical insight see also Section V. s(t) is the disturbance and w(t) is the correlated measurement with the disturbance. The primary (D), secondary (G) and reverse (positive coupling) (M) paths
4412
Global primary path
while y(t ˆ + 1), y(t), ˆ . . . are the measurements provided by the primary transducer1 . The unmeasurable a priori output of the secondary path will be denoted
Primary path Residual acceleration measurement
Measurement of the image of the disturbance
B∗ (q−1 ) zˆ0 (t + 1) = zˆ(t + 1|θˆ (t)) = G −1 u(t). ˆ AG (q )
(a) Global primary path
The a posteriori unmeasurable value of the output of the secondary path is denoted by
Residual acceleration measurement Primary path
(7)
+
zˆ(t + 1) = zˆ(t + 1|θˆ (t + 1)).
Positive feedback coupling
(8)
+
-1 Feedforward compensator
+
The measured primary signal (called also reference) satisfies the equation
Secondary path
+ Measurement of the image of the disturbance
y(t ˆ + 1) = w(t + 1) +
PAA Parameter adaptation algorithm
(b)
B∗M (q−1 ) u(t). ˆ AM (q−1 )
(9)
The measured residual error satisfies
Fig. 1. Feedforward AVC: (a) in open loop, (b) with adaptive feedforward compensator.
e0 (t + 1) = e(t + 1|θˆ (t)) = x(t + 1) + zˆ0 (t + 1).
(10)
The a priori adaptation error is represented in Fig. 1(b) are respectively characterized by the asymptotically stable transfer operators bX1 q−1 + ... + bXnB q−nBX BX (q−1 ) X , X(q ) = = AX (q−1 ) 1 + aX1 q−1 + ... + aXnA q−nAX −1
(1)
ν 0 (t + 1) = ν(t + 1|θˆ (t)) = −e0 (t + 1).
The a posteriori adaptation (residual) error (which is computed) will be given by ν(t + 1) = ν(t + 1|θˆ (t + 1)) = −x(t + 1) − zˆ(t + 1).
X
BX = q−1 B∗X ,
ˆ Mˆ and Dˆ denote the with ∀ X ∈ {D, G, M}. G, identified (estimated) models of G, M and D. The optimal feedforward filter (unknown) which will minimize the residual acceleration in the sense of a certain criterion is defined by N(q−1 ) =
R(q−1 ) r0 + r1 q−1 + . . . + rnR q−nR = . S(q−1 ) 1 + s1 q−1 + . . . + snS q−nS
(2)
ˆ −1 ) or N( ˆ θˆ , q−1 ) when The estimated filter is denoted by N(q ˆ q−1 ) it is a linear filter with constant coefficients or N(t, during estimation (adaptation) of its parameters. The input of the feedforward filter is denoted by y(t) ˆ and it corresponds to the measurement provided by the primary transducer (force or acceleration transducer in AVC or a microphone in ANC). In the absence of the compensation loop (open loop operation) y(t) ˆ = w(t). The a priori output of the estimated feedforward filter is given by uˆ0 (t + 1) = u(t ˆ + 1|θˆ (t)) = θˆ T (t)φ (t)
(3)
III. D EVELOPMENT AND A NALYSIS OF THE A LGORITHMS The block diagram of the adaptive feedforward compensator in the presence of an internal positive feedback associated with an AVC system is shown in Fig. (1(b)). The description, equations and notations of the various blocks and transfer functions have been presented in detail in [3] eqs. (1) - (22). Using Lemma 4.1 and eqs. (26) - (30) from [3], the a posteriori adaptation error, ν(t + 1), which is computed, is given by ν(t + 1) =
AM (q−1 )G(q−1 ) [θ − θˆ (t + 1)]T φ f (t), P(q−1 )L(q−1 )
T
θ =[s1 , . . . snS , r0 , . . . rnR ] = (4)
φ (t) =[−u(t), ˆ . . . − u(t ˆ − nS + 1), y(t ˆ + 1), . . . y(t ˆ − nR + 1)] =[φuTˆ (t), φyˆT (t)]
u(t ˆ + 1) = u(t ˆ + 1|θˆ (t + 1)) = θˆ T (t + 1)φ (t),
[θST ,
θRT ],
(14) (15)
θˆ (t) and φ (t) given in (4) and (5) respectively and φ f (t) = L(q−1 )φ (t),
(5)
and u(t), ˆ u(t ˆ − 1), . . . are the a posteriori outputs of the feedforward filter generated by
(13)
with P(q−1 ) =AM (q−1 )S(q−1 ) − BM (q−1 )R(q−1 ),
T
(12)
When using an estimated filter Nˆ with constant parameters: uˆ0 (t) = u(t), ˆ zˆ0 (t) = zˆ(t), and ν 0 (t) = ν(t).
where θˆ T (t) =[sˆ1 (t), . . . sˆnS (t), rˆ0 (t), . . . rˆnR (t)] =[θˆST (t), θˆRT (t)],
(11)
where L(q−1 ) =
(6) 4413
1 y(t ˆ + 1)
BL (q−1 ) AL (q−1 )
(16)
will be detailed later.
is available before adaptation of parameters starts at t + 1.
Eq. (13) has the standard form for an a posteriori adaptation error ([8]). The following ”Integral + Proportional” parameter adaptation algorithm (IP-PAA) is proposed: θˆI (t + 1) = θˆI (t) + ξ (t)FI (t)Φ(t)ν(t + 1), θˆP (t + 1) = FP (t)Φ(t)ν(t + 1), ν(t + 1) =
(17a) (17b)
ν 0 (t + 1)
, (17c) 1 + ΦT (t)(ξ (t)FI (t) + FP (t))Φ(t) FI (t)Φ(t)ΦT (t)FI (t) 1 FI (t) − λ (t) FI (t + 1) = , (17d) 1 λ1 (t) + ΦT (t)FI (t)Φ(t)
last case the SPR condition is satisfied provided that one has good estimations of AM , G, and P (for more details see [3]). Neglecting the non-commutativity of time varying operators, one has the following result: Theorem 3.1: Assuming that eq. (20) represents the evolution of the a posteriori adaptation error and that the IP-PAA (17a) - (17j) is used, one has lim ν(t + 1) = 0,
FP (t) = α(t)FI (t); α(t) > −0.5,
(17e)
[ν 0 (t + 1)2 ] = 0, t→∞ 1 + Φ(t)T F(t)Φ(t) ||Φ(t)|| is bounded,
F(t) = ξ (t)FI (t) + FP (t), λ2 (t) T ξ (t) = 1 + Φ (t)FP (t)Φ(t), λ1 (t) θˆ (t + 1) = θˆI (t + 1) + θˆP (t + 1),
(17f)
lim ν 0 (t + 1) = 0,
Φ(t) = φ f (t),
where ν(t + 1) is the adaptation error2 , λ1 (t) and λ2 (t) allow to obtain various profiles for the matrix adaptation gain FI (t) (see [8] for more details). By taking λ2 (t) ≡ 0 one obtains a constant adaptation gain matrix (and choosing FI = γI, γ > 0 one gets a scalar adaptation gain). For α(t) ≡ 0, one obtains the algorithm with integral adaptation gain introduced in [3]. Three choices for the filter L will be considered, leading to three different algorithms: Algorithm II
L=G L = Gˆ
Algorithm III
L=
Algorithm I
Aˆ M ˆ G Pˆ
(18)
Pˆ = Aˆ M Sˆ − Bˆ M Rˆ
(19)
where is an estimation of the characteristic polynomial of the internal feedback loop computed on the basis of available ˆ To use Algorithm estimates for the parameters of the filter N. III one has to start with Algorithm II where the SPR condition is in general not satisfied. Therefore, relaxing the SPR condition for Algorithm II is very important. A. Analysis of the Algorithms For Algorithms I, II, and III the equation for the a posteriori adaptation error has the form (eqs. (29) and (30) from [3]) ν(t + 1) = H(q−1 )[θ − θˆ (t + 1)]T Φ(t), AM (q−1 )G(q−1 ) , Φ = φ f = L(q−1 )φ . H(q ) = P(q−1 )L(q−1 ) −1
Thus for Algorithm II one has H(q−1 ) = Algorithm III one obtains 2 ν 0 (t + 1)
H(q−1 ) =
AM GPˆ . PAˆ M Gˆ
is the measured a priori adaptation error
AM G PGˆ
H 0 (q−1 ) = H(q−1 ) −
(21) and for
Note that in this
λ2 , max λ2 (t) ≤ λ2 < 2. t 2
(26)
is a SPR transfer function. The proof of (22) is given in Appendix I. For (23), (24), and (25), the proof follows [11], [3] and it is omitted. The proof in [10] for IP adaptation with time varying integral adaptation gain is given for ξ (t) = λ 1(t) + λ2 (t) T λ1 (t) Φ (t)FP (t)Φ(t).
1
To the knowledge of the authors, the proof for ξ (t) given in eq. (17g) is presented here for the first time. IV. R ELAXING THE P OSITIVE R EAL CONDITION An equivalent feedback system can be associated to the IP-PAA where the feedforward path is characterized by the transfer function H 0 (z−1 ). There is additional ”excess” of passivity in the feedback path (which depends upon the magnitude of the adaptation gains and the magnitude of Φ(t)) which can be transferred to the linear feedforward block in order to relax the SPR condition. This idea was prompted out in the context of recursive identification by Tomizuka and results have been given for the case of integral adaptation with constant gains and when the equivalent linear feedforward path is characterized by an all poles (no zeros) transfer function (see [7]). These results have been extended in [8] for IP adaptation with constant adaptation gain. Taking into account the poles-zeros structure of H(z−1 ), the results of [7], [8] should be extended for the situation described in this paper. One has the following result: Lemma 4.1: Given the discrete transfer function H(z−1 ) =
(20)
where
(25)
for any initial conditions θˆ (0), ν 0 (0), F(0), provided that
(17i) (17j)
(24)
t→∞
(17h)
0 < λ1 (t) ≤ 1, 0 ≤ λ2 (t) < 2, FI (0) > 0,
(23)
lim
λ2 (t)
(17g)
(22)
t→∞
B(z−1 ) b0 + b1 z−1 + . . . + bnB z−nB = , A(z−1 ) 1 + a1 z−1 + . . . + anA z−nA
(27)
under the hypothesis: H1) H(z−1 ) has all its zeros inside the unit circle, H2) b0 6= 0, H there exists a positive scalar gain K such that 1+KH is SPR. The proof of this lemma is presented in II. Using the above property, for the IP-PAA given by the eqs. (17a) - (17j) and (20) for λ2 (t) ≡ 0, λ1 (t) ≡ 1 (constant
4414
adaptation gain) and choosing K such such that one gets the equivalent feedback system
H 1+KH
is SPR,
H(z−1 ) y2 (t), (28) 1 + KH(z−1 ) θ˜I (t) = θˆI (t) − θ , (29) θ˜I (t + 1) = θ˜I (t) + ξ (t)FI Φ(t)ν(t + 1), (30) T T y2 (t) = Φ (t)θ˜I (t) + (Φ (t)F(t)Φ(t) − K)ν(t + 1), (31) ν(t + 1) = −
To prove the stability it remains to show that the new feedback path given by (31) is passive, i.e., it satisfies the Popov inequality t1
∑ y2 (t)u2 (t) ≥ −γ02 .
(32)
t=0
bellow plate M3 and is used for disturbance rejection. Two accelerometers positioned as in figure 2 measure the image of the disturbance and the residual acceleration e0 (t). The corresponding block diagrams in open loop operation and with the compensator system are shown in Figs. 1(a) and 1(b). VI. E XPERIMENTAL RESULTS A. System identification The procedure for identifying the models of the various paths has been described in [3]. Their frequency characteristics are shown in Fig. 3. The model orders for the secondary path (solid line) and the reverse path (dotted line) have been estimated to be: nBG = 14, nAG = 14, and nBM = 14, nAM = 14 respectively. The primary path model has been used only for simulations. The models show the presence of several low
Amplitude [dB]
The following theorem provides the necessary result. Theorem 4.1: The adaptive system described by eq. (13) 40 Secondary path and eqs. (17a) - (17j) for λ2 (t) ≡ 0 and λ1 (t) ≡ 1 is Reverse path 20 asymptotically stable provided that: Primary path H 0 T1) It exists a gain K such that 1+KH is SPR, T2) The adaptation gains FI and FP (t) and the observation −20 vector Φ(t) satisfy −40 t1 1 T 2 −60 ∑ Φ (t − 1) 2 FI + FP (t − 1) Φ(t − 1) − K ν (t) ≥ 0 t=0 −80 (33) 0 50 100 150 200 250 300 350 400 Frequency [Hz] for all t1 ≥ 0 or 1 Fig. 3. Frequency characteristics of the primary, secondary and reverse T Φ (t) FI + FP (t) Φ(t) > K > 0, (34) paths. 2 damped pairs of complex poles and complex zeros. Note for all t ≥ 0. that the primary path features a strong resonance at 108 Hz The proof of this theorem is given in Appendix III. exactly where the secondary path has a pair of low damped complex zeros (almost no gain). Therefore one can not expect V. A N ACTIVE VIBRATION CONTROL SYSTEM USING AN good attenuation around this frequency. INERTIAL ACTUATOR
B. Broadband disturbance rejection
Fig. 2.
An AVC system using a feedforward compensation - scheme.
Fig. 2 represents an AVC system using a measurement of the image of the disturbance and an inertial actuator for reducing the residual acceleration which has been used for real time experiments. The system is composed of three metal plates interconnected by springs. The one on top (M1) is equipped with an inertial actuator which generates the disturbance s(t) (Fig. 1). Another inertial actuator is located
The adaptive feedforward filter structure for all of the experiments has been nR = 3, nS = 4 (total of 8 parameters). This complexity does not allow to verify the ”perfect matching condition” (not enough parameters). A PRBS excitation on the global primary path will be considered as the disturbance. For the adaptive operation the Algorithm II has been used with scalar adaptation gain (λ1 (t) = 1, λ2 (t) = 0)3 . The experiments have been carried out by first applying the disturbance in open loop during 50 s and after that closing the loop with the adaptive feedforward algorithms. Time domain results obtained on the AVC system are shown in Fig. 4. The advantage of using an ”Integral + Proportional” PAA is an overall improvement of the transient behavior despite that the SPR condition on H(q−1 ) = APMGˆG is not satisfied as shown in Fig. 5 (the SPR condition is not satisfied around 83 Hz and around 116 Hz). Note that Fig. 5 corresponds to an estimation of this transfer function
4415
3 Note
that Algorithm II uses the same filtering as FuLMS algorithm.
development shows that the SPR condition can be relaxed and an improvement of the adaptation transients is obtained.
Residual acceleration [V]
Residual acceleration [V]
Plant output using broadband disturbance adaptive compensation after 50 seconds 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −0.5 −0.5 Integral + Proportional Integral −0.6 −0.6 40 50 60 70 80 40 50 60 70 80 Time [sec] Time [sec]
A PPENDIX I P ROOF OF RESULT (22)-T HEOREM 3.1
Fig. 4. Real time results obtained with Algorithm II using ”Integral” scalar adaptation gain (left) and ”Integral + Proportional” scalar adaptation gain (right).
One has to show that for the proposed algorithm there exist sequences of time varying positive semidefinite matrices P(t), Q(t), R(t), and a matrix sequence S(t) such that the equivalent feedback path associated with the algorithm satisfies Lemma 2 and the condition of Theorem 1 of [10]. Such a sequence is given below: P(t) = FI−1 (t), Q(t) =
150 120 60
def
def
0
T
−60 −90 −120 −150 0
Fig. 5.
50
100
150 200 250 Frequency [Hz]
300
350
400
(38)
λ22 (t)
fF (t) fF2P (t) λ1 (t) I λ2 (t) + λ2 (t) fF2P (t) + 2 fF (t) fFP (t) + 2 fFP (t). λ1 (t) I
R(t) = [2 − λ1 (t)] fFI (t) +
−30
(39)
A PPENDIX II P ROOF OF L EMMA 4.1
Phase of estimated H(z−1 ) for Algorithm II.
assuming Gˆ = G, Mˆ = M and P being estimated using equation (19) in which the parameters of Rˆ and Sˆ have been obtained by running the adaptation algorithm for 1500s. A variable α(t) in the PAA has been chosen, starting with an initial value of 200 and linearly decreasing to 100 (over a horizon of 25s). Fig. 6 shows the comparison between ”Integral” and ”Integral + Proportional” adaptation over an horizon of 1500s (Fig. 4 is a zoom of Fig. 6 covering only the first 30s after the introduction of the adaptive feedforward compensator). It is clear that ”Integral + Proportional” adaptation gives better results even on a long run. VII. C ONCLUSIONS The paper has shown that the ”Integral + Proportional” adaptation algorithms presented are useful in the context of adaptive feedforward vibration compensation. Theoretical
Proof: To analyse the strict positive realness of this transfer function, one has to check first that it’s real part is strictly positive. We then have: Re{
H(z−1 ) K · Re{H}2 + Re{H} + K · Im{H}2 } = . 1 + K · H(z−1 ) (1 + K · Re{H})2 + (K · Im{H})2 (40)
In eq. (40), the denominator is always strictly positive. Thus, the strict positive realness is satisfied if K is chosen such that the numerator of eq. (40) is also strictly positive. This is always true if K satisfies the relation K >−
Re{H(e− jω )} Re{H(e− jω )}2 + Im{H(e− jω )}2
, 0 ≤ ω ≤ π · fS ,
fS being the sampling frequency. Next, the stability of H/(1 + KH) is analyzed. Under hypothesis H2, the poles of H/(1 + KH) are given by the roots of the polynomial n
P(q−1 ) = 1 +
n
A B a p q−p + K ∑m=1 bm q−m ∑ p=1 1 + Kb0
(41)
and assuming K large enough such that Kbm a p , ∀m ∈ {1, . . . , nB }, p ∈ {1, . . . , nA }, P(q−1 ) ∼ = ( bm −m B 1 + ∑nm=1 if nB ≥ nA , b0 q ∼ = ap nA nB bm −m −p 1 + ∑m=1 b0 q + ∑ p=nB +1 1+Kb0 q if nB < nA .
Plant output using broadband disturbance adaptive compensation after 50 seconds 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −0.5 Integral −0.5 Integral + Proportional −0.6 −0.6 0 500 1000 1500 0 500 1000 1500 Time [sec] Time [sec] Residual acceleration [V]
Residual acceleration [V]
(37)
T
fFI (t) = Φ (t)FI (t)Φ(t), fFP (t) = Φ (t)FP (t)Φ(t),
30
−180
(36)
S(t) = [1 − λ1 (t)]Φ(t),
90
Phase [deg]
(35)
[1 − λ1 (t)]FI−1 (t),
Fig. 6. Real time results obtained with Algorithm II using ”Integral” scalar adaptation gain (left) and ”Integral + Proportional” scalar adaptation gain (right) over 1500s.
Thus for nB ≥ nA , the poles and the zeros of H/(1 + KH) become identical when K → ∞. For nB < nA , in addition to the poles identical to the zeros of B(q−1 ), nA − nB poles appear that go to zero as K → ∞. The hypothesis H1 has been introduced to assure the stability of the direct path when H2 is satisfied. Hypothesis H2 is necessary since if b0 = 0, H/(1 + KH) becomes unstable for large K.
4416
and (42), results
A PPENDIX III P ROOF OF T HEOREM 4.1 Proof: The proof is similar to that of Theorem 3.3 (p. +
-
-
1 θ˜IT (t + 1)Φ(t)ν(t + 1) = θ˜IT (t + 1)FI−1 θ˜I (t + 1)− 2 1 T 1 ˜T −1 ˜ − θI (t)FI θI (t) + Φ (t)FI Φ(t)ν 2 (t + 1), (49) 2 2 and summing up from t = 0 to t1 , one gets
+
+
+
1 θ˜IT (t + 1)FI−1 θ˜I (t) = θ˜IT (t + 1)FI−1 θ˜I (t + 1)+ 2 1 ˜T 1 −1 ˜ + θI (t)FI θI (t) − ΦT (t)FI Φ(t)ν 2 (t + 1). 2 2 Substituting the last equation back into (46) and using (48)
+ +
t1
Fig. 7. Equivalent feedback representation of the PAA with ”Integral + Proportional” adaptation with constant integral adaptation gain.
109) in [8] where Lemma 3.3 (p.110) is replaced by Lemma 4.1 of this paper. However the details of the proof of Theorem 3.3 in [8] are not given. For the sake of completeness, the details of the proof of Theorem 4.1 are given next. The proof is done by using Theorem 1 of [10]. The adaptive system can be rearranged into the one in Fig. 7, where θ˜I (t + 1) = θ˜I (t) + ξ (t)FI (t)Φ(t)ν(t + 1), y¯e2 (t) = ΦT (t)θ˜I (t) + ΦT (t)F(t)Φ(t)ν(t + 1),
t1
t1
t1
t=0
t=0
t=0
(42) (43)
(44)
Taking into consideration eqs. (42) and (43) y¯e2 (t)ue2 (t) = y¯e2 (t)ν(t + 1) = θ˜IT (t + 1)Φ(t)ν(t + 1)+ + ΦT (t)FP (t)Φ(t)ν 2 (t + 1).
(45)
The first term in the right hand side can be further expressed as (see also Lemma 3.2 of [8]) θ˜IT (t + 1)Φ(t)ν(t + 1) = θ˜IT (t + 1)FI−1 θ˜I (t + 1)− (46) − θ˜IT (t + 1)FI−1 θ˜I (t). On the other hand [θ˜I (t + 1) − θ˜I (t)]T FI−1 [θ˜I (t + 1) − θ˜I (t)] = = θ˜IT (t + 1)F −1 θ˜I (t + 1) + θ˜IT (t)F −1 θ˜I (t)− I
I
− 2θ˜IT (t + 1)FI−1 θ˜I (t) ≥ 0,
t=0
t1
t1 1 − K ∑ ν 2 (t + 1) − θ˜IT (0)FI−1 θ˜I (0). 2 t=0
(50)
From eq. (50) and the fact that FI is positive definite one concludes that t1
1
∑ ye2 (t)ue2 (t) ≥ − 2 θ˜IT (0)FI−1 θ˜I (0)
(51)
(47)
as long as K satisfies condition T 2 of the theorem, thus Popov’s inequality is satisfied and the adaptive system is asymptotically stable. R EFERENCES [1] C. Jacobson, C. Johnson, D. M. Cormick, and W. Sethares, “Stability of active noise control algorithms,” IEEE Signal Processing letters, vol. 8, no. 3, pp. 74–76, 2001. [2] A. Wang and W. Ren, “Convergence analysis of the filtered-u algorithm for active noise control,” Signal Processing, vol. 73, pp. 255– 266, 1999. [3] I. Landau, M. Alma, and T. Airimit¸oaie, “Adaptive feedforward compensation algorithms for active vibration control with mechanical coupling,” Automatica, vol. 47, no. 10, pp. 2185 – 2196, 2011. [4] I. Landau, T. Airimit¸oaie, and M. Alma, “An IIR Youla-Kucera parametrized adaptive feedforward compensator for active vibration control with mechanical coupling,” in Proceedings of the 50th IEEE Conference on Decision and Control, Orlando, USA, 2011. [5] A. Montazeri and J. Poshtan, “A computationally efficient adaptive IIR solution to active noise and vibration control systems,” IEEE Trans. on Automatic Control, vol. AC-55, pp. 2671 – 2676, 2010. [6] B. Anderson, R. Bitmead, C. Johnson, P. Kokotovic, R. Kosut, I. Mareels, L. Praly, and B. Riedle, Stability of adaptive systems. Cambridge Massachusetts , London, England: The M.I.T Press, 1986. [7] M. Tomizuka, “Parallel MRAS without compensation block,” Automatic Control, IEEE Transactions on, vol. 27, pp. 505 – 506, apr 1982. [8] I. D. Landau, R. Lozano, M. M’Saad, and A. Karimi, Adaptive control. London: Springer, 2nd ed., 2011. [9] I. D. Landau, Adaptive control, The model reference approach. New York and Basel: Marcel Dekker, Inc., 1st ed., 1979. [10] I. Landau and H. Silveira, “A stability theorem with applications to adaptive control,” Automatic Control, IEEE Transactions on, vol. 24, pp. 305 – 312, apr 1979. [11] I. Landau, “An extension of a stability theorem applicable to adaptive control,” Automatic Control, IEEE Transactions on, vol. 25, pp. 814 – 817, aug 1980.
from which, using FI−1 (t + 1) = λ1 (t)FI−1 (t) + λ2 (t)Φ(t)ΦT (t),
1 + ∑ Φ (t) FI + FP (t) Φ(t)ν 2 (t + 1)− 2 t=0 T
t=0
Under condition T 1, the linear feedforward block from u1 (t) to ν(t) belongs to the class L(0). Given the choice in adaptation gain (λ2 (t) ≡ 0, λ1 (t) ≡ 1), the necessary condition for asymptotic stability is only that the time-varying feedback block belongs to the class N(0) and, therefore, its input-output product verifies Popov’s inequality (32),
∑ ye2 (t)ue2 (t) = ∑ y¯e2 (t)ue2 (t) − K ∑ u2e2 (t) ≥ −γ02 .
1
∑ ye2 (t)ν(t + 1) = 2 θ˜IT (t1 + 1)FI−1 θ˜I (t1 + 1)+
(48) 4417
