Mathematics and Computers in Simulation 60 (2002) 303–314
Reconfigurable internal model control based on adaptive lattice filtering G. Nikolakopoulos, A. Tzes∗ Department of Electrical and Computer Engineering, University of Patras, Patras 26500, Greece
Abstract In this paper a reconfigurable adaptive control (RAC) algorithm is presented. The algorithm relies on lattice filtering and is suitable for linear discrete jump systems. The utilized lattice filter identifies recursively the system’s dynamics as an autoregressive model through the transfer function’s reflection coefficients. In addition, the system’s order is monitored through the magnitude of the estimated reflection coefficients. An internal model control (IMC) structure is employed, where the stable-inverse of the filter is utilized in the forward path in cascade with a gain compensator and a lowpass detuning filter. Upon sudden variations of the estimated system’s order the RA controller reconfigures its structure and calculates the command signal based on the IMC principle. Simulation studies are used to investigate the efficacy of the suggested scheme. © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Internal model control; Adaptive lattice filter; Reconfigurable control; Jump systems
1. Introduction The adaptive control problem for discrete jump systems has received significant attention over the last years [1]. Jump systems tend to alter their structure in rare instances and the adaptation mechanism needs to account for such modifications [2]. A typical adaptive controller [3] is composed of: (a) controller portion and (b) identifier module. The coupling between the identification scheme and the controller is natural within the internal model control (IMC) principle [4–6], in order to address the robustness and performance issues. The IM controller minimizes a cost which is a weighted function of the system’s sensitivity function and its multiplicative uncertainty. In most studies, a lowpass detuning filter is used in cascade with the IM controller in order to ensure the system’s robust stability. If the performance attributes of the system’s transient response are to be considered in the controller design process a learning scheme can be employed. Motivated by the “progressive learning” control design approach [7,8], the uncertainty associated with the system dynamics is learned progressively ∗
Corresponding author. E-mail addresses:
[email protected] (G. Nikolakopoulos),
[email protected] (A. Tzes). 0378-4754/02/$ – see front matter © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 2 ) 0 0 0 2 2 - 8
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through an identification scheme. This scheme employs a prefilter [9,10], whose characteristics depend on the system’s frequency spectrum and the adopted controller’s objectives. Furthermore, the structure of the selected model in the identification process is that of a lattice filter [11,12]. The identifier estimates the reflection coefficients associated with this particular structure. By proper monitoring of the magnitude of the reflection coefficients, the order of the system can be inferred as well as its stability attributes. Issues of concern in the IMC area involve: (a) the modification of the algorithm when the estimated model’s order varies and (b) the careful adjustment of the detuning filter for increasing the bandwidth of the closed loop system at the expense of compromising the system’s robust stability. In this study, an adaptive scheme relying on the IMC principle is employed for a class of autoregressive linear discrete time jump systems. The controller is reconfigured according to the identified model order provided by the lattice filter. The detuning filter is coupled to the prefilter for progressively increasing the bandwidth of the closed loop system. This paper is structured in the following manner. In Section 2, the problem statement is expressed. In Section 3, the adaptive robust control design based on the IMC principle and the relevant prefiltering is addressed. The proposed approach is applied in simulation studies in Section 4. Final remarks are offered in Section 5. 2. Problem statement Consider the autoregressive linear discrete time jump system with its dynamics expressed as: ˜ t (z)(1 + lm )u(t) + ν(t) y(t) = Gt (z)u(t) + ν(t) = G −d bd (t)z na = (1 + lm )u(t) + ν(t), 1 + i=1 ai (t)z−i
(1)
˜ t is the where z−1 corresponds to the delay operator, u(t) (y(t)) is the input (output) of the system, G “nominal” plant description, the lm term characterizes the uncertainty about the nominal plant, and the output measurements are corrupted by additive white noise ν(t). Under the assumption of an AR parameterization d is the delay term, ai (t) corresponds to the AR terms, and na the order of the system. The system is assumed to be stable and the delay term d is considered to be known a priori. The considered plant belongs to the class of jump systems, where the order, bd and AR coefficients can change only once over a large sliding time-window with a predetermined minimum length H , or [bd (t), a1 (t), . . . , ana (t)] = [bd , a1 , . . . , ana ],
t ∈ [t − H, t ],
where t − H is the last time that a “jump” in the system parameter vector occurred. The objective is to design an adaptive robust controller relying on the IMC principle to ensure the system’s robust stability in lieu of jumps in the parameter vector. 3. Adaptive IMC In this section, an adaptive IMC structure [13] is formed by coupling an identification scheme to a robust IM controller, as shown in Fig. 1. The IM controller, q(ejω ), is generated by the cascade composition of: (1) an H2 optimal controller q(e ˜ jω ) for the nominal plant and (2) a lowpass filter F (ejω ) which detunes the
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controller characteristics at high frequencies in order to extend the system’s robustness. The identification module relies on lattice filtering and estimates the plant dynamics by computing the corresponding reflection coefficients. The order of the system is approximated as the index of the reflection coefficient whose magnitude is below a threshold (close to 0). The convergence of the identifier is influenced by the prefiltering (L(z−1 )) of the incoming data to the lattice estimator. 3.1. H2 optimal controller design For robust performance [4], the control objective is to minimize the infinity norm of the system’s weighted sensitivity function (=1 − Gq) wp ∞ = sup|wp ( jω)| ω
˜ G ˜ −1 | ≤ lm }. The |wp |−1 represents an upper bound on for all members of plants G = {G : |(G − G) the sensitivity function, since |(jω)| ≤ |wp (jω)|−1 ∀ω if and only if supω (|ηl ˜ m | + |wp ˜ |) ≤ 1, where η˜ ˜ is the complementary sensitivity function for the nominal system G. ˜ (=1 − ˜ = Gq) The optimal controller selection problem is formulated as ˜ ˜ ˜ m |) = arg min sup(|wp (1 − Gq| + |Gqlm |) , (2) q = arg min sup(|wp ˜ | + |ηl q
q
ω
ω
where q is the IMC feedback controller as shown in Fig. 1. The philosophy behind the IMC design consists of two steps, and although the resulting controller has no inherent optimality characteristics, it provides a good engineering approximation to the optimal solution of (2). The first step amounts to designing a controller q˜ for good nominal performance so that ˜ q) q˜ = arg minwp ˜ 2 = arg minwp (1 − G (3) ˜ 2 . q˜
q˜
˜ q, ˜ and the optimal complementary sensitivity In this case, the optimal sensitivity becomes ¯ 1 − G ˜ function η¯ Gq. ˜ Furthermore for the case, where the weight wp reflects the particular input (wp = r(s)), the cost function (3) to be minimized is the l2 norm of the error wp ˜ 2 = er 2 , and the resulting controller is H2 optimal. The second step addresses the robust stability and performance issue. At high frequencies, when the multiplicative uncertainty lm exceeds unity, η¯ has to be rolled off. To achieve this action, q˜ is augmented
Fig. 1. RAC structure.
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(cascaded) by a lowpass filter F , as q qF ˜ . The order of F is such that q is proper, and its roll-off ˜ m ∞ < 1 is satisfied. The frequency is selected so that the robust stability constraint ηl ˜ m ∞ = Gql purpose of the filter F is to detune the controller, since it sacrifices performance for robustness. This ˜ = 1−G ˜ qF is justified, since the sensitivity ˜ = 1 − Gq ˜ (performance measure) is increased, while ˜ qF η˜ = G ˜ (robustness measure) decreases. ˜ Because of the G-system’s inherent minimum phase characteristics, the optimal solution to the minimization of the cost in (3) is independent of the weight and is equal to ˜ −1 . q˜ = G
(4)
3.2. Lattice form system identification ˜ is unknown, the estimated transfer function G ˆ will be used in the controller Since the nominal system G ˜ ˆ ˆ −1 , and the corresponding design process. Subsequently, the multiplicative error is em = (G − G)G −1 −1 ˜ ˆ sensitivity function ˜ = (1− η)(1+e ˆ ˆ , where ηˆ = GG . From Parseval’s theorem the l2 error-norm m η) is mapped into the frequency domain as: π 1/2 1 −1 2 er 2 = . (5) |(1 − η)(1 ˆ + em η) ˆ (r)| dω π 0 For the frequency range where |em η| ˆ 1 the control objective from (5) can be modified and bounded by π 1/2 ∞ 1/2 1 1 2 2 2 2 2 er 2 ≤ + |1 − η| ˆ |r| dω |1 − η| ˆ |em η| ˆ |r| dω . (6) π 0 π 0 The effect of the identified transfer function to the cost objective is at the second term of (6), which ˆ should be selected so as to minimize includes the contribution of the multiplicative error. Therefore, G the control relevant identification cost π 1/2 1 ˜ − G) ˆ G ˆ −1 η| ˆ = arg min |1 − η| ˆ 2 |(G ˆ 2 |r|2 dω . (7) G ˆ π 0 G 3.2.1. Data prefiltering The system model, to be identified, is assumed to have the following linear regression structure ˆ y(t|θ ˆ ) = G(z, θ)u(t) + xw (t),
(8)
where θ is the system parameter vector (i.e., the lattice filter’s reflection coefficients), and xw corresponds to white noise. Let the filtered prediction error sequence be filtered through a stable linear filter L(z): ˆ βF (t, θ) = L(z)β(t, θ) = L(z)[y(t) − y(t|θ)].
(9)
The utilized L-filter can be used to enhance or suppress certain properties of the model, since it acts as a frequency weighting factor. The purpose of the identifier is to provide a parameter vector, θ, that minimizes the following norm: N 1 θ = arg min [βF (t, θ)]2 . (10) θ N i=1
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After tedious computations, θ is inferred from π
2
2 ˜ ˆ θ = arg min |L| G − G Φu (ω) dω , ˆ G
307
(11)
0
where Φu (ω) is the spectrum of the control input. Comparing this term to the integrand quantity in (7) infers the optimal prefilter L as L(z) =
ˆ −1 (z)η(z)[r(z)] (1 − η(z)) ˆ G ˆ , −1 F (Φu (ω))
(12)
where F −1 corresponds to the inverse discrete Fourier transform. The computation of the optimal prefilter demands the knowledge of Φu (ω) and η(z), ˆ which is rather difficult to be obtained a priori. In this research effort, the selection of the prefilter is based primarily to affect the convergence of the estimated transfer function to the actual one in the frequency domain. If L(z−1 ) is selected as a lowpass ˜ (from filter, then the parameter vector, θ is computed in order to match the low frequency spectrum of G Eq. (11)). In our case, the cutoff of the lowpass filter is increased progressively as the estimator matches more closely the spectrum of the true system at the low frequency end. In order to ascertain the efficiency of the suggested scheme in lieu of limited subsystem variations, the prefilter is selected to be identical to that of the detuning filter (L(z−1 ) = F (z−1 )). 3.2.2. Lattice form system identification Upon computation of the L(z) prefilter, the input and output data streams are filtered prior to their processing by the adopted identification scheme. The identifier, shown in Fig. 2, rather than estimating the tapped weights in the direct transversal form of (1), computes in a recursive manner the reflection coefficients κm characterizing the system’s behavior. Under the assumption of an na -order system, its reflection coefficients are computed recursively based on the gradient adaptive lattice (GAL) algorithm [12,14] (summarized in Table 1) using the a posteriori estimation errors. The link between the direct and lattice form parameterizations can be found from the following rela tionships shown in Table 2 for the AR part (1 + D(z) = 1 + (Dna (z)/zna )). The primary advantage of this configuration is the trivial monitoring of the system’s stability, which is related to the magnitude of the reflection coefficients; the system is termed unstable when |κm | ≥ 1. Furthermore, the order of the system is estimated based on the magnitude of the reflection coefficients. The system’s order is truncated at µ : µ ∈ [0, na ] = arg {|κˆ µ | 0}.
Fig. 2. Lattice filter structure.
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Table 1 GAL algorithm Initialization phase fm (0) = bm (0) = 0, m = 1, . . . , na ξm−1 (0) = c 0+ , κˆ m (1) = 0 Recursive computation phase—for time n = 1, 2, . . . f0 (n) = b0 (n) = y(n) For AR-order m = 1, . . . , na fm (n) = fm−1 (n) + κˆ m∗ (n)bm−1 (n − 1) bm (n) = bm−1 (n − 1) + κˆ m∗ (n)fm−1 (n) ξm−1 (n) = βξm−1 (n − 1) + (1 − β)(|fm−1 (n)|2 + |bm−1 (n − 1)|2 ) ∗ [fm−1 (n)bm (n) + bm−1 (n − 1)fm∗ (n)] κˆ m (n + 1) = κˆ m (n) − ξm−1 (n)
Table 2 Lattice to autoregressive direct realization D1 (z) = κ1 Recursive computation—for AR-order m = 2, . . . , na Let Dm (z) = d0 + d1 z + · · · + dm zm ∗ Dm∗ (z) = dm−1 zm−1 + · · · + d0∗ Dm (z) = z[Dm−1 (z) + κm∗ Dm∗ (z)] + κm
3.3. Detuning filter design The instant of the first vanishing reflection coefficient affects the H2 controller’s attributes, since ˆ −1 = Dm (z)/bd z−d . Furthermore, the “detuning lowpass filter” F (ejω ) is modified according to q˜ = G the µ-order of the identified system. The F (z)-filter design philosophy is to devise a scheme which progressively increases the bandwidth of the closed loop system. Provided with an identified transfer function, the learning mechanism increases progressively the cutoff of filter F . Although this action increases the bandwidth of the closed loop system, it compromises the robust stability of the system. The robust stability of the system is satisfied if ˆ jω )q(e |ηl ˆ m | = |G(e ˜ jω )F (e jω )lm | < 1
for ω = 0.
Therefore, the cutoff of this filter is progressively increased until the previous robust stability index has approached 1. This progression results in a filter F ∗ with the largest bandwidth. If the bandwidth of the closed loop system is satisfactory, then this learning process terminates. Let the identified transfer function computed for a certain number µ of reflection coefficients be denoted ˆ µ . The proposed learning mechanism can be interpreted as: “for the identified system G ˆ µ , design a as G sequence of controllers qµ = q˜µ Fµ that progressively increase the bandwidth of the closed loop system”. In order to simplify the computational complexity of the algorithm, this filter is typically selected as a first-order lowpass discrete filter Fµ (z) =
(1 − eλµ )z , z − eλµ
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where λµ corresponds to the cutoff frequency. The tuning algorithm starts with a small λµ ∈ [0, 0.5), which is progressively increased λµ = λµ + λµ . The correction factor λµ (≥ 0) is adjusted based on the continuous monitoring of the squared tracking error over an M sample sliding window tt =t−M [r(t ) − y(t )]2 . If this error exceeds a certain number this can be an indication of marginal stability and the cutoff frequency is not increased. 4. Simulation studies The proposed reconfigurable adaptive IM controller is applied in simulation studies for controlling a ˜ system with a nominal transfer function G(z) = 6.25 × 10−6 /(z − 0.95)4 . The output measurements are corrupted with a white noise signal ν(t) such that the SNR is 80 dB. For the purpose of investigating the efficiency of the lattice filter in the identification process, the system is excited with a white noise input sequence. The data stream is prefiltered with a time invariant first-order lowpass filter with λµ = 0.45. The identified Bode diagrams of the model for varying na -order (na = 3, 4, 5) are shown in Figs. 3–5, respectively. The first (second) case na = 3 (4) represents an underparametrized (properly parametrized) case, while the last one reflects an overparametrized system (na = 5 > na = 4). In the presented figures, the frequency response of the system is displayed with varying data sample size (t = 100, 1000, 10,000 and 20,000). The convergence of the fourth- and fifth-order model is superior to that of the third-order model, since in the first two cases the system converges to the true plant within 100 iterations, while in the latter case more than 10,000 iterations are needed for the same purpose. The convergence of reflection coefficients (versus the logarithmic sample index) is depicted in Figs. 6–8 for model orders 3, 4 and 5, respectively.
Fig. 3. Identified third-order model frequency responses.
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Fig. 4. Identified fourth-order model frequency responses.
Fig. 5. Identified fifth-order model frequency responses.
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Fig. 6. Reflection coefficient convergence for the third-order model.
Fig. 7. Reflection coefficient convergence for the fourth-order model.
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Fig. 8. Reflection coefficient convergence for the fifth-order model.
Fig. 9. System response with adaptive IMC (third-order model).
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Fig. 10. System response with adaptive IMC (fourth-order model).
Fig. 11. System response with adaptive IMC (fifth-order model).
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The nominal reflection coefficients for the fourth-order AR system are κ¯ = [−0.9997, 0.9989, −0.9935, 0.8145]. During the first steps of the identification routine, the estimated model is unstable, since max κm ≥ 1, m = 1, . . . . In all examined cases, the identifier converges within 5% of the reflection coefficients’ final value in the first 200 samples. For the overdetermined case, κ5 → 0; from the advocated design it is inferred that there is no need to maintain such a high order and it is therefore truncated to na = 4. The overall suggested concept’s efficiency is investigated in the responses of the closed loop system with the adaptive IM controller for a system of third-, fourth- and fifth-order; Figs. 9–11 indicate the corresponding output signals (dashed line) when the reference input was a squared-pulse waveform. In all the cases, the error signals r(t) − y(t) and βF (t|θ), θ = [κ1 , . . . , κna ] converge rapidly to the anticipated values (i.e., MSE(r(t) − y(t)) → −80 dB). 5. Conclusion An adaptive IMC scheme was presented in this paper. The unknown autoregressive system dynamics are identified through a lattice filter which estimates the process’ reflection coefficients. The order of the system is inferred by monitoring the magnitude of the estimated coefficients. The IM controller is the inverse of the estimated transfer function in cascade with a detuning filter. Simulation studies are used to illustrate the efficiency of the proposed scheme. Acknowledgement This project was supported in part by U. of Patras’ Young Investor Fund “Caratheodory”. References [1] M. Mariton, Jump Linear Systems in Automatic Control, Marcel-Dekker, New York, 1990. [2] O. Costa, J.D. Val, J. Geromel, A convex programming approach to H2 control of discrete time Markovian jump systems, Int. J. Control 66 (1997) 557–580. [3] P. Wellstead, M. Zarrop, Self-tuning Systems: Control and Signal Processing, Wiley, New York, 1991. [4] M. Morari, E. Zafiriou, Robust Process Control, Prentice-Hall, Englewood Cliffs, NJ, 1989. [5] G. Silva, A. Datta, Adaptive internal model control: the discrete case, Int. J. Adapt. Control Signal Processing 15 (2001) 15–36. [6] A. Datta, L. Xing, Adaptive internal model control: h∞ optimization for stable plants, IEEE Trans. Automat. Control 44 (1999) 2130–2134. [7] B. Anderson, R. Kosut, Adaptive robust control: on-line learning, in: Proceedings of the IEEE Conference on Decision and Control, Vol. 1, Brighton, UK, December 1991, pp. 297–298. [8] W. Lee, B. Anderson, R. Kosut, I. Mareels, A new approach to adaptive robust control, Int. J. Adapt. Control Signal Processing 7 (2) (1993) 183–211. [9] Y. Zhu, P. van der Bosch, Optimal closed-loop identification test design for internal model control, Automatica 36 (2001) 1237–1241. [10] D. Rivera, J. Pollard, C. Gar´cia, Control-relevant prefiltering: a systematic design approach and case study, IEEE Trans. Automat. Control 37 (7) (1992) 964–974. [11] B. Friedlander, Lattice filters for adaptive processing, Proceedings of the IEEE 70 (1982) 829–867. [12] S.S. Haykin, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, NJ, 2001. [13] A. Datta, J. Ochoa, Adaptive internal model control: design and stability analysis, Automatica 32 (1996) 261–266. [14] M.L. Honig, D.G. Messerschmitt, Adaptive Filters: Structures, Algorithms, and Applications, Kluwer Academic Publishers, Hingham, MA, 1984.
