Robust Control Theory and Applications Part 17 pot

where σ min (·) and σ max (·) denote the smallest and largest singular values, respectively. Suppose that the TF matrix is acoustically symmetric so that H p,11 (ω)=H p,22 (ω) and H p,21 (ω)=H p,12 (ω).Wenowhave H H p (ω)H p (ω)=2   H p,11 (ω)   2  1cos (2πλ −1 Δ) cos(2πλ −1 Δ) 1  , (32) where Δ denotes the interaural path difference given by Δ 11 − Δ 12 . Singular values can be found from the following characteristic equation: (1 −k) 2 −cos 2 (2πλ −1 Δ)=0. (33) By the definition of robustness, the equalization system will be the most robust when cos (2πλ −1 Δ)=0(H p (ω) is minimized) and the least robust when cos(2πλ −1 Δ)=±1 (H p (ω) is maximized) Ward & Elko (1999). A similar analysis can be applied to acoustic energy density control. The composite transfer function between the two loudspeakers and the two microphones in the pressure and velocity fields becomes H ed (ω)= ⎡ ⎢ ⎢ ⎣ H p,11 (ω) H p,21 (ω) ( ρc)H v,11 (ω)(ρc)H v,21 (ω) H p,12 (ω) H p,22 (ω) ( ρc)H v,12 (ω)(ρc)H v,22 (ω) ⎤ ⎥ ⎥ ⎦ , (34) where H v,ml (ω) is the frequency-domain matrix corresponding to H v,ml . Note that the pressure and velocity at a point in space x =(x, y, z), −→ v (x),andp(x) are related via jωρ −→ v (x)=−∇p(x), (35) where ∇ represents a gradient. Using this relation, the velocity component for the x direction can be written as H v x ,ml (ω)= 1 ρc · Δx ml d H p,ml (ω), (36) where d and Δx ml denote the distance and the x component of the displacement vector between the mth loudspeaker and the lth control point, respectively. Note that the velocity component for the y and z directions can be expressed similarly. Now we have H H ed (ω)H ed (ω)=2  2 Q cos (2πλ −1 Δ) Q cos( 2πλ −1 Δ) 2  , (37) where Q = 1 + Δx 11 Δx 12 + Δy 11 Δy 12 + Δz 11 Δz 12 d 11 (d 11 + Δ) . (38) Singular values can be obtained from the following characteristic equation: (2 −k) 2 −  Q cos  2πλ −1 Δ  2 = 0. (39) From Eqs. (33) and (39), it can be noted that the maximum condition number of H p (ω) equals to infinity, while that of H ed (ω) is (2 + Q) /(2 −Q),whencos(2πλ −1 Δ)=±1. Eq. (38) also shows that the maximum condition number of the energy density field becomes smaller as Δ increases because Q approaches to 1. Now, by comparing the maximum condition numbers, 628 Robust Control, Theory and Applications Fig. 5. The reciprocal of the condition number. the robustness of the control system can be inferred. Fig. 5 shows the reciprocal condition number for the case where the loudspeaker is symmetrically placed at a 1 m and 30 ◦ relative to the head center. The reciprocal condition number of the pressure control approaches to zero, but the energy density control has the reciprocal condition numbers that are relatively significant for entire frequencies. Thus, it can be said that the equalization in the energy density field is more robust than the equalization in the pressure field. Fig. 6. Simulation environments. (a) Configuration for the simulation of a multichannel sound reproduction system. (b) Control points in the simulations. l 0 corresponds to the center of the listener’s head. 0 2000 4000 6000 8000 10000 12000 14000 16000 0 0.5 Frequency (Hz) V min / V max 0 2000 4000 6000 8000 10000 12000 14000 16000 0 0.5 1 Frequency (Hz) V min / V max (b) Energy density control 2 2 Q Q  + 629 Robust Inverse Filter Design Based on Energy Density Control 4. Performance Evaluation We present simulation results to validate energy density control. First, the robustness of an inverse filtering for multichannel sound reproduction system is evaluated by simulating the acoustic responses around the control points corresponding to the listener’s ears. The performance of the robustness is objectively described in terms of the spatial extent of the equalization zone. 4.1 S imulation result In this simulation, we assumed a multichannel sound reproduction system consisting of four sound sources (M = 4) as shown in Fig. 6(a). Details of the control points are depicted in Fig. 6(b). We assumed a free field radiation and the sampling frequency was 48 kHz. Impulse responses from the loudspeakers to the control points were modeled using 256-tap FIR filters (N h = 256), and equalization filters were designed using 256-tap FIR filters (N w = 256). The conventional LS method was tried by jointly equalizing the acoustic pressure at l 1 , l 2 , l 3 ,andl 4 points, and the energy density control was optimized only for the l 0 point. The delayed Dirac delta function was used for the desired response, i.e., d p,l 0 (n)=···= d p,l 4 (n)=δ(n −n 0 ). Center The control point (cm) frequency (0, 0) (0, 5) (2.5, 2.5) (5, 0) (5, 5) 500 Hz 0.06 -0.28 -0.13 -0.42 -0.28 1kHz 0.30 -1.39 -0.60 -1.91 -3.55 2kHz 1.26 -7.61 -2.76 -14.53 -10.25 Table 1. The error in dB for the pressure control system based on joint LS optimization at each center frequency. Center The control point (cm) frequency (0, 0) (0, 5) (2.5, 2.5) (5, 0) (5, 5) 500 Hz 0.00 0.25 0.09 -0.21 0.03 1kHz 0.00 0.25 0.06 -0.95 -0.76 2kHz 0.00 0.25 -0.69 -4.50 -4.58 Table 2. The error in dB for the energy density control system at each center frequency. We scanned the equalized responses in a 10 cm square region around the l 0 position, and results are shown in Fig. 7. Note that only the upper right square region was evaluated due to the symmetry. For the energy density control, velocity x and y were used. Velocity z was not used. As evident in Fig. 7, the energy density control shows a lower error level than the joint LS-based squared pressure control over the entire region of interest except at the points corresponding to l 2 (2 cm,0 cm) and l 4 (0 cm,2 cm), where the control microphones for the joint LS control were located. Next, an equalization error was measured as the difference between the desired and actual responses defined by C (dB)=10 log ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ω max ∑ ω=ω min   D(ω) − ˆ D(ω)   2 ω max ∑ ω=ω min | D( ω) | 2 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ , (40) 630 Robust Control, Theory and Applications Fig. 7. The spatial extent of equalization by controlling pressure based joint LS optimization and energy density. 10 2 10 4 -10 0 10 (0cm, 5cm) Response (dB) 10 2 10 4 -10 0 10 (1cm, 5cm) 10 2 10 4 -10 0 10 (2cm, 5cm) 10 2 10 4 -10 0 10 (3cm, 5cm) 10 2 10 4 -10 0 10 (4cm, 5cm) 10 2 10 4 -10 0 10 (5cm, 5cm) 10 2 10 4 -10 0 10 (0cm, 4cm) Response (dB) 10 2 10 4 -10 0 10 (1cm, 4cm) 10 2 10 4 -10 0 10 (2cm, 4cm) 10 2 10 4 -10 0 10 (3cm, 4cm) 10 2 10 4 -10 0 10 (4cm, 4cm) 10 2 10 4 -10 0 10 (5cm, 4cm) 10 2 10 4 -10 0 10 (0cm, 3cm) Response (dB) 10 2 10 4 -10 0 10 (1cm, 3cm) 10 2 10 4 -10 0 10 (2cm, 3cm) 10 2 10 4 -10 0 10 (3cm, 3cm) 10 2 10 4 -10 0 10 (4cm, 3cm) 10 2 10 4 -10 0 10 (5cm, 3cm) (0cm 2cm) (1cm 2cm) (2cm 2cm) (3cm 2cm) (4cm 2cm) (5cm 2cm) Pressure control (LS) Energy density control (LS) 10 2 10 4 -10 0 10 (0cm , 2cm) Response (dB) 10 2 10 4 -10 0 10 (1cm , 2cm) 10 2 10 4 -10 0 10 (2cm , 2cm) 10 2 10 4 -10 0 10 (3cm , 2cm) 10 2 10 4 -10 0 10 (4cm , 2cm) 10 2 10 4 -10 0 10 (5cm , 2cm) 10 2 10 4 -10 0 10 (0cm, 1cm) Response (dB) 10 2 10 4 -10 0 10 (1cm, 1cm) 10 2 10 4 -10 0 10 (2cm, 1cm) 10 2 10 4 -10 0 10 (3cm, 1cm) 10 2 10 4 -10 0 10 (4cm, 1cm) 10 2 10 4 -10 0 10 (5cm, 1cm) 10 2 10 4 -10 0 10 (0cm, 0cm) Response (dB) Frequency (Hz) 10 2 10 4 -10 0 10 (1cm, 0cm) Frequency (Hz) 10 2 10 4 -10 0 10 (2cm, 0cm) Frequency (Hz) 10 2 10 4 -10 0 10 (3cm, 0cm) Frequency (Hz) 10 2 10 4 -10 0 10 (4cm, 0cm) Frequency (Hz) 10 2 10 4 -10 0 10 (5cm, 0cm) Frequency (Hz) 631 Robust Inverse Filter Design Based on Energy Density Control where ω min and ω max denote the minimum and maximum frequency indices of interest, respectively. In order to compare the robustness of equalization, we evaluated the pressure level in the vicinity of the control points. The equalization errors are summarized in Tables 1 and 2. Results show that the energy density control has a significantly lower equalization error than the joint LS-based squared pressure control, especially at 2 kHz where there are 7 ∼ 10 dB differences. Fig. 8. A three-dimensional plot of the error surface for the pressure control (left column) and the energy density control (right column) at different center frequencies. Finally, three-dimensional contour plots of the equalization errors are presented in Fig. 8. Fig. 8(a) and (d) show both methods have similar equalization performance at 1 kHz due to the relatively long wavelength. However, Figs. 8 (a), (b), and (c) indicate that the error of the pressure control rapidly increases as the frequency increased. On the other hand, the energy density control provides a more stable equalization zone, which implies that the energy density control can overcome the observability problem to some extent. Thus, it can be concluded that the energy density control system can provide a wider zone of equalization than the pressure control system. 4.2 Implementation consideration It should be mentioned that it is necessary to have the acoustic velocity components to implement the energy density control system. It has been demonstrated that the 632 Robust Control, Theory and Applications two-microphone approach yields performance which is comparable to that of ideal energy density control in the field of the active noise control system Park & Sommerfeldt (1997). Thus, it is expected that the energy density control being implemented using the two-microphone approximation maintains the robustness of room equalization observed in the previous simulations. To examine this, we applied two microphone techniques, which were described in section 3.3, to determine the acoustic velocity along an axis. By using Eq. (28), simulations were conducted for the case of Δx = 2cm to evaluate the performance of the two-sensor implementation. Here, l 0 and l 2 are used for estimating the velocity component for x direction and l 0 and l 4 are used for estimating the velocity component for y direction; the velocity component for z direction was not applied. The results obtained by using the ideal velocity signal and two microphone technique are shown in Fig. 9. It can be concluded that the energy density system employing the two microphone technique provides comparable performance to the control system employing the ideal velocity sensor. Fig. 9. The performance of the energy density control algorithm being implemented using the two microphone technique. 5. Conclusion In this chapter, a method of designing equalization filters based on acoustic energy density was presented. In the proposed algorithm, the equalization filters are designed by minimizing the difference between the desired and produced energy densities at the control points. For the effective frequency range for the equalization, the energy density-based method provides more robust performance than the conventional squared pressure-based method. Theoretical analysis proves the robustness of the algorithm and simulation results showed that the proposed energy density-based method provides more robust performance than the conventional squared pressure-based method in terms of the spatial extent of the equalization zone. 633 Robust Inverse Filter Design Based on Energy Density Control 6.References Abe, K., Asano, F., Suzuki, Y. & Sone, T. (1997). Sound field reproduction by controlling the transfer functions from the source to multiple points in close proximity, IEICE Trans. Fundamentals E80-A(3): 574–581. Elliott, S. J. & Nelson, P. A. (1989). Multiple-point equalization in a room using adaptive digital filters, J. Audio Eng. Soc. 37(11): 899–907. Gardner, W. G. (1997). Head-tracked 3-d audio using loudspeakers, Proc. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New Paltz, NY, USA. Hodges, T., Nelson, P. A. & Elliot, S. J. (1990). The design of a precision digital integrator for use in an active vibration control system, Mech. Syst. Sign. Process. 4(4): 345–353. Kirkeby, O., Nelson, P. A., Hamada, H. & Orduna-Bustamante, F. (1998). Fast deconvolution of multichannel systems using regularization, IEEE Trans. on Speech and Audio Process. 6(2): 189–195. Mourjopoulos, J. (1994). Digital equalization of room acoustics, J. Audio Eng. Soc. 42(11): 884–900. Mourjopoulos, J. & Paraskevas, M. (1991). Pole-zero modelling of room transfer functions, J. Sound and Vib. 146: 281–302. Nelson, P. A., Bustamante, F. O. & Hamada, H. (1995). Inverse filter design and equalization zones in multichannel sound reproduction, IEEE Trans. on Speech and Audio Process. 3(3): 185–192. Nelson, P. A., Hamada, H. & Elliott, S. J. (1992). Adaptive inverse filters for stereophonic sound reproduction, IEEE Trans. on Signal Process. 40(7): 1621–1632. Park, Y. C. & Sommerfeldt, S. D. (1997). Global control of broadband noise fields using energy density control, J. Acoust. Soc. Am. 101: 350–359. Parkins, J. W., Sommerfeldt, S. D. & Tichy, J. (2000). Narrowband and broadband active control in an enclosure using the acoustic energy density, J. Acoust. Soc. Am. 108(1): 192–203. Rao, H. I. K., Mathews, V. J. & Park, Y C. (2007). A minimax approach for the joint design of acoustic crosstalk cancellation filters, IEEE Trans. on Audio, Speech and Language Process. 15(8): 2287–2298. Sommerfeldt, S. D. & Nashif, P. J. (1994). An adaptive filtered-x algorithm for energy-based active control, J. Acoust. Soc. Am. 96(1): 300–306. Sturm, J. F. (1999). Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones, Optim. Meth. Softw. 11-12: 625–653. Toole, F. E. & Olive, S. E. (1988). The modification of timbre by resonances: Perception and measurement, J. Audio Eng. Soc. 36: 122–141. Ward, D. B. (2000). Joint least squares optimization for robust acoustic crosstalk cancellation, IEEE Trans. on Speech and Audio Process. 8(2): 211–215. Ward, D. B. & Elko, G. W. (1999). Effect of loudspeaker position on the robustness of acoustic crosstalk cancellation, IEEE Signal Process. Lett. 6(5): 106–108. 634 Robust Control, Theory and Applications 30 Robust Control Approach for Combating the Bullwhip Effect in Periodic-Review Inventory Systems with Variable Lead-Time Przemysław Ignaciuk and Andrzej Bartoszewicz Institute of Automatic Control, Technical University of Łódź Poland 1. Introduction It is well known that cost-efficient management of production and goods distribution systems in varying market conditions requires implementation of an appropriate inventory control policy (Zipkin, 2000). Since the traditional approaches to inventory control, focused mainly on the statistical analysis of long-term variables and (static) optimization performed on averaged values of various cost components, are no longer sufficient in modern production-inventory systems, new solutions are being proposed. In particular, due to the resemblance of inventory management systems to engineering processes, the methods of control theory are perceived as a viable alternative to the traditional approaches. A summary of the initial control-theoretic proposals can be found in (Axsäter, 1985), whereas more recent results are discussed in (Ortega & Lin, 2004) and (Sarimveis et al., 2008). However, despite a considerable research effort, one of the utmost important, yet still unresolved (Geary et al., 2006) problems observed in supply chain is the bullwhip effect, which manifests itself as an amplification of demand variations in order quantities. We consider an inventory setting in which the stock at a distribution center is used to fulfill an unknown, time-varying demand imposed by customers and retailers. The stock is replenished from a supplier which delivers goods with delay according to the orders received from the distribution center. The design goal is to generate ordering decisions such that the entire demand can be satisfied from the stock stored at the distribution center, despite the latency in order procurement, referred to as lead-time delay. The latency may be subject to significant fluctuations according to the goods availability at the supplier and transportation time uncertainty. When demand is entirely fulfilled any cost associated with backorders, lost sales, and unsatisfied customers is eliminated. Although a number of researchers have recognized the need to explicitly consider the delay in the controller design and stability analysis of inventory management systems, e.g. Hoberg et al. (2007), robustness issues related to simultaneous delay and demand fluctuations remain to a large extent unexplored (Dolgui & Prodhon, 2007). A few examples constitute the work of Boukas et al. (2000), where an H ∞ -norm-based controller has been designed for a production- inventory system with uncertain processing time and input delay, and Blanchini et al. (2003), who concentrated on the stability analysis of a production system with uncertain demand and process setup. Both papers are devoted to the control of manufacturing Robust Control, Theory and Applications 636 systems, rather than optimization of goods flow in supply chain, and do not consider rate smoothening as an explicit design goal. On the contrary, in this work, we focus on the supply chain dynamics and provide formal methods for obtaining a smooth, non-oscillatory ordering signal, what is imperative for reducing the bullwhip effect (Dejonckheere et al., 2003). From the control system perspective we may identify three decisive factors responsible for poor dynamical performance of supply chains and the bullwhip effect: 1) abrupt order changes in response to demand fluctuations, typical for the traditional order-up-to inventory policies, as discussed in (Dejonckheere et al., 2003); 2) inherent delay between placing of an order and shipment arrival at the distribution center which may span several review periods; and finally, 3) unpredictable variations of lead-time delay. Therefore, to avoid (or combat) the bullwhip effect, the designed policy should smoothly react to the changes in market conditions, and generate order quantities which will not fluctuate excessively in subsequent review intervals even though demand exhibits large and unpredictable variations. This is achieved in this work by solving a dynamical optimization problem with quadratic performance index (Anderson & Moore, 1989). Next, in order to eliminate the negative influence of delay variations, a compensation technique is incorporated into the basic algorithm operation together with a saturation block to explicitly account for the supplier capacity limitations. It is shown that in the inventory system governed by the proposed policy the stock level never exceeds the assigned warehouse capacity, which means that the potential necessity for an expensive emergency storage outside the company premises is eliminated. At the same time the stock is never depleted, which implies the 100% service level. The controller demonstrates robustness to model uncertainties and bounded external disturbance. The applied compensation mechanism effectively throttles undesirable quantity fluctuations caused by lead-time changes and information distortion thus counteracting the bullwhip effect. 2. Problem formulation We consider an inventory system faced by an unknown, bounded, time-varying demand, in which the stock is replenished with delay from a supply source. Such setting, illustrated in Fig. 1, is frequently encountered in production-inventory systems where a common point (distribution center), linked to a factory or external, strategic supplier, is used to provide goods for another production stage or a distribution network. The task is to design a control strategy which, on one hand, will minimize lost service opportunities (occurring when there is insufficient stock at the distribution center to satisfy the current demand), and, on the other hand, will ensure smooth flow of goods despite model uncertainties and external disturbances. The principal obstacle in providing such control is the inherent delay between placing of an order at the supplier and goods arrival at the center that may be subject to significant fluctuations during the control process. Another factor which aggravates the situation is a possible inconsistency of the received shipments with regard to the sequence of orders. Indeed, it is not uncommon in practical situations to obtain the goods from an earlier order after the shipment arrival from a more recent one. In addition, we may encounter other types of disturbances affecting the replenishment process related to organizational issues and quality of information (Zomerdijk & de Vries, 2003) (e.g. when a shipment arrives on time but is registered in another review period, or when an incorrect order is issued from [...]... system behavior and interactions among the principal system variables (ordering signal, on-hand stock level and imposed demand) will be used to develop a discrete control strategy goverining the flow of goods between the supplier and the distribution center 640 Robust Control, Theory and Applications 3 Proposed inventory policy In this section, we formulate a new inventory management policy and discuss... Table 1 Controller parameters in Scenario 1 Fig 5 Generated orders Fig 6 Received shipments Scenario 2 In the second scenario, we investigate the controller behavior in the presence of highly variable stochastic demand Function d(·) following the normal distribution with mean dμ = 25 items and standard deviation dδ = 25 items, Dnorm(25, 25), is illustrated in Fig 8 650 Robust Control, Theory and Applications. .. delay following the normal distribution with mean 8 days and standard deviation 2 days Fig 10 Generated orders 652 Robust Control, Theory and Applications Fig 11 Received shipments Fig 12 On-hand stock 5 Conclusion In this chapter, we presented a robust supply policy for periodic-review inventory systems The policy is designed based on sound control- theoretic foundations with the aim of reducing the... G & de Vries, J (2003) An organizational perspective on inventory control: Theory and a case study International Journal of Production Economics, Vol 81–82, No 1, 173 –183, ISSN: 0377-2 217 31 Robust Control Approaches for Synchronization of Biochemical Oscillators Hector Puebla1 , Rogelio Hernandez Suarez2 , Eliseo Hernandez Martinez3 and Margarita M Gonzalez-Brambila4 1,3,4 Universidad Autónoma Metropolitana... offered by a dead-beat scheme 3.4 Robustness issues The order calculation performed according to (28) is based on the nominal delay which constitutes an estimate of the true (variable) lead-time set according to the contracting agreement with the supplier The controller designed for the nominal system is robust with 644 Robust Control, Theory and Applications respect to demand fluctuations, yet may generate... at l1T, and this value can be as large as umax Consequently, the sum l −n −1 ∑ j =l 1 u ( jT ) = u ( l1T ) ≤ umax From inequalities (3) and the condition ξ(lT) ≤ ξmax we obtain the following stock estimate y ( lT ) ≤ y d + εξ ( l1T ) + ξ ( lT ) + u ( l1T ) ≤ y d + εξmax + ξmax + umax = ymax , (41) 646 Robust Control, Theory and Applications which concludes the second part of the reasoning and completes... simulations we test the controller performance in response to the demand pattern illustrated in Fig 3, which shows a trend in the demand with abrupt seasonal changes It is assumed that lead-time fluctuates according to 648 Robust Control, Theory and Applications L ( k ) = ⎢ ⎣1 + δ sin ( 2 πkT / n ) ⎦ nT ⎥ = ⎢ ⎣1 + 0.25sin ( πk / 4 ) ⎦ 8⎥ , ⎤ ⎦ ⎣⎡ ⎤ ⎦ ⎣⎡ (51) where ⎣f⎦ denotes the integer part of f The actual... ≤ d ( kT ) ≤ dmax (2) Notice that this definition of demand is quite general and it accounts for any standard distribution typically analyzed in the considered problem If there is a sufficient number of items in the warehouse to satisfy the imposed demand, then the actually met demand h(kT) 638 Robust Control, Theory and Applications (the number of items sold to customers or sent to retailers in the... properties related to handling the flow of goods First, the nominal system is considered, and the controller parameters are selected by solving a linear-quadratic (LQ) optimization problem Afterwards, the influence of perturbation is analyzed and an enhanced, nonlinear control law is formulated which demonstrates robustness to delay and demand variations The key element in the improved controller structure... Patrinos, P.; Tarantilis, C D & Kiranoudis, C T (2008) Dynamic modeling and control of supply chain systems: A review Computers & Operations Research, Vol 35, No 11, 3530–3561, ISSN: 0305-0548 654 Robust Control, Theory and Applications Zabczyk, J (1974) Remarks on the control of discrete-time distributed parameter systems SIAM Journal on Control, Vol 12, No 4, 721–735, ISSN: 0036-1402 Zipkin, P H (2000) . loudspeaker position on the robustness of acoustic crosstalk cancellation, IEEE Signal Process. Lett. 6(5): 106–108. 634 Robust Control, Theory and Applications 30 Robust Control Approach for Combating. of a production system with uncertain demand and process setup. Both papers are devoted to the control of manufacturing Robust Control, Theory and Applications 636 systems, rather than. We get k ij = k 11 – Robust Control, Theory and Applications 642 (j – 1)w for j ≥ i (the upper part of K) and k ij = k 11 – (i – 1)w for j < i (the lower part of K). In matrix form