A Single-Input Two-Output Feedback Formulation for ANC Problems

Transcription

1 A Single-Input Two-Output Feedback Formulation for ANC Problems V. Toochinda z,t.klawitter z, C.V. Hollot y and Y. Chait z zmie Department, yece Department, University of Massachusetts, Amherst, MA 3 Abstract This paper explores inherent feedback limitations of active noise control in ducts by using a single-input, twooutput framework and observing properties of closedloop transfer functions. Performance is assessed using the plant and disturbance alignment angle. We show that the sound levels at the measurement microphone are ampli ed when attenuating acoustic energy at the error microphone. We also show that the stability margins can be improved over feedforward control using measurements from two sensors. so-called feedforward/feedback con guration. In Section 3 we will show that the placement of the secondary source, relative to the disturbance source, plays an important role in performance. In particular, we will demonstrate, via analysis and examination of experimental data collected in [], that noise attenuation at the error microphone must come at the expense of ampli cation at the measurement microphone. We believe this is the rst time this observation is made in the literature. Our second result, discussed in Section, argues that the role of feedback in ANC is not for noise attenuation, but for stability robustness. In Section we begin our discussion by describing the ANC duct setup and the design results of []. Introduction In recent years, the potential bene t of using active noise control (ANC) in commercial applications has driven recent academic research; e.g., see [] and []. In contrast to passive techniques, a typical ANC scheme uses additional secondary sources and adaptive algorithms to cancel noise from the original primary source by, roughly speaking, introducing \anti-noise" - an exact but out-of-phase copy of the noise. The level of cancellation depends critically on the ability to produce such anti-noise in the face of uncertain system dynamics and noise properties. Such is the motivation for the introduction of adaptive cancellation techniques [] as well as the more recent feedforward/feedback methods [3,, 6]. This paper is a continuation of some of our previous work [3]-[6] where we explore the use of non-adaptive, xed- lter schemes for ANC in ducts. Our motivation for using xed- lters lies in the simplicity of implementation and availability of tools for analyzing stability and performance. Other work using the non-adaptive approach can be found in [7] and [8] and the references contained therein. The main goal of this paper is to analyze a nonadaptive ANC design reported in [], within the framework of single-input, two-output (SITO) feedback control previously developed in [9]. This design used two sensors (measurement and error microphones) and a The rst author is supported by Faculty of Engineering, Kasetsart University, Thailand. vtoochin@acad.umass.edu Description of ANC setup and design results Figure illustrates a basic con guration of an active noise control problem. It consists of a duct with two loudspeakers and two microphones mounted on it. The speaker located upstream simulates a disturbance source that injects acoustic \noise" into the duct. A measurement microphone detects the disturbance near the source and the downstream error microphone measures the level of noise cancellation at a point in the duct where noise attenuation is desired. The ANC system uses the information provided by these two microphones to generate a signal and send it to the canceling loudspeaker. The objective of the controller is to minimize the acoustic energy at the error microphone. In the ANC literature, the action taken on the measurement microphone signal y m is referred to as \feedforward" control, while action taken on y e is called \feedback" control. While such terminology can be ambiguous in the presence of acoustic feedback, we will nevertheless retain this terminology for sake of consistency. The ANC con guration used in [] is similar to that shown in Figure. The measurement microphone is colocated with the disturbance source, and the cancelling speaker is colocated with the error microphone. The duct-length is about one meter with a diameter of. meters. The two microphones are separated by The term feedforward comes from the fact that the measurement microphone is located upstream i.e. closer to the disturbance source. Unlike a conventional feedforward control, however, the acoustic feedback from the control speaker could a ect closed-loop stability.

2 measurement microphone error microphone the microphones and placement of the cancelling speaker. disturbance source d(t) y m (t) y e (t) ANC controller u(t) canceling loudspeaker Figure : ANC system in a duct. approximately.8 meters of duct. Motivation for this placement of measurement microphone and cancelling speaker came from [8], where only y m was considered as input to their ANC controller. Interestingly, we will see in Section 3 (see Remark ) that it may be preferrable to reconsider colocating the cancelling speaker with the disturbance { a con guration not recommended in [8]. With the con guration shown in Figure, a formal design of feedforward and feedback controllers was executed in [] using both H synthesis and QFT loopshaping techniques. A comparison of the experimental open and closed-loop performance is shown in Figure. This gure plots the magnitude frequency response of y e due to sinusoidal excitation at d. Note that this ANC design achieves broadband attenuation from approximately 8 - Hz with no ampli cation below Hz. Our subsequent analysis is driven by a desire to understand the roles of feedforward and feedback in this particular design. Magnitude (db) Open loop Closed loop The feedback controller C e did not contribute to the level of noise attenuation achieved at y e { this was primarily accomplished by the feedforward controller C m. Indeed, the attenuation curve in Figure is only slightly perturbed when we switched-out C e. However, the robustness of this performance to variations in C m were significantly improved by the presence of the feedback element C e. 3 Performance analysis In this section we analyze the performance of the ANC design in []. We start by treating the duct (or plant) as a SITO system P(s) where the input is the canceling speaker input u and the outputs are the microphone signals y m and y e, respectively. Conversely, we view the controller as a two-input, single-output system C(s) with inputs y m and y e and output u. Their interconnection forms the feedback control system shown in Figure 3. The objective of this section is to analyze the performance in y m and y e. We will show that y e is attenuated at the expense of ampli ed y m. Most importantly, this nding will be shown independent of the value or structure of the controllers (as long as y e is attenuated). It will depend only on the con guration of microphones and canceling speakers. To start, consider Figure 3, where (s), P eu (s), P md (s), and (s) are the transfer functions from the disturbance speaker to the error microphone, the canceling speaker to the error microphone, the disturbance speaker to the measurement microphone, and the canceling speaker to the measurement microphone, respectively. Let C m (s) denotethefeedforward controller and C e (s) thefeedback controller. Further, de ne the plant, controller, and disturbance transfer functions as: Figure : Comparison of open loop and closed-loop frequency response C(s) P(s) d Pd(s) Pmd Our empical observations were twofold: Cm Σ u Pmu Ped Σ d m y m. While noise attenuation was achieved at y e,as shown in Figure, ampli cation occurs at y m.in the next section, we will show why this occurs and prove that this ampli cation is not a function of the structure or value of the controllers, but is due to the duct dynamics coupled with the location of Ce Figure 3: ANC as a single input, two output feedback system Peu Σ d e ye

3 P(s) = Pmu (s) P eu (s) ; C(s) = C m (s) C e (s) : P d (s) = Pmd (s) (s) Assume zero input and initial conditions for the plant transfer function, the open loop response is then ym (s) y(s) OL = = P y e (s) d (s)d(s): Associated with this feedback system are several important transfer functions. These are the input and output loop transfer functions: L I (s) =C(s)P(s) and L O (s) = P(s)C(s), the input and output sensitivity functions: S I (s) = ( + L I (s)) and S O (s) = (I + L O (s)), and the input and output complementary sensitivity functions: T I (s) =L I (s)( + L I (s)) and T O (s) =L O (s)(i + L O (s)). The dimension of transfer functions at the plant input are x, while those at the plant output are x. In our context, the closed-loop response of interest is: dm (s) y(s) CL = S O (s) = S d e (s) O (s)p d (s)d(s): () We then de ne the attenuation factor as (!) = jjy(j!)jj CL jjy(j!)jj OL = jjs O(j!)P d (j!)jj : jjp d (j!)jj Inthesequelwewillshowthattherearesituations, experienced in the setup [] for which (!) mustbe greater than one. First, we de ne the notion of alignment angles introduced in [9]. De nition : The plant-controller alignment angle (at frequency!) is Á pc (j!) =cos jc(j!)p(j!)j () jjc(j!jjjjp(j!)jj while the plant-disturbance alignment angle is Á pd (j!) =cos jp H (j!)p d (j!)j : (3) jjp(j!jjjjp d (j!)jj The plant and controller (plant and disturbance) are said to be perfectly aligned if Á pc (j!) = ± (Á pd (j!) = ± ), and completely misaligned if Á pc (j!) = 9 ± (Á pd (j!) = 9 ± ). From [Proposition 9, 9], we have the following upper and lower bounds on the attenuation factor: (!) p sin Á pd (j!) +(cosá pd (j!)js I (j!)j +siná pd (j!)jt I (j!)j tan Ápc(j!)) () (!) p sin Á pd (j!) +j cos Á pd (j!)js I (j!)j sin Á pd (j!)jt I (j!)j tan Ápc(j!)j : (5) We now state the main result of this section. Its proof can be found in Appendix A. Proposition : If jy e (j!)j =and Á pd (j! o )=9 ±, then s jy m (j!)j jjp d (j!)jj = (!) + (j!) (j!) : (6) From Proposition we see that under perfect cancellation of y e (j!), y m (j!) must be ampli ed if the plant and disturbance transfer functions are completely misaligned. Furthermore, if j (j!)j À, then jy m (j!)j could be unacceptably large. Example : Using data from [], we plot Á pd (j!), jped(j!)j jp md(j!)j and the lower bound (5) in Figure. At! =¼(7) rad/sec, Á pd (j!) ¼ 9 ±,and j(j!)j experiences its peak. From Proposition, we would then predict that the closed loop response jy m (j!)j is large. This is veri ed in Figure 5. Thus, while jy e (j!)j was attenuated in the closed loop, it appears to come at the expense of amplifying jy m (j!)j. φ pd 5 (deg) Ped Pmd Lower bound (5) Figure : Á pd (j!); jped(j!)j, and lower bound (5) Remark : It is interesting to observe from Figures and 5 that both y m (j!) and y e (j!) are small at frequencies where the plant and disturbance are wellaligned; i.e., Á pd (j!) ¼ ±. Clearly, at these frequencies both the upper bound () and lower bound (5) collapse to js I (j!)j and the disturbance attenuation performance becomes a sensitivity minimization problem. For example, it may be unacceptable to have the disturbance ampli ed at any point in an HVAC system. The duct is a simpli ed model of such a system.

4 y m y e 5 5 Open loop Closed loop Open loop 8 Closed loop Figure 5: Open loop and closed loop comparison of y m (j!) andy e (j!) In other words, when Á pd (j!) = ±, the disturbance a ects the outputs in the same ay as does the control signal, and that this is favorable for disturbance rejection. Otherwise, one can only choose one of the two outputs for disturbance attenuation, and the control signal used to do so will act as a disturbance to the other output. Stability margins In the last section we concentrated on performance analysis and presented a situation where jy e (j!)j was attenuated at the expense of an ampli ed jy m (j!)j. Speci cally, we saw that jy m (j!)j peaks at the frequency where Á pd (j!) ¼ 9 ± and the ratio jped(j!)j jp md(j!)j has a maximum. So, we could expect a smaller stability margin (or a large jjs O (j!)jj) atthisfrequency. Therefore, our subsequent analysis will devote special attention to the condition Á pd (j!) ¼ 9 ± and j(j!)j À occurring when! = ¼(7) rad/sec. In this section we study the output sensitivity S O (j!), whose size jjs O (j!)jj provides one measure of stability margin; e.g. see []. Case : (feedforward only; C(j!) =[C m (j!) ]) In this case, the system's closed-loop response is given by " # S I (j!) S O (j!)= Peu(j!) P T : (7) mu(j!) I(j!) Now, suppose jy e (j!)j =. Then, from Lemmas and in Appendix A we have jt I (j!)j = jp ed(j!) (j!)j jp md (j!)p eu (j!)j ¼ (j!) (j!) : (8) Thus, with j(j!)j À, it follows from the fundamental algebraic constraint S I (j!)+t I (j!) =that js I (j!)j À. This in turn implies that jjs O (j!)jj À. Thus, whenever we use only feedforward control, C(j!) = [C m (j!) ], attenuation of jy e (j!)j at a frequency where both j(j!)j jp md À andplantdisturbance are misaligned necessarily leads to a re- (j!)j duced stability margin. Case : (feedforward and feedback control; C(j!) = [C m (j!) C e (j!)]) Using straightforward manipulations, the output sensitivity and complementary sensitivity functions can be written as and S O (s) =I S I (s)l O (s) T O (s) =S I (s)l O (s) where " S O (j!) Pmu(j!) P S O (j!)= T # eu(j!) O(j!) Peu(j!) P T mu(j!) O(j!) S O (j!) (9) and where S Oij (j!) andt Oij (j!) are the (i,j)th element of S O (j!) andt O (j!), respectively. Note the similarities and di erences to (7). The proof of the next propositon can be found in Appendix B. Proposition : If jy e (j!)j =,then Pmu(j!) P jt O (j!)j = eu(j!) Pmd(j!) (j!) + Ce(j!) C m(j!) : () Proposition shows that for perfect cancellation, jt O (j!)j depends not only on plant and disturbance transfer functions but also on both controllers. Therefore, unlike Case, the ratio j(j!)j jp eu (j!)j does not impose aconstraintonjt O (j!)j when y e (j!)=: Since S O (j!)+t O (j!) =,thenjt O (j!)j À implies js O (j!)j À, which, in using (9), leads to jjs O (j!)jj À and small stability margins. Note also that jt O (j!)j À implies poor robust stability to multiplicative uncertainty in. Example : Returningtotheexperimentsin[],we plot the magnitude ratios j(j!)j jp eu(j!)j and j(j!)j in Figure 6. The magnitude of each element of the output sensitivity S O (j!) is shown in Figure 7 for the case C(j!) =[C m (j!) ]: At! =¼(7) rad/sec, Á pd (j!) ¼ 9 ± (see Figure ), and jped(j!)j =3:7(from Figure 6). So, by (8), jt I (j!)j ¼(3:7) =3:7. From the algebraic constraint, js I (j!)j is commensurately large. The data yields jjs O jj ¼ 3, which compares favorably with the preceeding analysis. Now consider C(j!) =[C m (j!) C e (j!)]. Figure 8 shows the magnitude of the four elements of S O (j!).

5 5 S O.5 S O tor con gurations so that we can choose the setup that gives the best overall performance S O S O 5 Figure 6: Magnitude plots of S O(j!) forcasec(j!) = [C m(j!) ] 8 6 S O S O 6 Appendices A.ProofofProposition:First we need some lemmas. Lemma : Given!, y e (j!)=if and only if ³ T O (j!) P eu(j!) (j!) T Ped (j!) O(j!) P md (j!) =: () For the special case when C e (j!)=: T I (j!)= (j!) (j!) P md (j!)p eu (j!) : () 5 S O.5 5 S O Proof: From () we have Figure 7: Magnitude plots of S O(j!) forcasec(j!) = [C m(j!) C e(j!)] We can compute jt O (j!)j directly using Proposition, though it is easier to see from (9) that jt O (j!)j = j (j!)j jp js eu(j!)j O(j!)j. At! =¼(7) rad/sec, the data from Figure 6 and Figure 8 gives jpeu(j!)j j (j!)j =: and js O (j!)j =:7, resulting in jt O (j!)j = :7 : =:5. This implies that the size of js O (j!)j is comparable. This is con rmed in Figure 8, where js O (j!)j = jjs O jj ¼ 6; a signi cant improvement over jjs O jj ¼ 3 using just feedforward control. 5 Conclusions In this paper we have analyzed stability and performance of ANC in a duct using the SITO formulation. We derived a lower bound on closed-loop sound pressure level that depends only on open-loop transfer functions. We proved that using only feedforward controller imposes inherent limitations on the achievable closedloop performance. We also show that this limitation, in terms of stability margin, can be alleviated when both feedforward and feedback controllers are used. At this pointwehavenotdiscussedhowtodesignthecontrollers to achieve the performance objective. This will be a topic for further research. Another interesting research direction is to study the relation of plant - disturbance alignment angle to the sensor and actua- y e (j!)=s O (j!)p md (j!)+s O (j!) (j!) = T O (j!)p md (j!)+( T O (j!)) (j!) = T O (j!)p md (j!)+ P eu(j!) (j!) T O(j!) (j!): Letting y e (j!) = yields (). Equation () follows from C e = which, in turn, implies T O (j!)=and T O (j!)= Peu(j!) P T mu(j!) I(j!). Lemma : Suppose Á pd =9 ±.Then, and (j!) P eu (j!) = (j!) P md (j!) j (j!)j jp eu (j!)j independent of the controller 3. = j(j!)j (3) () Proof: Substitute Á pd =9 ± into (3). We need the following notation (see [9]) for stating Lemma 3: ~P(j!) ~P? (j!) ~C(j!) ~C? (j!) tan Á pc (j!) = P(j!)=jjP(j!)jj = [ p (j!) p (j!)]=jjp(j!)jj = C(j!)=jjC(j!)jj = [ c (j!) c (j!)] T =jjc(j!)jj = j~ C?H (j!) P(j!)j ~ j C(j!) ~ P(j!)j ~ = j~ C(j!) P ~?H (j!)j j C(j!) ~ P(j!)j ~ 3 Overbar denotes complex conjugate.

6 Lemma 3: Suppose Á pd =9 ± and y e (j!)=.then, jt I (j!)j tan Á pc (j!)= j(j!)j : (5) Proof: We will only show for the more general case C(j!) =[C m (j!) C e (j!)]. The situation C(j!) = [C m (j!) ] would then follow directly by setting C e (j!) =. The statement in Lemma 3 is equivalent to y e (j!)=; Á pd =9 ± ) jt I(j!)j tan Á pc (j!) (j!) (j!) =: To simplify notation, the dependence on! is omitted. Then, jt I j tan Á pc = jt I j j~ C P ~?H j j C ~ Pj ~ = jt I j j C ¹ m P eu + C e Pmu ¹ j ; jc m + C e P eu j jt I j tan Á pc = P ¹ eu T O + ¹ T O P eu : Therefore, jt Ij tan Á pc = P ¹ eu T O + ¹ T O P eu : Applying Lemma and rearranging terms gives jt Ij tan Á pc = P ¹ eu P ¹Peu ed + + ¹ T P eu P md P md O : Now,at Á pd =9 ±, we use Lemma to obtain jt Ij tan Á pc = ¹P eu ¹ Pmu P + ¹ eu Pmu ¹ + ¹ T P eu ¹P eu ¹P eu O P eu = ¹ P eu P eu =: We can now prove Proposition. Without loss of generality, assume jd(j!)j =. From (), y e (j!) = implies jy m (j!)j jjp d (j!)jj = jjs O(j!)P d (j!)jj = (!): jjp d (j!)jj Applying Lemma 3 and Á pd =9 ± to the lower bound (5) gives s (!) + (j!) (j!) and (6) follows. B. Proof of Proposition : Since T O = Peu T O, then, T O = P eu Ped = T O C ep eu T O C m P md P eu Ped P md P eu : Rearranging gives + C e T O = C m P md P md P eu and () follows. References [] S.M.KuoandD.R.Morgan,Active Noise Control Systems, Wiley-Interscience, New York, 996. [] P.A. Nelson and S. J. Elliott, Active Control of Sound, Academic Press, London, 99. [3] P.G. Metha, Y. Zheng, C.V. Hollot and Y. Chait, \Active Noise Control in Ducts: Feedforward/Feedback Design by Blending H and QFT Methods,"Proceedings of the 3th IFAC World Congress, San Francisco, 996. [] T. Klawitter, Linear Active Duct Noise Control: Feedforward/Feedback Controllers Designed with H and Quantitative Feedback Theory, M.S. Thesis, University of Massachusetts, Amherst,. [5] L. Li, Experimental Inverstigation of LTI Feedforward Active Noise Control Design From Experimental Frequency Responses, M.S. Thesis, University of Massachusetts, Amherst, 999. [6] P.G. Metha, Fixed-Filter Designs for Active Noise Control in Ducts, M.S. Thesis, University of Massachusetts, Amherst, 996. [7] A.J. Hull, C.J. Radcli e, and S. C. Southward, \Global Active Noise Control of a One Dimensional Acoustic Duct Using a Feedback Controller,"J. Dyn. Sys. Meas. Contr., Vol. 5, pp. 88-9, 993. [8] J. Hong and D.S. Bernstein, \Bode Integral Constraints, Colocation, and Spillover in Active Noise and Vibration Control," IEEE Trans. Control Systems Technology, Vol. 6(), pp. -, 998. [9] J.S. Freudenberg and R.H. Middleton, \Properties of Single Input, Two Output Feed back Systems," International Journal of Control, Vol. 7(6) pp. 6-65, 999. [] J.S. Freudenberg, C.V. Hollot, and D.P.Looze, \A First Graduate Course in Feedback Control," Course Notes, EECS 565, University of Michigan, Winter,.