NCJ MarApr INSIDES-2 - CiteSeerX

Robust broadband control of acoustic noise in ducts: a passivity-based approach A. G. Kelkara) and H. R. Potab) (Received 2001 November 19; revised 2003 March 04; accepted 2003 March 07) An active feedback controller design methodology based on passivity-based robust control techniques is presented for broad-band noise control in an acoustic duct. The controller design methodology is demonstrated on an experimental one-dimensional (1-D) acoustic duct facility. The experimental results exhibit the effectiveness of the controller in suppressing acoustic noise levels over a broad frequency range without destabilizing high frequency dynamics of the system. The controller design is shown to be robust to unmodelled dynamics and parametric uncertainties. A finite dimensional mathematical model is derived for a 1-D acoustic duct using analytical as well as system identification techniques. It is shown that the theoretically determined model agrees very well with the experimentally identified system model. The control design methodology exploits inherent robustness of passivity-based controllers and selective mode attenuation capability of resonant mode controllers. The controller is easy to implement as it uses only output feedback. Moreover, the controller is also low-order, robust, broadband, and has guaranteed stability. © 2003 Institute of Noise Control Engineering. Primary subject classification: 37.7; Secondary subject classification: 38.2

1. INTRODUCTION Control of acoustic systems is a challenging problem for several reasons. Some primary reasons include computational complexity resulting from very high-order models, nonminimum phase behavior introduced by finite dimensional approximations and transportation delays, uncertainties introduced by non-uniform boundary conditions, and acoustic interaction with the dynamics of the enclosure structure. Until recently, most active noise control techniques focused on feedforward cancellation. It is only in the last few years that attempts have been made to use feedback for active noise control. Work by Hu and co-workers is remarkable for its thoroughness in setting up the system model, the theoretical analysis, and providing experimental demonstrations.1-3 The work by Bernstein and co-workers also provides a solid analysis and application of feedback control.4 Other examples of feedback control approaches are direct-rate feedback5 and pole placement.6 Much work continues towards the design and analysis of feedback control methods for achieving broadband reduction which is also robust to uncertainties. Acoustic ducts have certain dynamic characteristics that make it difficult to design an active feedback controller. Firstly, the model has no natural roll-off at high frequencies and it is modally very rich. Also, the frequency response is characterized by resonant peaks which dominate the dynamics. In the presence of these characteristics, any uncertainty in the system model has a significant affect on closed-loop stability. Much of the design of feedback controllers for acoustic systems is based on models identified using experimental frequency response. Acoustic ducts in the form of heating, ventilation, air-conditioning, and exhaust ducts form a significant category of systems which need active noise control. An ability to derive a)

Mechanical Engineering, Iowa State University, Ames, IA 50011, U.S.A.; E-mail: [email protected] b) School of Electrical Engineering, University of New South Wales, ADFA Canberra ACT 2600, Australia; [email protected] Noise Control Eng. J. 51 (2), 2003 Mar–Apr

analytical models for this class of systems will be most helpful in developing controllers for active noise control. A simple method using symbolic computation is presented in Ref. 7 to derive models for virtually any configuration of an acoustic duct. The analytically derived model for an acoustic duct is linear, time-invariant, and infinite-dimensional. For most control designs, however, a finite-dimensional approximation is needed. In this paper, a finite-dimensional approximation model is obtained using an assumed modes approach. A detailed discussion of the modelling and identification of acoustic ducts can be found in Ref. 7. The control design methodology used in this paper is based on passivity theory. An experimental validation of the passivity-based robust control design methodology based on the ideas given in Refs. 8-12 is presented. It is shown that controllers designed using passivity theory can achieve broadband reduction of noise. It is also shown that the controller design is not only robust to unmodelled dynamics (higher order modes) and modelling errors, but also to parametric variations. The experimental verification demonstrates the efficacy of the controller. To facilitate understanding of passivity-based control design as presented in later sections of this paper a brief review of passivity-based control is given next. 2. BRIEF REVIEW OF PASSIVITY-BASED CONTROL Passivity is an important property of dynamic systems. A large class of physical systems, such as flexible space structures with collocated and compatible actuators and sensors, can be classified as being naturally passive. The important passivity result states that a passive system can be robustly stabilized by any strictly passive feedback controller, despite unmodelled dynamics and parametric uncertainties. This important stability characteristic has attracted attention of control designers to © 2003 Institute of Noise Control Engineering

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passivity-based design methods. Robust stabilization and control using passivity techniques has been one of the most active research topics in the past few decades. A number of stability results exist in the literature for passive systems. The next section presents a few selected definitions and stability results that are relevant to this paper. A. Passivity of linear systems For finite-dimensional linear, time-invariant (LTI) systems, passivity is equivalent to “positive realness” of the transfer function.13,14 The concept of strict positive realness has also been defined in the literature, and is closely related to strict passivity. Let G(s) denote an m x m matrix whose elements are proper rational functions of the complex variables. G(s) is said to be stable if all its elements are analytic in Re(s) ≥ 0. Let the conjugate-transpose of a complex matrix H be denoted by H*. Definition 1: An m x m rational matrix G(s) is said to be positive real (PR) if (i) all elements of G(s) are analytic in Re(s) > 0; (ii) G(s) + G*(s) ≥ 0 in Re(s) > 0; or equivalently, (iia) poles on the imaginary axis are simple and have nonnegative-definite residues, and (iib) G(jω) + G*(jω) ≥ 0 for ωε (–∞,∞). There are various definitions of strictly positive real (SPR) systems found in the literature.9 Given below is the definition of a class of SPR systems, namely, marginally strictly positivereal (MSPR) systems. Definition 2:15 An m x m rational matrix G(s) is said to be marginally strictly positive real (MSPR) if it is positive real, G(jω) + G*(jω) > 0 for ωε (–∞,∞). Definition 2 gives the least restrictive class of SPR systems. If G(s) is MSPR, it can be expressed as: G(s) = G1(s) + G2(s), where G2(s) is weak SPR9 and all the poles of G1(s) (in the Smith-McMillan sense) are purely imaginary.15 The stability theorem for feedback interconnection of PR and MSPR system is given next without proof. Stability Theorem:15 The closed-loop system consisting of negative feedback interconnection of G1(s) and G2(s) (Fig. 1) is globally asymptotically stable if G1(s) is PR, G2(s) is MSPR, and none of the purely imaginary poles of G2(s) is a transmission zero of G1(s).15 Note that in the above theorem, systems G1(s) and G2(s) can be interchanged. Some nonlinear extensions of these results are also obtained in Ref. 16. Passivity-based controllers based

on these fundamental stability results have proved to be highly effective in robustly controlling inherently passive linear as well as nonlinear systems. Many physical systems, however, are not inherently passive, and passivity-based control methods do not extend directly to such systems. An acoustic system under consideration is one such example. One possible method for making these non-passive systems amenable to passivity-based control is to passify such systems (i.e., rendering them passive) using suitable compensation. If the compensated system is ensured to be robustly passive despite plant uncertainties, it can be robustly stabilized by any MSPR controller (refer to stability theorem given above). In Ref. 10, various passification techniques were presented and some numerical examples were given demonstrating the use of such techniques. A brief review of passification methods is given next. B. PASSIFICATION METHODS Four different passifucation methods, series, feedback, feedforward, and hybrid passification were given in Ref. 10 for finite-dimensional linear time-invariant non-passive systems. Figure 2 shows five types of passification configurations which include the above mentioned four plus the passification technique based on actuator-sensor blending. For the acoustic duct considered in this paper, a combination of feedforward and series compensation was required to passify the system. For nonminimum phase systems, the first step in passification is to render the system minimum-phase by feedforward compensation and then use additional compensation, if necessary, to render the resulting minimum phase plant positive-real. Once passified, the system can be controlled by any MSPR or weakly SPR (WSPR) controller.16 One important thing to be noted here is that, in the case of inherently

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Fig. 1– Closed-loop system.

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Fig. 2– Methods of passification.

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The “true” model of the plant is obtained experimentally from a system identification approach using a Stanford Research 785 spectrum analyzer and sine-sweep excitation in the frequency band up to 1 kHZ. Figure 4 shows the experimentally obtained open-loop frequency response of the system. The experimentally obtained frequency response data was used to derive the finite dimensional approximate model of the system which includes modes up to 500 Hz. Both, discretetime and continuous-time system identification algorithms were used to identify the system. The results obtained using these two algorithms were found to be almost identical. Figure 5 shows the system identification results obtained using the continuous-time algorithm. For analytical development, a finite dimensional analytical model was also derived from the infinite dimensional system model using the assumed modes method. Although the use of system identification algorithms tends to be a more efficient way to obtain a finite dimensional model of the plant, it is important to develop an analytical model and validate its accuracy by comparing it to experimental data. The

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passive systems, the use of an MSPR controller guarantees stability robustness to unmodelled dynamics and parametric uncertainties; however, in the case of non-passive systems which are rendered passive using passifying compensation, the stability robustness depends on the robustness of passification. That is, the problem of robust stability is transformed into the problem of robust passification. In Ref. 11, a number of frequency-domain sufficient conditions were derived to check the robustness of passification. The acoustic plant under consideration is the 1-D acoustic duct facility (Fig. 3) located in the Controls Research Lab at Iowa State University. The analytical model for this duct is obtained using a finite dimensional approximation based on the assumed modes approach.7 This finite-dimensional model has non-minimum phase zeros introduced by quartic symmetry needed for realizing zero dynamics of the plant to match the true frequency response of the system. The system identification algorithms also yield a plant model with such undesirable non-minimum phase characteristics. Obviously, such a plant model is not suitable for passivity-based control as it does not satisfy passivity conditions. However, the passification techniques10 outlined in section 2.B can be used in conjunction with the control techniques presented in Ref. 16 to design passivity-based control for such a system. The next section presents a robust controller design using passivity techniques for the acoustic system under consideration.

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3. ROBUST CONTROLLER DESIGN FOR AN ACOUSTIC DUCT: A PASSIVITY-BASED APPROACH

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Fig. 4– Experimentally obtained frequency response.

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The passivity-based design techniques discussed in section 2.B will be used to synthesize a robustly stabilizing controller for the 1-D acoustic duct under consideration. The open-loop system under consideration is a 46th order ISU acoustic-duct model including the dynamics of the speaker and microphone (Fig. 3). The physical dimensions of the ISU duct are: length 3.66 m (12 ft) and circular cross section of 0.28 m (11 in) diameter. The duct is equipped with speakers and microphones. The length of the duct can also be altered through a telescopic mechanism. One of the speakers is used as a disturbance and another is used as a control speaker. The microphone located inside of the duct is used as a feedback sensor.

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Fig. 3– 1-D acoustic duct facility. Noise Control Eng. J. 51 (2), 2003 Mar–Apr

Fig. 5– Experimental and identified responses. 99

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where, xp(t) is the (np x 1) state vector, yperf (t) is the (q x 1) performance output vector, yp(t) is the (l x 1) sensor output vector, and u(t) is the (m x 1) control input vector. For the system under consideration, two configurations were considered. In the first configuration, it was assumed that the control and disturbance signal are entering into the system through the same channel and the sensor output is the same as the performance output, i.e., Cperf = Cp (i.e., q = l), Dperf = Dp. In the second configuration, it was assumed that the disturbance speaker is different from the control speaker. The open-loop Bode plot of the system (Po(s)) is shown in Fig. 7. The open-loop system is non-minimum phase, and therefore, not inherently passive. This characteristic of non-minimum phase

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Fig. 6– Experimental and assumed modes-based analytical model responses. 100


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y1 (t ) = Cp x p (t ) + D1u(t )

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behavior is representative of all finite-dimensional acoustic systems with non-collocated speakers and microphones. Since such a plant is not inherently passive, passivity-based control techniques cannot extend directly to this system. However, as stated previously, even for non-passive systems, it is possible to exploit the inherent stability robustness property of passivitybased controllers if such a system can be rendered passive via suitable compensation. Once the system is rendered passive, any weak SPR or marginally SPR controller can be used to stabilize the closed-loop system.9 Moreover, as stated previously, if the passification is robust, stability is also robust. Sufficient conditions for robust passification were given in Ref. 11 which can be used to test if the compensated system remains passive under perturbation given the upper bound on the uncertainty. Recently, LMI-based techniques were also developed for robust passification.12 These techniques are especially useful for multi-input multi-output systems. For the ISU acoustic duct system under consideration, a hybrid (feedforward plus series) passification technique was used to passify the system.10 The control system block diagram is shown in Fig. 8.

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Fig. 7– Open-loop Bode plot of the design model.

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The controller design was accomplished in various steps. The first step in the design was to render the open-loop system minimum phase and then passive via a combination of suitable compensators. In this case, a simple feedforward constant-gain passifier (Cff (s) = Dff ) was found to be adequate to render the open-loop system minimum-phase and also passive. The passified system is thus given by

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A. Robust passification of the plant

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reason being, the high fidelity analytical model allows a control engineer to derive models for various system configurations without actually reconfiguring the experimental set up. The details of analytical modelling of acoustic systems can be found in Ref. 7. Figure 6 shows a comparison of the Bode plots obtained from experimental data and the assumed modes analytical model. As can be seen from the figure, the match between the two responses is very good. The analytical model often requires tuning of inherent damping of the modes since there is no known analytical technique to accurately model the modal damping. For controller design purposes a low-order plant model is considered, which includes acoustic modes up to 500 Hz. This gives a 22nd order nominal plant model. The modes above 500 Hz, which constitute the unmodelled dynamics of the plant, are considered as additive uncertainty. In addition, the plant is also assumed to have structured uncertainty due to modelling inaccuracies in the acoustic mode frequencies and damping. Let the open-loop system (i.e., the nominal plant model) be given by

(1) (2)

where D1 = Dp + Dff and Dff is the constant-gain feedforward term. The constant-gain feedforward controller that simultaneously rendered the system minimum-phase and

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passive was: Cfb(s): = Dff = 2:0. The Bode plot of the passified nominal system is shown in Fig. 9. As seen in Fig. 9, the phase of the system is restricted to ±90° implying that the system is passive. Please note that the passification is also guaranteed for the true model (see Fig. 10) and not just for the nominal model. Please also note that Figs. 9 and 10 show the Bode plots of the modified plant dynamics from input u(t) to output y1(t). The actual system of interest from a performance point of view is from input u(t) to output yperf (t). The purpose behind passification (i.e., modifying system dynamics such that certain mathematical conditions are satisfied by the input and the modified output) is that the design of the feedback controller can focus on performance enhancement alone as long as it satisfies WSPR/MSPR properties since the stability of the feedback loop is guaranteed automatically due to passivity. This technique can be viewed as a method by which one can independently target the stability and performance requirements in the different parts of the controller design. Having passified the system, any WSPR/MSPR controller can now be designed to robustly stabilize the closed-loop system. Passification of the true model guarantees closedloop stability with any WSPR controller. Moreover, this stability is robust to unmodelled dynamics and parametric uncertainties as long as passivity of the system is maintained under perturbations. The parametric uncertainty considered

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Fig. 10– Bode Plot of passified true plant (Po(s) + (Cff(s)).

here is the perturbations in all acoustic mode frequencies in the design model caused by the change in the length of the acoustic duct. This uncertainty is realized in the experimental set up by extending the telescopic duct by 0.305 m (1 ft.). It is to be noted that the perturbations occur simultaneously at all frequencies. The perturbed plant is checked again for passivity. If the perturbed plant is found to violate passivity, the passifying compensator is redesigned to ensure robust passivity. In the present case, the perturbed plant was found to remain passive with the feedforward passifier designed for the nominal plant. Having ensured the passivity of the perturbed model, the next step is to design a WSPR/MSPR controller to meet the desired performance.

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1. Design of resonant-mode controller In the second step of the design, a series compensator was first designed which essentially consists of parallel combinations of resonant mode controllers.8 Each of these resonant mode controllers is a second order controller designed specifically to suppress a particular resonant mode of the system. This resonant-mode series compensator is given

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where the controller matrices As, Bs, Cs, Ds have the form

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 A1 0 As =  0 0 

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0 0  B1   B  0 0 , Bs =  2  O 0  M  B  0 Ar   r

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Fig. 9– Bode Plot of passified nominal plant (Po(s) + (Cff(s)). Noise Control Eng. J. 51 (2), 2003 Mar–Apr

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As seen in Eq. (4), the series compensator is simply the combination of r resonant mode controllers (Ai, Bi, Ci, Di, i = 1, …, r) in parallel. The transfer function of each resonant101

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Fig. 11– Bode plot of passified nominal plant with series compensation.

Each of the Ki (s) compensators is a second-order controller. Such resonant controllers (Ki (s)) are shown to be very effective in damping resonant modes in flexible structures, as well.17 Resonant-mode controllers have a resonant mode frequency exactly at one of the resonant modes of the system that is intended to be damped. When such a controller is used in the passivity-based framework employed here, the zeros of the controller are designed to ensure that the combination of the plant and series compensator remains passive in the presence of structured as well as unstructured plant uncertainty. A combined system comprising passified plant and resonant compensator is given by ~.

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mode controller is then given by: Ki(s) = Ci(sI – Ai)-1Bi + Di. For the system under consideration, it was decided to target only the first 6 resonant-modes of the system. That is, the resonant-mode controller was designed to suppress all resonant modes up to 500 Hz. The resonant-mode controllers (Ki(s), i = 1,2,…,6) selected are given below:

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Fig. 12– Bode plot of true plant with passifier and series compensation.

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2. Design of WSPR/MSPR feedback controller The final step of the design is to choose a feedback controller Cfb (s) such that it satisfies WSPR/MSPR conditions.9 The general form of the feedback controller is given by

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In Eq. (11), D and are defined as: = and ~ For the acoustic system under consideration the simplest type of constant-gain feedback controller satisfying WSPR conditions was designed. The feedback gain that gave satisfactory results was found to be: Cfb (s): = Dfb = 0.1. A comparison of the open- and closed loop Bode plots for the design model as well as the true plant model is given in Figs. 13 and 14. 3. Robustness to unmodelled dynamics and parametric uncertainties In order to test the controller under simultaneous perturbations in structured and unstructured parameters, the controller was designed assuming 10% - 20% perturbation for the frequencies in the design model. These parametric perturbations were injected in addition to the unmodelled uncertainty which included modes above 500 Hz. For experimental validation, the actual plant perturbations were obtained by increasing the acoustic duct length by 0.3048 m (one ft.). This change in the duct parameter caused parametric variations in the model in the form of changes in the frequencies of all the modes. The unmodelled dynamics consisted of high frequency modes (i.e., modes above 500 Hz) which were not used in the design model. 4. Overall controller The overall controller is a combination of feedforward passifier, resonant-mode compensator, and feedback compensator. The state-space equations for the overall controller are given by .

u(t ) = Ck xk + Dk1 y p + Dk 2uref

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where

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and D* is given by D* = 1 + DfbDsDff. If (Afb, Bfb, Cfb, Dfb) satisfy WSPR conditions, (Ak, Bk, Ck, Dk) asymptotically stabilizes the plant. Moreover, the stability is robust to unmodelled dynamics ~ ~ ~ ~ and parametric uncertainties as long as remain passive under perturbed plant conditions. Please note that, although with respect to the output y 2(t), the resonant controllers are in the feedforward path, with respect to the plant output yperf(t) they are in the feedback path. The most important thing to be noted is that the overall controller, which is a combination of feedforward passifier, resonant compensator, and feedback compensator, is a low-order, output-feedback controller and therefore easy to implement. Moreover, the feedback gain of the controller Dfb can be increased as high as necessary without any concern for closed-loop instability. This is a great advantage the proposed controller has over many other controllers presented in the literature which are prone to instability for higher values of gains. The Bode plot of the overall controller is given in Fig. 15.

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4. EXPERIMENTAL VALIDATION �

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Fig. 13– Simulated closed-loop frequency response for nominal case. Noise Control Eng. J. 51 (2), 2003 Mar–Apr

The controller design described in Section 3 was validated experimentally using the ISU acoustic duct facility shown in Fig. 3. A schematic of the experimental set-up is shown in Fig. 16. As shown in Fig. 16, the acoustic duct system with 103

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Fig. 15– Bode Plot of overall controller.

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speaker A as the actuator and the microphone located in the middle of the duct as the sensor is not inherently passive. As discussed in previous section, this system is rendered passive by a combination of feedforward controller Cff (s) and series compensator Cs(s). This passified plant is shown in the dotted box in Fig. 16. As mentioned previously, the feedback controller is simply the constant-gain controller which satisfies SPR conditions. For the first set of experiments, speaker B was kept inactive and the closed-loop system was tested with speaker A excited with sine sweep signal. In the second set of experiments, it was assumed that the external disturbance N(s) (generated by speaker B) can be sensed and was used for feedback. The experimental closed-loop frequency response of the system for the first set of experiments is given in Fig. 17. As seen in the figure, a reduction of up to 15dB is obtained in the frequency range of interest. Also, the uncontrolled modes are not destabilized. The resonant mode controller with a lead compensator was designed to maintain closed loop stability in the presence of modelling errors and parametric uncertainty. Parametric uncertainty was introduced into the system by changing the length of the duct as described previously. The duct is made to have telescopic configuration so that its length can be varied in predetermined steps. Please note that, under such plant perturbations, the resonant-mode controller poles do not match the plant poles. However, this does not pose any ��

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problem for closed-loop system stability since the passivity of the plant (the true model) under such perturbations is ensured by proper selection of the zero dynamics in the resonant-mode controller. The worst effect that can be caused under such perturbations is a deterioration in the performance. Figure 18 shows the experimental closed-loop Bode plot of the system under such perturbation. As seen from this figure, although the performance has deteriorated compared to the nominal case, the reduction in the peaks is still satisfactory. The most important thing to note is that these plant perturbations do not cause the controller to destabilize any uncontrolled modes. For the second set of experiments, when speaker B is used as a persistent sinusoidal disturbance, the experimental closed-

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Fig. 17– Experimental Bode of the closed-loop system for the nominal model.

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Fig. 18– Experimental Bode plot of the closed-loop system for the perturbed model.

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under plant perturbations. Analytical, as well as experimental results were presented which show the effectiveness of such controllers. A multi-input multi-output extension of such controllers is expected to yield better results and is currently under investigation.

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6. ACKNOWLEDGMENT

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The author acknowledges NSF support through grant nos. CMS:9713846 and CMS:9703092.

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7. REFERENCES 1

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Fig. 19– Experimental closed-loop response with persistent disturbance.

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loop response for the nominal case was obtained as shown in Fig. 19. As seen from the figure, the performance for this case was slightly inferior to the case when speaker A was used for control as well as disturbance input. Nevertheless, in both cases, the noise level reduction was very satisfactory. Future work will address the case when the control and disturbance signals enter through separate channels and there is no access to disturbance signal for feedback.

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5. CONCLUSIONS

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Robust broadband control of acoustic noise in a 1-D duct configuration using passivity based controllers was presented. Analytical as well as experimental results were given which validate the robust control design methodology based on passivity theory. The acoustic system under consideration was a 3.66 m (12 ft.) long acoustic duct with circular cross section of 0.28 m (11 in.) diameter. A finite-dimensional approximation of this infinite-dimensional system was obtained using analytical as well as experimental identification techniques. It was shown that the models obtained by both methods were equally good and either of these models could be used as the design model. Typical characteristics of such acoustic systems include under-damped resonant modes and non-minimum phase behavior which makes these systems inherently nonpassive. The system was rendered passive using passification techniques and it was ensured that the passification was robust to modelling errors and parametric uncertainties. Robust passification was used to ensure the robust closed-loop stability


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