Thermoacoustic Instability: Model-based Optimal Control Designs and ...
Thermoacoustic Instability: Model-based Optimal Control Designs and Experimental Validation A. M. Annaswamyy, M. Fleifily , J.W. Rumseyz, R. Prasanthx, J.P. Hathouty , and A.F. Ghoniemy
Abstract Active control of thermoacoustic instability has been increasingly sought after in the past two decades to suppress pressure oscillations while maintaining other performance objectives such as low NOx emission, high efficiency and power density. Recently, we have developed a feedback model of a premixed laminar combustor which captures several dominant features in the combustion process such as heat release dynamics, multiple acoustic modes, and actuator effects [1]. In this paper, we study the performance of optimal control designs using the model in [1] with additional effects of mean heat and mean flow, actuator dynamics, and input saturation. These designs are verified experimentally using a 1kW bench-top combustor rig and a 0.2W loudspeaker over a range of flow rates and equivalence ratios. Our results show that the proposed controllers, which are designed using a two-mode finite dimensional model, suppress the thermoacoustic instability significantly faster than those obtained using empirical approaches in similar experimental set-ups without creating secondary resonances, and guarantee stability robustness.
1 Introduction In several applications such as propulsion, power generation, and heating, processes that involve continuous combustion are encountered. One of the main characteristics of these processes is a dynamic behavior denoted as thermoacoustic instability. In most cases, the instability occurs due to a coupling between the unsteady components of pressure and heat release rate and manifests in the form of growing pressure oscillations. Often these pressure oscillations become more severe as the operating condition of the combustors change to meet specific performance criteria such as The work reported here was supported in part by the National Science Foundation under grant No. ECS-9296070. The third author was supported in part by an NSF Graduate Fellowship. y Department of Mechanical Engineering, MIT, Cambridge, MA 02139 z Arthur D. Little Inc., Cambridge, MA 02140 x Scientific Systems Inc., Woburn, MA
1
operating lean to reduce NOx formation, increasing the thermal output, or reducing the size of the combustor. Pressure oscillations are undesirable since they lead to excessive vibrations resulting in mechanical failures, high levels of acoustic noise, high localized burn rates, and possibly component melting. While passive approaches such as changing the flame anchoring point, installing baffles and acoustic dampers, etc., have been sought to counter the instability, a desire to operate over a wide range of conditions without running the risk of self-destruction, and maintaining various performance measures at desirable levels, has led to exploring active control as a possible strategy for achieving the desired performance. Several experimental results have been reported over the past decade for controlling thermally driven acoustic oscillations using active methods [2]- [12]. Active control has also been attempted for realizing other objectives such as efficiency (complete combustion), high performance (increased thermal output), and low NOx formation [13, 14]. In much of these efforts, the experimental controllers are implemented using analog electronic circuitry whose components are designed so as to provide the functionalities of a filter, phase-shifter, and an amplifier, and their parameters are determined by trial-and-error so as to add the requisite phase. Typically, the results from these experiments have demonstrated that the dominant thermoacoustic instability can be suppressed. In many cases, however, secondary peaks at frequencies which were not excited in the uncontrolled combustor appear (for example, [4, 7, 9]). Also, as operating conditions such as equivalence ratio and the flow rate change, the controller would fail in suppressing the primary instability as well [7]. Active-adaptive control strategies have been attempted in [9, 12] so as to expand the range of operating conditions. Typically, these investigations have employed an adaptive filter and an LMS-algorithm [15] for adjusting its coefficients. Studies have also been reported in [16] using an observer-based approach wherein a real-time identification of the unstable modes is proposed to cope with uncertainties in a combustion process. An alternative prescriptive approach for designing active controllers is to use a model of the combustion process by employing the conservation equations and constitutive relations that govern the acoustics and combustion dynamics. This not only allows one to obtain fundamental insights into the underlying mechanisms and quantify the system properties in relation to various physical parameters but also allows the development of a systematic controls methodology that incorporates features of optimization and robustness, and enables an enhanced range of operation. Attempts have been made in this direction in [4],[17]-[19], where the effect of acoustics is characterized, but the combustion dynamics is not modeled. In Ref. [4] a model-based control design is carried out using a model with a single acoustic mode at which the combustor is unstable. In [17]-[19], multiple acoustic modes are included but the coupling between acoustic modes is neglected, and simulation studies are reported using data obtained from a solid rocket motor [20]. In [21] an active controller is proposed using system identification at a stable operating point and the -synthesis 2
control procedure. We have recently developed a physically-based finite-dimensional model of a continuous combustion process [22, 23] and a model-based control methodology [1, 24]. The model includes flame kinematics derived assuming that the the flame is laminar and anchored on a perforated plate [25], acoustics with longitudinal modes, and a loudspeaker as an actuator. In [22] and [23], various properties of the model are derived, including the effect of coupling between acoustic modes when a heat source and an active control source are present, and the cause of secondary peaks that occur in the experimental investigations of active control. In [1] and [24], model-based control designs based on the LQG-LTR approach and adaptation are proposed, and their advantages over empirical control designs are discussed and validated using numerical studies of finite-dimensional models. The main goal of this paper is to carry out an extensive study of model-based optimal control designs and their experimental validation. The underlying model is an extension of that in [1] and includes the effects of mean flow and mean heat additions, actuator dynamics, and input saturation all of which have a significant impact on the efficacy of the control design. The optimal control designs are based on the LQG/LTR and H1 approaches. The closed-loop performance using both these controllers is compared through experimental studies and their robustness properties are characterized. The experimental investigations are carried out using a bench-top rig which exhibits several features that are commonly encountered in combustion processes such as limit-cycles, bifurcation, and hysteresis, a condenser microphone as a sensor, and a loudspeaker as an actuator. The control designs are also validated using a PDE model of the acoustics. Section 2 presents the input-output model of the combustor starting from the conservation equations and the flame surface kinematics. Section 3 presents the active control designs which are then verified experimentally using a bench-top combustor rig and numerically using a PDE model of the combustion acoustics.
2 A Physics-based Dynamic Model of a Premixed Combustor Thermoacoustic instability is generated due to the feedback between combustion and acoustics. That is, the heat release source responds dynamically to acoustic perturbations, and the acoustic oscillations are excited by the unsteady heat release rate. In [25, 22], a dynamic model of an one-dimensional rig was developed which captures the dominant interactions between these two subsystems, starting from the conservation equations of mass, momentum, and energy, and kinematics of a laminar flame. The acoustics was modeled primarily by considering longitudinal modes, and linear dynamics. The kinematics of a premixed laminar flame was modeled in [25] assuming that it was stabilized in a low velocity region, such as behind a perforated disk, and 3
flameholder
? Reactants xa
;; @@ @ ;
-Products xf
xs microphone
loudspeaker
Figure 1: Schematic of the combustor with a side-mounted loudspeaker. representing the flame as a thin sheet moving with the local convective velocity plus a small constant burning velocity in a normal direction to its surface relative to the reactants. The following assumptions were made in the derivation of the model (see Fig. 1 for a schematic): (A1) Effects of viscosity and heat conduction are negligible. (A2) Acoustic effects are one-dimensional. (A3) the flame zone is spatially localized, at x = xf (see Figure 1) with a heat release rate per unit area, qf0 . (A4) Perturbations about the mean are small. (A5) The premixed flame has a conical mean flame surface with a small apex angle. (A6) A loudspeaker is side-mounted with r as the ratio of cross-sectional areas of the loudspeaker and the combustor, xa is the location of the loudspeaker and xl is the displacement of its diaphragm. (A7) The pressure perturbation p0 can be approximated as
p0 (x; t)
=
p
n X
x t
(1)
kx ;
(2)
i=1
i( ) i( )
where p is the mean pressure, using basis functions
x
i ( ) = sin ( i + i0 )
where ki and i0 are determined by the boundary conditions, and correspond to the wave numbers and spatial mode shapes, respectively, of the unforced wave equation [26, 27, 28]. (A8) Effects of mean-flow and mean-heat addition are negligible. 4
(A9) The loudspeaker dynamics can be neglected so that x¨l the loudspeaker.
=
kai, where i is the input current into
Using the above assumptions, the conservation equations can be used to obtain the following equations governing the underlying acoustic and heat-release dynamics:
p
n X i=1
x ;c p
i ( ) ¨i
2
n X i=1
x t
i( ) i( )
=
:0 qf + p r x¨ l
( ; 1)
(3)
: (t) d i ;x + a q0 (t) + x˙ H ;x ; x(4) f r l f a 0 f i
i=1 ki dx :0 q f = ;!f qf0 ; gf u0f (5) where is the specific heat ratio, p is the mean pressure, c is the mean speed of sound, ao = ( ; 1)= p, represents the effect of the flow velocity before and ahead of the flame at xf , i.e., uf (t) = (1 ; )u;f + u+f 0 < < 1; df is the diameter of the flame1 , ! !2 4Su d p !f = d ; gf = D nf u ∆qr ; p ∆qr is the heat release rate per unit mass of the mixture, D is the diameter of the flameholder, u is
u0f (t)
=
n 1 1X
2
the density of the premixed reactants, and H() is Heaviside function.
We now extend the model in Eqs. (3)–(5) when both assumptions (A8) and (A9) are not valid, both of which are restrictive and affect the performance of the active controller. Assumption (A8) is unrealistic since both mean-flow and mean-heat effects are always present in a combustor. More
importantly, the basis functions (mode shapes) i () depend on the mean temperature field in the combustor. This effect is quantified in Section 2.1. That the actuator dynamics can be neglected is not realistic either, since the natural frequencies of a speaker are often of the same order of magnitude as the unstable frequencies of the combustor. We include the dynamics of the actuator in this paper as well.
2.1
Relaxation of Assumptions (A8) and (A9)
2.1.1 Effect of Mean Flow and Mean-Heat Additions In a typical combustion system, the mean flow velocity is nonzero and there is a non-negligible amount of mean heat release, which causes a significant change in the velocity as well as density 1
For a laminar flame stabilized on a perforated disc with nf holes, the flame base corresponding to each hole tends to increase due to the entrainment from the neighboring locations. We incorporate this effect by assuming that df = dp , where dp is the diameter of the perforation of the flame holder, and (1.2,2).
2
5
and temperature of the hot gases. It can be shown that due to the mean heat, c(x), u(x), and (x) experience a step increase at x = xf , and are constants otherwise [22]. Denoting the speed of sound c(x) = c1 for x xf and c(x) = c2 for x > xf , the following change of coordinates
c1 ;x ; x H ;x ; x (6) f f c2 is needed. This implies that points with x > xf shrink in the transformed cordinate z , since c2 > c1 . This implies that the equivalent length decreases from L to Le , where Le = L xLf + cc1 1 ; xLf : 2 z =x;
1;
We now discuss the effect of the mean heat addition on the mode shapes. In the absence of mean heat addition, if the fundamental mode is (x) = sinfkx + g, using the relation in (6), one can show that with mean heat addition, it becomes (x) = sinfke x + e g where
ke = L2
e
4
=
ke = cc : L2 1 2
e
kc; 4
for 0 x xf
e =
1 ; cc12
kh; e = + ke
=
xf
for xf
<xL
and is a constant that depends on the boundary conditions of the combustor. For example, for an open-open combustor, = 2 and for a closed-open combustor, = 4. Similar comparisons can be derived for higher modes with and without mean heat addition. Figure 2 shows a comparison between the fundamental mode shape with and without mean heat addition, for a combustor that is open at both ends. 2 , it follows that the Since the wave number with no mean heat addition is given by k = L effective wave number in the cold section is larger since kc > k and the wave number is smaller in the hot section since kh < k . It is interesting to note that the average wave number kav is unchanged from k since 1ZL
kav = L
0
kedx = k:
The change in the wave number, in turn, affects the acoustic frequency as well as the relative location of the flame. In both the cold and hot sections, the acoustic frequency is increased due to heat addition. If ! is the frequency without heat addition,
!e since Le
=
kec > !
< L.
On the other hand, the relative flame location is moved toward the downstream end, since ;
xf
=
;
sin khxf + 6
=
;
sin kxef +
1 0.9 0.8 0.7
ψ (x)
0.6 0.5
"−" With mean heat "−." Without mean heat
0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Dimensionless distance x/L
0.8
0.9
1
Figure 2: Fundamental wave with and without mean heat addition for a combustor that is open at both ends.
where
x c f 1 1; L 1 ; c2 xf + c1 1 ; xf L c2 L
xef = xf + xf ;
=
The most important effect of heat addition on the instability properties is due to the change from xf to xef . If xf is close to a node or an antinode of the mode shape, heat addition can cause a transition from a stable mode to an unstable mode or vice versa. When it comes to control design therefore, a robust performance with respect to c2 may be nontrivial to establish since perturbations in c2 may change the number of unstable poles. An additional point needs to be made regarding the mode shapes and their dependence on boundary conditions. Since the perturbation in the heat release does not affect the mode shapes substantially, we adopt the unperturbed mode shapes (defined in Eq. (2)) as our basis functions for the pressure. ki and i0 can be calculated in a straight forward manner for ideal boundary conditions where the ends are “fully open” or “fully closed”. For non-ideal boundary conditions, the mode shapes can still be derived assuming that the impedance between the velocity and the pressure is jZ where Z is real, as shown in Appendix A. The case where Z is complex addresses the effect of dissipation from the boundaries [29]. Since we have neglected the internal dissipation in the combustor due to viscous effects in our model, it is reasonable to neglect the dissipation from 7
the boundaries as well.
2.1.2 Loudspeaker Dynamics In order to ensure an accurate control design, the effect of the loudspeaker dynamics governing the relationship between ua , the voltage applied to the loudspeaker, and the diaphragm acceleration x¨ l , must be taken into account. Neglecting effects of magnetic inductance, for small input amplitudes, this model can be derived as 2 Gl (s) = m s2 +k1sb s + k
l
l
(7)
l
where ml , bl , and kl represent the mass, friction, and stiffness properties, respectively, of the loudspeaker, and k1 is a calibration gain. Additional dynamics can arise from the housing used to focus the acoustics of the loudspeaker onto to the combustor, such as a funnel or a waveguide [8]. This housing typically encloses some volume and can act as a Helmholtz resonator with a certain damping and resonant frequency which could overlap with the acoustic range of the combustor, making the task of designing a controller more difficult. It may be important to design this housing so as to ensure minimal attenuation and minimize the introduction of additional dynamics. Also, in order to ensure that the model in (7) is valid, care needs to be taken such that the input voltage does not saturate the speaker.
2.2
An Extended Dynamic Model of the Combustor
The discussions in Section 2.1, together with the change in the mode-shape, effective acoustic length, and the actuator dynamics, lead us to the following dynamic model:
¨i + !i i + 2
n X j =1
dij : j y
: q0f +bf qf0
ue0f ml x¨ l + bl x˙ l + kl xl
=
= = = =
bi q: 0f +bM qf0 + bc x¨ l i
n X i=1
i
cc i;
(8) (9)
i
!f gf ue0f ; n ; X ci : i +cM i + kao r x˙ l i=1 k1ua i
(10) (11) (12)
where y = p0 (xs ; t)=p, the normalized unsteady pressure component, is the output, ua is the input voltage to the speaker, xa , xs , and xf are the locations of the actuator, the sensor, and the flame respectively, kao = 0 if xa > xf and unity otherwise, xl is the position of the speaker diaphragm, 8
M is the Mach number of the mean-flow, and the parameters ; 1 d i;
a0 Mc d i ;x ; b = r (x ) ; 0 bi = a x b i xf ; ci = f ; M = ; 2 E
ki dx E 1 dx f c E i a Z L Mc1 Z L d ; c = ; Mc1 ;x ; 2 E = dx; c = ( x ) ; d = c i s ij i E 0 i j M
i f 0 ; !i = cki; bf = !f 1 ; a0 gf ; 2 (0; 0:5); 2 (1:2; 2): i
i
e
e
i
i
In the above model, the model for heat-release dynamics was derived starting from the flame kinematics relations and making simplifications based on the flame geometry and the structure of the perforated disk as a flame stabilization mechanism [25]. The model indicates that qf0 is yet another state variable of the combustor system and must be taken into account in the analysis and control synthesis. Since for premixed flames, Su 1. For the combustor with dominant (unstable) frequency at 488 Hz, we took the initial pole locations at 600 Hz. The final set of weights are: 100 5(s + 1600 ) W1 = ss++1200 ; W 2 = 4(s + 200 )
The H1 controller obtained with these weights resulted in:
GD=D(s) =;337:32
;
;
(s+14:9)(s 666:33)(s+3770)(s 8689)[(s+203:5)2+994:32 ][(s+182)2 +18132 ] (s+11:6)(s+360:73)(s+628:32)[(s+592:1)2+888:72 ][(s+303)2 +1997:62 ][(s+146:91)2 +3050:82 ]
21
It can be easily seen that this controller exhibits certain desirable features such as adding gain and phase at the unstable combustor frequency, large magnitudes at low frequencies. The Bode plot of the control loop transfer function in Figure 12 shows smaller magnitudes indicating that a large stability robustness bound. In experiments, it was found that (see Figure 14) this controller is rather sluggish and settling time for pressure oscillations were of the order of 120 msec. 1000
Loudspeaker Accl. (m/sec^2)
800 600 400 200 0 −200 −400 −600 −800 −1000 1.45
1.5
1.55 Time (sec)
1.6
1.65
1.5
1.55 Time (sec)
1.6
1.65
100 80 60
Pressure (Pa)
40 20 0 −20 −40 −60 −80 −100 1.45
Figure 14: Pressure response and control input with D/D configuration, and H1 control: Experimental results for = 0:7. Controller was turned on at 1500 msec. A more aggressive H1 controller was obtained by choosing W1 and W2 using closed-loop transfer functions obtained from the LQG-LTR controller. The order of the controller was 21, which after balanced truncation, was computed to be + 537:3337)[(s + 289:5)2 + 10212 ](s ; 1598)[(s ; 28:522 + 20522 ] ;11616 (s +(s342 :4)[(s + 316)2 + 7192][(s + 259:6)2 + 24652][(s + 639:9)2 + 31472] ;
and resulted in a settling time of 65msec.
3.3
Discussion
The main contribution of the model-based approach used in the control designs discussed in this section is the optimization framework that it provides. In the LQG-LTR design, the cost function J in (15) is minimized, whereas in the H1 method, the peak value of kTz2w k1 is minimized. A 22
direct consequence of such a systematic design procedure is the fast suppression of pressure in a 1 kW combustor using a 0.2W speaker with a minimal control effort (for example, 3 mW peak electrical power in the LQG-LTR control design) and a guaranteed margin of stability robustness (for example, in the H1 design). The model-based approach enabled pressure suppression over a range of equivalence ratios (0.69-0.74) and flow rates (267mL/s-400mL/s) without resulting in any secondary peaks. Beyond these ranges, the linearity of the heat release dynamics, and more importantly, that of the actuator dynamics failed, thereby making the control design inadequate. Combustor rigs of comparable power densities have been experimentally investigated in [5] and [7], both of which used an empirical phase-shift controller. In [5], pressure suppression is achieved in 80 msec. using a 10W speaker and a peak electrical power of 16mW. In [7], a 30W speaker is used to suppress the pressure where the closed-loop system exhibits secondary peaks at 240Hz and 550Hz. The results in sections 3.1 and 3.2 also show that various properties of the actuator can be naturally accommodated in the design procedure. Given the open-loop instability of the combustion system, a dominant constraint in the control design is the input amplitude. As mentioned earlier the LQG-LTR procedure accommodates this in a straight forward manner, and as indicated by Figures 8 and 14, results in a better performance than the H1 controller. It may be possible to realize a similar performance with a H1 approach as well through successive iterations of W1 and W2. But these may be large in number, and hence result in a significant computational burden. The model-based control designs also provide quantitative measures of the robustness of the controlled combustor. As seen above, both the LQG-LTR and H1 controllers (a) are successful in pressure stabilization, and (b) provide a certain amount of stability robustness. While the H1 controller guarantees a bound a priori, the robustness achievable from an LQG-LTR cannot be precomputed. One can, however, iterate on the selection of a suitable W1 to enhance stability robustness.
3.4
Simulation results using the PDE model
In this section, we verify the LQG-LTR control design using the PDE model of the combustor acoustics given by
@ 2 p0 ; c2 @ 2 p0 @t2 1 @z2
!
@ 2 p0 + 2Mc1 @z@t
=
( ; 1)
@p0 + Mc @p0 + m c1 @u0 1 @t @z M @z
=
(
@qf0 @t
@qf0 + Mc1 @z
; 1)qf0 + p r va
23
!
@va + p r @t
!
(18)
(19)
where (x)u(x) = m, and the flame model in (5), for various actuator-sensor configurations. The PDEs were simulated using the Split-Coefficient-Matrix method [35] with a Courant number CN = 0:85, which is defined as CN = c∆∆tx where ∆x and ∆t denote the step sizes in t and x, respectively. CN needs to be chosen to be less than unity to ensure convergence of the numerical solution of the PDE when it has feedback control inputs. The effect of adding CN is equivalent to adding damping to the system. To simulate a similar effect in the two-mode model, a damping term was added to both modes with = 0:0033. This value was chosen so that the uncontrolled combustor exhibits similar responses using the PDE model as well as the two-mode model. One can view the addition of the damping term in both these models to represent any passive damping that may be present in the combustor. The output equation and the actuator dynamics as in (9) and (7) were simulated with the same values for the parameters as in Section 2.3. We assumed that the loudspeaker was side-mounted and that the actuator-sensor location was D/D. We observed that both controllers are effective in stabilizing the system at D/D. We also evaluated the performances of both the controllers as the sensor location was moved away from the actuator. We found that for all sensor-actuator locations, the LQG controller resulted in almost the same performance when evaluated using the PDE model or the two-mode model. The corresponding pressure responses for A/D location, where the deviation between the PDE model and the two-mode model is a maximum, are shown in Figure 15. We also observed that at the D/D location, the pressure response obtained using the PDE model and the LQG controller resulted in a similar performance to that obtained in the experiment, with a settling time of 40 msec. The pressure response using the H1 controller led to a similar performance as in Figure 14 as well.
4 Summary In this paper, we studied the performance of model-based controllers for suppressing thermoacoustic instability in a premixed laminar combustor with a loudspeaker as an actuator and a microphone as a sensor. Two control designs were discussed to suppress the pressure oscillations using (i) the LQG-LTR method and (ii) the H1 approach. Their performance was validated experimentally using a 1kW benchtop rig, and their robustness properties were discussed. The results indicated that the LQG-LTR provides a better performance, while the H1 provides pre-computable stability robustness bounds. Both control designs were effective though they were based on a linear model, indicating that the limit-cycle behavior of the nonlinearity did not affect the performance of the linear design. The experimental results reported here represent the first of its kind where a predictive modelbased control design was used for combustion control. The results show that a systematic optimal 24
(a) With the PDE model and LQG controller 150
Pressure (Pa)
100 50 0 −50 −100 −150
0
10
20
30
40
50 Time (msec)
60
70
80
90
100
80
90
100
(b) With the Two−mode model and LQG controller 150
Pressure (Pa)
100 50 0 −50 −100 −150
0
10
20
30
40
50 Time (msec)
60
70
Figure 15: Pressure responses obtained using both the PDE model and the two-mode model for LQG control for A/D location.
control design can be carried out for suppression of thermoacoustic instability and can lead to a faster settling time and a reduced controlled effort, both of which are attractive features for purposes of commercial implementation. Various features of the available actuator technology such as control bandwidth and input saturation can be readily incorporated into the design to ensure efficient utilization of the control energy. Measures of robustness of closed-loop system stability can be derived, quantifying the requisite model accuracy. Bounds on the operating range over which satisfactory performance can be realized can be ascertained using the proposed modelbased approach. We are currently evaluating nonlinear control strategies that take into account the structure of the nonlinearities in the system to expand the scope of operation.
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[4] G.J. Bloxsidge, A.P. Dowling, N. Hooper, and P.J. Langhorne. “Active control of an acoustically driven combustion instability”. Journal of Theoretical and Applied Mechanics, supplement to vol. 6, 1987. [5] T. Poinsot, F. Bourienne, S. Candel, and E. Esposito. “Suppression of combustion instabilities by active control”. Journal of Propulsion and Power, 5(1):14–20, 1989. [6] P.J. Langhorne, A.P. Dowling, and N. Hooper. Practical active control system for combustion oscillations. Journal of Propulsion and Power, 6(3):324–333, 1990. [7] A. Gulati and R. Mani. Active control of unsteady combustion-induced oscillations. Journal of Propulsion and Power, 8(5):1109–1115, 1992. [8] K.R. McManus, U. Vandsburger, and C.T. Bowman. “Combustor performance enhancement through direct shear layer excitation”. Combustion and Flame, 82:75–92, 1990. [9] G. Billoud, M.A. Galland, C. Huynh Huu, and S. Candel. Adaptive active control of combustion instabilities. Combust. Sci. and Tech., 81:257–283, 1992. [10] S. Sivasegaram and J.H. Whitelaw. “Active control of oscillations in combustors with several frequency modes”. In Proceedings of the ASME Winter Annual Meeting, Anaheim, CA, 1992. [11] E. Gutmark, T.P. Parr, K.J. Wilson, D.M. Hanson-Parr, and K.C. Schadow. “Closed-loop control in a flame and a dump combustor”. IEEE Control Systems, 13:73–78, April 1993. [12] A. Kemal and C.T. Bowman. Active adaptive control of combustion. In Proceedings of the IEEE Conference on Control Applications, pages 667–672, Albany, NY, 1995. [13] K.T. Padmanabhan, C.T. Bowman, and J.D. Powell. “An adaptive optimal combustion control strategy”. In 25th International Symposium on Combustion, 1995. [14] D. St. John and G.S. Samuelson. Active, optimal control of a model, industrial, natural gas-fired burner. In Proc. 25th Int. Symp. Combustion, 1994. [15] B. Widrow and S.D. Stearns. Adaptive signal processing. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1985. [16] Y. Neumeier and B.T. Zinn. Active control of combustion instabilities using real time identification of unstable combustor modes. In Proceedings of the IEEE Conference on Control Applications, pages 691–698, Albany, NY, 1995. [17] Y.-T. Fung, V. Yang, and A. Sinha. Active control of combustion instabilities with distributed actuators. Combust. Sci. and Tech., 78:217–245, 1991. [18] Y-T. Fung and V. Yang. Active control of nonlinear pressure oscillations in combustion chambers. Journal of Propulsion and Power, Vol. 8, No. 6:1282–1289, 1992. [19] V. Yang, A. Sinha, and Y.-T. Fung. “State feedback control of longitudinal combustion instabilities”. Journal of Propulsion and Power, 8, 1992. [20] F.E.C. Culick and V. Yang. “Prediction of the stability of unsteady motions in solid-propellant rocket motors”. In Nonsteady Burning and Combustion Stability of Solid Propellants, edited by L. DeLuca and M. Summerfield, 143:719-780, Progress in Astronautics and Aeronautics, AIAA, NY, 1992.
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[21] J.E. Tierno and J.C. Doyle. “Multimode active stabilization of a Rijke tube”. In DSC-Vol. 38. ASME Winter Annual Meeting, 1992. [22] A.M. Annaswamy, M. Fleifil, J.P. Hathout, and A.F. Ghoniem. “Impact of linear coupling on the design of active controllers for thermoacoustic instability ”. Combustion Science and Technology, 128:131–180, December 1997. [23] M. Fleifil, A.M. Annaswamy, J.P. Hathout, and A.F. Ghoniem. “The origin of secondary peaks with active control of thermoacoustic instability”. Combustion Science and Technology, 133:227–260, 1998. [24] A.M. Annaswamy, O. El-Rifai, M. Fleifil, and A.F. Ghoniem. “A model-based self-tuning controller for thermoacoustic instability”. Combustion Science and Technology, 135:213–240, 1998. [25] M. Fleifil, A.M. Annaswamy, Z. Ghoniem, and A.F. Ghoniem. “Response of a laminar premixed flame to flow oscillations: A kinematic model and thermoacoustic instability result”. Combustion and Flame, 106:487–510, 1996. [26] T.Myint-u and L. Debnath. Partial Differential Equations for Scientists and Engineers. Elsevier Science Publishing Co., New York, NY, 1987. [27] B.T. Zinn and M.E. Lores. Application of the galerkin method in the solution of nonlinear axial combustion instability problems in liquid rockets. CST, 4:269–278, 1972. [28] F.E.C. Culick. “Nonlinear behavior of acoustic waves in combustion chambers”. Acta Astronautica, 3:715–756, 1976. [29] A.P. Dowling and J.E. Ffowcs Williams. Sound and Sources of Sound. Ellis Horwood Limited, West Sussex, PO191EB, England, 1983. [30] G. Stein and M. Athans. The LQG/LTR procedure for multivariable feedback control design. IEEE Transactions on Automatic Control, 32:105–114, February 1987. [31] D. A. Wilson. Convolution and hankel operator norms for linear systems. IEEE Transactions on Automatic Control, 34(1):94–97, 1989. [32] M. A. Rotea. The generalized
H2 control problem. Automatica, 29(2):373–385, 1993.
[33] J. Doyle, B. Francis, and A. Tannenbaum. Feedback Control Theory. Macmillan International, Sydney, 1992. [34] R. Y. Chiang and M. G. Safonov. Robust control toolbox : user’s guide. The MathWorks, Inc., South Natick, MA, 1988. [35] R.J. LeVeque. Numerical methods for conservation laws. Birkhauser Verlag, Basel, 1992.
Appendix A We show that the solution p0 (x; t) to the unperturbed wave equation
@ 2 p0 ; c2 @ 2 p0 @t2 @z2 27
=
0
(20)
is of the form
p0 (x; t)
=
Re
h
(x) ej!t
i
x
i ( ) = sin (
where
kx + ) ;
(21)
in the presence of non-ideal boundary conditions given by
u0 (zn )
p0
jZn (!) (zn ) ;
=
n = 1; 2
(22)
where z1 and z2 denote the inlet and outlet, respectively, and Z1 and Z2 are real functions of ! . Assuming that p0
where k
=
=
Re(pˆ(z)ej!t ), (20) can be simplified as d2pˆ + k2pˆ = 0 dz2
(23)
!=c, whose solution is of the form pˆ (z)
aejkz + be;jkz
=
(24)
where a and b are two unknown parameters to be determined. The boundary conditions in (22) can be expressed in terms of the pressure as follows: The unperturbed momentum equation
0
@u @t
+
@p0 @z
=
0
(25)
can be integrated to obtain
j dpˆ ej!t: ! dz Combining Eq.(26) with Eq. (22), we obtain that, for n = 1; 2, dpˆ (z ) = ; ! pˆ (z ) : dz n Zn (!) n u0
=
(26)
(27)
Applying (27) to Eq.(24), a linear system of equations can be obtained as
jkejkz a ; jke;jkz b n
n
=
; Z !(!) ejkz a + e;jkz b ; h
i
n
n
n
Nonzero solutions for a and b can be found from (28) if !
!
n = 1; 2:
(28)
2 2 c !L !L c c 1+ (29) Z1 (!) Z2 (!) sin c + z1 (!) ; z2 (!) cos c = 0 which can be solved to obtain admissible values of ! . Using (28) to get a relation between a and b, we obtain that " # ! 1 pb (z) = v (30) !2 kcos (kz ) + Z (! ) sin (kz ) u 1 u ! t 2 k + Z (!) 1
28
Eq. (30) shows that pb(z ) is real, and hence, it follows that of (20) where
k
=
!; c
tan
=
z
pz
( ) = b( )
k Z (!): ! 1
and that (21) is the solution
(31)
The above derivation shows that for general boundary conditions determined by Z1 and Z2 in Eq. (22), the mode shape is of the form (21) where k and are given by Eq. (31). Eq. (31) also shows that when Z1 = 1 the mode shape is cos(kz ) which corresponds to the case when the inlet is “fully closed”, and when Z1 = 0 the mode shape is sin(kz ) when the inlet is “fully open”. It should also be noted that Z2 (! ) affects the solution of Eq. (29) and therefore the wave number k of the mode shape.
29
Abstract Active control of thermoacoustic instability has been increasingly sought after in the past two decades to suppress pressure oscillations while maintaining other performance objectives such as low NOx emission, high efficiency and power density. Recently, we have developed a feedback model of a premixed laminar combustor which captures several dominant features in the combustion process such as heat release dynamics, multiple acoustic modes, and actuator effects [1]. In this paper, we study the performance of optimal control designs using the model in [1] with additional effects of mean heat and mean flow, actuator dynamics, and input saturation. These designs are verified experimentally using a 1kW bench-top combustor rig and a 0.2W loudspeaker over a range of flow rates and equivalence ratios. Our results show that the proposed controllers, which are designed using a two-mode finite dimensional model, suppress the thermoacoustic instability significantly faster than those obtained using empirical approaches in similar experimental set-ups without creating secondary resonances, and guarantee stability robustness.
1 Introduction In several applications such as propulsion, power generation, and heating, processes that involve continuous combustion are encountered. One of the main characteristics of these processes is a dynamic behavior denoted as thermoacoustic instability. In most cases, the instability occurs due to a coupling between the unsteady components of pressure and heat release rate and manifests in the form of growing pressure oscillations. Often these pressure oscillations become more severe as the operating condition of the combustors change to meet specific performance criteria such as The work reported here was supported in part by the National Science Foundation under grant No. ECS-9296070. The third author was supported in part by an NSF Graduate Fellowship. y Department of Mechanical Engineering, MIT, Cambridge, MA 02139 z Arthur D. Little Inc., Cambridge, MA 02140 x Scientific Systems Inc., Woburn, MA
1
operating lean to reduce NOx formation, increasing the thermal output, or reducing the size of the combustor. Pressure oscillations are undesirable since they lead to excessive vibrations resulting in mechanical failures, high levels of acoustic noise, high localized burn rates, and possibly component melting. While passive approaches such as changing the flame anchoring point, installing baffles and acoustic dampers, etc., have been sought to counter the instability, a desire to operate over a wide range of conditions without running the risk of self-destruction, and maintaining various performance measures at desirable levels, has led to exploring active control as a possible strategy for achieving the desired performance. Several experimental results have been reported over the past decade for controlling thermally driven acoustic oscillations using active methods [2]- [12]. Active control has also been attempted for realizing other objectives such as efficiency (complete combustion), high performance (increased thermal output), and low NOx formation [13, 14]. In much of these efforts, the experimental controllers are implemented using analog electronic circuitry whose components are designed so as to provide the functionalities of a filter, phase-shifter, and an amplifier, and their parameters are determined by trial-and-error so as to add the requisite phase. Typically, the results from these experiments have demonstrated that the dominant thermoacoustic instability can be suppressed. In many cases, however, secondary peaks at frequencies which were not excited in the uncontrolled combustor appear (for example, [4, 7, 9]). Also, as operating conditions such as equivalence ratio and the flow rate change, the controller would fail in suppressing the primary instability as well [7]. Active-adaptive control strategies have been attempted in [9, 12] so as to expand the range of operating conditions. Typically, these investigations have employed an adaptive filter and an LMS-algorithm [15] for adjusting its coefficients. Studies have also been reported in [16] using an observer-based approach wherein a real-time identification of the unstable modes is proposed to cope with uncertainties in a combustion process. An alternative prescriptive approach for designing active controllers is to use a model of the combustion process by employing the conservation equations and constitutive relations that govern the acoustics and combustion dynamics. This not only allows one to obtain fundamental insights into the underlying mechanisms and quantify the system properties in relation to various physical parameters but also allows the development of a systematic controls methodology that incorporates features of optimization and robustness, and enables an enhanced range of operation. Attempts have been made in this direction in [4],[17]-[19], where the effect of acoustics is characterized, but the combustion dynamics is not modeled. In Ref. [4] a model-based control design is carried out using a model with a single acoustic mode at which the combustor is unstable. In [17]-[19], multiple acoustic modes are included but the coupling between acoustic modes is neglected, and simulation studies are reported using data obtained from a solid rocket motor [20]. In [21] an active controller is proposed using system identification at a stable operating point and the -synthesis 2
control procedure. We have recently developed a physically-based finite-dimensional model of a continuous combustion process [22, 23] and a model-based control methodology [1, 24]. The model includes flame kinematics derived assuming that the the flame is laminar and anchored on a perforated plate [25], acoustics with longitudinal modes, and a loudspeaker as an actuator. In [22] and [23], various properties of the model are derived, including the effect of coupling between acoustic modes when a heat source and an active control source are present, and the cause of secondary peaks that occur in the experimental investigations of active control. In [1] and [24], model-based control designs based on the LQG-LTR approach and adaptation are proposed, and their advantages over empirical control designs are discussed and validated using numerical studies of finite-dimensional models. The main goal of this paper is to carry out an extensive study of model-based optimal control designs and their experimental validation. The underlying model is an extension of that in [1] and includes the effects of mean flow and mean heat additions, actuator dynamics, and input saturation all of which have a significant impact on the efficacy of the control design. The optimal control designs are based on the LQG/LTR and H1 approaches. The closed-loop performance using both these controllers is compared through experimental studies and their robustness properties are characterized. The experimental investigations are carried out using a bench-top rig which exhibits several features that are commonly encountered in combustion processes such as limit-cycles, bifurcation, and hysteresis, a condenser microphone as a sensor, and a loudspeaker as an actuator. The control designs are also validated using a PDE model of the acoustics. Section 2 presents the input-output model of the combustor starting from the conservation equations and the flame surface kinematics. Section 3 presents the active control designs which are then verified experimentally using a bench-top combustor rig and numerically using a PDE model of the combustion acoustics.
2 A Physics-based Dynamic Model of a Premixed Combustor Thermoacoustic instability is generated due to the feedback between combustion and acoustics. That is, the heat release source responds dynamically to acoustic perturbations, and the acoustic oscillations are excited by the unsteady heat release rate. In [25, 22], a dynamic model of an one-dimensional rig was developed which captures the dominant interactions between these two subsystems, starting from the conservation equations of mass, momentum, and energy, and kinematics of a laminar flame. The acoustics was modeled primarily by considering longitudinal modes, and linear dynamics. The kinematics of a premixed laminar flame was modeled in [25] assuming that it was stabilized in a low velocity region, such as behind a perforated disk, and 3
flameholder
? Reactants xa
;; @@ @ ;
-Products xf
xs microphone
loudspeaker
Figure 1: Schematic of the combustor with a side-mounted loudspeaker. representing the flame as a thin sheet moving with the local convective velocity plus a small constant burning velocity in a normal direction to its surface relative to the reactants. The following assumptions were made in the derivation of the model (see Fig. 1 for a schematic): (A1) Effects of viscosity and heat conduction are negligible. (A2) Acoustic effects are one-dimensional. (A3) the flame zone is spatially localized, at x = xf (see Figure 1) with a heat release rate per unit area, qf0 . (A4) Perturbations about the mean are small. (A5) The premixed flame has a conical mean flame surface with a small apex angle. (A6) A loudspeaker is side-mounted with r as the ratio of cross-sectional areas of the loudspeaker and the combustor, xa is the location of the loudspeaker and xl is the displacement of its diaphragm. (A7) The pressure perturbation p0 can be approximated as
p0 (x; t)
=
p
n X
x t
(1)
kx ;
(2)
i=1
i( ) i( )
where p is the mean pressure, using basis functions
x
i ( ) = sin ( i + i0 )
where ki and i0 are determined by the boundary conditions, and correspond to the wave numbers and spatial mode shapes, respectively, of the unforced wave equation [26, 27, 28]. (A8) Effects of mean-flow and mean-heat addition are negligible. 4
(A9) The loudspeaker dynamics can be neglected so that x¨l the loudspeaker.
=
kai, where i is the input current into
Using the above assumptions, the conservation equations can be used to obtain the following equations governing the underlying acoustic and heat-release dynamics:
p
n X i=1
x ;c p
i ( ) ¨i
2
n X i=1
x t
i( ) i( )
=
:0 qf + p r x¨ l
( ; 1)
(3)
: (t) d i ;x + a q0 (t) + x˙ H ;x ; x(4) f r l f a 0 f i
i=1 ki dx :0 q f = ;!f qf0 ; gf u0f (5) where is the specific heat ratio, p is the mean pressure, c is the mean speed of sound, ao = ( ; 1)= p, represents the effect of the flow velocity before and ahead of the flame at xf , i.e., uf (t) = (1 ; )u;f + u+f 0 < < 1; df is the diameter of the flame1 , ! !2 4Su d p !f = d ; gf = D nf u ∆qr ; p ∆qr is the heat release rate per unit mass of the mixture, D is the diameter of the flameholder, u is
u0f (t)
=
n 1 1X
2
the density of the premixed reactants, and H() is Heaviside function.
We now extend the model in Eqs. (3)–(5) when both assumptions (A8) and (A9) are not valid, both of which are restrictive and affect the performance of the active controller. Assumption (A8) is unrealistic since both mean-flow and mean-heat effects are always present in a combustor. More
importantly, the basis functions (mode shapes) i () depend on the mean temperature field in the combustor. This effect is quantified in Section 2.1. That the actuator dynamics can be neglected is not realistic either, since the natural frequencies of a speaker are often of the same order of magnitude as the unstable frequencies of the combustor. We include the dynamics of the actuator in this paper as well.
2.1
Relaxation of Assumptions (A8) and (A9)
2.1.1 Effect of Mean Flow and Mean-Heat Additions In a typical combustion system, the mean flow velocity is nonzero and there is a non-negligible amount of mean heat release, which causes a significant change in the velocity as well as density 1
For a laminar flame stabilized on a perforated disc with nf holes, the flame base corresponding to each hole tends to increase due to the entrainment from the neighboring locations. We incorporate this effect by assuming that df = dp , where dp is the diameter of the perforation of the flame holder, and (1.2,2).
2
5
and temperature of the hot gases. It can be shown that due to the mean heat, c(x), u(x), and (x) experience a step increase at x = xf , and are constants otherwise [22]. Denoting the speed of sound c(x) = c1 for x xf and c(x) = c2 for x > xf , the following change of coordinates
c1 ;x ; x H ;x ; x (6) f f c2 is needed. This implies that points with x > xf shrink in the transformed cordinate z , since c2 > c1 . This implies that the equivalent length decreases from L to Le , where Le = L xLf + cc1 1 ; xLf : 2 z =x;
1;
We now discuss the effect of the mean heat addition on the mode shapes. In the absence of mean heat addition, if the fundamental mode is (x) = sinfkx + g, using the relation in (6), one can show that with mean heat addition, it becomes (x) = sinfke x + e g where
ke = L2
e
4
=
ke = cc : L2 1 2
e
kc; 4
for 0 x xf
e =
1 ; cc12
kh; e = + ke
=
xf
for xf
<xL
and is a constant that depends on the boundary conditions of the combustor. For example, for an open-open combustor, = 2 and for a closed-open combustor, = 4. Similar comparisons can be derived for higher modes with and without mean heat addition. Figure 2 shows a comparison between the fundamental mode shape with and without mean heat addition, for a combustor that is open at both ends. 2 , it follows that the Since the wave number with no mean heat addition is given by k = L effective wave number in the cold section is larger since kc > k and the wave number is smaller in the hot section since kh < k . It is interesting to note that the average wave number kav is unchanged from k since 1ZL
kav = L
0
kedx = k:
The change in the wave number, in turn, affects the acoustic frequency as well as the relative location of the flame. In both the cold and hot sections, the acoustic frequency is increased due to heat addition. If ! is the frequency without heat addition,
!e since Le
=
kec > !
< L.
On the other hand, the relative flame location is moved toward the downstream end, since ;
xf
=
;
sin khxf + 6
=
;
sin kxef +
1 0.9 0.8 0.7
ψ (x)
0.6 0.5
"−" With mean heat "−." Without mean heat
0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Dimensionless distance x/L
0.8
0.9
1
Figure 2: Fundamental wave with and without mean heat addition for a combustor that is open at both ends.
where
x c f 1 1; L 1 ; c2 xf + c1 1 ; xf L c2 L
xef = xf + xf ;
=
The most important effect of heat addition on the instability properties is due to the change from xf to xef . If xf is close to a node or an antinode of the mode shape, heat addition can cause a transition from a stable mode to an unstable mode or vice versa. When it comes to control design therefore, a robust performance with respect to c2 may be nontrivial to establish since perturbations in c2 may change the number of unstable poles. An additional point needs to be made regarding the mode shapes and their dependence on boundary conditions. Since the perturbation in the heat release does not affect the mode shapes substantially, we adopt the unperturbed mode shapes (defined in Eq. (2)) as our basis functions for the pressure. ki and i0 can be calculated in a straight forward manner for ideal boundary conditions where the ends are “fully open” or “fully closed”. For non-ideal boundary conditions, the mode shapes can still be derived assuming that the impedance between the velocity and the pressure is jZ where Z is real, as shown in Appendix A. The case where Z is complex addresses the effect of dissipation from the boundaries [29]. Since we have neglected the internal dissipation in the combustor due to viscous effects in our model, it is reasonable to neglect the dissipation from 7
the boundaries as well.
2.1.2 Loudspeaker Dynamics In order to ensure an accurate control design, the effect of the loudspeaker dynamics governing the relationship between ua , the voltage applied to the loudspeaker, and the diaphragm acceleration x¨ l , must be taken into account. Neglecting effects of magnetic inductance, for small input amplitudes, this model can be derived as 2 Gl (s) = m s2 +k1sb s + k
l
l
(7)
l
where ml , bl , and kl represent the mass, friction, and stiffness properties, respectively, of the loudspeaker, and k1 is a calibration gain. Additional dynamics can arise from the housing used to focus the acoustics of the loudspeaker onto to the combustor, such as a funnel or a waveguide [8]. This housing typically encloses some volume and can act as a Helmholtz resonator with a certain damping and resonant frequency which could overlap with the acoustic range of the combustor, making the task of designing a controller more difficult. It may be important to design this housing so as to ensure minimal attenuation and minimize the introduction of additional dynamics. Also, in order to ensure that the model in (7) is valid, care needs to be taken such that the input voltage does not saturate the speaker.
2.2
An Extended Dynamic Model of the Combustor
The discussions in Section 2.1, together with the change in the mode-shape, effective acoustic length, and the actuator dynamics, lead us to the following dynamic model:
¨i + !i i + 2
n X j =1
dij : j y
: q0f +bf qf0
ue0f ml x¨ l + bl x˙ l + kl xl
=
= = = =
bi q: 0f +bM qf0 + bc x¨ l i
n X i=1
i
cc i;
(8) (9)
i
!f gf ue0f ; n ; X ci : i +cM i + kao r x˙ l i=1 k1ua i
(10) (11) (12)
where y = p0 (xs ; t)=p, the normalized unsteady pressure component, is the output, ua is the input voltage to the speaker, xa , xs , and xf are the locations of the actuator, the sensor, and the flame respectively, kao = 0 if xa > xf and unity otherwise, xl is the position of the speaker diaphragm, 8
M is the Mach number of the mean-flow, and the parameters ; 1 d i;
a0 Mc d i ;x ; b = r (x ) ; 0 bi = a x b i xf ; ci = f ; M = ; 2 E
ki dx E 1 dx f c E i a Z L Mc1 Z L d ; c = ; Mc1 ;x ; 2 E = dx; c = ( x ) ; d = c i s ij i E 0 i j M
i f 0 ; !i = cki; bf = !f 1 ; a0 gf ; 2 (0; 0:5); 2 (1:2; 2): i
i
e
e
i
i
In the above model, the model for heat-release dynamics was derived starting from the flame kinematics relations and making simplifications based on the flame geometry and the structure of the perforated disk as a flame stabilization mechanism [25]. The model indicates that qf0 is yet another state variable of the combustor system and must be taken into account in the analysis and control synthesis. Since for premixed flames, Su 1. For the combustor with dominant (unstable) frequency at 488 Hz, we took the initial pole locations at 600 Hz. The final set of weights are: 100 5(s + 1600 ) W1 = ss++1200 ; W 2 = 4(s + 200 )
The H1 controller obtained with these weights resulted in:
GD=D(s) =;337:32
;
;
(s+14:9)(s 666:33)(s+3770)(s 8689)[(s+203:5)2+994:32 ][(s+182)2 +18132 ] (s+11:6)(s+360:73)(s+628:32)[(s+592:1)2+888:72 ][(s+303)2 +1997:62 ][(s+146:91)2 +3050:82 ]
21
It can be easily seen that this controller exhibits certain desirable features such as adding gain and phase at the unstable combustor frequency, large magnitudes at low frequencies. The Bode plot of the control loop transfer function in Figure 12 shows smaller magnitudes indicating that a large stability robustness bound. In experiments, it was found that (see Figure 14) this controller is rather sluggish and settling time for pressure oscillations were of the order of 120 msec. 1000
Loudspeaker Accl. (m/sec^2)
800 600 400 200 0 −200 −400 −600 −800 −1000 1.45
1.5
1.55 Time (sec)
1.6
1.65
1.5
1.55 Time (sec)
1.6
1.65
100 80 60
Pressure (Pa)
40 20 0 −20 −40 −60 −80 −100 1.45
Figure 14: Pressure response and control input with D/D configuration, and H1 control: Experimental results for = 0:7. Controller was turned on at 1500 msec. A more aggressive H1 controller was obtained by choosing W1 and W2 using closed-loop transfer functions obtained from the LQG-LTR controller. The order of the controller was 21, which after balanced truncation, was computed to be + 537:3337)[(s + 289:5)2 + 10212 ](s ; 1598)[(s ; 28:522 + 20522 ] ;11616 (s +(s342 :4)[(s + 316)2 + 7192][(s + 259:6)2 + 24652][(s + 639:9)2 + 31472] ;
and resulted in a settling time of 65msec.
3.3
Discussion
The main contribution of the model-based approach used in the control designs discussed in this section is the optimization framework that it provides. In the LQG-LTR design, the cost function J in (15) is minimized, whereas in the H1 method, the peak value of kTz2w k1 is minimized. A 22
direct consequence of such a systematic design procedure is the fast suppression of pressure in a 1 kW combustor using a 0.2W speaker with a minimal control effort (for example, 3 mW peak electrical power in the LQG-LTR control design) and a guaranteed margin of stability robustness (for example, in the H1 design). The model-based approach enabled pressure suppression over a range of equivalence ratios (0.69-0.74) and flow rates (267mL/s-400mL/s) without resulting in any secondary peaks. Beyond these ranges, the linearity of the heat release dynamics, and more importantly, that of the actuator dynamics failed, thereby making the control design inadequate. Combustor rigs of comparable power densities have been experimentally investigated in [5] and [7], both of which used an empirical phase-shift controller. In [5], pressure suppression is achieved in 80 msec. using a 10W speaker and a peak electrical power of 16mW. In [7], a 30W speaker is used to suppress the pressure where the closed-loop system exhibits secondary peaks at 240Hz and 550Hz. The results in sections 3.1 and 3.2 also show that various properties of the actuator can be naturally accommodated in the design procedure. Given the open-loop instability of the combustion system, a dominant constraint in the control design is the input amplitude. As mentioned earlier the LQG-LTR procedure accommodates this in a straight forward manner, and as indicated by Figures 8 and 14, results in a better performance than the H1 controller. It may be possible to realize a similar performance with a H1 approach as well through successive iterations of W1 and W2. But these may be large in number, and hence result in a significant computational burden. The model-based control designs also provide quantitative measures of the robustness of the controlled combustor. As seen above, both the LQG-LTR and H1 controllers (a) are successful in pressure stabilization, and (b) provide a certain amount of stability robustness. While the H1 controller guarantees a bound a priori, the robustness achievable from an LQG-LTR cannot be precomputed. One can, however, iterate on the selection of a suitable W1 to enhance stability robustness.
3.4
Simulation results using the PDE model
In this section, we verify the LQG-LTR control design using the PDE model of the combustor acoustics given by
@ 2 p0 ; c2 @ 2 p0 @t2 1 @z2
!
@ 2 p0 + 2Mc1 @z@t
=
( ; 1)
@p0 + Mc @p0 + m c1 @u0 1 @t @z M @z
=
(
@qf0 @t
@qf0 + Mc1 @z
; 1)qf0 + p r va
23
!
@va + p r @t
!
(18)
(19)
where (x)u(x) = m, and the flame model in (5), for various actuator-sensor configurations. The PDEs were simulated using the Split-Coefficient-Matrix method [35] with a Courant number CN = 0:85, which is defined as CN = c∆∆tx where ∆x and ∆t denote the step sizes in t and x, respectively. CN needs to be chosen to be less than unity to ensure convergence of the numerical solution of the PDE when it has feedback control inputs. The effect of adding CN is equivalent to adding damping to the system. To simulate a similar effect in the two-mode model, a damping term was added to both modes with = 0:0033. This value was chosen so that the uncontrolled combustor exhibits similar responses using the PDE model as well as the two-mode model. One can view the addition of the damping term in both these models to represent any passive damping that may be present in the combustor. The output equation and the actuator dynamics as in (9) and (7) were simulated with the same values for the parameters as in Section 2.3. We assumed that the loudspeaker was side-mounted and that the actuator-sensor location was D/D. We observed that both controllers are effective in stabilizing the system at D/D. We also evaluated the performances of both the controllers as the sensor location was moved away from the actuator. We found that for all sensor-actuator locations, the LQG controller resulted in almost the same performance when evaluated using the PDE model or the two-mode model. The corresponding pressure responses for A/D location, where the deviation between the PDE model and the two-mode model is a maximum, are shown in Figure 15. We also observed that at the D/D location, the pressure response obtained using the PDE model and the LQG controller resulted in a similar performance to that obtained in the experiment, with a settling time of 40 msec. The pressure response using the H1 controller led to a similar performance as in Figure 14 as well.
4 Summary In this paper, we studied the performance of model-based controllers for suppressing thermoacoustic instability in a premixed laminar combustor with a loudspeaker as an actuator and a microphone as a sensor. Two control designs were discussed to suppress the pressure oscillations using (i) the LQG-LTR method and (ii) the H1 approach. Their performance was validated experimentally using a 1kW benchtop rig, and their robustness properties were discussed. The results indicated that the LQG-LTR provides a better performance, while the H1 provides pre-computable stability robustness bounds. Both control designs were effective though they were based on a linear model, indicating that the limit-cycle behavior of the nonlinearity did not affect the performance of the linear design. The experimental results reported here represent the first of its kind where a predictive modelbased control design was used for combustion control. The results show that a systematic optimal 24
(a) With the PDE model and LQG controller 150
Pressure (Pa)
100 50 0 −50 −100 −150
0
10
20
30
40
50 Time (msec)
60
70
80
90
100
80
90
100
(b) With the Two−mode model and LQG controller 150
Pressure (Pa)
100 50 0 −50 −100 −150
0
10
20
30
40
50 Time (msec)
60
70
Figure 15: Pressure responses obtained using both the PDE model and the two-mode model for LQG control for A/D location.
control design can be carried out for suppression of thermoacoustic instability and can lead to a faster settling time and a reduced controlled effort, both of which are attractive features for purposes of commercial implementation. Various features of the available actuator technology such as control bandwidth and input saturation can be readily incorporated into the design to ensure efficient utilization of the control energy. Measures of robustness of closed-loop system stability can be derived, quantifying the requisite model accuracy. Bounds on the operating range over which satisfactory performance can be realized can be ascertained using the proposed modelbased approach. We are currently evaluating nonlinear control strategies that take into account the structure of the nonlinearities in the system to expand the scope of operation.
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[4] G.J. Bloxsidge, A.P. Dowling, N. Hooper, and P.J. Langhorne. “Active control of an acoustically driven combustion instability”. Journal of Theoretical and Applied Mechanics, supplement to vol. 6, 1987. [5] T. Poinsot, F. Bourienne, S. Candel, and E. Esposito. “Suppression of combustion instabilities by active control”. Journal of Propulsion and Power, 5(1):14–20, 1989. [6] P.J. Langhorne, A.P. Dowling, and N. Hooper. Practical active control system for combustion oscillations. Journal of Propulsion and Power, 6(3):324–333, 1990. [7] A. Gulati and R. Mani. Active control of unsteady combustion-induced oscillations. Journal of Propulsion and Power, 8(5):1109–1115, 1992. [8] K.R. McManus, U. Vandsburger, and C.T. Bowman. “Combustor performance enhancement through direct shear layer excitation”. Combustion and Flame, 82:75–92, 1990. [9] G. Billoud, M.A. Galland, C. Huynh Huu, and S. Candel. Adaptive active control of combustion instabilities. Combust. Sci. and Tech., 81:257–283, 1992. [10] S. Sivasegaram and J.H. Whitelaw. “Active control of oscillations in combustors with several frequency modes”. In Proceedings of the ASME Winter Annual Meeting, Anaheim, CA, 1992. [11] E. Gutmark, T.P. Parr, K.J. Wilson, D.M. Hanson-Parr, and K.C. Schadow. “Closed-loop control in a flame and a dump combustor”. IEEE Control Systems, 13:73–78, April 1993. [12] A. Kemal and C.T. Bowman. Active adaptive control of combustion. In Proceedings of the IEEE Conference on Control Applications, pages 667–672, Albany, NY, 1995. [13] K.T. Padmanabhan, C.T. Bowman, and J.D. Powell. “An adaptive optimal combustion control strategy”. In 25th International Symposium on Combustion, 1995. [14] D. St. John and G.S. Samuelson. Active, optimal control of a model, industrial, natural gas-fired burner. In Proc. 25th Int. Symp. Combustion, 1994. [15] B. Widrow and S.D. Stearns. Adaptive signal processing. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1985. [16] Y. Neumeier and B.T. Zinn. Active control of combustion instabilities using real time identification of unstable combustor modes. In Proceedings of the IEEE Conference on Control Applications, pages 691–698, Albany, NY, 1995. [17] Y.-T. Fung, V. Yang, and A. Sinha. Active control of combustion instabilities with distributed actuators. Combust. Sci. and Tech., 78:217–245, 1991. [18] Y-T. Fung and V. Yang. Active control of nonlinear pressure oscillations in combustion chambers. Journal of Propulsion and Power, Vol. 8, No. 6:1282–1289, 1992. [19] V. Yang, A. Sinha, and Y.-T. Fung. “State feedback control of longitudinal combustion instabilities”. Journal of Propulsion and Power, 8, 1992. [20] F.E.C. Culick and V. Yang. “Prediction of the stability of unsteady motions in solid-propellant rocket motors”. In Nonsteady Burning and Combustion Stability of Solid Propellants, edited by L. DeLuca and M. Summerfield, 143:719-780, Progress in Astronautics and Aeronautics, AIAA, NY, 1992.
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[21] J.E. Tierno and J.C. Doyle. “Multimode active stabilization of a Rijke tube”. In DSC-Vol. 38. ASME Winter Annual Meeting, 1992. [22] A.M. Annaswamy, M. Fleifil, J.P. Hathout, and A.F. Ghoniem. “Impact of linear coupling on the design of active controllers for thermoacoustic instability ”. Combustion Science and Technology, 128:131–180, December 1997. [23] M. Fleifil, A.M. Annaswamy, J.P. Hathout, and A.F. Ghoniem. “The origin of secondary peaks with active control of thermoacoustic instability”. Combustion Science and Technology, 133:227–260, 1998. [24] A.M. Annaswamy, O. El-Rifai, M. Fleifil, and A.F. Ghoniem. “A model-based self-tuning controller for thermoacoustic instability”. Combustion Science and Technology, 135:213–240, 1998. [25] M. Fleifil, A.M. Annaswamy, Z. Ghoniem, and A.F. Ghoniem. “Response of a laminar premixed flame to flow oscillations: A kinematic model and thermoacoustic instability result”. Combustion and Flame, 106:487–510, 1996. [26] T.Myint-u and L. Debnath. Partial Differential Equations for Scientists and Engineers. Elsevier Science Publishing Co., New York, NY, 1987. [27] B.T. Zinn and M.E. Lores. Application of the galerkin method in the solution of nonlinear axial combustion instability problems in liquid rockets. CST, 4:269–278, 1972. [28] F.E.C. Culick. “Nonlinear behavior of acoustic waves in combustion chambers”. Acta Astronautica, 3:715–756, 1976. [29] A.P. Dowling and J.E. Ffowcs Williams. Sound and Sources of Sound. Ellis Horwood Limited, West Sussex, PO191EB, England, 1983. [30] G. Stein and M. Athans. The LQG/LTR procedure for multivariable feedback control design. IEEE Transactions on Automatic Control, 32:105–114, February 1987. [31] D. A. Wilson. Convolution and hankel operator norms for linear systems. IEEE Transactions on Automatic Control, 34(1):94–97, 1989. [32] M. A. Rotea. The generalized
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Appendix A We show that the solution p0 (x; t) to the unperturbed wave equation
@ 2 p0 ; c2 @ 2 p0 @t2 @z2 27
=
0
(20)
is of the form
p0 (x; t)
=
Re
h
(x) ej!t
i
x
i ( ) = sin (
where
kx + ) ;
(21)
in the presence of non-ideal boundary conditions given by
u0 (zn )
p0
jZn (!) (zn ) ;
=
n = 1; 2
(22)
where z1 and z2 denote the inlet and outlet, respectively, and Z1 and Z2 are real functions of ! . Assuming that p0
where k
=
=
Re(pˆ(z)ej!t ), (20) can be simplified as d2pˆ + k2pˆ = 0 dz2
(23)
!=c, whose solution is of the form pˆ (z)
aejkz + be;jkz
=
(24)
where a and b are two unknown parameters to be determined. The boundary conditions in (22) can be expressed in terms of the pressure as follows: The unperturbed momentum equation
0
@u @t
+
@p0 @z
=
0
(25)
can be integrated to obtain
j dpˆ ej!t: ! dz Combining Eq.(26) with Eq. (22), we obtain that, for n = 1; 2, dpˆ (z ) = ; ! pˆ (z ) : dz n Zn (!) n u0
=
(26)
(27)
Applying (27) to Eq.(24), a linear system of equations can be obtained as
jkejkz a ; jke;jkz b n
n
=
; Z !(!) ejkz a + e;jkz b ; h
i
n
n
n
Nonzero solutions for a and b can be found from (28) if !
!
n = 1; 2:
(28)
2 2 c !L !L c c 1+ (29) Z1 (!) Z2 (!) sin c + z1 (!) ; z2 (!) cos c = 0 which can be solved to obtain admissible values of ! . Using (28) to get a relation between a and b, we obtain that " # ! 1 pb (z) = v (30) !2 kcos (kz ) + Z (! ) sin (kz ) u 1 u ! t 2 k + Z (!) 1
28
Eq. (30) shows that pb(z ) is real, and hence, it follows that of (20) where
k
=
!; c
tan
=
z
pz
( ) = b( )
k Z (!): ! 1
and that (21) is the solution
(31)
The above derivation shows that for general boundary conditions determined by Z1 and Z2 in Eq. (22), the mode shape is of the form (21) where k and are given by Eq. (31). Eq. (31) also shows that when Z1 = 1 the mode shape is cos(kz ) which corresponds to the case when the inlet is “fully closed”, and when Z1 = 0 the mode shape is sin(kz ) when the inlet is “fully open”. It should also be noted that Z2 (! ) affects the solution of Eq. (29) and therefore the wave number k of the mode shape.
29
