Open access peer-reviewed chapter

Adaptive Nonlinear Regulation Control of Thermoacoustic Oscillations in Rijke-Type Systems

By William MacKunis, Mahmut Reyhanoglu and Krishna Bhavithavya Kidambi

Submitted: September 23rd 2016Reviewed: August 23rd 2017Published: December 20th 2017

DOI: 10.5772/intechopen.70683

Downloaded: 333

Abstract

Adaptive nonlinear control of self-excited oscillations in Rijke-type thermoacoustic systems is considered. To demonstrate the methodology, a well-accepted thermoacoustic dynamic model is introduced, which includes arrays of sensors and monopole-like actuators. To facilitate the derivation of the adaptive control law, the dynamic model is recast as a set of nonlinear ordinary differential equations, which are amenable to control design. The control-oriented nonlinear model includes unknown, unmeasurable, nonvanishing disturbances in addition to parametric uncertainty in both the thermoacoustic dynamic model and the actuator dynamic model. To compensate for the unmodeled disturbances in the dynamic model, a robust nonlinear feedback term is included in the control law. One of the primary challenges in the control design is the presence of input-multiplicative parametric uncertainty in the dynamic model for the control actuator. This challenge is mitigated through innovative algebraic manipulation in the regulation error system derivation along with a Lyapunov-based adaptive control law. To address practical implementation considerations, where sensor measurements of the complete state are not available for feedback, a detailed analysis is provided to demonstrate that system observability can be ensured through judicious placement of pressure (and/or velocity) sensors. Based on this observability condition, a sliding-mode observer design is presented, which is shown to estimate the unmeasurable states using only the available sensor measurements. A detailed Lyapunov-based stability analysis is provided to prove that the proposed closed-loop active thermoacoustic control system achieves asymptotic (zero steady-state error) regulation of multiple thermoacoustic modes in the presence of the aforementioned model uncertainty. Numerical Monte Carlo-type simulation results are also provided, which demonstrate the performance of the proposed closed-loop control system under various sets of operating conditions.

Keywords

  • thermoacoustics
  • robust
  • adaptive
  • nonlinear control

1. Introduction

Rijke-type instability is a widely investigated example of a thermoacoustic phenomenon, which describes the generation of potentially unstable pressure oscillations that results from the dynamic coupling between unsteady heat transfer and acoustics [1, 2]. The resulting oscillations in Rijke-type systems can degrade performance and even cause structural damage in combustion systems. Based on this fact, thermoacoustic instability is a primary challenge that must be addressed in the design and manufacture of land-based gas turbines and aircraft engines [311]. Other applications for which thermoacoustic oscillations are a concern include boilers, furnaces, ramjet engines, and rocket motors. The myriad practical engineering applications impacted by Rijke-type instability necessitate the design of reliable control systems to regulate the potentially catastrophic effects of thermoacoustic oscillations.

Control design methods for thermoacoustic oscillation suppression systems can be separated into two main categories: passive control and active control approaches. Passive control methods [1218] can employ acoustic dampers, such as Helmholtz resonators [13] or acoustic liners [12], or they can be achieved by physically redesigning the system by changing the location of the heat source, for example. Passive approaches offer the virtues of simplicity and inexpensive maintenance; however, the performance of passive control methods can only be ensured over a relatively narrow range of operating conditions [4]. To expand the usable range of operating conditions, active control methods offer the capability to automatically adjust the level of control actuation in response to sensor stimuli.

Active control methods are usually implemented in closed-loop configurations, where sensor measurements are utilized in a feedback loop to automatically drive the input signal to the actuators. Figure 1 provides an example of functional schematic of a closed-loop thermoacoustic oscillation suppression system. The two primary strategies for achieving closed-loop active control of thermoacoustic oscillations include (1) using a monopole-like acoustic source such as a loudspeaker to control the acoustic field [19] or (2) using a secondary fuel injector to control the unsteady heat release rate [20, 21]. Several active control approaches to suppress thermoacoustic oscillations have been presented in recent research literature.

Figure 1.

A functional implementation diagram of a thermoacoustic oscillation control system, including a microphone for sensing and a loudspeaker for actuation.

Standard linear control systems for thermoacoustic oscillation suppression are based on stabilizing the closed-loop system through causing the dominant eigenmodes to exponentially decay. However, for realistic thermoacoustic systems where the eigenmodes are nonorthogonal, controlling only the dominant eigenmode can result in the excitation of other modes as a result of the coupling between the acoustic modes. To address this challenge, a transient growth controller is presented in [2224], which achieves strict dissipativity. Experimental or numerical empirical methods for thermoacoustic oscillation control have been widely considered, but more systematic approaches such as robust and adaptive control have gained popularity in more recent research. Active linear control methods have been widely investigated for applications considering simplified thermoacoustic dynamic models [1, 25]. However, by leveraging the tools of nonlinear control, effective suppression control of thermoacoustic oscillations can be achieved over a wide range of operating conditions and dynamic model uncertainty.

Physically speaking, thermoacoustic oscillation suppression can be achieved by disrupting the inherent dynamic coupling between the unsteady heat release and the acoustic waves. By designing active control systems to alter the interaction between the acoustic waves and the unsteady heat release, the amplitude of the thermoacoustic oscillations can be forced to decrease, instead of increase. Additional challenges in designing control systems for thermoacoustic oscillations can be incurred as a result of parametric model uncertainty and unmodeled operating conditions. The recent result in [26] presents a nonlinear active control method, which is proven to asymptotically regulate thermoacoustic oscillations in Rijke-type systems that do not include parametric model uncertainty and unmodeled nonlinearities. The design of active closed-loop control systems for thermoacoustic oscillation suppression that achieve reliable performance over a wide range of operating conditions and model uncertainty remains very much an open problem.

In this chapter, an observer-based nonlinear active closed-loop control method is presented, which achieves asymptotic suppression of self-excited thermoacoustic oscillations in a Rijke-type system, where the system dynamic model includes unmodeled nonlinearities and parametric uncertainty in the system dynamics and actuator dynamics. To achieve the result, a well-accepted thermoacoustic model is utilized, which employs arrays of sensors and monopole-like actuators. To facilitate the control design, the original dynamic equations are recast in a control-amenable form, which explicitly includes the effects of unmodeled, nonvanishing external disturbances and linear time delay. A sliding-mode observer-based nonlinear control law is then derived to regulate oscillations in the thermoacoustic system. A primary challenge in the control design is the presence of input-multiplicative parametric uncertainty in the control-oriented model. This challenge is handled through innovative algebraic manipulation in the regulation error system derivation along with a Lyapunov-based adaptive law. A rigorous Lyapunov-based stability analysis is used to prove that the closed-loop system achieves asymptotic regulation of a thermoacoustic system consisting of multiple modes. Numerical Monte Carlo-type simulation results are also provided, which demonstrate the performance of the proposed closed-loop active thermoacoustic oscillation suppression system.

2. Thermoacoustic system model

The thermoacoustic system model that will be utilized in this chapter consists of a horizontal Rijke tube with multiple actuators. The model is identical to that studied in our previous work in [2224, 27].

Consider the system shown in Figure 2 , where the actuators are modeled as multiple monopole-like moving pistons. It will be assumed that K ≥ 1 actuators are available for control purposes. To facilitate the following observer and control design and analysis, a block diagram is also provided in Figure 3 .

Figure 2.

A control-oriented schematic of a combustion system with actuators modeled as monopole-like moving pistons.

Figure 3.

A block diagram illustrating the main components of the proposed robust and adaptive thermoacoustic oscillation control system.

To facilitate the subsequent model development, nondimensional system variables are defined as

u=u˜u0,p=p˜γM0p0,Q̇s=Q˜̇sγp0u0,E1
x=x˜L0,t=t˜c0L0,δxxfL0=δ˜x˜x˜f,E2

where the above tilde notation denotes the dimensional quantities and the subscript 0 denotes the mean values. In Eqs. (1) and (2), x ∈ Rdenotes the location along the duct (the actuators are located at xak  ∈ R, for k = 1 ,  …  , K, and the heat source is located at xf ), t ∈ R≥0 denotes nondimensional time, p(x, t) ∈ Ris the acoustic pressure, u(x, t) ∈ Rdenotes the velocity, M ∈ Ris the Mach number, c ∈ Ris the speed of sound, L 0 ∈ Ris the length of the duct, and γ ∈ Ris the ratio of specific heats.

By using the nondimensionalized variables defined in Eqs. (1) and (2), the thermoacoustic system with K actuators can be expressed as

ut+px=0,E3
pt+ζp+ux=γ1Q̇sδxxf+γk=1Kαakvakδxxak,E4

In the expressions in Eqs. (3) and (4), ζ ∈ Rdenotes a damping coefficient; thus, the term ζp physically expresses losses resulting from the effects of friction and thermo-viscous damping, and αak  ∈ Rrepresents a dimensionless area ratio that can be explicitly defined as αak  = Sak /S for k = 1 ,  …  , K. The nondimensional heat release rate Q̇sRis explicitly defined as [19].

Q̇s=K13+uftτ13,E5

where

K=2LwTwT¯03u0p0πλcvρ0dw2.E6

In Eq. (6), dw , Lw , and Tw  ∈ Rdenote the diameter, length, and temperature of the heated wire, respectively; ρ ∈ Rdenotes air density; T ∈ Ris temperature; λ is thermal conductivity; cv  ∈ Rdenotes the specific heat capacity at constant volume; and τ ∈ Rrepresents the time delay between the initial velocity field perturbation (i.e., the actuation) and the resulting effect on the heat release. Readers are referred to [22] for details on the numerical values of the physical parameters used for the thermoacoustic model being considered in this chapter. The gas is assumed to be inviscid, perfect, and nonconductive.

The acoustic pressure p and velocity u inside the duct can be expressed as a superposition of the duct natural modes as

pxt=j=1Nsinjπxηjt,E7
uxt=j=1Ncosjπxηjt,E8

where N ∈  denotes the number of modes considered in the numerical discretization.

The actuation signal vak  ∈ Rof the kth monopole-like source (e.g., a loudspeaker) [28] can now be expressed as

vak=Rkuxak+Skpxak,E9

where R k and SkRare dimensionless control parameters of the actuators. After using the expressions in Eqs. (4)(8), the discretized governing equations are obtained as

η¨j+ηj+ζjη̇j=2γ1Q̇sxftτsinxf2γk=1Kαakvaxaktsinxak.E10

In Eq. (10), ζ represents the overall damping in the system [29, 30]. For the model under consideration here, the terms p and uxare taken to be zero at the ends of the duct. Moreover, it is assumed that no acoustic energy is dissipated in the thermal and viscous boundary layers at the duct walls [2931].

3. Control-oriented model derivation

To facilitate the presentation of the main ideas, we consider a thermoacoustic system with two modes (i.e., N = 2). However, the theoretical development presented here can be directly extended to address N modes.

In the following discussion, the vector of modes (i.e., the state vector) will be annotated as η(t) ≜ [η 1(t), η 2(t)] T , and define

Ψx=cosπxcos2πxT,E11
Φx=sinπxsin2πxT.E12

Assuming that ∣uf  (t − τ)∣ < 1/3, the heat release rate can be approximated as

Q̇sxftτ3K2ΨfTηtτ,E13

where Ψ f  ≜ Ψ(xf ).

By following a derivation procedure similar to that presented in [24], the dynamics of the duct natural modes can be expressed as

Mη¨+Dγ1K3τΦfΨfTη̇+M1+γ1K3ΦfΨfTηhηη̇τd=Bv,E14

where

M=diag1π12π,D=diagζ1πζ22π.E15

In Eq. (14), η(t) = [η 1(t), η 2(t)] T  ∈ R2 is a vector containing the natural modes, hηη̇R2is an unknown nonlinear function, and τd (t) ∈ R2 is a general unknown bounded disturbance. To facilitate the control development in the following analysis, the dynamic equation in Eq. (14) is rewritten in the control-oriented form:

Mη¨+Cdη̇+Kηhηη̇τd=Bv,E16

where the uncertain constant terms C d and K ∈ R2 × 2 are defined as

Cd=Dγ1K3τΦfΨfT,E17
K=M1+γ1K3ΦfΨfT.E18

Also, in Eq. (16), the uncertain constant control input gain matrix BR2×2is defined via the relationship:

Bv=2γk=1KαakRkΦxakΨTxakη+2γk=1KαakSkΦxakΦTxakMη̇E19

where v(t) ∈ R2 is a subsequently defined auxiliary control signal.

Remark 1 Note that Eq. (19) highlights one of the primary challenges in the control design presented in this chapter. Specifically, the input-multiplicative parametric uncertainty in Bpresents a nontrivial control design challenge, which will be mitigated in the proposed control method through the use of a novel Lyapunov-based adaptive law.

Assumption 1 The unknown nonlinear disturbance τd (t) satisfies

τdtζ,t0,E20

where ζ ∈ Rdenotes a positive bounding parameter.

4. Control development

In this section, a rigorous regulation error system development will be utilized to develop a nonlinear control system, which will be proven to effectively compensate for the inherent parametric uncertainty in the dynamic model of the thermoacoustic system in addition to the uncertain actuator model. Moreover, the proposed controller compensates for unmodeled, norm-bounded disturbances present in the dynamic model (e.g., the disturbances could represent unmodeled nonlinearities resulting from time delays due to the finite heat release rate).

4.1. Open-loop error system

The robust and adaptive nonlinear control design presented here is motivated by the desire to eliminate the transient growth of acoustical energy in a thermoacoustic dynamic system. To present the control design methodology, we consider a simplified N = 2 mode scenario, which will be shown to regulate the modes η 1(t) and η 2(t) to zero in the sense that

ηt,η̇t0.E21

To mathematically describe the regulation control objective, an auxiliary regulation error signal r(t) ∈ R2 is defined as

r=η̇+αη,E22

where α ∈ Rdenotes a positive, constant control gain. After taking the time derivative of Eq. (22), multiplying the result by M, and using Eq. (16), the regulation error dynamics can be expressed as

Mṙ=Ddη̇Kη+hηη̇+Bv+τd.E23

To address the case where the constant matrices D d  = C d  + M α, K, and Bare uncertain, the dynamics can be linearly parameterized as

Y1θ1=Ddη̇Kη,E24
Y2θ2=Bv.E25

In Eqs. (24) and (25), Y1ηη̇R2×p1and Y 2(v) ∈ R2 × p 2 are measurable regression matrices, and θ 1 ∈ Rp 1 and θ 2 ∈ Rp 2 are vectors containing the uncertain constant parameters in D d , K, and B. The constants p 1 and p 2 ∈  denote the number of uncertain parameters in the vectors θ 1 and θ 2, respectively.

To facilitate the subsequent Lyapunov-based adaptive control law development to compensate for the input-multiplicative uncertain matrix B, an estimate θ̂2tRp2of the uncertain vector θ 2 is defined via the linear parameterization:

Y2θ̂2=B̂v.E26

In Eq. (26), B̂t2×2denotes a time-varying estimate of the uncertain constant matrix B. By adding and subtracting the term B̂tvtin Eq. (23) and using Eqs. (24) and (26), the open-loop error dynamics can be expressed as

Mṙ=Y1θ1+hηη̇+Y2θ˜2+B̂v+τd,E27

where θ˜2tRp2denotes the parameter estimate mismatch, which is defined as

θ˜2θ2θ̂2.E28

The error dynamics in Eq. (27) are now in a form amenable for the design of a robust and adaptive control law, which compensates for the parametric uncertainty and unmodeled nonlinearities present in the system dynamics.

Assumption 2 The unknown nonlinear term hηη̇can be upper bounded as

hηη̇ρzz,E29

where ρ(·) ∈ Ris a positive, globally invertible nondecreasing function and z(t) ∈ R4 is defined as

zt=ηTtrTtT.E30

In Eq. (29), ‖·‖ denotes the standard Euclidean norm of the vector argument.

Assumption 2 is mild in the sense that inequality (29) is satisfied for a wide range of nonlinear function hηη̇.

4.2. Closed-loop error system

Based on the open-loop error system in Eq. (27), the control input v(t) is designed as

v=B̂1Y1θ̂1ks+1rβsgnrη,E31

where ks  ∈ Rdenotes a positive, constant control gain and β ∈ R2 × 2 is a positive-definite, diagonal control gain matrix. In Eq. (31), sgn(·) denotes a vector form of the standard signum function.

After substituting the control input expression in Eq. (34) into the open-loop dynamics in Eq. (27), the closed-loop error system is obtained as

Mṙ=Y1θ˜1+Y2θ˜2+hηη̇ηks+1rβsgnr+τd,E32

where θ˜1tRp1is the parameter estimate mismatch defined as

θ˜1θ1θ̂1.E33

Based on Eq. (32) and the subsequent stability analysis, the parameter estimates θ̂1tand θ̂2tare generated online according to the adaptive laws:

θ̂˙1=projΓ1YTr,θ̂˙2=projΓ2Y2Tr,E34

where Γ1 ∈ Rp 1 × p 1 and Γ2 ∈ Rp 2 × p 2 are positive-definite adaptation gains.

Remark 2 The function proj(·) in Eq. (34) denotes a normal projection algorithm, which ensures that the following inequalities are satisfied:

θ¯1θ̂1θ¯1,θ¯2θ̂2θ¯2,E35

where θ¯1,θ¯1,θ¯2andθ¯2Rrepresent known, constant lower and upper bounds of the elements of θ̂1tand θ̂2t, respectively. In the current result, the use of the proj(·) function is primarily motivated by the desire to avoid singularities in the matrix estimate and facilitate the matrix inverse calculation in Eq. (31).

To facilitate the following stability analysis, the control gain matrix β will be selected to satisfy the sufficient condition:

λminβ>ζ,E36

where ζ is introduced in Eq. (20) and λ min{·} denotes the minimum eigenvalue of the argument.

5. Stability analysis

Theorem 1 The control law in Eq. (31) with adaptive laws defined as in Eq. (34) ensures asymptotic regulation of the thermoacoustic modes η 1(t) and η 2(t) in the sense that

ηt0astE37

provided that ks is selected as sufficiently large (see the subsequent stability proof) and β is selected to satisfy inequality (36).

Proof. Let Vηrθ̂1θ̂2tRbe defined as the nonnegative function:

Vt12ηTη+12rTMr+12θ˜1TΓ11θ˜1+12θ˜2TΓ21θ˜2.E38

After taking the time derivative of Eq. (38) and using Eq. (32), V̇tcan be expressed as

V̇t=ηTrαη+rThηη̇ηks+1rβsgnr+τd+rTY1θ˜1+Y2θ˜2θ˜1TΓ11θ̂˙1θ˜2TΓ21θ̂˙2,E39

where Eq. (22) was utilized. After substituting the adaptive laws in Eq. (34) and canceling common terms, V̇tcan be expressed as

V̇t=αηTη+rThηη̇ks+1rβsgnr+τd.E40

By using inequalities of Eqs. (20) and (29), the expression in Eq. (40) can be upper bounded as

V̇tαη2ksr2ρzzrr2βrTsgnr+ζr.E41

After completing the squares for the parenthetic terms in Eq. (41), the upper bound on V̇tcan be expressed as

V̇tαη2r2βr+ζrksrρz2ksz2+ρ2z4ksz2,E42

where the fact that r T  sgn(r) = |r| was utilized. After using inequality (36), the upper bound in Eq. (42) can be expressed as

V̇tλ0ρ2z4ksz2,E43

where λ 0 ≜ min {α, 1} and the triangle inequality (i.e., |r| ≥ ‖r‖ ∀r ∈  R n ) was utilized. Based on Eq. (43), V̇cz2, for some positive constant c, inside the set R, where Ris defined as

Rzz<ρ12λ0ks.E44

The expressions in Eqs. (38) and (43) can be used to prove that η(t), r(t), θ˜1t, and θ˜2tLin Rand Eq. (22) can then be used to prove that η̇tLin R. Given that η(t) and r(t) ∈ L in R, Eq. (31) can be used along with Eq. (35) to prove that v(t) ∈ L in R. Since η(t), η̇t, and v(t) ∈ L in R, Y1ηη̇and Y 2(v) ∈ L in R. Given that η(t), r(t), η̇t, Y1ηη̇, and Y 2(v) ∈ L in R, Eq. (32) can be used along with Eq. (35) to show that ṙtLin R. Since η̇tandṙtLin R, η(t) and r(t) are uniformly continuous in R. It then follows from Eq. (30) that z(t) is uniformly continuous in R. Given that η(t), r(t), θ˜1t, and θ˜2tLin R, V(t) ∈ L in R, and Eq. (43) can be integrated to prove that 0zt2dtLin R. Thus, z(t) ∈ L ∩ L2 in R. Barbalat’s lemma can now be invoked to prove that ‖z(t)‖→0 as t→∞. Hence, ‖η(t)‖→0 as t→∞ in R, where the set Rcan be made arbitrarily large by increasing the control gain ks —a semiglobal result.

6. Sliding-mode observer design

In practical thermoacoustic systems, the full state of the dynamic system is not directly measurable, and so it must be estimated through direct sensor measurements of velocity and pressure. This section presents an observer design, which is utilized to estimate the complete state of the system. The necessary observability condition can easily be satisfied through judicious sensor placement.

Let x=ηTη̇TMTdenote the state. For simplicity in the subsequent development, only one sensor is assumed to be available at location xs . Then, the system can be rewritten as

ẋt=Axt+Bxtτ+Gut,E45
y=Cx,E46

where

A=0M1M1D,B=00W0,G=0I,E47

where

W=3γ1KΦfΨfT,E48

and the output matrix C is determined by the sensor choice and its location. The output equation for the velocity sensor case is given by

y=uxst=ΨsTη,E49

and thus

C=ΨsT01×2.E50

For the pressure sensor case, we have

y=pxst=ΦsTη̇,E51

and hence

C=01×2ΦsTM..E52

It is assumed that the sensor location xs is chosen such that the systems (45) and (46) subject to τ = 0 are observable, i.e.,

rankCTETCTET2CTET3CT=4,E53

where E = A + B.

It can be shown that there exists a coordinate transformation of the forms z = [z 1, z 2] T  = Tx, z 1 ∈ R, and z 2 ∈ R3, such that the systems (45) and (46) in the new variables take the following form:

żt=A˜zt+B˜ztτ+G˜ut,E54
yt=z1t,E55

where A˜=TAT1, B˜=TBT1, and G˜=TG.

The estimate ẑof the the state z will now be generated via the observer equation

ẑ̇t=A˜ẑt+B˜ẑtτ+G˜ut+Lsgnytẑ1t.E56

Then, the error dynamics z¯=zẑcan be written as

z¯̇t=A˜z¯t+B˜z¯tτLsgnz¯1t.E57

Partition the system above as

z¯̇1t=A˜11z¯1t+A˜12z¯2t+B˜11z¯1tτ+B˜12z¯2tτL1sgnz¯1t,E58
z¯̇2t=A˜21z¯1t+A˜22z¯2t+B˜21z¯1tτ+B˜22z¯2tτL2sgnz¯1t.E59

The following result can now be stated:

Theorem 2 Let E˜12=A˜12+B˜12and E˜22=A˜22+B˜22, and assume that the pair E˜22E˜12is observable. Then, the observer gain L 1 can be chosen such that the system (57) is asymptotically stable for all τ ∈ [0, τmax ] with τmax defined by:

τmax=12λmaxQ1TFTPHQ1HTPFQ11,E60

where F=E˜22L2E˜12/L1, H=B˜22L2B˜12/L1, and Q is any symmetric, positive-definite matrix such that P is a symmetric, positive-definite matrix P solution of the Lyapunov matrix equation:

FTP+PF=Q,E61

and Q 1 is the square root of the matrix Q, i.e.,

Q1TQ1=Q.E62

Proof. From Eq. (58), sliding mode exists in an area:

L1>A˜11z¯1t+A˜12z¯2t+B˜11z¯1tτ+B˜12z¯2tτ.E63

Condition (63) guarantees sliding in Eq. (58) along the manifold z¯1=0; thus η̂η. According to the equivalent control method [32], the system in sliding mode behaves as if L1sgnz¯1is replaced by its equivalent value L1sgnz¯1eqwhich can be calculated from subsystem (58) assuming z¯1=0and z¯̇1=0. Hence,

L1sgnz¯1eq=A˜12z¯2t+B˜12z¯2tτ.E64

Substitution of Eq. (64) into Eq. (59) yields

z¯̇2=A˜22L2A˜12/L1z¯2t+B˜22L2B˜12/L1z¯2tτ.E65

Using the fact that the pair E˜22E˜12/L1is observable, the observer gain L 2 can be chosen such that the eigenvalues of the matrices E˜22L2E˜12/L1have negative real parts. Thus, the subsystem (65) is asymptotically stable for τ ≤ τ max. This implies that z¯0 and ẑt=zt, and hence, x̂=x.

7. Simulation results

A numerical simulation was created for two modes and two actuators (i.e., N = K = 2) to demonstrate the performance of the control law described by Eqs. (26), (31), and (34). The simulation utilizes the dynamics described in Eq. (16) for a system with two modes. The physical parameters used in the simulation are given in Table 1 .

ρ1.025 kg/m3λ0.0328 W/m K
cv719 J/kg Kγ1.4
L 01 mLw2.5 m
c344 m/su 00.3 m/s
T 0295 KTw1680 K
dw0.5 × 10−3 mS1.56 × 10−3 m
P 08.69 × 104 Paα a 1  = α a 20.01
ζ 10.0440ζ 20.1657

Table 1.

Physical parameters.

The initial conditions for the modes were selected as η 1(0)  =  0.07 and η 2(0) = 0. The initial conditions of the velocity and pressure parameters at the location of the heat source are u(xf , 0) = − 0.1646 and p(xf , 0) = 0 for xf = 0.7. Figure 4 shows the open-loop velocity and pressure perturbations at the heat source in the absence of control actuation.

Figure 4.

Time response of the velocity u(t) and pressure p(t) at the heat source location during open-loop (uncontrolled) operation.

In closed-loop operation, the adaptation gain matrices used in the simulation were selected as Γ 1 = 0.1I 8 and Γ 2 = 0.001I 4. The value for the gain α was selected as 1, and the control gain matrices β and k s were selected as β = diag {2.43, 2.1} ,  k s  = diag {4.5, 2.4}. The results of 20 Monte Carlo-type simulations for the closed-loop operation are shown in Figures 5 10 . Figure 5 shows the time evolution of the velocity and pressure values at the heat source location during closed-loop controller operation. Figure 6 shows the time history of the modes η 1(t) and η 2(t) during closed-loop operation. The results clearly show the capability of the proposed robust and adaptive control law to drive the states to zero. The commanded control signals are shown in Figure 7 . The control actuation remains within the reasonable limits throughout closed-loop operation. Figures 8 10 show the time responses of the elements of the parameter estimate vectors θ̂1tand θ̂2tduring closed-loop controller operation.

Figure 5.

Time response of the velocity u(t) and pressure p(t) during closed-loop controller operation at the heat source location.

Figure 6.

Time response of the oscillation modes η 1(t) and η 2(t) during closed-loop controller operation.

Figure 7.

Commanded control inputs ν 1(t) and ν 2(t) during closed-loop controller operation.

Figure 8.

Time response of the adaptive parameter estimates θ ̂ 11 t , θ ̂ 12 t , θ ̂ 13 t , and θ ̂ 14 t during closed-loop controller operation.

Figure 9.

Time response of the adaptive parameter estimates θ ̂ 15 t , θ ̂ 16 t , θ ̂ 17 t , and θ ̂ 18 t during closed-loop controller operation.

Figure 10.

Time response of the adaptive parameter estimates θ ̂ 21 t , θ ̂ 22 t , θ ̂ 23 t , and θ ̂ 24 t during closed-loop controller operation.

8. Conclusion

A robust and adaptive nonlinear control method is presented, which asymptotically regulates thermoacoustic oscillations in a Rijke-type system in the presence of dynamic model uncertainty and unknown disturbances. To demonstrate the methodology, a well-accepted thermoacoustic dynamic model is introduced, which includes arrays of sensors and monopole-like actuators. To facilitate the derivation of the adaptive control law, the dynamic model is recast as a set of nonlinear ordinary differential equations, which are amenable to control design. To compensate for the unmodeled disturbances in the dynamic model, a robust nonlinear feedback term is included in the control law. One of the primary challenges in the control design is the presence of input-multiplicative parametric uncertainty in the dynamic model for the control actuator. This challenge is mitigated through innovative algebraic manipulation in the regulation error system derivation along with a Lyapunov-based adaptive control law. To address practical implementation considerations, where sensor measurements of the complete state are not available for feedback, a detailed analysis is provided to demonstrate that system observability can be ensured through judicious placement of pressure (and/or velocity) sensors. A sliding-mode observer design is developed, which is shown to estimate the unmeasurable states using only the available sensor measurements. A detailed Lyapunov-based stability analysis is provided to prove that the proposed closed-loop active thermoacoustic control system achieves asymptotic (zero steady-state error) regulation of multiple thermoacoustic modes in the presence of the aforementioned model uncertainty. Numerical Monte Carlo-type simulation results are also provided, which demonstrate the performance of the proposed closed-loop control system under 20 different sets of operating conditions.

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William MacKunis, Mahmut Reyhanoglu and Krishna Bhavithavya Kidambi (December 20th 2017). Adaptive Nonlinear Regulation Control of Thermoacoustic Oscillations in Rijke-Type Systems, Adaptive Robust Control Systems, Le Anh Tuan, IntechOpen, DOI: 10.5772/intechopen.70683. Available from:

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