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This work investigates the active control of an unstable Rijke tube using robust output model predictive control (RMPC). As internal model a polytopic linear system with constraints is assumed to account for uncertainties. For guaranteed stability, a linear state feedback controller is designed using linear matrix inequalities and used within a feedback formulation of the model predictive controller. For state estimation a robust gain-scheduled observer is developed. It is shown that the proposed RMPC ensures robust stability under constraints over the considered operating range.

Modern gas turbines have to comply with increasingly stringent emission requirements for

Active control of thermoacoustic instabilities (in parts taken from [

As a consequence high-pressure oscillations take place in the combustor which are detrimental for performance, emissions and durability of the combustor components. To avoid these drawbacks, two main directions are possible, namely, passive and active control, to stabilize the thermoacoustic system [

In this paper active control is investigated, that is, the closed-loop control of the thermoacoustic system. Closed-loop control uses a feedback path consisting of sensors, a control law and actuators to control a system, cf. Figure

In summary, the following demands have to be fulfilled by an active control system:

robust stabilization under constraints;

handling MIMO;

fault-tolerance.

To cope with these demands, robust output model predictive control (RMPC) is investigated in this work. A common physical modelling approach is applied to the Rijke tube, and from this a simplistic linear polytopic model is derived to model the system dynamics with (assumed) parameter uncertainties. The model is used to determine a robust linear state feedback controller by the use of linear matrix inequalities (LMI). This controller is incorporated in an approach presented in [

The paper is organised as follows. Section

A schematic drawing of the Rijke tube setup is shown in Figure

Dimensions of the Rijke tube setup.

Size | Value |
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Pos. 1: | |

Pos. 2: | |

Pos. 3: | |

Pos. 4: |

Schematic drawing of the Rijke tube setup.

The acoustic model is a one-dimensional acoustic network consisting of simple geometric components, which are analytically tractable. The underlying assumption, which is common in thermoacoustic community [

Signal-oriented model of the Rijke tube in MATLAB/SIMULINK.

Since only low frequencies are of interest the flame zone is short compared to the acoustic wavelength. Thus, the flame zone represents a discontinuity for the acoustic waves. An approach derived in [

Due to the resulting interaction between acoustics and combustion, the thermoacoustic instability reaches a limit cycle after a period of exponential growth which is a clear indicator for nonlinear effects in the system. Since only linear acoustics are expected in the operational range of a combustor, the nonlinear effects are supposed to be in the combustion itself. The main non-linear effect of the combustion is possibly a saturation in the heat release. Besides sophisticated models relating the flame surface to the heat release via the

For simulation purpose, the presented model is implemented in signal oriented form in Matlab/SIMULINK as shown in Figure

Model parameters for the physical model.

Parameter | Value |
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−0.89 | |

−0.89 | |

918 |

Figure

Comparison of frequency response of measurements (green) and analytical model (blue).

Model predictive control (MPC) is a control technique that utilizes modern optimization algorithms for control by using a predictive optimization formulation and algorithms for constrained optimization online. In a receding horizon policy the cost function:

In order to explicitly take into account model uncertainties within the MPC framework, we consider a linear polytopic model of the Rijke tube in the following form:

This LPV system can be transformed into a polytopic system (

Because an open-loop prediction of the systems state trajectory, as it is standard in most MPC formulations, cannot account for the reduced sensitivity to disturbances or modelling errors due to a feedback controller, we use the approach of incorporating an internal state feedback controller

Formulations for guaranteed stability within the MPC framework can be considered as a mature topic nowadays. A standard approach is to utilize the cost function

In the literature, this technique of guaranteed stability is sometimes called dual-mode control. The optimizer has to steer the system into the terminal region

As terminal region

Since the actual state

The proposed robust MPC with its internal polytopic model and the robust state observer is able to stabilize the Rijke tube robustly by solving a quadratic program (QP) online. Critical for the application are the number of constraints in the QP as a result of the feedback MPC formulation. For a prediction horizon of

RMPC of the unstable Rijke tube with different length in simulation. The RMPC has a prediction horizon of

RMPC of the unstable Rijke tube with different length in the experiment. The RMPC has a prediction horizon of

An analytical model for a Rijke tube has been applied which is able to reproduce the stability map of the thermoacoustic setup as well as the dynamic behaviour over different lengths of the tube. Therefore, it is an ideal test bench for the robust control of unstable thermoacoustic systems, especially the real-time capability of MPC algorithms. It is used to derive a simplistic linear parameter varying system which represents the unstable modes. Using this system, it is shown that a robust output MPC can be designed that is capable of steering the system robustly to the origin under constraints. The proposed RMPC is a good compromise between conservatism and computational load when considering the very fast system dynamics of a thermoacoustic system and as a consequence the short calculation times needed for the application online. Thus, RMPC is a promising approach in order to fulfill the demands for an active control system in modern gas turbines. The robust stabilization and constraint handling are shown in this paper. In addition, a MIMO setup and fault-tolerant control can be incorporated quiet naturally into the MPC framework. Finally, the importance of physical understanding and modelling for the estimation of unavoidable uncertainties is demonstrated for the robust control of thermoacoustic instabilities.

The authors gratefully acknowledge the contribution of the Deutsche Forschungsgemeinschaft through the Collaborative Research Center 686 “Model-Based Control of Homogenized Low-Temperature Combustion”.