1.3.2 Model Predictive Control
According to the overview of Findeisen et al. [2003], model predictive control for linear constrained systems has been successfully applied as a useful control solution for many practical applications. It is expected that the use of non-linear models in the predictive control framework, leading to non-non-linear model predictive control, results in improved control performance. However giving theoretical, computational and implementational problems. For example the guarantee of closed-loop stability in output feedback non-linear MPC, when using observers for state recovery.
In Morari and Lee [1999] it is concluded that closed-loop stability of MPC algorithms has been studied extensively and addressed satisfactorily from a the-oretical point of view, if not from a practical (implementation) point of view.
Contrary to the linear case, however, feasibility and the possible mismatch be-tween the open-loop performance objective and the actual closed-loop perfor-mance are largely unresolved research issues in non-linear MPC. An additional difficulty is that the optimization problems to be solved on-line are generally non-linear programs without extensive features, which implies that convergence to a global optimum cannot be assured. For the quadratic programs arising in the linear case this is guaranteed.
A very interesting work is Lazar [2006], where Model Predictive Control of hybrid systems is used effectively, stabilization is proven and successful im-plementation of the designed MPC controller can be expected, because the on-line computation time and effort can be reduced significantly using an ex-plicit method. Therefore this approach is chosen here in combination with the bounded feedback control system discussed in Section 1.3.1.
1.4 Goals of research
In this work a centrifugal compression system is modeled with a two state Gre-itzer lumped parameter model, this system and model used are taken from the work of Willems [2000]. Active control is used to stabilize surge limit cycles in a desired set-point. The first goal here is to reach a mass-flow which is a small as possible. The purpose of this is to obtain a large as possible stable operating region and thus giving a large as possible safety margin when the compressor operates at maximum performance (maximum pressure point in the compressor map). Next to this attention is paid to disturbance rejection and robustness of the closed-loop.
First a one-sided controller is designed, as described in Willems [2000]. Out-put and state-feedback are both investigated using this controller. Both a linear and hybrid Model Predictive Controller are designed on the one-sided control system to investigate if improvements in the stable operating region, distur-bance rejection and robustness of the closed-loop system can be achieved.
The work has the following structure:
1.4 Goals of research 8
In Chapter 2 the two state Greitzer lumped parameter model is presented, together with a one-sided controller and closed-loop simulations are performed, these are numerically compared to the ones in Willems [2000]. Both output and state-feedback are used here. Also two scenarios are introduced in this Chapter, in which robustness and disturbance rejection of the closed-loop will be investigated. These same scenarios are also adopted to investigate the linear and hybrid Model Predictive Controllers.
The design of a linear Model Predictive Controller is discussed in Chapter 3 and simulations are performed.
Chapter 4 introduces a hybrid Model Predictive Control strategy. The design of a hybrid Model Predictive Controller is described here and simulated. All the different types of control in this work are compared and conclusions are drawn in Chapter 5.
Chapter 2
Active Surge Control
In this chapter the used compressor model with the one-sided controller de-scribed in (Willems [2000, Section 3.3.3, p. 38]) is presented and explained.
Simulations are performed of deep surge control for the output and feedback controlled cases. Also robustness and disturbance rejection of the closed-loop systems are investigated. The results are compared with Willems.
2.1 Greitzer lumped compressor model
The model of the centrifugal compression system which is used is given in Figure 2.1. It is represented by a duct in which the compressor works that discharges in a large volume (plenum). The compressed fluid flows via the plenum through the throttle and control valve into the atmosphere.
Figure 2.1: Compression system (Willems [2000]).
2.1 Greitzer lumped compressor model 10
To describe the dynamic behavior of the examined compression system, the Greitzer lumped parameter model (Greitzer [1976]) is used, this model is origi-nally designed for axial compressors and was proven to be usable for centrifugal compressors as well (Hansen et al. [1981]).
The following assumptions are made in Greitzer [1976]: 1) the flow in the ducts is one-dimensional and incompressible 2) in the plenum the pressure is uniformly distributed and the gas velocity is neglected 3) the temperature ratio of the plenum and ambient is assumed to be near unity: therefore an energy balance is not required 4) the influence of the rotor speed variations on the system behavior is neglected
The dimensionless mass-flow ϕ, dimensionless pressure difference Ψ and di-mensionless time et are defined as
ϕ= m˙
ρaAcUt Ψ = △P
1
2ρaUt2 et= tωH (2.1) with the Helmholtz frequency
ωH= a r Ac
VPLC (2.2)
here ˙mis the mass-flow,△P the pressure difference between the pressure in the system and the ambient pressure, ρa the air density at ambient conditions, Ac
the compressor duct area, a the speed of sound, Ut the rotor tip speed, Vp the plenum volume and Lc the equivalent compressor duct length.
The following set of dimensionless equations that describe the non-linear compression system are
The equations for the behavior of the dimensionless mass-flow ϕc in the com-pressor duct and ϕtin the throttle duct are essentially the momentum equations for each duct. Ψc is the dimensionless pressure rise across the compressor and Ψt gives the dimensionless pressure drop across the throttle. The equation for the pressure rise in the plenum ψ gives the mass conservation in the plenum.
The expression for the dimensionless pressure rise across the compressor Ψcis a first order transient response model with time constant eτ and Ψc,ss the steady-state dimensionless compressor pressure rise given in the compressor map. In