H&-robust optimal kontrol tekniğinin kullanılması ile T 700 turbo motorunun güç türbini hızının kontrolü

Abstract

ÖZET Günümüzde Silahlı Kuvvetler1 in en önemli unsurlarından olan Apache ve Blackhawk helikopterlerinin manevra kabiliyeti, emniyeti ve verimliliği açısından bu helikopterlerde kullanılan GE-T700 turbo motorunun güç türbini hızının kontrolü hayati öneme sahiptir. Güç türbini hızında meydana gelebilecek değişiklikler motora zarar verebileceği ve düzensiz çalışmaya neden olabileceği için iyi bir kontrol sistemi zaruridir. Bu çalışmada öncelikle GE-T700' ün yapısı, kullanımı ve matematik modeli üzerinde durulmuştur. Çünkü iyi bir kontrol sistemi elde etmenin yolu, sistemin iyi bir şekilde tanılanmasından geçmektedir. Daha sonra H` -robust kontrol tekniği fazla detaya girilmeden anlatılmıştır. Bu teknik ile kontrolör, durum uzayı modeli şeklinde elde edilmekte ve dolayısıyla geri beslemeli bir kontrol sistemi ile bu kontrolör kolayca tatbik edilebilmektedir. Matlab paket programı yardımıyla yapılan simülasyon çalışmaları ile kontrolörün, sisteme etkiyen yük bozucularına karşı ne kadar etkili olduğu eğrilerle gösterilmekte ve sonuçlar irdelenmektedir.

SUMMARY POWER TURBINE SPEED CONTROL OF THE GE-T700 TURBO ENGINE USING THE H` -ROBUST OPTIMAL CONTROL TECHNIQUE In this study, a mathematical model was developed using the data obtained from the nonlinear simulation of the GE-T700 turbo engine and an HOT -Robust Controller was designed for this model and then its performance against disturbances was tested. Also for three different GE-T700 models, H` -Controllers were designed and tested. FIVE-STAGE LOW FUEL PRESSURE AXIAL FLOW FUELFLOW INPUT FLOWTHROUGH ANNULAR COMPRESSOR I COMBUSTION CHAMBER SINGLE-STAGE TWO-STAGE AXIAL TWO-STAGE OUTPUT SHAFT CENTRIFUGAL FLOW HIGH-PRESSURE AXIAL FLOW COMPRESSOR TURBINE POWER TURBINE Figure 1 Cross Section of a GE-T700 Turboshaft Engine The GE-T700 is a turboshaft engine in pairs in the Army's Apache and Blackhawk helicopters. This 1600 HP-class, modular, two-spool engine shown in Figure 1 consists of a gas generator section and a free power turbine. The gas generator section is made up of a five-stage axial and a single-stage centrifugal compressor, a low fuel pressure flowthrough annular combustion chamber, and an air-cooled, two-stage, axial-flow, high-pressure turbine. The free power turbine is a two-stage, uncooled, axial-flow type. There exists a one-way coupling between the power turbine and the gas generator. VIFigure 2 A Conventional Helicopter A conventional helicopter, as shown in Figure 2, utilizes a single main rotor, primarily for lift, and a tail rotor for torque reaction and directional control in the yaw degree of freedom. The main and tail rotor systems are directly coupled to two turboshaft engines through gear reduction sets and shafting. A helicopter pilot does not control the power turbine speed. During the helicopter operates, the power turbine speed must be almost constant. This constant speed is providedby the available control system. The helicopter pilot commands an altitude change by moving the collective stick which alters the collective pitch of the main rotor blades. A change in the blade pitch angle causes a change in load as lift is increased or decreased. When the helicopter tries to climb, implying that torque must be increased, the control system increases the fuel flow to the engines. The rotor applies a load to the turbine as long as they are engaged. A command to descend causes the control system to reduce fuel flow, thereby reducing the torque applied to the rotor from the shaft. The rotor will spin freely along with the shaft, but will only be driven by the shaft when friction slows the rotor down. This leads to a situation where the load depends on the direction of motion of the helicopter. A command to climb causes a significant load and the associated excursions in the state variables. On the other hand, a command to descend causes sudden drop in load because the rotor's inertia keeps it spinning at the desired velocity even though fuel flow is reduced, meaning shaft torque is decreased. The incorporation of a fast or tight engine speed control capable of rejection rotor system load disturbances will be resulted in increased helicopter maneuvering capability. This increased maneuvering capability must be accomplished, however, without exciting coupled engine/rotor systems complaint dynamics. vuThe control system must insure the following requirements for the turboshaft engine: 1) Maintain constant power-turbine speed in the presence of load disturbances, 2) Not provide input energy to excite coupled engine/drive train resonant modes, 3) Maintain safe turbine operating conditions, including adequate stall margin, and hard operational constraints such as temperature, speed and torque, 4) Operate at peak efficiency. Turbine engine dynamics are described by complex nonlinear equations. For a linear controller design, the nonlinear dynamic model must be linearized about an equilibrium operating point. As with most turbine engines, the assumption is made that the pressure and temperature dynamics appearing in the flow equations are fast compared to the gas generator and free turbine dynamics in the low frequency region. Therefore, their dynamics can be neglected and they are included in the model only as outputs. It was decided that the SISO structure was appropriate from the implementation side, namely cost and complexity. The combined engine-rotor system can be modeled by a state-space model in the following form: x = A x + B u y = Cx + Du ^ A A where u = WF (Fuel Flow or Trim Signal), y = Np (Power Turbine Speed) Fuel flow is used to regulate the power turbine speed. In this study, for 4 different models, H^ -controllers were tested. The first model used in this study does not consider the main rotor system dynamics. The second and third models belonging to the Apache and Blackhawk air frames respectively, include the main rotor dynamics and ignore the effects of the tail rotor dynaimcs. The last model was obtained using the least-squares method on the nonlinear simulation data of the GE-T700 turbo engine. Only for this last model, the input u is trim signal, which goes to the electronic fuel unit, instead of fuel flow. Time series model of the system is as follows : A(z ') x(t) = Bfz-1) u(t - 1) + M (2) where A(z`') = I + A, z`1 + A2 z`2+...+An z`n B(z-!) = 80+8, z-1 + B2 z`2+...+Bm z`m (3) For m=0 and n=l the discrete state space model is obtained as VUlx(k +1) = AD x(k) + BD u(k) + M (4) In this case, for general expression X = P §, the parameter matrix is as P = [AD Bd M] (5) Where X = X(2) X(3) x(N + 1). <P0) <p(2) * -9(N)., cp(k) = [x(k) u(k) l] (6) In the least squares method, the loss function is given as 2k=l (7) Where e(k) = x(k) - x(k) dJ x(k) = P q>(k - 1) (8) To minimize J, - - = 0 is written and the parameter matrix is solved as : p = (*r4>rf x (9) Compensators are designed to satisfy requirements for steady state error, transient response, stability margin, or closed loop pole locations. Meeting all objectives is usually difficult because of various tradeoffs that have to be made, and the limitations of the design techniques. Classical design is applied on SISO systems successfully, but fails when modelling errors and disturbances occur. By LQ technique, only one controller (a filter gain) is obtained. In this technique, poles are not selected, instead of that Q and R optimization parameters are selected. In case of modelling uncertainty, an estimator design is needed. Early pioneers of control, particularly Bode and Horowitz, studied and delineated most of the properties of feedback. In the early sixties, with the birth of modern control, optimality and design of optimal control systems became the dominant paradigm. The LQG paradigm failed to meet the main objectives of control system designer. The major problem with the LQG solution was lack of robustness. During the eighties, much of the attention was shifted back to feedback properties and frequency-domain techniques, and their generalization to multivariable systems. The loop transfer recovery (LQG/LTR) and HB techniques IXmaintain LQG machinery, but modify the design procedure to address some of the shortcomings of the original LQG approach. The ultimate goal of a control system designer is to build a system that will work in the real environment. Because the real environment may change with time, the control system must be able to withstand these variations. Even if the environment does not change, another fact of life is the issue of model uncertainty. Any mathematical representation of a system often involves simplifying assumptions. Nonlinearities are either unknown and hence unmodeled, or modeled and later ignored to simplify analysis. Different components of systems are sometimes modeled by constant gains, even though they may have dynamics or nonlinearities. Dynamic structures have complicated high frequency dynamics that are often ignored at the design stage. Because control systems are typically designed using much-simplified models of systems, they may not work on the real plant in real environments. The particular property that a control system must possess for it to operate properly in realistic situations is called `Robustness`. Mathematically, this means that the controller must perform satisfactory not just for one plant, but for a family of plants. If, in addition, the system is to satisfy performance specifications such as steady state tracking, disturbance rejection, and speed of response requirements, it is said the system possesses `Robust Performance`. The problem of designing controllers that satisfy robust stability and performance requirements is called `Robust Control`. The underlying concept within control theory that has made it into a field of science is feedback. The study of feedback and its properties is responsible for the rapid growth of this field. There are two important properties that a feedback system possesses that an open loop system cannot have. These are sensitivity and disturbance rejection. By sensitivity it is meant that feedback reduces the sensitivity of the closed loop system with respect to uncertainties or variations in elements located in the forward path of the system. Disturbance rejection refers to the fact that feedback can eliminate or reduce the effects of unwanted disturbances occuring within the feedback loop. An open loop system can also eliminate certain disturbances, but it requires full knowledge of the disturbance, which is not always available. This is impossible in a real environment. Feedback is also used to stabilize unstable systems, but feedback itself is frequently the cause of instability. Because of this, it must be carefully in designing a control system. A stable system is not the final objective. However, stability must be maintained despite model uncertainties. Model uncertainty is generally divided into two categories: structured uncertainty and unstructured uncertainty. Structured uncertainty assumes that the uncertainty is modeled, but unstructured uncertainty assumes less knowledge of the system. Unstructured uncertainy can be modeled in different ways. In this study, additive uncertainty and multiplicative uncertainty will be discussed. Additive uncertainty and multiplicative uncertainty are given as follows:A.(s) = G(s)-G(s) A. (s) = G(s) - G(s) G(s) (10) where G(8) = (l + Am(s))G(s) 01) It will be said that the compansator robustly stabilizes the system if the closed loop system remains stable for the true plant G(s). Procedure of H^ -Robust Contoller Design : 1. First, the state-space matrices of the model are determined. 2. The weighting functions Wi penalizing error signal e and W3 penalizing plant output y are defined. Then the plant G(s) is augemented with weighting functions Wi and W3 (design dpecifications) to form an `augemented plant` P(s) as shown in Figure 3. Augmented Plant P(s) U1 U2 yi Y2 Controller Figure 3 Augemented Plant Augemented plant is calculated as follows : P(s) = B2 D 12 '22. (12) XIwhere the matrices which have `g` subscript are state-space matrices of the system in w-domain, the matrices which have `Wi` and `W3` subscripts are state-space matrices of Wi (s) and W3 (s). 3. The Controller must stabilize the augemented plant P(s) and satisfy the following expression: Tylul = supamax(Tylul(jco)) <; 1 (13) The following assumptions must be done for P(s) : - P(s) is stabilizable. - (Ci, A) ve (C2, A) are detectable. - D12TD12 = I - D21 D21T = I - Dn = 0 andD22 = 0 C (Controller) Figure 4 To obtaine the H,,, -robust controller, first, the following Riccati equations in matrix form must be solved: )L = Ric A - B2 D12 D*2 C, y 2 B, B* - B2 Dl2 B* -(A - B2 Dl2 D* C)T _ <C, Y` = Rid f(A - B, D* D2l C2)T y~2 Cj C - C/ D21 C '21 -`21 ^2 '21 ^2 -(A - B, »I D21 Cj where B, = B, (I - D* D3I ), C, = (T - D12 D,T, ) C, (14) (15) (16) XIIIt can be provided that there exists a stabilizing compensator if and only if there exist positive semi-definite solutions to the two Riccati equations and the following conditions: p(Xa)YJ<Y2 (17) where p is an operator which represents the largest eigenvalue of X^X,, and y is the cost coefficient. The controller in Figure 4 is shown as C = F(J, 0) Where Q is any transfer function that satisfy IqJ^ < y. Additionally the controller C must satisfy F(P, C)L < y. Transfer function J in Figure 4 is obtained as follows : -{Q+y^Bfxj z^k, zm+y'X.ffPb) 0 I 1 0 (18) Ke = Y. C2T + B, D2T, (19) where Kc = B2T X` + D,T2 C, Ke = Y. C2T + B, D2T, Z. = (I-y-2Y.X.r While obtaining the controller, H`, existence are performed as follows: i) IsDıı small enough? ii) Solving X,,, Riccati - No Hamiltonian imaginer-axis roots? - solution Xx > 0 ? iii) Solving Y^ Riccati - No Hamiltonian imaginer-axis roots? - solution Y^ > 0 ? iv) Maxeig(Xa,YJ<l ? To obtaine a suitable contoller, all of the answers of the questions above must be OK, if not, the procedure must be repeated changing Wi, W3 and y. Xlll