HYBRID CONTROLLER FOR GUST LOAD ALLEVIATION AND RIDE COMFORT IMPROVEMENT USING DIRECT LIFT CONTROL FLAPS

Transcription

1 3 rd EUROPEAN CONFERENCE FOR AEROSPACE SCIENCES (EUCASS) HYBRID CONTROLLER FOR GUST LOAD ALLEVIATION AND RIDE COMFORT IMPROVEMENT USING DIRECT LIFT CONTROL FLAPS Andreas Wildschek*, Rudolf Maier*, Martin Hrocik**, Toas Hanis**, Alexander Schirrer***, Martin Kozek***, Christian Westerayer***, and Mark Heedi*** *EADS Innovation Works, Munich, 81663, Gerany **Czech Technical University, Prague, 16627, Czech Republic ***Vienna University of Technology, Vienna, 1040, Austria ABSTRACT In this paper two robust H feedback control s are designed, one for active wing bending daping, and one for active daping of the wing and hull bending odes. For the latter, not only syetrically coanded ailerons are used, but also the elevator and direct lift control (DLC) flaps. The control objective of these feedback s is the reduction of fatigue of the wing roots, as well as the iproveent of ride cofort. Two different H control design ethods, DK iteration and HIFOO, are applied and copared. For the additional copensation of turbulence excited peak loads, the active wing bending daper is augented by an adaptive feed-forward controller which uses the odified output of an alpha probe ounted at the front fuselage as reference signal. Nueric siulations with a state-space odel of the syetric dynaics of a large airliner are perfored for validation of the controllers perforances. 1. INTRODUCTION Turbulent atosphere, gusts, and anoeuvres significantly excite aircraft rigid body otions and, especially on large airliners, also structural vibrations, which leads to increased structural loads and reduced ride cofort. Today, large transport aircraft are coonly equipped with gust load alleviation systes. The objective of these control systes is the reduction of atospheric turbulence excited dynaic loads, as well as an increase of passenger cofort and handling qualities. Robust feedback of structural accelerations to aerodynaic control surfaces has been proposed for active vibration daping in the past [1], [2]. A significant technological step forward is the work of HAHN & KOENIG [3] who successfully reduced turbulence excited vertical accelerations on the DLR Advanced Technologies Testing Aircraft (ATTAS) by using a feed-forward controller and direct lift control (DLC) flaps. Thereby, the ai was an iproveent of passenger cofort. A odified alpha probe signal provided the reference for vertical turbulence. HECKER & HAHN [4] propose a siilar feed-forward approach for gust load alleviation and ride cofort iproveent on a large flexible airliner. In WILDSCHEK [5] an adaptation algorith is proposed for the feed-forward copensation of wing bending vibrations in order to ake the control perforance robust against plant uncertainties. Continuatively, a hybrid controller, i.e. the augentation of active wing bending daping by feed-forward control is suggested for axiu alleviation of dynaic wing loads. Recently, said approach has been extended to the additional alleviation of turbulence excited pitch oscillations by using the elevators as actuators [6]. It is concluded, however, that the availability of additional DLC flaps would enable the siultaneous reduction of wing loads and iproveent of ride cofort. In this paper a hybrid controller is designed for gust load alleviation and iproveent of ride cofort on a large airliner using such DLC flaps in addition to the priary control surfaces such as elevators and ailerons. Copyright 2009 by EADS. Published by the EUCASS association with perission.

2 EUCASS AIRLINER MODEL WITH DIRECT LIFT CONTROL FLAPS State-space odels of syetric dynaics of a large airliner for 4 different ass configurations and two different centre of gravity () positions are used in this paper for the design of a hybrid controller for gust load alleviation and cofort iproveent. Said odels are in principle siilar to the ones used in JEANNEAU ET AL. [2] and in WILDSCHEK [5], [6]. However, in addition to syetrically coanded inner ailerons and the elevator, 3 pairs of direct lift control (DLC) flaps are odelled using the Doublet Lattice Method, copare Figure 1. For the following investigation the Mach nuber and altitude are considered to be available as gain scheduling paraeters. Thus, the hybrid controller is only designed for the cruise Mach nuber and altitude. All actuator dynaics are odelled by 2 nd order low pass filters. Sensor delays and analogue filters are also considered. Moreover, a pitch daper consisting of a wash-out filter and a low pass filter is applied to the airliner odel before design of the hybrid controller. The outputs of the odels coprise pitch rate q, vertical wing root bending oent Mx WR, vertical root bending oent of the horizontal tail plane Mx HTP, as well as vertical accelerations at several positions. The control objective of the robust inner feedback control loop (which will be used for the hybrid controller) is the iniisation of the H nor of a odal wing bending acceleration signal Nz as also proposed in JEANNEAU ET AL. [2] in order to reduce wing root fatigue loads. Said signal Nz is defined to be half of the su of the vertical accelerations of the two wings Nz LW, Nz RW inus the vertical acceleration of the centre of gravity Nz, see eq. (1) and Figure 1. This approach allows the separation of the vertical wing bending fro the rigid body otion in the easureent. Nz NzLW + NzRW ) = Nz 2 ( (1) For a second robust H feedback controller design that seeks also active daping of the hull bending ode (which was found to be the critical for ride cofort), vertical accelerations at the pilot station Nz front, at the, and at the rear fuselage Nz are used to define a odal hull bending acceleration signal Nz hull : rear Nz hull Nz front + Nzrear ) = Nz 2 ( (2) For control design and evaluation in regards to ride cofort a cofort filter is applied to vertical fuselage accelerations as proposed in ISO2631, copare Figure 2 for a bode agnitude plot of this filter. Nz front Bode Diagra Nz Nz LOE Nz ROE DLC inner DLC iddle DLC outer Nz rear Figure 1: Nz acceleration sensors, DLC flap pairs. Magnitude (db) Frequency (rad/sec) Figure 2: ISO2631 cofort filter for Nz. 2

3 A. Wildschek, R. Maier, M. Hrocik, T. Hanis, A. Schirrer, M. Kozek, C. Westerayer, and M. Heedi HYBRID CONTROLLER FOR GUST LOAD ALLEVIATION AND RIDE COMFORT IMPROVEMENT USING DLC FLAPS 3. DESIGN OF THE INNER FEEDBACK LOOP Two odern approaches were considered for the design of the robust inner feedback loop, naely the DK iteration and a fixed-order H optial control design using the HIFOO (H Fixed-Order Optiization) toolbox (see GUMUSSOY ET AL. [8]). n z u W pu w u P z y w u W W d G W pz y z M K W a a Figure 3: Generalized plant representation Figure 4: Design syste for DK iteration K 3.1. DK iteration The DK iteration procedure is applied to obtain robust control s that guarantee both robust stability and robust perforance for an explicitly odelled set of uncertainties. The set of uncertain plants is forulated via the generalized P- interconnection structure coon in robust control design (see Figure 3). The derivation of the ethod is outlined in detail in SKOGESTAD AND POSTLETHWAITE [9] and utilizes the notion of the structured singular value µ in ters of the µ-analysis of robust stability / perforance. The robust controller design algorith alternately coputes the H -optial controller design proble (K-step) and the optiization for applied scaling atrices D to adapt for the considered and critical uncertainties. The algorith seeks to iniize (bounds for) µ using the objective function 1 ( DMD ) in inf K D D, (3) where M is the noinal closed loop with controller K and D is the set of all atrices D fulfilling 1 D D = ). Results were obtained for the aircraft odel with the following control goals and choice of inputs and outputs: The chosen design task was active vibration daping of the aircraft subject to excitation by gust and turbulence. The priary goal was the daping of the first wing bending ode at a frequency of about 9 rad/s (1.4 Hz). The exogenous vertical input w is odelled as a white-noise signal filtered by a onediensional von Kárán turbulence spectru. To obtain robustness with respect to easureent noise, an additional noise input n is odelled. The exogenous (perforance) output zm is the wing root bending oent. The first wing bending ode is also strongly distinct in the transfer function fro the input to the wing bending acceleration signal Nz, and is therefore also a possible perforance output. However, if anoeuvre load control is a further design goal, a load perforance output is advantageous to the odal sensor. The easureent output (controller input) y is the wing bending acceleration signal Nz. As control input u the syetrically coanded inner ailerons are used, showing the strongest effect of all available control surfaces on the wing bending ode. 3

4 EUCASS The plant odel is furtherore extended by uncertainties for actuator deviation (ultiplicative input uncertainty), neglected dynaics between validation and design odels (additive uncertainty), as well as perforance design weights. The resulting design syste is depicted in Figure Choosing design weights and tuning perforance The chosen weights are cobinations of band-pass and notch-filters around the wing bending frequencies. In order to design a controller to attenuate the first wing bending ode, the perforance weight is high at that frequency, while the additive uncertainty weight is reduced there using a notch filter. The second wing bending ode lies in soe ass cases very close to the hull bending ode (at around 17 rad/s or 2.7 Hz), which requires a weight paraeter search to optiize for both robustness and perforance for all validation ass/ cases. It was found that an uncertainty peak at 16.7rad/s successfully yields a controller which is robust and perfors well in all validation cases. Moreover, actuator uncertainties were odelled as ultiplicative input uncertainty with constant agnitude of W=0.05, and actuator agnitude liitation was obtained by control input perforance weights (unity for Wpu for control input agnitude liitation and a tuned design weight Wpz). Measureent noise is odelled by white noise n, and the turbulence excitation is ipleented using white noise w and the turbulence spectru shaping in Wd. OL CL coparison of wing bending ode frequency in rad/s The resulting SISO controller is robust for all ass/ cases and successfully attenuates the wing bending ode and, to a sall extent, the second wing bending ode (at around 17 rad/s or 2.7 Hz), while having little or no influence at other frequencies. Due to the resulting high controller order, subsequent controller order reduction is required before ipleentation. It also becae evident that the inner ailerons suffer large phase uncertainty with respect to ass/ variations in the frequency range of the second wing bending and hull bending ode ( rad/s or Hz), so a MIMO approach was also taken in the following Fixed-order H optial control design using HIFOO (H Fixed-Order Optiization) In order to directly obtain a robust feedback controller of pre-specified order, the H Fixed-Order Optiization (HIFOO) toolbox is used, outlined in detail in GUMUSSOY ET AL. [8]. The HIFOO control design ethod searches for locally optial solutions of a non-sooth optiization proble that is built to incorporate iniization objectives and constraints for ultiple plants. First, the controller order is fixed at the outset, allowing for low-order controller design. Second, no Lyapunov or lifting variables are introduced to deal with the conflicting specifications. The resulting optiization proble is forulated on the controller coefficients only, resulting in a typically sall-diensional non-sooth non-convex optiization proble that does not require the solution of large convex sub-probles, relieving the coputational burden typical for Lyapunov LMI techniques. Because finding the global iniu of this optiization proble ay be hard, an algorith that searches only for local inia is used. While no guarantee can be given on the result quality of this algorith, in practice it is often possible to deterine a satisfying controller efficiently. A MIMO low-order H -optial controller was coputed aiing at the following goals and requireents: Robust daping of the wing bending ode (WBM) and hull bending ode (HBM) for reduction of fatigue loads and for cofort iproveent Low-order controller, robust with respect to the entire set of validation plants agnitude 1st wing bending ode Figure 5: Coparison of the wing root bending oent transfer functions for all validation plants, open loop (blue lines) and closed loop (red lines) using the robust DK iteration controller 4

5 A. Wildschek, R. Maier, M. Hrocik, T. Hanis, A. Schirrer, M. Kozek, C. Westerayer, and M. Heedi HYBRID CONTROLLER FOR GUST LOAD ALLEVIATION AND RIDE COMFORT IMPROVEMENT USING DLC FLAPS The control architecture is chosen as MIMO feedback control with 2 perforance outputs (wing root bending oent MxWR, horizontal tail plane bending oent MxHTP), 2 easureent outputs (i.e., controller inputs the odal wing bending sensor Nz and the odal hull bending sensor Nzhull) and all actuators being coanded by the controller Perforance criteria definition The weighting filters used to represent perforance criteria are defined according to the requireents and constrains for the control as listed above. The selected filters are ostly cobinations of band-pass and notch-filters to ensure WBM and HBM suppression. The shape of the perforance criteria can by slightly odified for a particular plant (according to natural frequency shift and changes in ode daping) in order to exactly fulfil the perforance for all ass/ cases. The resulting controller achieves considerable reductions of the first WBM and the HBM at very low controller order. However, this MIMO concept requires further careful fine-tuning in order to reduce spill-over into other odes. Design Method DK iteration Fixed-order HIFOO design Robustness explicit uncertainty odel (robustness for the defined uncertain plant set) vertex robustness (robustness for a nuber of design plants) Coputational effort coputationally fast (consecutive convex coputations) coputationally fast (for local optiization) / high (for ore relevant results) Controller design high control order (plant order + 2 x D scales order) arbitrarily low controller order Table 1: Properties of the applied robust control design ethods 4. DESIGN OF THE ADAPTIVE FEED-FORWARD AUGMENTATION The feed-forward controller uses the elevator, the syetrically coanded ailerons, and the syetrically coanded DLC flaps as actuators. The control objective of the feed-forward augentation is the iniisation of a quadratic cost function J, defined as: J ( ISO Nz ) Nzhull + = Nz (4) The transfer function of the cofort filter denoted ISO is used to weigh the vertical acceleration Nz according to relevancy for cofort. This has the additional advantage that the low frequency range where the transfer path fro DLC flaps to Nz is copletely uncertain is cut off. In order to avoid an adaptation towards unwanted copensation of pilot inputs, vertical accelerations are iniized in ters of deviations fro pilot coands Nz : pilot Nz = Nz Nz (5) pilot The feed-forward controller is adapted online in order to iprove its perforance robustness against plant uncertainties, as described in WILDSCHEK & MAIER [6]. Using only one reference sensor (i.e. the alpha probe), the discrete-tie feed-forward control for the th actuator u at tie step n is: T T u ( n) = h ( n) α ( n) = α ( n) h ( n) = 1,2, 3 (6) with th FIR (Finite Ipulse Response) control filter: h n) = h ( n), h ( n),..., h [ ( n ] T ( 0 1 N 1 ) = 1,2,3 (7) 5

6 EUCASS Thereby, h0 ( n), h1 ( n),..., h 1 ( n) N are the coefficients of the th FIR control filter, and N denotes the control filter length, which is assued to be equal for all three controllers for the sake of straightforwardness of notation. α(n) is the vector of the sapled reference signal at tie step n: α ( n ) = α ( n), α ( n 1),..., α ( n N + 1) (8) [ ] T The frequency doain steepest descent update for the th FIR control filter is: h [ Rˆ *( n, f ) E ( n, f )] = L ( n) h ( n 1) c IDFT l k l k = l 1 + Thereby, IDFT {... } + denotes the causal share of the Inverse Discrete Fourier Transfor of the quantity inside {... } with f k denoting the discrete frequency. The superscript * denotes coplex conjugation and c is the convergence coefficient. Contrary to [6], but as illustrated in ELLIOT [7], only one coon convergence coefficient is chosen in order to avoid distortion of the gradient of the cost function J. Then R ˆ l( n, f k ) is the Discrete Fourier Transfor (DFT) of the latest 2N-point segent of the sapled estiated filtered reference signal rˆ l, which is the sapled reference signal α filtered by the transfer path Ĝ l. Thereby, Ĝ l is an estiate of the plant s transfer path fro the th control coand u to the l th error signal e l, which is denoted G l. Furtherore, E l ( n, f k ) is the 2N-point DFT of the latest N point segent of e l padded with N zeros. Finally, only the causal share of the quantity inside the brackets {... } is used. This approach is called overlap-save ethod and prevents circular correlation. 5. SIMULATION RESULTS Nueric siulations were perfored with state space odels of the syetric dynaics of a large airliner. An alpha probe odelled at the aircraft s front fuselage was used as a reference sensor for feed-forward control. For the siulations, the reference signal, i.e. the odified alpha probe signal, is denoted α. α si ( t ) = α ( t ) + α ( t ) = α ( t ) α 0( t ) (10) ground It is coposed of two parts, naely α which is generated by von-kárán-filtered white noise and represents atospheric turbulence, and α ground which is the output of the state space odel representing the oveent of the alpha probe s ounting node in the air strea due to aircraft rigid body otions and structural vibrations. The required reference signal is α. Therefore, any significant α ground has to be copensated in the alpha probe output, such as proposed in [3] and [4]. The ain share of α ground stes fro the aircraft rigid body otion which are ainly excited by pilot inputs. Since in the nueric siulations no pilot inputs were considered, α ground was so sall that it can be neglected in the derivation of the adaptation algorith. The unfiltered alpha probe easureent is denoted α air. It is assued that the ean value α 0 is reoved, e.g. by a high pass filter. The turbulence excitation of the aircraft was odelled as span-wise constant angle of attack variation α. w air α w = α + αν (11) Thereby an angle of attack α ν is added to the observable share of the turbulence excitation α in order to represent the coherence degradation between reference easureent and turbulence exciting the wing. For sall angles α = vz VTAS (in radians) holds. Thereby, the vertical flow velocity v z shall represent the one-diensional von Kárán turbulence spectru. For the tie-doain siulations the von Kárán turbulence spectru is approxiated by filtering a white noise signal by a 3 rd order filter with the integral scale length chosen as 762 eter. In order to achieve 75% coherence between reference easureent α si and the aircraft excitation (which is an appropriate estiate for the excitation of the first syetric vertical wing bending on a large airliner as explained in [5]), α ν is von-kárán-filtered white noise which is uncorrelated to α (different initial seeds in the siulation) with a agnitude ratio of α α = 1 3 v si (9) 6

7 A. Wildschek, R. Maier, M. Hrocik, T. Hanis, A. Schirrer, M. Kozek, C. Westerayer, and M. Heedi HYBRID CONTROLLER FOR GUST LOAD ALLEVIATION AND RIDE COMFORT IMPROVEMENT USING DLC FLAPS Figure 6: Modelling of turbulence excitation and turbulence easureent with 75% coherence between reference easureent and aircraft excitation [5]. In Figure 7 agnitudes of the outputs of the airliner odel at the design ass case/ position are illustrated with turned off controller (blue lines), with a HIFOO feedback controller aiing at active daping of the wing bending and hull bending ode (red lines), and with a hybrid controller, i.e. inner feedback controller fro Nz to the inner ailerons designed by DK iteration, ainly for active wing bending daping, augented by an adaptive feed-forward controller that coands the inner ailerons, the elevator and the DLC flaps (green lines). All controllers achieve a reduction of Nz at the first syetric vertical wing bending frequency (i.e. between 1 and 1.5 Hz), and thus also reduce the wing root bending oent Mx WR in this frequency range. However, this does not necessarily iply that peak loads are reduced. The pitch rate q and the tail plane bending oent Mx HTP reain largely untouched. The HIFOO feedback controller also attenuates the hull bending ode significantly (top right plot), thus reducing Nz, as well as the vertical accelerations in the front and the back of the fuselage weighted by the ISO2631 cofort filter (denoted ISO Nz front and ISO Nz rear ) especially in the frequency range of the first vertical hull bending ode, i.e. between 3 Hz and 4 Hz. This leads to a significant iproveent of ride cofort. Active daping of the hull bending ode sees to be the ost proising approach for ride cofort iproveent in this frequency range. Here the ain challenges are the uncertainty of the plant dynaics of these frequencies and the high coputational burden for real-tie control. The hybrid controller (green lines) shows only little iproveent of ride cofort in the frequency range between 3 Hz and 4 Hz. Additional daping of the hull bending ode ost probably will iprove this situation. The ain advantage of the additional feed-forward path used in the hybrid control concept however is the reduction of peak loads in the wing roots, see Figure 8, which, in theory, allows for a lighter design of the wing structure. Note that in this case the feed-forward controller becoes safety critical. The tie response to von Kárán turbulence excitation shows that the peak loads of the wing root bending oent Mx WR are reduced without increasing the peaks of the horizontal tail plane bending oent Mx HTP. Additionally, the axia of pitch rate q and vertical accelerations in the fuselage are reduced by feed-forward control. The ain challenge with feed-forward control is to provide a reliable reference signal. On the large airliner investigated in this paper wing bending vibration acceleration agnitude can be reduced to 50% by pure feed-forward control when an alpha probe ounted at the front fuselage is used as a reference sensor (as also shown in [5]). 7

8 EUCASS Nz Nz hull q Control off HIFOO feedback Hybrid controller Nz Mx WR ISO*Nz front Mx HTP Frequency, Hz ISO*Nz rear Frequency, Hz Figure 7: Magnitudes of aircraft odel outputs with different controllers for one ass/ case. 8

9 A. Wildschek, R. Maier, M. Hrocik, T. Hanis, A. Schirrer, M. Kozek, C. Westerayer, and M. Heedi HYBRID CONTROLLER FOR GUST LOAD ALLEVIATION AND RIDE COMFORT IMPROVEMENT USING DLC FLAPS Nz Nz front Control off Control on q Nz Mx WR Nz rear Mx HTP Cofort Tie,s Tie, s Figure 8: Reduction of peak loads with the converged adaptive feed-forward controller. 6. CONCLUSIONS Hybrid control, that is, a cobination of robust feedback wing bending daping and adaptive feed-forward copensation of turbulence excitation is proposed for gust load alleviation. The overall objective is the reduction of the dynaic wing root bending oent and fatigue loads, as well as an iproveent of ride cofort, which is evaluated with an ISO2631 cofort filter. All available control surfaces (elevators, syetrically coanded ailerons, and syetrically coanded direct lift control (DLC) flaps) serve as 9

10 EUCASS actuators. The design of the inner feedback loop for active wing bending daping is perfored by DK iteration. The proposed adaptive feed-forward concept is applied to the feedback-controlled closed loop syste. The adaptation iproves the perforance robustness of the feed-forward path against plant uncertainties. Peak loads in the wing root are reduced by the feed-forward path, thus opening the possibility of a lighter design of the wing structure. Cofort iproveent is very good in the frequency range of first vertical wing bending, but low in the frequency range of the hull bending ode. Note also that the resulting high order of the robust H feedback controller still needs to be reduced before ipleentation. In order to iprove cofort also in the frequency range of the hull bending ode an alternative robust MIMO feedback controller for active daping of the wing and hull bending ode using all control surfaces is designed. This tie also a low controller order is sought which can be achieved by using the H fixed order optiization (HIFOO) toolbox. It can be shown that active daping of the hull bending ode indeed is the ost efficient eans for ride cofort iproveent. These two approaches are validated in nueric siulations with state space odels of the longitudinal dynaics of a large airliner. The excitation is odelled as one-diensional von Kárán turbulence. A next step would be to tune the HIFOO feedback controller to reduce spill-over and to augent it with the adaptive feed-forward concept for additional reduction of wing root peak loads. Manoeuvre load alleviation is currently under investigation. In a given pitch-up anoeuvre, for exaple, the inner DLC flap pair is deflected downwards, so that less (upward) deflection of the elevator is required in order to perfor said anoeuvre. This procedure is assued to draatically decrease the anoeuvre wing root bending oent. Control s for anoeuvre load alleviation are in the design phase. However, no results are yet available. REFERENCES [1] JOHNSTON, J. F., ET AL., Accelerated Developent and Flight Evaluation of Active Controls Concepts for Subsonic Transport Aircraft Volue 1 Load Alleviation/Extended Span Developent and Flight Test. NASA CR , pages 2-11, [2] JEANNEAU, M., AVERSA, N., DELANNOY, S., HOCKENHULL, M., Awiator's study of a Wing Load Control: Design and Flight-test Results. 16th IFAC Syposiu on Autoatic Control in Aerospace, St. Petersburg (RUSSIA), June [3] HAHN, K.-U. KOENIG, R., ATTAS flight test and siulation results of the advanced gust anageent syste LARS. In AIAA Atospheric Flight Mechanics Conference, Hilton Head Island, SC, Aug 10-12, [4] HECKER, S., HAHN, K.-U., Advanced Gust Load Alleviation Syste for Large Flexible Aircraft, 1st CEAS European Air & Space Conference, Berlin, Gerany, Septeber, [5] WILDSCHEK, A., An Adaptive Feed-Forward Controller for Active Wing Bending Vibration Alleviation on Large Transport Aircraft, Dissertation, Technische Universität München, [6] WILDSCHEK, A., MAIER, R., Integrated Adaptive Feed-forward Control of Atospheric Turbulence excited Rigid Body Motions and Structural Vibrations on a Large Transport Aircraft. In Aerican Control Conference 2008, Seattle, Washington, June 11-13, [7] ELLIOTT, S. J., Signal Processing for Active Control, Acadeic Press, London, 2001, page 247 ff. [8] GUMUSSOY, S., HENRION, D., MILLSTONE, M., OVERTON, M., Multiobjective Robust Control with HIFOO 2.0, to be registered as a LAAS-CNRS Research Report, January IFAC Syposiu on Robust Control Design, Haifa, Israel, June 16-18, [9] SKOGESTAD, S., POSTLETHWAITE, I., Multivariable Feedback Control Analysis and Design, John Wiley & Sons Ltd,