- Email: [email protected]
Improved digital tracking controller design for pilot-scale unmanned helicopter
Available online at www.sciencedirect.com
Journal of the Franklin Institute 349 (2012) 42–58 www.elsevier.com/locate/jfranklin
Improved digital tracking controller design for pilot-scale unmanned helicopter Magdi S. Mahmouda,n, Arief B. Koesdwiadyb a
Systems Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 985, Dhahran 31261, Saudi Arabia b Systems Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 5067, Dhahran 31261, Saudi Arabia Received 20 April 2011; received in revised form 1 October 2011; accepted 6 October 2011 Available online 15 October 2011
Abstract In this paper, methods for improved design of digital tracking controller for a pilot-scale unmanned helicopter are considered. By discretizing the linearized helicopter model, the linear quadratic with integral (LQI) capability is investigated and applied in order to develop an efficient tracking system including a state-feedback plus integral action. The helicopter velocities are used to formulate a prescribed position reference tracking trajectory. When both process and measurement noises are present, a Kalman filter (KF) is combined with the LQI to form a linear quadratic Gaussian with integral (LQGI) tracking system. Simulation studies illuminate both the capability of the controller design and the accuracy of the estimator. Next, H2, H1 and mixed H2 =H1 controls are designed and the results between methods are produced and compared. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction In recent years, we have witnessed wide variety of techniques and applications of control theory to several systems engineering areas robust control [1–3], including PID control [4], time-varying systems [5,6], networked control systems [7,8], autonomous systems [9–11], Internet control [12], attitude stabilization of rigid spacecraft [13–15], to name a few. The primary focus of this work is on a class of autonomous systems [9]. In this regard, n
Corresponding author. E-mail addresses: [email protected], [email protected] (M.S. Mahmoud), [email protected] (A.B. Koesdwiady). 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.10.003
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
43
unmanned aerial vehicle (UAV) with helicopter-like capabilities such as vertical take-off and landing is becoming popular in systems studies. The interest in helicopter-based unmanned aerial vehicle (HUAV) has central focus for both military and civil applications. It is well-known that helicopter is a non-linear complex system with unstable nature and, in turn, these characteristics render the design in flight control systems a difficult task. Pilotscale unmanned helicopter constitute by now common platforms for HUAV control development with many great advances been made since the last decade. Compared with traditional full-size helicopters, a pilot-scale helicopter tend to be naturally more maneuverable and responsive [10]. Some related results are reported in [16–18]. System identification has been used as a modeling technique to derive linear models for control design and to study the vehicle flying qualities. Also, system identification is utilized for the validation and refinement of detailed non-linear first-principle models. There are several reported applications of system identification techniques to modeling of pilot-scale helicopters, including the model identification of YAMAHA R-50 [10], and X-Cell [19] for flight control, and a 6-DOF dynamic modeling of Raptor-50 V2 for simulations [9]. In this paper, the pilot-scale unmanned helicopter dynamics reported in [10] will be used in the sequel as the preliminary vehicle for control system design. Most HUAVs used classical control system such as single-loop PD systems. The tuning for controller parameters usually performed manually for certain operating point such as hover and cruise mode. This condition giving a wide opportunity for multi-variable controller synthesis method to be implemented. In previous work [20], linear quadratic trajectory tracking system with derivative on the error for HUAV conducted in continuous-time mode, and reports encouraging preliminary results. There are numerous predicted utilizations for HUAV either individually or working as a team, including surveillance, search and rescue, and mobile sensor networks. For this uses, HUAV needs to be able to navigate to the desired destination through desired trajectory, the position control as well as velocity control are performed for this scenario. A controller that can accommodate this scenario is needed, linear quadratic integral (LQI) [21] tracking control is proposed, and for practical needs in the presence of process and measurements noise, linear quadratic Gaussian integral (LQGI) tracking control is proposed with the assumption that the noise is white. This motivates the study of this paper by looking at effective methods for controlling pilot-scale HUAVs. Looked at in this light, this paper provides improved digital controller design and simulations using the powerful tools of MATLAB and SIMULINK. Computer simulation in discrete-time is conducted with certain sampling time will allow direct implementation on the practical controller. The dynamics of helicopter on cruise is used for the trajectory tracking system [10]. Then a linear quadratic Gaussian with integral (LQGI) control is designed with the assumption that sensor available is only for velocities and corrupted with white noise. The ensuing measurements are used to perform state estimation using Kalman filter (KF). For the controller design based on H2, H1 and mixed H2 =H1 performance, the simulations are performed in continuous-time mode. The results are presented and a comparison among design methods is performed. 2. Dynamics of pilot-scale helicopter For controller synthesis or controller optimization, the dynamic models usually have strict requirements. Essentially, the model must capture the effects that influence the
44
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
performance of the system. This implies that for helicopter they must explicitly account for rotor and fuselage coupling effects. At the same time the model must be simple enough to be insightful and practical for the control synthesis. The basic equations of motion for a linear model of the helicopter dynamics are derived from the Newton–Euler equations for a rigid body that is free to simultaneously rotate and translate in all six degree of freedom.
The dynamics of helicopter is defined as follows [10]: rotor derivatives Xa, Xb are used to express rotor forces, and flapping spring-derivatives Lb, Ma to express rotor moments. Xu, Yv, Lu, Lv, Mu, Mv are speed derivatives to express general aerodynamics effect [22,23]. u_ ¼ ðw0 q þ v0 rÞgy þ Xu u þ Xa a v_ ¼ ðu0 r þ w0 pÞ þ gf þ Yv v þ Yb b p_ ¼ Lu u þ Lv v þ Lb b q_ ¼ Mu u þ Mv v þ Ma a
ð1Þ
u, v, and w are velocity components for longitudinal, lateral, and vertical movements (ft/s), while u0, v0, and w0 are the values of u, v, w at trim condition. p, q, and r are roll, pitch, and yaw rates, rad s1. The rigid body equation from Newton–Euler for vertical dynamic is w_ ¼ ðv0 p þ u0 qÞ þ Zw w þ Zcol dcol
ð2Þ
The centrifugal forces represented in the terms v0 and u0, which is relevant only for cruise conditions. Rotor equations of motions for the longitudinal and lateral flapping are two first order differential equations tf a_ ¼ atf q þ Aa a þ Alat dlat þ Ac c þ Alon dlon tf b_ ¼ btf p þ Ba a þ Blat dlat þ Bd d þ Blon dlon
ð3Þ
a,b are main motor longitudinal and lateral flapping motions (rad), and c,d are stabilizer bar longitudinal and lateral flapping motions (rad). tf and ts are main rotor and stabilizer time constants. The longitudinal and lateral stabilizer bar dynamic equation presented in
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
45
the similar way as the main rotor ts c_ ¼ cts q þ Clon dlon ts d_ ¼ dts p þ Dlat dlat
ð4Þ
In this case, the augmented yaw dynamics is the first order bare airframe dynamics with a yaw rate feedback represented by a simple first order low-pass filter rfb Kr ¼ r s þ Krfb
ð5Þ
where the corresponding differential equations used in the state-space model r_ ¼ Nr r þ Nped ðdped rfb Þ r_fb ¼ Kr rKrfb rfb
ð6Þ
The full state-space equation of the model is given as x_ ¼ Ax þ Bu x ¼ ½u, v, w, p, q, r, f, y, tf a, tf b, ts c, ts d, rfb t u ¼ ½dlat , dlon , dped , dcol t ð7Þ where dlat is lateral deflection of main rotor, dlon is longitudinal deflection of main rotor, dped is pitch deflection of tail rotor blade, and dcol is pitch deflection of main rotor blade, all in radians. A helicopter responds differently in hover flight than it does in cruise flight. Since this paper is only concerned with tracking a desired trajectory, the helicopter in cruise flight response will be used. Typical data values are rotor speed¼ 850 r min1, tip speed¼ 499 ft/s, dry weight¼ 97 lb, instrumented (full)¼ 150 lb, and flight autonomy¼ 30 min. Remark 2.1. The main goal of this paper is to design a discrete-time linear-quadratic integral (LQI) and linear-quadratic Gaussian integral (LQGI) tracking systems and to apply the elaborated systems in the controller of a pilot-scale helicopter. 2.1. Discrete-time model A discrete model, to be used hereafter, is derived from the continuous model of pilotscale helicopter with 6-DOF. The experiments to develop the non-linear dynamics model of pilot-scale helicopter were conducted on Carnegie Mellon’s helicopter-based UAV (HUAV based on Yamaha R-50 model scale helicopter with 10 ft rotor diameter) [10]. Both hover and cruise flight conditions have been treated and all important effects have been captured. The conversion from continuous-time model to discrete-time model is performed using c2d command in MATLAB. The continuous-time model is discretized by incorporating a zero-order hold on the inputs and sampling at a rate of 0.01 s. A statespace representation for discrete-time model is described by x½k þ 1 ¼ Fx½k þ Gu½k
ð8Þ
y½k ¼ Hx½k
ð9Þ
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
46
Longitudinal Velocity
Response
2 0 −2 −4
1
1.05
1.1
1.15
1.2
1.25 Time
1.3
1.35
1.4
1.45
1.5
1.35
1.4
1.45
1.5
1.35
1.4
1.45
1.5
Lateral Velocity
Response
10
5
0
1
1.05
1.1
1.15
1.2
1.25 Time
1.3
Vertical Velocity
Response
0 −10 −20 −30
1
1.05
1.1
1.15
1.2
1.25 Time
Continuous
1.3
Discrete
Fig. 1. Discrete versus continuous open loop response.
To confirm that the discrete-time model have similar behavior as the continuous-time model, the validity test of the model must be performed. The test conducted in open-loop mode with the step signal used in all the inputs. The response captured only on the three states u, v, w since this paper addresses the problem only on position tracking. From Fig. 1, it can be seen that the response shown by discrete-time model is similar to the continuous model. It means the discrete time model is valid to be used as the representation of the dynamics system of 6-DOF pilot-scale helicopter.
Remark 2.2. It should be noted that the simulation results in Fig. 1 showed that adopted discretization scheme effectively reproduces the continuous model and hence provides a good starting point in preparation for the controller design.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
47
3. Controller design In what follows, we present methods for controller design of the developed model of HUAV.
3.1. Linear quadratic integral tracking We consider a linear quadratic regulator (LQR) with tracking capability. Thus, it is required that the output y to track a reference trajectory yr . To achieve this, we use both the state x and the integral of error between reference and output z [24] thereby eliminating steady state errors when tracking constant signals [21]. This system is termed linear quadratic integral (LQI) tracking system. Z t zðtÞ ¼ ðyr yÞ dt, z_ ðtÞ ¼ yr ðtÞyðtÞ ð10Þ With sampling time Ts , we get z½k þ 1 ¼ z½k þ Ts ðyr ½ny½nÞ
ð11Þ
In our design, the number of the states x is 13, with four inputs u, and three outputs y that measured for tracking system as well as the reference yr . Augmenting the state x½k to the error z½k, we have xaug ½k ¼ ½x½kz½kt , where xaug ½k þ 1 ¼ Faug xaug ½k þ Gaug u½k þ r½k y½k ¼ Haug xaug ½k " # F 0133 Faug ¼ , Ts H I33 Haug ¼ ½H 033 ,
Gaug ¼
G
0
r½k ¼ ½0131 ytr Tts t
ð12Þ
The performance index is 1 1X ðxaug ½kt Qxaug ½k þ ut ½kRu½kÞ 2k¼1 0rQtaug ¼ Qaug , Rt ¼ R40
J¼
Qaug ¼ diag½Gt Qy G Qz
ð13Þ
Qy , Qz are the state and error weighting matrices, respectively. The linear quadratic controller is the unique, optimal full state feedback control law u ¼ KLQI xaug ,
KLQI ¼ R1 Gtaug S
ð14Þ
with 0rSt ¼ S is the solution to the DARE 0 ¼ Ftaug SFaug SFtaug SGaug ðGtaug SGaug þ RÞ1 Gtaug SFaug þ Q KLQI ¼ ½Kx Kz where Kx is the gain for state-feedback, and Kz is the gain for error.
ð15Þ
48
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
3.2. Linear quadratic Gaussian integral tracking A linear quadratic state feedback problem was solved which assumes that all states are available for feedback. In the sequel, a Kalman filter (KF) is to be designed to estimate the unmeasured states. Under the standard Gaussian noise assumption [25], a linear quadratic Gaussian (LQG) regulator is considered is formed where the LQI tracking and KF are to be designed independently. 3.2.1. Kalman filter design Since the velocity of the HUAV corrupted by noise is the only measured variable, the ^ remaining states have to be estimated through a KF in real time in real time. Let x½k be the state estimate of the state x½k where the plant model is x½k þ 1 ¼ Fx½k þ Gu½k y½k ¼ Hx½k
ð16Þ
The KF dynamics are given as follows ^ þ 1 ¼ Fx½k ^ þ Gu½k þ Kf ðy½kHx½kÞ ^ x½k
ð17Þ
The Kalman gain Kf is given by [25] Kf ¼ PHt ðHPHt þ VÞ1
ð18Þ
where V is the measurement noise spectral density matrix and P is the steady state error covariance matrix given by the solution of a discrete steady state Riccatti equation, P ¼ FPFt FPHt ðHPHt þ VÞ1 HPFt þ W
ð19Þ
where W is the process noise spectral density matrix. 3.2.2. LQGI design Once the Kalman gain is evaluated for the desired specifications, the optimal stateestimate feedback control is calculated. 3.3. H2, H1 , and mixed H2 =H1 tracking design In this section, H2 and =H1 optimal control problems are addressed [26,27]. The plant dynamics represented by 3 2 A B1 B2 B 7 6 0 D1u 7 6 C1 D1 7 ð20Þ P2=1 ¼ 6 6 C2 0 0 D2u 7 5 4 C Dy1 0 0 The goal of H2 optimal control is to maintain the H2 norm of the closed-loop transfer function JT2 J2 from w2 to z2, while the goal of the H1 optimal control is to maintain the RMS gain (H1 norm) of the closed-loop transfer function JT2 J1 from w1 to z1 . Mixed H2 =H1 control design, the control objective is to minimize an H2 =H1 trade of criterion of
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
49
the form aJT1 J21 þ bJT2 J22
ð21Þ
4. Simulations and analysis of results 4.1. LQI tracking system The LQI-design is evaluated through MATLAB simulation. The cited goal is to observe the HUAV performance on position tracking a predefined reference trajectory autonomously in the light of cruising mode [10]. In simulation, the focus will be on tracking three variables: longitudinal velocity, lateral velocity and vertical velocity, the other variables performance is neglected. To interpret the results of the simulation physically, coordinate transformations are needed between body coordinate to local horizon coordinate system using transformation matrix given as 2 3 cycc cysc sy 6 7 TbI ¼ 4 sfsycccfsc sfsysc þ cfcc sfcy 5 ð22Þ cfsycc þ sfsc
cfsyscsfcc
40
cfcy
40
30 20
20
10
0
0 −20
−10
−40
−20 −30 −40
−60 0
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
30
60
20
40
10
20
0
0
−10
−20
−20
−40
−30
−60
−40
−80 0
20
40
60
80
100
Fig. 2. Control input profile.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
50
where cð:Þ ¼ cosð:Þ, and sð:Þ ¼ sinð:Þ. After transformation from body coordinate to local horizon performed, integral operation is needed to represent the simulation results in position coordinate. Simulation experiments are conducted in two stages: in the first stage, the simulation goal is to seek for the proper weighting matrices that will be used in the second stage. The trade-off between tracking performance and control effort within a prescribed range is used to determine the weighting matrices. Thus, we run simulations for longitudinal, lateral, and vertical velocities to follow some targeted velocities, and compare the performance for different weighting matrices. These simulations also come in accordance with the control effort suitable for real flight situation. The limitation of the control effort of the HUAV is governed by the maximum allowable deflection of its control surfaces, as given by 301rdlat r301,
301rdlon r301
301rdped r301,
301rdcol r301
ð23Þ
These limits will be effective in determining the control weighting matrix R. For the state weighting Q, longitudinal velocity is as important as lateral and vertical velocity, so 70 60 50 40 0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
10 0 −10 −20
20 15 10 5 0
Fig. 3. Velocities response.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
20
40
60
80
100
−30
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
20
40
60
80
100
−30
20
40
60
80
100
20
40
60
80
100
Fig. 4. LQI control signal.
100 90 80 70 60 50 40 30 20 10 0 250
200
100
150
50
100
0
50 0
−50
51
−50 −100
Fig. 5. LQI control tracking predefined trajectory.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
52
20 10 0 −10 −20
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
50
0
−50
20 10 0 −10 −20
Fig. 6. State estimation using Kalman filter.
the weighting matrix component for all three states is set to equal. The results of extensive simulation experiments have concluded that the appropriate weighting matrices for the given control objectives are Qy ¼ diag½0:1 0:1 0:1, R ¼ diag½2 2 35 1
Qz ¼ diag½0:1 0:1 0:1 ð24Þ
These choices of the weighting matrices are found to handle the velocities tracking successfully with good performance and ensure that the control efforts stay between the limits as shown in Fig. 2. The corresponding variations of velocities are given in Fig. 3. In the second stage of simulation, the HUAV is controlled to follow a predefined trajectory that is taken as the helix-shape trajectory. The control input history depicted in Fig. 4 shows that the control effort stays within the prescribed limits, and it succeeds in providing acceptable tracking performance as can be seen in Fig. 5. Simulation results lead us to conclude that the weight assignments play a crucial role in the optimization process. In this study, the weight of the errors were assigned equal values. This means that all the errors were considered to yield equal contribution in the optimization process.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
53
200 200
150 100
100
50 0
0
−50 −100
−100 −150 −200
−200 0
20
40
60
80
100
60
60
40
40
20
20
0
0
−20
−20
−40
−40
−60
−60 0
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
Fig. 7. LQGI control signal.
120 100 80 60 40 20 0 −20 250 200
100
150
50
100
0
50 −50
0 −50 −100
Fig. 8. LQGI control tracking predefined trajectory.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
54
4.2. LQGI tracking system The similar procedure with the LQI control tracking system is conducted in LQGI simulation. For the state estimation part, HUAV model is subjected to process and measurement noises to observe the capability of the designed Kalman filter (KF). In what follows, standard deviation of measurement noise is taken as 1 and standard deviation of measurement noise is taken as 9 104. Initially, Fig. 6 shows the behavior of KF for the estimates of the longitudinal, lateral and vertical velocities. As can be seen in Fig. 6, the x^ has good replica of the actual state and state with noise. Although the sensors highly corrupted with noise, the KF still can produce good y^ behavior compared to the measurements without noise. These results indicate that x^ have a good chance to be used as an input for the feedback gain KLQI and performs LQGI control tracking system. The weighting matrices used in this simulation correspond to the previous LQI simulations as well as the gain KLQI in order to maintain the accuracy of the x^ produced by the Kalman filter. The optimal control produced by KLQI and x^ can be seen in Fig. 7. Control efforts produced by the LQGI tracking system are beyond the given limits, which means that these controls are not suitable for the prescribed control objectives. With this optimal control, LQGI can provide acceptable tracking performance as can be seen in Fig. 8, although the performance is degrading when compared to the LQI controller. These results are acceptable since there are errors between the estimated state x^ and actual state x.
30 20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
0
20
40
60
80
100
0
20
40
60
80
100
Fig. 9. H2 control effort.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
55
0
20
40
60
80
100
0
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
Fig. 10. H1 control effort.
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30 0
20
40
60
80
100
−30 0
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
0
Fig. 11. Mixed H2 =H1 control effort.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
56
60 40 20 0 0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
50 40 30 20 10 0
50 40 30 20 10 0
Fig. 12. H2, H1 , and mixed H2 =H1 tracking performances.
4.3. H2, H1 , and mixed H2 =H1 In this section, the performances of the velocities-tracking produced by H2, H1 , and mixed H2 =H1 are compared. The control signals are observed such that they are within the limits given in Eq. (23). First simulation is performed for H2 tracking control and the ensuing control signals produced by this controller are found to stay within the given limits as can be seen in Fig. 9. Next H1 simulation is executed and then followed the mixed H2 =H1 method. Control signals produced by these controllers are found also to stay between the prescribed range, as can be seen in Figs. 10 and 11, respectively. It is readily seen that velocities tracking performed by H2 controller is quite acceptable, see Fig. 12, although there is an overshoot in the longitudinal and lateral velocities. Tracking performance for H1 controller is also acceptable, however the steady-state error for longitudinal and vertical velocities is larger compared to H2 controller. Lateral velocity tracking performance for H1 is almost perfect, with steady-state error being smaller as compared to another controller. For mixed H2 =H1 controller, the longitudinal velocity tracking shows unacceptable performance, the deviation is about two times for the time ranges 10–20 s and 70–100 s. Lateral and vertical velocities tracking control showed
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
57
acceptable tracking performance and comparable with another controller. This is quite evident from Fig. 12. 5. Conclusions Design of discrete-time LQI and LQGI tracking systems have been elaborated and applied in the controller of a pilot-scale helicopter. Simulations have shown that weight assignments play a significant role in the optimization process, which has been obtained empirically based on the control objectives. It has been established that tracking system can bring the helicopter to follow a specified trajectory with weighting matrices assignment to adjust the trade-off between tracking performance and control input expenditure. In the presence of process noise and measurement noise, Kalman filter has shown good performance and product of the estimation have been utilized to perform LQGI tracking control system. Next, H2 , H1 , and mixed H2 =H1 controllers have been performed and typical simulation results have emphasized the capability of the designed controller to satisfy the cited objectives. Acknowledgments The authors would like to thank the reviewers for their constructive suggestions and comments on our submission. Also, the authors would like to extend their appreciation to the deanship for scientific research (DSR) at KFUPM for research support through research group project RG 1105-1. References [1] P. Khargonekar, M. Rotea, Mixed H2 =H1 control—a convex optimization approach, IEEE Transactions on Automatic Control 36 (1991) 824–837. [2] H. Kwakernaak, Robust control and H1 -optimization: tutorial, Automatica 29 (2) (1993) 255–273. [3] G. Yang, J. Wang, Y. Soh, Reliable H1 controller design for linear systems, Automatica 37 (5) (2001) 717–725. [4] T.K. Liua, S.H. Chen, J.H. Chou, C.Y. Chen, Regional eigenvalue-clustering robustness of linear uncertain multivariable output feedback PID control systems, Journal of the Franklin Institute 346 (3) (2009) 253–266. [5] M.S. Mahmoud, New results on linear parameter-varying time-delay systems, Journal of the Franklin Institute 341 (7) (2004) 675–703. [6] M.S. Mahmoud, Y. Shi, F.M. AL-Sunni, Mixed H2 =H1 control of uncertain jumping time-delay systems, Journal of the Franklin Institute 345 (5) (2008) 536–552. [7] Z. Mao, B. Jiang, P. Shi, Observer based fault-tolerant control for a class of nonlinear networked control systems, Journal of the Franklin Institute 347 (6) (2010) 940–956. [8] C.X. Yang, Z.H. Guan, J. Huang, Stochastic fault tolerant control of networked control systems, Journal of the Franklin Institute 346 (10) (2009) 1006–1020. [9] S. Bhandari, R. Colgren, P. Lederbogen, S. Kowalchuk, Six-DOF dynamic modeling and flight testing of a UAV helicopter, AIAA Modeling and Simulation Technologies Conference, vol. 2, 2005, pp. 992–1008. [10] B. Mettler, T. Kanade, M. Tischler, System identification modeling of a model-scale helicopter, Citeseer, 2000. [11] H. Shim, T. Koo, F. Hoffmann, S. Sastry, A comprehensive study of control design for an autonomous helicopter, Proceedings of the 37th IEEE Conference on Decision and Control, vol. 4, 2002, pp. 3653–3658. [12] M.S. Mahmoud, H.N. Nounou, Y. Xia, Dissipative control for Internet-based switching systems, Journal of the Franklin Institute 347 (1) (2010) 154–172. [13] Y. Xia, Z. Zhu, M. Fu, S. Wang, Attitude tracking of rigid spacecraft with bounded disturbances, IEEE Transactions on Industrial Electronics 58 (2) (2011) 647–659.
58
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
[14] Y. Xia, Z. Zhu, M. Fu, Back-stepping sliding mode control for missile systems based on extended state observer, IET Control Theory and Applications 5 (1) (2011) 93–102. [15] Z. Zhu, Y. Xia, M. Fu, Attitude stabilization of rigid spacecraft with finite-time convergence, International Journal of Robust and Nonlinear Control 21 (2011) 686–702. [16] Y. Yin, P. Shi, F. Liu, Gain scheduled PI tracking control on stochastic nonlinear systems with partially known transition probabilities, Journal of the Franklin Institute 348 (4) (2011) 685–702. [17] Z. Wu, J. Yang, P. Shi, Adaptive tracking for stochastic nonlinear systems with Markovian switching, IEEE Transactions on Automatic Control 55 (9) (2010) 2135–2141. [18] C. Hua, X. Guan, P. Shi, Robust output feedback tracking control for time-delayed nonlinear systems using neural network, IEEE Transactions on Neural Networks 18 (2) (2007) 495–505. [19] V. Gavrilets, B. Mettler, E. Feron, Dynamic model for a miniature aerobatic helicopter, in: Proceedings of the AIAA Guidance Navigation and Control Conference, Montreal, Canada, 2001. [20] A. Budiyono, S. Wibowo, Optimal tracking controller design for a small scale helicopter, Journal of Bionic Engineering 4 (4) (2007) 271–280. [21] P. Young, J. Willems, An approach to the linear multivariable servomechanism problem, International Journal of Control 15 (5) (1972) 961–979. [22] R. Prouty, Helicopter Performance, Stability, and Control, RE Krieger Pub. Co, 1990. [23] P. Pesek, J. Mosna, LQG tracking problem and its tracking robustness, in: Proceedings of the 11th Mediterranean Conference on Control and Automation, 2001, pp. 1–5. [24] D. Naidu, Optimal Control Systems, CRC Press, 2003. [25] D. Simon, Optimal state estimation: Kalman, H1 and nonlinear approaches, John Wiley and Sons, 2006. [26] D. Arzelier, D. Georgia, S. Gumussoy, D. Henrion, H2 for HIFOO, 2010. [27] J. Burke, D. Henrion, A. Lewis, M. Overton, HIFOO—a MATLAB package for fixed-order controller design and H1 optimization, in: Proceedings of the Fifth IFAC Symposium on Robust Control Design, Toulouse, 2006.
Journal of the Franklin Institute 349 (2012) 42–58 www.elsevier.com/locate/jfranklin
Improved digital tracking controller design for pilot-scale unmanned helicopter Magdi S. Mahmouda,n, Arief B. Koesdwiadyb a
Systems Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 985, Dhahran 31261, Saudi Arabia b Systems Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 5067, Dhahran 31261, Saudi Arabia Received 20 April 2011; received in revised form 1 October 2011; accepted 6 October 2011 Available online 15 October 2011
Abstract In this paper, methods for improved design of digital tracking controller for a pilot-scale unmanned helicopter are considered. By discretizing the linearized helicopter model, the linear quadratic with integral (LQI) capability is investigated and applied in order to develop an efficient tracking system including a state-feedback plus integral action. The helicopter velocities are used to formulate a prescribed position reference tracking trajectory. When both process and measurement noises are present, a Kalman filter (KF) is combined with the LQI to form a linear quadratic Gaussian with integral (LQGI) tracking system. Simulation studies illuminate both the capability of the controller design and the accuracy of the estimator. Next, H2, H1 and mixed H2 =H1 controls are designed and the results between methods are produced and compared. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction In recent years, we have witnessed wide variety of techniques and applications of control theory to several systems engineering areas robust control [1–3], including PID control [4], time-varying systems [5,6], networked control systems [7,8], autonomous systems [9–11], Internet control [12], attitude stabilization of rigid spacecraft [13–15], to name a few. The primary focus of this work is on a class of autonomous systems [9]. In this regard, n
Corresponding author. E-mail addresses: [email protected], [email protected] (M.S. Mahmoud), [email protected] (A.B. Koesdwiady). 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.10.003
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
43
unmanned aerial vehicle (UAV) with helicopter-like capabilities such as vertical take-off and landing is becoming popular in systems studies. The interest in helicopter-based unmanned aerial vehicle (HUAV) has central focus for both military and civil applications. It is well-known that helicopter is a non-linear complex system with unstable nature and, in turn, these characteristics render the design in flight control systems a difficult task. Pilotscale unmanned helicopter constitute by now common platforms for HUAV control development with many great advances been made since the last decade. Compared with traditional full-size helicopters, a pilot-scale helicopter tend to be naturally more maneuverable and responsive [10]. Some related results are reported in [16–18]. System identification has been used as a modeling technique to derive linear models for control design and to study the vehicle flying qualities. Also, system identification is utilized for the validation and refinement of detailed non-linear first-principle models. There are several reported applications of system identification techniques to modeling of pilot-scale helicopters, including the model identification of YAMAHA R-50 [10], and X-Cell [19] for flight control, and a 6-DOF dynamic modeling of Raptor-50 V2 for simulations [9]. In this paper, the pilot-scale unmanned helicopter dynamics reported in [10] will be used in the sequel as the preliminary vehicle for control system design. Most HUAVs used classical control system such as single-loop PD systems. The tuning for controller parameters usually performed manually for certain operating point such as hover and cruise mode. This condition giving a wide opportunity for multi-variable controller synthesis method to be implemented. In previous work [20], linear quadratic trajectory tracking system with derivative on the error for HUAV conducted in continuous-time mode, and reports encouraging preliminary results. There are numerous predicted utilizations for HUAV either individually or working as a team, including surveillance, search and rescue, and mobile sensor networks. For this uses, HUAV needs to be able to navigate to the desired destination through desired trajectory, the position control as well as velocity control are performed for this scenario. A controller that can accommodate this scenario is needed, linear quadratic integral (LQI) [21] tracking control is proposed, and for practical needs in the presence of process and measurements noise, linear quadratic Gaussian integral (LQGI) tracking control is proposed with the assumption that the noise is white. This motivates the study of this paper by looking at effective methods for controlling pilot-scale HUAVs. Looked at in this light, this paper provides improved digital controller design and simulations using the powerful tools of MATLAB and SIMULINK. Computer simulation in discrete-time is conducted with certain sampling time will allow direct implementation on the practical controller. The dynamics of helicopter on cruise is used for the trajectory tracking system [10]. Then a linear quadratic Gaussian with integral (LQGI) control is designed with the assumption that sensor available is only for velocities and corrupted with white noise. The ensuing measurements are used to perform state estimation using Kalman filter (KF). For the controller design based on H2, H1 and mixed H2 =H1 performance, the simulations are performed in continuous-time mode. The results are presented and a comparison among design methods is performed. 2. Dynamics of pilot-scale helicopter For controller synthesis or controller optimization, the dynamic models usually have strict requirements. Essentially, the model must capture the effects that influence the
44
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
performance of the system. This implies that for helicopter they must explicitly account for rotor and fuselage coupling effects. At the same time the model must be simple enough to be insightful and practical for the control synthesis. The basic equations of motion for a linear model of the helicopter dynamics are derived from the Newton–Euler equations for a rigid body that is free to simultaneously rotate and translate in all six degree of freedom.
The dynamics of helicopter is defined as follows [10]: rotor derivatives Xa, Xb are used to express rotor forces, and flapping spring-derivatives Lb, Ma to express rotor moments. Xu, Yv, Lu, Lv, Mu, Mv are speed derivatives to express general aerodynamics effect [22,23]. u_ ¼ ðw0 q þ v0 rÞgy þ Xu u þ Xa a v_ ¼ ðu0 r þ w0 pÞ þ gf þ Yv v þ Yb b p_ ¼ Lu u þ Lv v þ Lb b q_ ¼ Mu u þ Mv v þ Ma a
ð1Þ
u, v, and w are velocity components for longitudinal, lateral, and vertical movements (ft/s), while u0, v0, and w0 are the values of u, v, w at trim condition. p, q, and r are roll, pitch, and yaw rates, rad s1. The rigid body equation from Newton–Euler for vertical dynamic is w_ ¼ ðv0 p þ u0 qÞ þ Zw w þ Zcol dcol
ð2Þ
The centrifugal forces represented in the terms v0 and u0, which is relevant only for cruise conditions. Rotor equations of motions for the longitudinal and lateral flapping are two first order differential equations tf a_ ¼ atf q þ Aa a þ Alat dlat þ Ac c þ Alon dlon tf b_ ¼ btf p þ Ba a þ Blat dlat þ Bd d þ Blon dlon
ð3Þ
a,b are main motor longitudinal and lateral flapping motions (rad), and c,d are stabilizer bar longitudinal and lateral flapping motions (rad). tf and ts are main rotor and stabilizer time constants. The longitudinal and lateral stabilizer bar dynamic equation presented in
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
45
the similar way as the main rotor ts c_ ¼ cts q þ Clon dlon ts d_ ¼ dts p þ Dlat dlat
ð4Þ
In this case, the augmented yaw dynamics is the first order bare airframe dynamics with a yaw rate feedback represented by a simple first order low-pass filter rfb Kr ¼ r s þ Krfb
ð5Þ
where the corresponding differential equations used in the state-space model r_ ¼ Nr r þ Nped ðdped rfb Þ r_fb ¼ Kr rKrfb rfb
ð6Þ
The full state-space equation of the model is given as x_ ¼ Ax þ Bu x ¼ ½u, v, w, p, q, r, f, y, tf a, tf b, ts c, ts d, rfb t u ¼ ½dlat , dlon , dped , dcol t ð7Þ where dlat is lateral deflection of main rotor, dlon is longitudinal deflection of main rotor, dped is pitch deflection of tail rotor blade, and dcol is pitch deflection of main rotor blade, all in radians. A helicopter responds differently in hover flight than it does in cruise flight. Since this paper is only concerned with tracking a desired trajectory, the helicopter in cruise flight response will be used. Typical data values are rotor speed¼ 850 r min1, tip speed¼ 499 ft/s, dry weight¼ 97 lb, instrumented (full)¼ 150 lb, and flight autonomy¼ 30 min. Remark 2.1. The main goal of this paper is to design a discrete-time linear-quadratic integral (LQI) and linear-quadratic Gaussian integral (LQGI) tracking systems and to apply the elaborated systems in the controller of a pilot-scale helicopter. 2.1. Discrete-time model A discrete model, to be used hereafter, is derived from the continuous model of pilotscale helicopter with 6-DOF. The experiments to develop the non-linear dynamics model of pilot-scale helicopter were conducted on Carnegie Mellon’s helicopter-based UAV (HUAV based on Yamaha R-50 model scale helicopter with 10 ft rotor diameter) [10]. Both hover and cruise flight conditions have been treated and all important effects have been captured. The conversion from continuous-time model to discrete-time model is performed using c2d command in MATLAB. The continuous-time model is discretized by incorporating a zero-order hold on the inputs and sampling at a rate of 0.01 s. A statespace representation for discrete-time model is described by x½k þ 1 ¼ Fx½k þ Gu½k
ð8Þ
y½k ¼ Hx½k
ð9Þ
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
46
Longitudinal Velocity
Response
2 0 −2 −4
1
1.05
1.1
1.15
1.2
1.25 Time
1.3
1.35
1.4
1.45
1.5
1.35
1.4
1.45
1.5
1.35
1.4
1.45
1.5
Lateral Velocity
Response
10
5
0
1
1.05
1.1
1.15
1.2
1.25 Time
1.3
Vertical Velocity
Response
0 −10 −20 −30
1
1.05
1.1
1.15
1.2
1.25 Time
Continuous
1.3
Discrete
Fig. 1. Discrete versus continuous open loop response.
To confirm that the discrete-time model have similar behavior as the continuous-time model, the validity test of the model must be performed. The test conducted in open-loop mode with the step signal used in all the inputs. The response captured only on the three states u, v, w since this paper addresses the problem only on position tracking. From Fig. 1, it can be seen that the response shown by discrete-time model is similar to the continuous model. It means the discrete time model is valid to be used as the representation of the dynamics system of 6-DOF pilot-scale helicopter.
Remark 2.2. It should be noted that the simulation results in Fig. 1 showed that adopted discretization scheme effectively reproduces the continuous model and hence provides a good starting point in preparation for the controller design.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
47
3. Controller design In what follows, we present methods for controller design of the developed model of HUAV.
3.1. Linear quadratic integral tracking We consider a linear quadratic regulator (LQR) with tracking capability. Thus, it is required that the output y to track a reference trajectory yr . To achieve this, we use both the state x and the integral of error between reference and output z [24] thereby eliminating steady state errors when tracking constant signals [21]. This system is termed linear quadratic integral (LQI) tracking system. Z t zðtÞ ¼ ðyr yÞ dt, z_ ðtÞ ¼ yr ðtÞyðtÞ ð10Þ With sampling time Ts , we get z½k þ 1 ¼ z½k þ Ts ðyr ½ny½nÞ
ð11Þ
In our design, the number of the states x is 13, with four inputs u, and three outputs y that measured for tracking system as well as the reference yr . Augmenting the state x½k to the error z½k, we have xaug ½k ¼ ½x½kz½kt , where xaug ½k þ 1 ¼ Faug xaug ½k þ Gaug u½k þ r½k y½k ¼ Haug xaug ½k " # F 0133 Faug ¼ , Ts H I33 Haug ¼ ½H 033 ,
Gaug ¼
G
0
r½k ¼ ½0131 ytr Tts t
ð12Þ
The performance index is 1 1X ðxaug ½kt Qxaug ½k þ ut ½kRu½kÞ 2k¼1 0rQtaug ¼ Qaug , Rt ¼ R40
J¼
Qaug ¼ diag½Gt Qy G Qz
ð13Þ
Qy , Qz are the state and error weighting matrices, respectively. The linear quadratic controller is the unique, optimal full state feedback control law u ¼ KLQI xaug ,
KLQI ¼ R1 Gtaug S
ð14Þ
with 0rSt ¼ S is the solution to the DARE 0 ¼ Ftaug SFaug SFtaug SGaug ðGtaug SGaug þ RÞ1 Gtaug SFaug þ Q KLQI ¼ ½Kx Kz where Kx is the gain for state-feedback, and Kz is the gain for error.
ð15Þ
48
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
3.2. Linear quadratic Gaussian integral tracking A linear quadratic state feedback problem was solved which assumes that all states are available for feedback. In the sequel, a Kalman filter (KF) is to be designed to estimate the unmeasured states. Under the standard Gaussian noise assumption [25], a linear quadratic Gaussian (LQG) regulator is considered is formed where the LQI tracking and KF are to be designed independently. 3.2.1. Kalman filter design Since the velocity of the HUAV corrupted by noise is the only measured variable, the ^ remaining states have to be estimated through a KF in real time in real time. Let x½k be the state estimate of the state x½k where the plant model is x½k þ 1 ¼ Fx½k þ Gu½k y½k ¼ Hx½k
ð16Þ
The KF dynamics are given as follows ^ þ 1 ¼ Fx½k ^ þ Gu½k þ Kf ðy½kHx½kÞ ^ x½k
ð17Þ
The Kalman gain Kf is given by [25] Kf ¼ PHt ðHPHt þ VÞ1
ð18Þ
where V is the measurement noise spectral density matrix and P is the steady state error covariance matrix given by the solution of a discrete steady state Riccatti equation, P ¼ FPFt FPHt ðHPHt þ VÞ1 HPFt þ W
ð19Þ
where W is the process noise spectral density matrix. 3.2.2. LQGI design Once the Kalman gain is evaluated for the desired specifications, the optimal stateestimate feedback control is calculated. 3.3. H2, H1 , and mixed H2 =H1 tracking design In this section, H2 and =H1 optimal control problems are addressed [26,27]. The plant dynamics represented by 3 2 A B1 B2 B 7 6 0 D1u 7 6 C1 D1 7 ð20Þ P2=1 ¼ 6 6 C2 0 0 D2u 7 5 4 C Dy1 0 0 The goal of H2 optimal control is to maintain the H2 norm of the closed-loop transfer function JT2 J2 from w2 to z2, while the goal of the H1 optimal control is to maintain the RMS gain (H1 norm) of the closed-loop transfer function JT2 J1 from w1 to z1 . Mixed H2 =H1 control design, the control objective is to minimize an H2 =H1 trade of criterion of
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
49
the form aJT1 J21 þ bJT2 J22
ð21Þ
4. Simulations and analysis of results 4.1. LQI tracking system The LQI-design is evaluated through MATLAB simulation. The cited goal is to observe the HUAV performance on position tracking a predefined reference trajectory autonomously in the light of cruising mode [10]. In simulation, the focus will be on tracking three variables: longitudinal velocity, lateral velocity and vertical velocity, the other variables performance is neglected. To interpret the results of the simulation physically, coordinate transformations are needed between body coordinate to local horizon coordinate system using transformation matrix given as 2 3 cycc cysc sy 6 7 TbI ¼ 4 sfsycccfsc sfsysc þ cfcc sfcy 5 ð22Þ cfsycc þ sfsc
cfsyscsfcc
40
cfcy
40
30 20
20
10
0
0 −20
−10
−40
−20 −30 −40
−60 0
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
30
60
20
40
10
20
0
0
−10
−20
−20
−40
−30
−60
−40
−80 0
20
40
60
80
100
Fig. 2. Control input profile.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
50
where cð:Þ ¼ cosð:Þ, and sð:Þ ¼ sinð:Þ. After transformation from body coordinate to local horizon performed, integral operation is needed to represent the simulation results in position coordinate. Simulation experiments are conducted in two stages: in the first stage, the simulation goal is to seek for the proper weighting matrices that will be used in the second stage. The trade-off between tracking performance and control effort within a prescribed range is used to determine the weighting matrices. Thus, we run simulations for longitudinal, lateral, and vertical velocities to follow some targeted velocities, and compare the performance for different weighting matrices. These simulations also come in accordance with the control effort suitable for real flight situation. The limitation of the control effort of the HUAV is governed by the maximum allowable deflection of its control surfaces, as given by 301rdlat r301,
301rdlon r301
301rdped r301,
301rdcol r301
ð23Þ
These limits will be effective in determining the control weighting matrix R. For the state weighting Q, longitudinal velocity is as important as lateral and vertical velocity, so 70 60 50 40 0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
10 0 −10 −20
20 15 10 5 0
Fig. 3. Velocities response.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
20
40
60
80
100
−30
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
20
40
60
80
100
−30
20
40
60
80
100
20
40
60
80
100
Fig. 4. LQI control signal.
100 90 80 70 60 50 40 30 20 10 0 250
200
100
150
50
100
0
50 0
−50
51
−50 −100
Fig. 5. LQI control tracking predefined trajectory.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
52
20 10 0 −10 −20
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
50
0
−50
20 10 0 −10 −20
Fig. 6. State estimation using Kalman filter.
the weighting matrix component for all three states is set to equal. The results of extensive simulation experiments have concluded that the appropriate weighting matrices for the given control objectives are Qy ¼ diag½0:1 0:1 0:1, R ¼ diag½2 2 35 1
Qz ¼ diag½0:1 0:1 0:1 ð24Þ
These choices of the weighting matrices are found to handle the velocities tracking successfully with good performance and ensure that the control efforts stay between the limits as shown in Fig. 2. The corresponding variations of velocities are given in Fig. 3. In the second stage of simulation, the HUAV is controlled to follow a predefined trajectory that is taken as the helix-shape trajectory. The control input history depicted in Fig. 4 shows that the control effort stays within the prescribed limits, and it succeeds in providing acceptable tracking performance as can be seen in Fig. 5. Simulation results lead us to conclude that the weight assignments play a crucial role in the optimization process. In this study, the weight of the errors were assigned equal values. This means that all the errors were considered to yield equal contribution in the optimization process.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
53
200 200
150 100
100
50 0
0
−50 −100
−100 −150 −200
−200 0
20
40
60
80
100
60
60
40
40
20
20
0
0
−20
−20
−40
−40
−60
−60 0
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
Fig. 7. LQGI control signal.
120 100 80 60 40 20 0 −20 250 200
100
150
50
100
0
50 −50
0 −50 −100
Fig. 8. LQGI control tracking predefined trajectory.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
54
4.2. LQGI tracking system The similar procedure with the LQI control tracking system is conducted in LQGI simulation. For the state estimation part, HUAV model is subjected to process and measurement noises to observe the capability of the designed Kalman filter (KF). In what follows, standard deviation of measurement noise is taken as 1 and standard deviation of measurement noise is taken as 9 104. Initially, Fig. 6 shows the behavior of KF for the estimates of the longitudinal, lateral and vertical velocities. As can be seen in Fig. 6, the x^ has good replica of the actual state and state with noise. Although the sensors highly corrupted with noise, the KF still can produce good y^ behavior compared to the measurements without noise. These results indicate that x^ have a good chance to be used as an input for the feedback gain KLQI and performs LQGI control tracking system. The weighting matrices used in this simulation correspond to the previous LQI simulations as well as the gain KLQI in order to maintain the accuracy of the x^ produced by the Kalman filter. The optimal control produced by KLQI and x^ can be seen in Fig. 7. Control efforts produced by the LQGI tracking system are beyond the given limits, which means that these controls are not suitable for the prescribed control objectives. With this optimal control, LQGI can provide acceptable tracking performance as can be seen in Fig. 8, although the performance is degrading when compared to the LQI controller. These results are acceptable since there are errors between the estimated state x^ and actual state x.
30 20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
0
20
40
60
80
100
0
20
40
60
80
100
Fig. 9. H2 control effort.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
55
0
20
40
60
80
100
0
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
Fig. 10. H1 control effort.
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30 0
20
40
60
80
100
−30 0
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
0
20
40
60
80
100
−30
0
Fig. 11. Mixed H2 =H1 control effort.
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
56
60 40 20 0 0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
50 40 30 20 10 0
50 40 30 20 10 0
Fig. 12. H2, H1 , and mixed H2 =H1 tracking performances.
4.3. H2, H1 , and mixed H2 =H1 In this section, the performances of the velocities-tracking produced by H2, H1 , and mixed H2 =H1 are compared. The control signals are observed such that they are within the limits given in Eq. (23). First simulation is performed for H2 tracking control and the ensuing control signals produced by this controller are found to stay within the given limits as can be seen in Fig. 9. Next H1 simulation is executed and then followed the mixed H2 =H1 method. Control signals produced by these controllers are found also to stay between the prescribed range, as can be seen in Figs. 10 and 11, respectively. It is readily seen that velocities tracking performed by H2 controller is quite acceptable, see Fig. 12, although there is an overshoot in the longitudinal and lateral velocities. Tracking performance for H1 controller is also acceptable, however the steady-state error for longitudinal and vertical velocities is larger compared to H2 controller. Lateral velocity tracking performance for H1 is almost perfect, with steady-state error being smaller as compared to another controller. For mixed H2 =H1 controller, the longitudinal velocity tracking shows unacceptable performance, the deviation is about two times for the time ranges 10–20 s and 70–100 s. Lateral and vertical velocities tracking control showed
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
57
acceptable tracking performance and comparable with another controller. This is quite evident from Fig. 12. 5. Conclusions Design of discrete-time LQI and LQGI tracking systems have been elaborated and applied in the controller of a pilot-scale helicopter. Simulations have shown that weight assignments play a significant role in the optimization process, which has been obtained empirically based on the control objectives. It has been established that tracking system can bring the helicopter to follow a specified trajectory with weighting matrices assignment to adjust the trade-off between tracking performance and control input expenditure. In the presence of process noise and measurement noise, Kalman filter has shown good performance and product of the estimation have been utilized to perform LQGI tracking control system. Next, H2 , H1 , and mixed H2 =H1 controllers have been performed and typical simulation results have emphasized the capability of the designed controller to satisfy the cited objectives. Acknowledgments The authors would like to thank the reviewers for their constructive suggestions and comments on our submission. Also, the authors would like to extend their appreciation to the deanship for scientific research (DSR) at KFUPM for research support through research group project RG 1105-1. References [1] P. Khargonekar, M. Rotea, Mixed H2 =H1 control—a convex optimization approach, IEEE Transactions on Automatic Control 36 (1991) 824–837. [2] H. Kwakernaak, Robust control and H1 -optimization: tutorial, Automatica 29 (2) (1993) 255–273. [3] G. Yang, J. Wang, Y. Soh, Reliable H1 controller design for linear systems, Automatica 37 (5) (2001) 717–725. [4] T.K. Liua, S.H. Chen, J.H. Chou, C.Y. Chen, Regional eigenvalue-clustering robustness of linear uncertain multivariable output feedback PID control systems, Journal of the Franklin Institute 346 (3) (2009) 253–266. [5] M.S. Mahmoud, New results on linear parameter-varying time-delay systems, Journal of the Franklin Institute 341 (7) (2004) 675–703. [6] M.S. Mahmoud, Y. Shi, F.M. AL-Sunni, Mixed H2 =H1 control of uncertain jumping time-delay systems, Journal of the Franklin Institute 345 (5) (2008) 536–552. [7] Z. Mao, B. Jiang, P. Shi, Observer based fault-tolerant control for a class of nonlinear networked control systems, Journal of the Franklin Institute 347 (6) (2010) 940–956. [8] C.X. Yang, Z.H. Guan, J. Huang, Stochastic fault tolerant control of networked control systems, Journal of the Franklin Institute 346 (10) (2009) 1006–1020. [9] S. Bhandari, R. Colgren, P. Lederbogen, S. Kowalchuk, Six-DOF dynamic modeling and flight testing of a UAV helicopter, AIAA Modeling and Simulation Technologies Conference, vol. 2, 2005, pp. 992–1008. [10] B. Mettler, T. Kanade, M. Tischler, System identification modeling of a model-scale helicopter, Citeseer, 2000. [11] H. Shim, T. Koo, F. Hoffmann, S. Sastry, A comprehensive study of control design for an autonomous helicopter, Proceedings of the 37th IEEE Conference on Decision and Control, vol. 4, 2002, pp. 3653–3658. [12] M.S. Mahmoud, H.N. Nounou, Y. Xia, Dissipative control for Internet-based switching systems, Journal of the Franklin Institute 347 (1) (2010) 154–172. [13] Y. Xia, Z. Zhu, M. Fu, S. Wang, Attitude tracking of rigid spacecraft with bounded disturbances, IEEE Transactions on Industrial Electronics 58 (2) (2011) 647–659.
58
M.S. Mahmoud, A.B. Koesdwiady / Journal of the Franklin Institute 349 (2012) 42–58
[14] Y. Xia, Z. Zhu, M. Fu, Back-stepping sliding mode control for missile systems based on extended state observer, IET Control Theory and Applications 5 (1) (2011) 93–102. [15] Z. Zhu, Y. Xia, M. Fu, Attitude stabilization of rigid spacecraft with finite-time convergence, International Journal of Robust and Nonlinear Control 21 (2011) 686–702. [16] Y. Yin, P. Shi, F. Liu, Gain scheduled PI tracking control on stochastic nonlinear systems with partially known transition probabilities, Journal of the Franklin Institute 348 (4) (2011) 685–702. [17] Z. Wu, J. Yang, P. Shi, Adaptive tracking for stochastic nonlinear systems with Markovian switching, IEEE Transactions on Automatic Control 55 (9) (2010) 2135–2141. [18] C. Hua, X. Guan, P. Shi, Robust output feedback tracking control for time-delayed nonlinear systems using neural network, IEEE Transactions on Neural Networks 18 (2) (2007) 495–505. [19] V. Gavrilets, B. Mettler, E. Feron, Dynamic model for a miniature aerobatic helicopter, in: Proceedings of the AIAA Guidance Navigation and Control Conference, Montreal, Canada, 2001. [20] A. Budiyono, S. Wibowo, Optimal tracking controller design for a small scale helicopter, Journal of Bionic Engineering 4 (4) (2007) 271–280. [21] P. Young, J. Willems, An approach to the linear multivariable servomechanism problem, International Journal of Control 15 (5) (1972) 961–979. [22] R. Prouty, Helicopter Performance, Stability, and Control, RE Krieger Pub. Co, 1990. [23] P. Pesek, J. Mosna, LQG tracking problem and its tracking robustness, in: Proceedings of the 11th Mediterranean Conference on Control and Automation, 2001, pp. 1–5. [24] D. Naidu, Optimal Control Systems, CRC Press, 2003. [25] D. Simon, Optimal state estimation: Kalman, H1 and nonlinear approaches, John Wiley and Sons, 2006. [26] D. Arzelier, D. Georgia, S. Gumussoy, D. Henrion, H2 for HIFOO, 2010. [27] J. Burke, D. Henrion, A. Lewis, M. Overton, HIFOO—a MATLAB package for fixed-order controller design and H1 optimization, in: Proceedings of the Fifth IFAC Symposium on Robust Control Design, Toulouse, 2006.