- Email: [email protected]
A nonlinear internal model control using backstepping method for attitude control of a helicopter model
IFAC Workshop on Adaptation and Learning in Control and Signal Processing, and IFAC Workshop on Periodic Control Systems, Yokohama, Japan, August 30 – September 1, 2004
A NONLINEAR INTERNAL MODEL CONTROL USING BACKSTEPPING METHOD FOR ATTITUDE CONTROL OF A HELICOPTER MODEL Kazuya Yoshioka ∗ Shiro Masuda ∗ ∗
Tokyo Metropolitan Institute of Technology, 6-6, Asahigaoka, Hino, Tokyo, Japan
Abstract: The backstepping method is to design nonlinear control systems to follow a given reference signal by iterative recursive procedure. Although it is an effective approach to design stable closed loop systems for nonlinear controlled systems, it has drawbacks to become extremely complex when the relative order of the controlled system is large. In order to make the design procedure more tractable, we need to approximate the controlled system to a low order system. However, the approximated error between the model and the real process due to model reduction might bring significant performance degradation. Hence, this paper proposes a nonlinear internal model control (IMC) incoporated with the inverse of a low order system designed from the backstepping method. In the proposed method, the backstepping design procedure becomes simple because it can be done for a low order system. Furthermore, the peformance degradation due to model reduction can be compensated from an IMC structure. The proposed control method is applied to the attitude control for a helicopter model, and we compare the proposed method with the usual backstepping method through simulation experiments. From the comparative study, it follows that the proposed one is especially effective for improving the characteristic of input signals. Keywords: Backstepping method, Internal model control
1. INTRODUCTION The backstepping method(Kanellakopoulos et al., 1991; Krstic et al., 1995) is to design nonlinear control systems to follow a given reference signal by iterative recursive procedure. Although it is an effective approach to design stable closed loop systems for nonlinear controlled systems, it has drawbacks to become extremely complex when the relative order of the controlled system is large. In order to make the design procedure more tractable, we need to approximate the controlled system to a low order system. However, the approximated error between the model and the real
process due to model reduction might bring significant performance degradation. Hence, this paper proposes a nonlinear internal model control (IMC) incoporated with the inverse of a low order system designed from the backstepping method. The IMC is a popular control technique in the chemical process control field, and various research results have been reported so far (Morari and Zafiriou, 1989). However, these researches have mainly focused on the linear systems. Hence, the proposed method is cosidered to be a new device for the IMC design method for nonlinear systems.
717
In the proposed method, the backstepping design procedure becomes simple because it can be done for a low order system. Furthermore, the peformance degradation due to model reduction can be compensated from an IMC structure. The proposed control method is applied to the attitude control for a helicopter model(Humusoft, 1992; Tanabe and Ichikawa, 2000), and we compare the proposed method with the usual backstepping method through simulation experiments. In the case of applying a usual backstepping method, a desired tracking property can be achieved even if the moment of rotational inertia is unknown. However, it has drawbacks that the trajectory of the input signal is oscillatory and that the input signal initially swelled significantly(Watanabe and Masuda, 2002). This is because the controlled helicopter model is the fourth order system and we must repeat the recursive procedure for designing Lyapunov function four times; hence the design procedure of the control system complex. On the other hand, in the proposed method, the system model is approximated to the second order system. Hence, the backstepping procedure is repeated only two times. From the comparative study, it follows that the proposed one is especially effective for improving the characteristic of input signals. The paper is organaized as follows. Section 2 gives the design method for the proposed nonlinear IMC based on the backstepping method. Section 3 shows the simulation result of the application of the proposed nonlinear IMC to the attitude control of the helicopter model. Section 4 gives concluding remarks.
2. NON-LINEAR IMC BASED ON BACKSTEPPING METHOD
x˙ m(l−1)
x˙ ml = θ0 β(xm )u + ϕml (xm )T θ
The order l of the system model is less than n; that implies the system model Eq.(3) ∼ Eq.(6) are a low ordered system model reduced from the controlled system Eq.(1) and Eq.(2).
2.2 IMC control structure
Fig. 1. Basic IMC structure The basic IMC control structure is is depicted in Fig. 2.2. In the control structure, the difference between the real process and internal model outputs is a feedback signal for modifying a given reference signal. Since the same input signal is given to both the real process and internal model, the feedback signal becomes zero when there is neither disturbance nor mismatch between the process and its model, and it becomes active only when there exist disturbances and/or model-process mismatch. The modified reference input is equal to the reference input minus the feedback signal. Hence, the following equation is satisfied. yrm = yr − (x1 − xm1 ) ⇔ yrm − xm1 = yr − x1
Consider the following n-the order nonlinear system having a single input, single output system. (1) (2)
where x ∈ Rn , u, y ∈ R are state variables, and control input, and control output, repectively. f(·) : Rn × R → R is Lipschitz in x and u. h(·) : Rn → R is piecewise continuous function. While Eq.(1) and Eq.(2) are the controlled real system, we consider the following l-th order system model for an internal model. x˙ m1 = xm2 + ϕm1 (xm1)T θ x˙ m2 = xm3 + ϕm2 (xm1, xm2 )T θ
(3) (4)
(6)
where xm is the state variable vector of internal model whose element are xm1 , xm2, · · · , xml, and θ is a parameter vector. ϕm1 , ϕm2 , · · · , ϕml are smooth nonlinear function.
2.1 The controlled system and internal model
x˙ = f(x, u) y = h(x)
.. . = xml + ϕm(l−1) (xm1 , · · · , xml)T θ (5)
(7) (8)
Therefore, if the controller could be designed so that the model output xm1 is equal to the yrm , the process output x1 would also be equal to the reference signal.
2.3 Nonlinear Internal Model Control When the process model is a linear system, the design of the inverse of the process model is not so difficult. However, when it is a non-linear system, it requires a certain device for constructing the inverse of the system model. In this research, we will introduce the backstepping method (Kanellakopoulos et al., 1991; Krstic
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3. APPLICATION TO A HELICOPTER MODEL.
Fig. 2. Block Diagram of Nonlinear Internal Model Control
Fig. 3. Schematic diagram of the Helicopter Model
et al., 1995) for designing the inverse of the nonlinear system model. The block diagram of the proposed method is depicted in Fig. 2. The inverse of the non-linear system model Eq.(3) ∼ Eq.(6) is designed according as applying a usual backstepping method to the non-linear internal model as shown in the next equations.
ua =
1 (l−1) (l) ) + ymr ] [αn(xm , θ, y¯mr θ0 β(xm ) (9) (10)
(i−1) xmi − ymr
− αi−1 zi = αi = −zi−1 − ci zi − wiT θ i−1 ∂αi−1 ∂αi−1 (k) ( xm,k+1 + (k−1) ymr ) + ∂xmk ∂ymr k=1 ∂αi−1 − ki | wi |2 zi + Γ τi ∂θ i−1 ∂αk−1 + (11) Γ wi zk ∂θ k=2
τi = τi−1 + wi zi i−1 ∂αi−1 ϕk wi = ϕi − ∂xmk
(12)
In this research, we design an attitude control system, which keeps pitch angle (ψ) of the helicopter model (depicted in Fig. 3) constant. The model is a two-degree-of-freedom (TDF) control system, which supports the helicopter controlling the pitch angle (ψ) and the yaw angle (φ) with the main propeller and the side propeller driven by the DC motor. However, in this research, the yaw angle is fixed and only the pitch angle is controlled. We define the vertical direction of the pitch angle ψ by 0. I is the moment of inertia of the helicopter body around horizontal axis, Tg is the gravitation torque, u is the input voltage to motor, B is the viscous-friction coefficient, a and b are parameters for elevation driving torque, and θ is the angle about difference between supporting point and gravity point height, T is time constant for characteristic of motor. Here, y = x1 is observation output, and state ˙ x3 = ud , x4 = u˙ d . variables are x1 = ψ, x2 = ψ, Then, the state space description of the dynamical equation of the helicopter model is described as follwous:
(13)
x˙ 1 x˙ 2 x˙ 3 x˙ 4
k=1
T
¯ mi = [xm1 , · · · , xmi] , x T (i) ¯ (i) y mr = ymr , y˙ mr , · · · , ymr
(14) (15)
i = 1, 2, · · · , l where ua is the control input in the proposed nonlinear IMC. Noting that the above control law is designed from the internal model instead of the real process model. Hence, the state variable xm can be utilized. On the other hand, it is not straightforward to calculate the high order derivative of ymr because the high order derivative of the feedback signal is also required. Using the approximated differential compensator can avoid the difficulty.
= x2 = p(f(x1 , x2) + g(x3 )) = x4 = −k1 x3 − k2 x4 + k1 u
Here, p, k1 , k2 , f(x1 , x2 ), g(x3 ) are denoted 1 p= I 1 k1 = 2 T 2 k2 = T f(x1 , x2 ) = −Bx2 − Tg sin(x1 + θ) g(x3 ) = ax33 + bx3
(16) (17) (18) (19) as: (20) (21) (22) (23) (24)
In this research, we design the input u tracking the reference signal yr on the attitude control model
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regarding pitch angle direction of the helicopter shown in Eq.(16) ∼ Eq.(19). We assume that the moment of inertia of the helicopter body around horizontal axis I is the only unknown parameter. Further, the reference signal yr is processed by a four-dimensional filter. Concretely, we assume that Tr is a time constant, and yr is shown as follows: 1 yr (s) = r(s) (sTr + 1)4 1 = 4 4 r(s) Tr s + 4Tr3 s3 + 6Tr2 s2 + 4Tr s + 1
result of the input trajectory, and Fig. 6 is the simulation result of the reference signal and the plant output. From these simulation results, the tracking property shows an excellent one, and the chracteristic of the input trajectory is a reasonal one. For the comparative study, Fig. 4 shows the simulation result of the input trajectory using a usual backstepping method. From Fig. 4, we can see that the input trajectory is oscillatory and initially swelled significantly. Therefore, it follows that the proposed one is especially effective for improving the characteristic of input signals.
Further, we used the following parameters shown in (Tanabe and Ichikawa, 2000) as parameters. I = 0.00308 B = 0.00358 Tg = 0.0782 θ = 0.21 a = 0.0650 b = 0.0722 T = 0.2 Tr = 0.5
input 0.74
0.735
u
0.73
0.725
0.72
0
5
10
15
20
25
30
35
40
45
50
t
Here, although I is an unknown parameter, a nominal value is assumed to be given. In the simulation experiment, Im = 0.0308 (which is ten 1 times of the real value), pm = are given as a Im nominal parameter value.
Fig. 4. Cntrol input trajectory using conventional adaptive backstepping method where two parameters are adjusted and adaptive gains are 1. input 0.74
0.735
0.73
0.725
When the proposed non-linear IMC is applied to the helicopter model, we set up the internal model whose dimension is lowered to two described in the following way.
0.72
0.715
0.71
0.705
where Im and pm are the nominal value for the parameters I and p respectively. The internal model Eq.(25) and Eq.(26) can be transformed into the representation shown in Eq.(3) ∼ Eq.(6) using the following equations. θ=p ϕm1 (xm1 )T = 0 ϕm2 (xm1 , xm2)T = f(xm1 , xm2) xm3 = g(ua )θ
(27) (28) (29) (30)
15
10
5
0
t
Fig. 5. Control input trajectory using proposed nonlinear internal model control output [y] and reference signal [yr] 1.7
1.6
1.5
r
(25) (26)
[y] and [y ]
x˙ m1 = xm2 x˙ m2 = pm (f(xm1 , xm2) + g(ua ))
1.4
1.3
1.2
1.1
1
0
5
15
10
t
Therefore, the design procedure given in section 2 can be applied to the attitude control of helicopter model.
Fig. 6. Control output using proposed nonlinear internal model control and the reference signal
4. SIMULATIONS
5. CONCLUSIONS
Simulation results using the proposed non-linear IMC control law represented in the section 2, are shown in Fig. 5 and Fig. 6. Fig. 5 is the simulation
In this research, we proposed nonlinear internal model control (IMC) using backstepping method and applied the method to the attitude control
720
model for a helicopter. Then, we confirmed the method was especially effective for improving the input characteristics. As mentioned above, it was shown that the proposed method applying backstepping method to IMC was effective. However, it has a disadvantage that the nominal value of parameters is necessary for the design of the controller in the proposed method. Hence, as future work, we are trying to design the control law by applying an adaptation mechanism.
REFERENCES Humu-soft (1992). The user manual, CE150 Helicopter Model. Humu-soft. Kanellakopoulos, I., P. V. Kokotovic and A. S. Morse (1991). Systematic design of adaptive controllers for feedback linearizable systems. IEEE Transactions on Automatic Control 36, 1241–1253. Krstic, M., I. Kannellakopoulos and P. Kokotovic (1995). Nonlinear and adaptive Control Design. Jhon, Wiley & Sons, Inc.. London. Morari, M. and E. Zafiriou (1989). Robust Process Control. Prentice-Hall. Tanabe, M. and A. Ichikawa (2000). An attitude control for a helicopter model using the strict linearization method (in japanese). In: Proc. of the 29th SICE sympoium on control theory. SICE. Watanabe, K. and S. Masuda (2002). An application of adaptive backstepping method for attitude control of a helicopter model (in japanese). In: Proc. of the industry applications society. the institute of electrical engineers of Japan. Japan.
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A NONLINEAR INTERNAL MODEL CONTROL USING BACKSTEPPING METHOD FOR ATTITUDE CONTROL OF A HELICOPTER MODEL Kazuya Yoshioka ∗ Shiro Masuda ∗ ∗
Tokyo Metropolitan Institute of Technology, 6-6, Asahigaoka, Hino, Tokyo, Japan
Abstract: The backstepping method is to design nonlinear control systems to follow a given reference signal by iterative recursive procedure. Although it is an effective approach to design stable closed loop systems for nonlinear controlled systems, it has drawbacks to become extremely complex when the relative order of the controlled system is large. In order to make the design procedure more tractable, we need to approximate the controlled system to a low order system. However, the approximated error between the model and the real process due to model reduction might bring significant performance degradation. Hence, this paper proposes a nonlinear internal model control (IMC) incoporated with the inverse of a low order system designed from the backstepping method. In the proposed method, the backstepping design procedure becomes simple because it can be done for a low order system. Furthermore, the peformance degradation due to model reduction can be compensated from an IMC structure. The proposed control method is applied to the attitude control for a helicopter model, and we compare the proposed method with the usual backstepping method through simulation experiments. From the comparative study, it follows that the proposed one is especially effective for improving the characteristic of input signals. Keywords: Backstepping method, Internal model control
1. INTRODUCTION The backstepping method(Kanellakopoulos et al., 1991; Krstic et al., 1995) is to design nonlinear control systems to follow a given reference signal by iterative recursive procedure. Although it is an effective approach to design stable closed loop systems for nonlinear controlled systems, it has drawbacks to become extremely complex when the relative order of the controlled system is large. In order to make the design procedure more tractable, we need to approximate the controlled system to a low order system. However, the approximated error between the model and the real
process due to model reduction might bring significant performance degradation. Hence, this paper proposes a nonlinear internal model control (IMC) incoporated with the inverse of a low order system designed from the backstepping method. The IMC is a popular control technique in the chemical process control field, and various research results have been reported so far (Morari and Zafiriou, 1989). However, these researches have mainly focused on the linear systems. Hence, the proposed method is cosidered to be a new device for the IMC design method for nonlinear systems.
717
In the proposed method, the backstepping design procedure becomes simple because it can be done for a low order system. Furthermore, the peformance degradation due to model reduction can be compensated from an IMC structure. The proposed control method is applied to the attitude control for a helicopter model(Humusoft, 1992; Tanabe and Ichikawa, 2000), and we compare the proposed method with the usual backstepping method through simulation experiments. In the case of applying a usual backstepping method, a desired tracking property can be achieved even if the moment of rotational inertia is unknown. However, it has drawbacks that the trajectory of the input signal is oscillatory and that the input signal initially swelled significantly(Watanabe and Masuda, 2002). This is because the controlled helicopter model is the fourth order system and we must repeat the recursive procedure for designing Lyapunov function four times; hence the design procedure of the control system complex. On the other hand, in the proposed method, the system model is approximated to the second order system. Hence, the backstepping procedure is repeated only two times. From the comparative study, it follows that the proposed one is especially effective for improving the characteristic of input signals. The paper is organaized as follows. Section 2 gives the design method for the proposed nonlinear IMC based on the backstepping method. Section 3 shows the simulation result of the application of the proposed nonlinear IMC to the attitude control of the helicopter model. Section 4 gives concluding remarks.
2. NON-LINEAR IMC BASED ON BACKSTEPPING METHOD
x˙ m(l−1)
x˙ ml = θ0 β(xm )u + ϕml (xm )T θ
The order l of the system model is less than n; that implies the system model Eq.(3) ∼ Eq.(6) are a low ordered system model reduced from the controlled system Eq.(1) and Eq.(2).
2.2 IMC control structure
Fig. 1. Basic IMC structure The basic IMC control structure is is depicted in Fig. 2.2. In the control structure, the difference between the real process and internal model outputs is a feedback signal for modifying a given reference signal. Since the same input signal is given to both the real process and internal model, the feedback signal becomes zero when there is neither disturbance nor mismatch between the process and its model, and it becomes active only when there exist disturbances and/or model-process mismatch. The modified reference input is equal to the reference input minus the feedback signal. Hence, the following equation is satisfied. yrm = yr − (x1 − xm1 ) ⇔ yrm − xm1 = yr − x1
Consider the following n-the order nonlinear system having a single input, single output system. (1) (2)
where x ∈ Rn , u, y ∈ R are state variables, and control input, and control output, repectively. f(·) : Rn × R → R is Lipschitz in x and u. h(·) : Rn → R is piecewise continuous function. While Eq.(1) and Eq.(2) are the controlled real system, we consider the following l-th order system model for an internal model. x˙ m1 = xm2 + ϕm1 (xm1)T θ x˙ m2 = xm3 + ϕm2 (xm1, xm2 )T θ
(3) (4)
(6)
where xm is the state variable vector of internal model whose element are xm1 , xm2, · · · , xml, and θ is a parameter vector. ϕm1 , ϕm2 , · · · , ϕml are smooth nonlinear function.
2.1 The controlled system and internal model
x˙ = f(x, u) y = h(x)
.. . = xml + ϕm(l−1) (xm1 , · · · , xml)T θ (5)
(7) (8)
Therefore, if the controller could be designed so that the model output xm1 is equal to the yrm , the process output x1 would also be equal to the reference signal.
2.3 Nonlinear Internal Model Control When the process model is a linear system, the design of the inverse of the process model is not so difficult. However, when it is a non-linear system, it requires a certain device for constructing the inverse of the system model. In this research, we will introduce the backstepping method (Kanellakopoulos et al., 1991; Krstic
718
3. APPLICATION TO A HELICOPTER MODEL.
Fig. 2. Block Diagram of Nonlinear Internal Model Control
Fig. 3. Schematic diagram of the Helicopter Model
et al., 1995) for designing the inverse of the nonlinear system model. The block diagram of the proposed method is depicted in Fig. 2. The inverse of the non-linear system model Eq.(3) ∼ Eq.(6) is designed according as applying a usual backstepping method to the non-linear internal model as shown in the next equations.
ua =
1 (l−1) (l) ) + ymr ] [αn(xm , θ, y¯mr θ0 β(xm ) (9) (10)
(i−1) xmi − ymr
− αi−1 zi = αi = −zi−1 − ci zi − wiT θ i−1 ∂αi−1 ∂αi−1 (k) ( xm,k+1 + (k−1) ymr ) + ∂xmk ∂ymr k=1 ∂αi−1 − ki | wi |2 zi + Γ τi ∂θ i−1 ∂αk−1 + (11) Γ wi zk ∂θ k=2
τi = τi−1 + wi zi i−1 ∂αi−1 ϕk wi = ϕi − ∂xmk
(12)
In this research, we design an attitude control system, which keeps pitch angle (ψ) of the helicopter model (depicted in Fig. 3) constant. The model is a two-degree-of-freedom (TDF) control system, which supports the helicopter controlling the pitch angle (ψ) and the yaw angle (φ) with the main propeller and the side propeller driven by the DC motor. However, in this research, the yaw angle is fixed and only the pitch angle is controlled. We define the vertical direction of the pitch angle ψ by 0. I is the moment of inertia of the helicopter body around horizontal axis, Tg is the gravitation torque, u is the input voltage to motor, B is the viscous-friction coefficient, a and b are parameters for elevation driving torque, and θ is the angle about difference between supporting point and gravity point height, T is time constant for characteristic of motor. Here, y = x1 is observation output, and state ˙ x3 = ud , x4 = u˙ d . variables are x1 = ψ, x2 = ψ, Then, the state space description of the dynamical equation of the helicopter model is described as follwous:
(13)
x˙ 1 x˙ 2 x˙ 3 x˙ 4
k=1
T
¯ mi = [xm1 , · · · , xmi] , x T (i) ¯ (i) y mr = ymr , y˙ mr , · · · , ymr
(14) (15)
i = 1, 2, · · · , l where ua is the control input in the proposed nonlinear IMC. Noting that the above control law is designed from the internal model instead of the real process model. Hence, the state variable xm can be utilized. On the other hand, it is not straightforward to calculate the high order derivative of ymr because the high order derivative of the feedback signal is also required. Using the approximated differential compensator can avoid the difficulty.
= x2 = p(f(x1 , x2) + g(x3 )) = x4 = −k1 x3 − k2 x4 + k1 u
Here, p, k1 , k2 , f(x1 , x2 ), g(x3 ) are denoted 1 p= I 1 k1 = 2 T 2 k2 = T f(x1 , x2 ) = −Bx2 − Tg sin(x1 + θ) g(x3 ) = ax33 + bx3
(16) (17) (18) (19) as: (20) (21) (22) (23) (24)
In this research, we design the input u tracking the reference signal yr on the attitude control model
719
regarding pitch angle direction of the helicopter shown in Eq.(16) ∼ Eq.(19). We assume that the moment of inertia of the helicopter body around horizontal axis I is the only unknown parameter. Further, the reference signal yr is processed by a four-dimensional filter. Concretely, we assume that Tr is a time constant, and yr is shown as follows: 1 yr (s) = r(s) (sTr + 1)4 1 = 4 4 r(s) Tr s + 4Tr3 s3 + 6Tr2 s2 + 4Tr s + 1
result of the input trajectory, and Fig. 6 is the simulation result of the reference signal and the plant output. From these simulation results, the tracking property shows an excellent one, and the chracteristic of the input trajectory is a reasonal one. For the comparative study, Fig. 4 shows the simulation result of the input trajectory using a usual backstepping method. From Fig. 4, we can see that the input trajectory is oscillatory and initially swelled significantly. Therefore, it follows that the proposed one is especially effective for improving the characteristic of input signals.
Further, we used the following parameters shown in (Tanabe and Ichikawa, 2000) as parameters. I = 0.00308 B = 0.00358 Tg = 0.0782 θ = 0.21 a = 0.0650 b = 0.0722 T = 0.2 Tr = 0.5
input 0.74
0.735
u
0.73
0.725
0.72
0
5
10
15
20
25
30
35
40
45
50
t
Here, although I is an unknown parameter, a nominal value is assumed to be given. In the simulation experiment, Im = 0.0308 (which is ten 1 times of the real value), pm = are given as a Im nominal parameter value.
Fig. 4. Cntrol input trajectory using conventional adaptive backstepping method where two parameters are adjusted and adaptive gains are 1. input 0.74
0.735
0.73
0.725
When the proposed non-linear IMC is applied to the helicopter model, we set up the internal model whose dimension is lowered to two described in the following way.
0.72
0.715
0.71
0.705
where Im and pm are the nominal value for the parameters I and p respectively. The internal model Eq.(25) and Eq.(26) can be transformed into the representation shown in Eq.(3) ∼ Eq.(6) using the following equations. θ=p ϕm1 (xm1 )T = 0 ϕm2 (xm1 , xm2)T = f(xm1 , xm2) xm3 = g(ua )θ
(27) (28) (29) (30)
15
10
5
0
t
Fig. 5. Control input trajectory using proposed nonlinear internal model control output [y] and reference signal [yr] 1.7
1.6
1.5
r
(25) (26)
[y] and [y ]
x˙ m1 = xm2 x˙ m2 = pm (f(xm1 , xm2) + g(ua ))
1.4
1.3
1.2
1.1
1
0
5
15
10
t
Therefore, the design procedure given in section 2 can be applied to the attitude control of helicopter model.
Fig. 6. Control output using proposed nonlinear internal model control and the reference signal
4. SIMULATIONS
5. CONCLUSIONS
Simulation results using the proposed non-linear IMC control law represented in the section 2, are shown in Fig. 5 and Fig. 6. Fig. 5 is the simulation
In this research, we proposed nonlinear internal model control (IMC) using backstepping method and applied the method to the attitude control
720
model for a helicopter. Then, we confirmed the method was especially effective for improving the input characteristics. As mentioned above, it was shown that the proposed method applying backstepping method to IMC was effective. However, it has a disadvantage that the nominal value of parameters is necessary for the design of the controller in the proposed method. Hence, as future work, we are trying to design the control law by applying an adaptation mechanism.
REFERENCES Humu-soft (1992). The user manual, CE150 Helicopter Model. Humu-soft. Kanellakopoulos, I., P. V. Kokotovic and A. S. Morse (1991). Systematic design of adaptive controllers for feedback linearizable systems. IEEE Transactions on Automatic Control 36, 1241–1253. Krstic, M., I. Kannellakopoulos and P. Kokotovic (1995). Nonlinear and adaptive Control Design. Jhon, Wiley & Sons, Inc.. London. Morari, M. and E. Zafiriou (1989). Robust Process Control. Prentice-Hall. Tanabe, M. and A. Ichikawa (2000). An attitude control for a helicopter model using the strict linearization method (in japanese). In: Proc. of the 29th SICE sympoium on control theory. SICE. Watanabe, K. and S. Masuda (2002). An application of adaptive backstepping method for attitude control of a helicopter model (in japanese). In: Proc. of the industry applications society. the institute of electrical engineers of Japan. Japan.
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