In this paper, a nonlinear robust adaptive controller is proposed for gear transmission servo system (GTS) containing a sandwiched deadzone due to improper gear meshing. The controller is robust to dynamic uncertainties and can compensate the effect caused by the sandwiched nonlinearity which is separated from the control input through drive compliance. The proposed design methodology does not require an adaptive inverse deadzone function and does not require the knowledge of its parameter and only the knowledge of upper bounds is required.

1. Introduction

2. Problem Statement: Incorporating Drive-Train Compliance in the Design

In many applications, compliance is present in the drive system. This leads to another level of complication since the deadzone nonlinearity is present both on the motor and the load side and is not matched with the control input. The control design developed in  can be extended to the case when there is a compliance in the drive-train with the deadzone acting on both the actuator and load ends. The dynamics of a GTS system with sandwiched deadzone may be written as

J m d 2 θ m d t 2 + c m d θ m d t = τ − DZ ( θ r , δ , m ) (1)

J l d 2 θ l d t 2 + c l d θ l d t = N 0 ⋅ DZ ( θ r , δ , m ) , (2)

where J l , c l and θ l are the moment of inertia, viscous friction coefficient, and angle position on the load side respectively; similarly, J m , c m and θ m are the perspective variables on the driving side. In addition, m is the rigidity constant, and 2 δ is the width of the deadzone spacing. Moreover, N 0 defines the gear ratio, and τ is the driving torque. Meanwhile, θ r is the relative displacement parameter is given by

θ r = θ m − N 0 θ l , (3)

is utilized in the deadzone function DZ ( ⋅ ) , commonly written as

DZ ( θ r , δ , m ) = { m ( θ r − δ ) ,   if   θ r > δ 0 ,                           if   − δ < θ r < δ m ( θ r + δ ) ,   if   θ r < − δ (4)

In general, another useful representation of the deadzone function is written as

DZ ( θ ) = m [ θ r − Sat ( θ r ) ] , (5)

where Sat ( ⋅ ) represents the saturation function defined as

Sat ( θ r ) = { δ ,       if   θ r > δ θ r ,     if   − δ < θ r < δ − δ ,   if   θ r < − δ (6)

The function ψ ( θ r ) is defined to represent the normalized saturation function as follows

ψ ( θ r ) = { 1 ,                   if   θ r > δ θ r / δ ,   if   − δ < θ r < δ − 1 ,             if   θ r < − δ (7)

Hence, the deadzone nonlinearity function may be written more compactly as

DZ ( θ ) = m ( θ r − δ ψ ( θ r ) ) . (8)

An important and useful quality of the normalized saturation function ‖ ψ ( θ r ) ‖ that will be employed in the ensuing compensator design is stated in the following property:

Property 1. If at t = t 0 , the normalized saturation function ψ ( θ r ( t 0 ) ) ∈ [ − 1,1 ] then | ψ ( θ r ) | ≤ 1 ⇒ ∀ t ≥ 0 .

Basically, the property sets an upper bounds equal to δ on the saturation function Sat ( θ r ) because | ψ ( θ r ) | ≤ 1 . Meanwhile, writing the state space representation for the GTS system described in (1) and (2) combined with equations (3) and (8) yields

θ ˙ l = ω l

ω ˙ l = 1 J l { − c l ω l + m ( θ m − N 0 θ l ) − m δ ψ ( θ r ) }

θ ˙ m = ω m

ω ˙ m = 1 J m { − c m ω m + τ − m ( θ m − N 0 θ l ) + m δ ψ ( θ r ) } . (9)

It is noteworthy that the system dynamics given by (9) are not in any of the standard canonical forms for which the backstepping methodology is readily applicable. However, through the application of the saturation function property combined with the strengthened results of Lemma RANDM  and , the compensator design is carried out along similar lines as the systematic design procedure presented.

3. Applying Adaptive Backstepping Control Method

Based on the results presented in , and by looking at the load gear subsystem dynamics

θ ˙ l = ω l

ω ˙ l = − c l J l ω l + m J l ⋅ { ( θ m − N 0 θ l ) − δ ψ ( θ r ) } , (10)

can be rewritten as

x ˙ = [ θ ˙ l ω ˙ l ] = A x + B { θ m − δ ψ ( θ r ) } (11)

where

x = [ θ l ω l ] ,   A = [ 0 1 − m N 0 J l − c l J l ] ,   B = [ 0 m J l ] ,

moreover, the normalized saturation function ψ ( θ r ) is treated as a bounded unmeasurable disturbance by virtue of Property (1). The procedure begins with designing the control law by assuming that θ m is the virtual control for the load subsystem 33. Since θ m is not the actual control, we define the error between θ m and the virtual control developed, and design ω m appearing in the dynamics for this error such that the error goes to zero asymptotically. Thus, virtual controls are designed for θ m and ω m , and finally the actual control input τ is designed for the overall system. The details of the procedure are presented in the following steps:

Step 0: Note that the load side subsystem dynamics are not in the output canonical form (since the saturation function ψ ( θ r ) is not measurable) for which the technique of backstepping can be applied directly. Initially, the motor angle θ m is treated as the virtual control for the x = [ θ l ω l ] subsystem. In this step, the result presented in  which is basically a strengthend version of the previously proven Lemma RANDM in   is employed. The lemma is modified though for the special regulation case for the sake of clarity. Furthermore, based on the result of , the virtual control law for the subsystem 33 is written as

θ m * = − α B T P x − β ^ B T P x − ρ l Tanh [ ( a + b t ) B T P x ] (12)

where α , a , b > 0 , ρ l ≥ 2 δ ; while P is a positive definite, symmetric solution to the Algebraic Riccati Equation (ARE)

A T P + P A − 2 α ^ P B B T P + Q = 0 (13)

with Q a symmetric positive definite matrix, and β ^ is the adaptation parameter with dynamics given by

β ^ ˙ = Γ ‖ B T P x ‖ 2 (14)

with the constant Γ > 0 . The properties of the virtual control law are stated in the following theorem:

Theorem 4.1: The closed-loop subsystem (11) under the smooth, robust virtual controller (12) and the adaptation (14), is globally asymptotically stable. Furthermore, the position error asymptotically converges to zero.

Proof: The stability properties of the above virtual control law using the Lyapunov function

V 0 = x T P x + 1 2 Γ − 1 ( β ^ − β * ) 2 . (15)

Differentiating V 0 along the trajectories of the closed-loop system, we obtain

V ˙ 0 = x ˙ T P x + x T P x ˙ + ( β ^ − β * ) β ^ ˙ = [ A x + B ( θ m − δ ψ ( θ r ) ) ] T P x     + x T P [ A x + B ( θ m − δ ψ ( θ r ) ) ] + ( β ^ − β * ) ‖ B T P x ‖ 2 . (16)

Replacing θ m = θ ˜ m − θ m * and applying the virtual control law θ m * given in (12) and the adaptation law (14) yields

V ˙ 0 = [ A x + B ( − α B T P x − β ^ B T P x − ρ l Tanh [ ( a + b t ) B T P x ] − δ ψ ( θ r ) ) ] T P x     + 2 x T P B θ ˜ m + x T P [ A x + B ( − α B T P x − β ^ B T P x     − ρ l Tanh [ ( a + b t ) B T P x ] − δ ψ ( θ r ) ) ] + ( β ^ − β * ) ‖ B T P x ‖ 2 = x T A T P X − α x T P B B T P X + x T P A x + ( β ^ − β * ) ‖ B T P x ‖ 2 − β ^ x T P B B T P x     + 2 ( x T P B ) θ ˜ m − 2 ρ l ( x T P B ) Tanh [ ( a + b t ) B T P x ] − 2 δ x T P B ψ ( θ r ) . (17)

In the last equation, the term β ^ x T P B B T P x is equal to the term β ^ ‖ B T P x ‖ 2 and therefore the two terms are eliminated. Once again, by rearranging the terms in the last equation results in

V ˙ 0 = x T ( A T P + P A − α P B B T P ) x − β * ‖ B T P x ‖ 2 + 2 x T P B θ ˜ m             − 2 x T P B { ρ l Tanh [ ( a + b t ) B T P x ] − δ ψ ( θ r ) } . (18)

Collecting terms and utilizing the solution of the ARE (13) results in

V ˙ 0 = − x T Q x − β * ‖ B T P x ‖ 2 + 2 x T P B θ ˜ m       − 2 x T P B [ ρ l Tanh [ ( a + b t ) B T P x ] + δ ψ ( θ r ) ] . (19)

Applying this bound obtains

V ˙ 0 ≤ − λ min ( Q ) ‖ x ‖ 2 − β * ‖ x T P B ‖ 2 + 2 x T P B θ ˜ m       − 2 x T P B { ρ l Tanh [ ( a + b t ) B T P x ] + δ ψ ( θ r ) } . (20)

The first two terms are negative while the third term will be taken care in the next step of the design process. As for the last term in (20), to ensure that V ˙ 0 ≤ 0 we have to examine the condition needed to properly negate its overall outcome. To begin with the term can be bounded as

− 2 x T P B { ρ l Tanh [ ( a + b t ) B T P x ] + δ ψ ( θ r ) } = − ρ l x T P B { δ ρ l ψ ( θ r ) + Tanh [ ( a + b t ) B T P x ] } .

In order for the term to be negative the following inequality, along with the fact that the upper bounds on the normalized saturation function (7) as ‖ ψ ( θ r ) ‖ ≤ 1 , and the Tan hyperbolic function has an upper bound of 1, yields the condition

δ ρ l ‖ ψ ( θ r ) ‖ < ‖ Tanh [ ( a + b t ) B T P x ] ‖

δ ρ l < 1.

Hence, by properly selecting the gain parameter ρ l to be greater than δ ensures V ˙ 0 ≤ 0 making the subsystem asymptotically stable; thereby, concluding the proof of Theorem 4.1.

Step 1: Since θ m is not the actual control, an error θ ˜ m = θ m − θ m * is established and differentiated to obtain the dynamics for θ ˜ m which can be written as:

θ ˜ ˙ m = ω m − ∂ θ m * ∂ x [ A x + B { θ m − m δ ψ ( θ r ) } ] − ∂ θ m * ∂ t − ∂ θ m * ∂ β ^ β ^ ˙ (21)

≜ ω m − χ 1 ( x , β ^ ) − γ 1 ( x , β ^ ) m δ ψ ( θ r ) . (22)

To derive the stabilizing virtual controller for the extended subsystem that inculdes [ x θ ˜ m ] a newly formed Lyapunov function is proposed. A virtual contoller will be porposed to ensure the over all stability properties of the subsystem. Consider

V 1 = V 0 + 1 2 θ ˜ m 2 = x T P x + 1 2 Γ − 1 ( β ^ − β * ) 2 + 1 2 θ ˜ m 2 .

Differentiating V 1 along the trajectories of the [ x θ ˜ m ] subsystem, one obtains

V ˙ 1 = V ˙ 0 + θ ˜ m ⋅ θ ˜ ˙ m = V ˙ 0 + θ ˜ m ( θ ˙ m − θ ˙ m * ) + 2 x T P B θ ˜ m (23)

= V ˙ 0 + θ ˜ m ⋅ { ω m − ω m * + 2 x T P B } , (24)

where

ω m * = θ ˙ m * = ∂ θ m * ∂ x x ˙ + ∂ θ m * ∂ β ^ β ^ ˙ + ∂ θ m * ∂ t ≜ χ 1 ( x , β ^ ) + γ 1 ( x , β ^ ) Sat ( θ m − N 0 θ l ) . (25)

Refer to the appendix for complete derivations of the functions χ 1 ( x , β ^ ) and γ 1 ( x , β ^ ) . Subsequently, ω m is now viewed as the virtual control for the [ x θ ˜ m ] subsystem. The virtual control for ω m * is given by

ω m * = − σ 0 θ ˜ m − 2 x T P B + χ 1 ( x , β ^ ) − ρ l γ 1 ( x , β ^ ) Tanh [ ( a + b t ) γ 1 ( x , β ^ ) θ ˜ m ] (26)

where σ 0 > 0 .

V ˙ 1 ≤ V ˙ 0 − σ 0 θ ˜ m 2 − θ ˜ m [ ρ l γ 1 ( x , β ^ ) Tanh [ ( a + b t ) γ 1 ( x , β ^ ) θ ˜ m ] + γ 1 ( x , β ^ ) δ ψ ( θ r ) ] . (27)

the last term can be handled in a similar manner as in the conclusion of the proof of Theorem 4.1. By rewriting the last term as

− θ ˜ m ρ l γ 1 ( x , β ^ ) [ Tanh [ ( a + b t ) γ 1 ( x , β ^ ) θ ˜ m ] + δ ρ l ψ ( θ r ) ] ,

− ‖ θ ˜ m ρ l γ 1 ( x , β ^ ) ‖ [ Tanh [ ( a + b t ) ‖ γ 1 ( x , β ^ ) θ ˜ m ‖ ] + δ ρ l ‖ ψ ( θ r ) ‖ ] ,

shows that by ensuring that the gain parameter ρ l to be greater than δ ensures V ˙ 1 ≤ 0 . Employing the same arguments as in the proof of Theorem 4.1, it follows that ‖ x T P B ‖ and θ ˜ m converge asymptotically to zero. However, since ω m given in Equation (26) is not the actual control, the error is defined as

ω ˜ m = ω m − ω m * ,

and the dynamics of ω ˜ m can be obtained through differentiation as

ω ˙ m * = ∂ ω m * ∂ x x ˙ + ∂ ω m * ∂ β ^ β ^ ˙ + ∂ ω m * ∂ t ≜ χ 2 ( x , θ ˜ m , β ^ ) + γ 2 ( x , θ ˜ m , β ^ ) Sat ( θ m − θ l ) (28)

ω ˜ ˙ m = 1 J m { τ − c m ω m − ( θ m − N 0 θ l ) + δ ψ ( θ r ) }               − χ 2 ( x , θ ˜ m , β ^ ) − γ 2 ( x , θ ˜ m , β ^ ) δ ψ ( θ r ) . (29)

where χ 2 and γ 2 are completely derived and presented in the appendix. The actual control τ now appears in ω ˜ m dynamics which is designed in the next step.

Step 2: The Lyapunov function used for analysis of the overall closed-loop system is given by

V 2 = V 1 + 1 2 ω ˜ m 2 = x T P x + 1 2 Γ − 1 ( β ^ − β * ) 2 + 1 2 θ ˜ m 2 + 1 2 ω ˜ m 2 .

The following robust control law is developed for the overall system:

τ = − σ 2 ω ˜ m − ( θ m − N 0 θ l ) − ρ m Tanh [ ( a + b t ) ω ˜ m ] + χ 2 ( x , θ ˜ m , β ^ )       + ρ l γ 2 ( x , θ ˜ m , β ^ ) Tanh [ ( a + b t ) γ 2 ( x , θ ˜ m , β ^ ) ω ˜ m ] , (30)

where σ 2 > 0 and the ρ m > 2 δ is incorporated to compensate for the deadzone effect on the actuator side. Differentiating V 2 along the trajectories of the overall system yields

V ˙ 2 ≤ − λ min ( Q ) ‖ x ‖ 2 − β * ( x T P B ) 2 − σ θ ˜ m 2 − σ 2 ω ˜ m 2       − ω ˜ m { ρ m Tanh [ ( a + b t ) ω ˜ m ] − δ ψ ( θ r ) }       − ω ˜ m { ρ l γ 2 ( x , θ ˜ m , β ^ ) Tanh [ ( a + b t ) γ 2 ( x , θ ˜ m , β ^ ) ω ˜ m ]       − γ 2 ( x , θ ˜ m , β ^ ) d ψ ( θ r ) } .

Citing the same arguments used in the proof of Theorem 4.1, both gain parameters ρ l and ρ m must satisfy the following two conditions:

ρ l > 2 δ ρ m > 2 δ

to ensure all terms in V 2 < 0 . Accordingly, the terms ‖ x T P B ‖ , θ ˜ m , and ω ˜ m converge asymptotically to zero. In conclusion, the result of the proof is summarized in the following theorem.

Theorem 4.2. Asymptotic convergence to zero for the load position errors for the system (9) under the robust smooth controller (30) with the adaptation update law (14) is achieved and only upper bounds on the deadzone spacing is required.

4. Simulation Studies

Simulations were performed to demonstrate the efficacy of the designed controller (30). The parameters for a typical GTS system, described by the dynamic system Equations (9), are presented in Table 1.

In addition, Listed in Table 2 are the constant parameters for the designed controller. These parameters provide a good basis for simulation studies to elucidate the efficacy of our control design methodology. Although, the parameters reflect a standard gear transmission system it is not linked to any real experimental test bed.

Based on the parameters listed in Table 1 and Table 2, the load gear subsystem dynamics

θ ˙ l = ω l

ω ˙ l = − 0.24 ω l + 0.40 { ( θ m − 5.0 θ l ) − δ ψ ( θ r ) } (31)

System parameters
Parameters of the GTS
Inertia of the MotorJ m0.01 kg·m2
Inertia of the LoadJ l0.5 kg·m2
Gear RatioN 05; none
Controller design parameters used in the simulations
ParametersValue
α1.0
σ 01.0
σ 21.0
ρ l2.0
ρ m2.0
( a , b )(10, 100)
Γ500

θ ˙ m = ω m

ω ˙ m = − 12.0 ω m + τ − 0.40 { ( θ m − 5.0 θ l ) + δ ψ ( θ r ) } , (32)

which can compactly be rewritten as

x ˙ = [ θ ˙ l ω ˙ l ] = A x + B { θ m − 0.02 ψ ( θ r ) } (33)

where

x = [ θ l ω l ] ,   A = [ 0 1 − 2 − 0.24 ] ,   B = [ 0 0.40 ] .

Moreover, the normalized saturation function ψ ( θ r ) is treated as a bounded unmeasurable disturbance by virtue of property 3. It is worth recalling that, for our control design, only an upper bound on the deadzone spacing δ is required. The simulations were carried out with following robust control law:

τ = − σ 2 ω ˜ m + ( θ m − N 0 θ l ) − ρ m Tanh [ ( a + b t ) ω ˜ m ] + χ 2 ( x , θ ˜ m , β ^ )           + ρ l γ 2 ( x , θ ˜ m , β ^ ) Tanh [ ( a + b t ) γ 2 ( x , θ ˜ m , β ^ ) ω ˜ m ] . (34)

5. Conclusion

A robust adaptive backstepping compensator was employed for a dynamic system consisting of GTS drives which contain a compliant transmission with a deadzone nonlinearity. The new robust controller was designed to control the system with discontinuous deadzone in a sandwiched setting. The proposed controller did not require the deadzone nonlinearity’s parameters. Only an upper estimate of the deadzone spacing is required. The advocated control design methodology has been proven to be robust to variations in the deadzone and inertia parameters and does not require exact knowledge of any of the deadzone parameters. Without the aforementioned model based cancellation, we are still achieving a good performance enhancement. Furthermore, the robustness of the controller to variations in the system parameters is being validated through extensive simulation and the results will be presented as they become finalized. For future work, extending the challenge to the case where the state dynamics are considered to be unmeasurable requiring a dynamic state observer to be designed.

Acknowledgements

The authors wish to acknowledge the support of the Public Authority for Applied Education and Training; (Project No. TS-16-009). Furthermore, the authors appreciate Dr. Lavanya Murugan’s assistance in the simulations and analytical proofing of the theory development.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Alkhateeb, N.J., Ebraheem, H.K., Sultan, E.K. and Alrahsed, B.M. (2021) Adaptive Backstepping Compensation of Drives with Sandwiched Deadzone Nonlinearity. Intelligent Control and Automation, 12, 73-89. https://doi.org/10.4236/ica.2021.123005

Appendix

Step 0:

θ m * = − α B T P x − β ^ B T P x − ρ l Tanh [ ( a + b t ) B T P x ] (A.35)

θ ˜ ˙ m = θ ˙ m − θ ˙ m * = ω m − χ 1 ( x , β ^ ) − γ 1 B θ m − δ ψ ( θ r ) (A.36)

θ ˙ m * = ∂ θ m * ∂ x x ˙ + ∂ θ m * ∂ β ^ β ^ ˙ + ∂ θ m * ∂ t (A.37)

∂ θ m * ∂ x = − α B T P − β ^ B T P − ρ l Sech 2 [ ( a + b t ) B T P x ] ⋅ ( a + b t ) B T P (A.38)

∂ θ m * ∂ β ^ = − B T P x (A.39)

∂ θ m * ∂ t = − ρ l Sech 2 [ ( a + b t ) B T P x ] ⋅ b B T P x (A.40)

χ 1 = { − α − β ^ − ρ ( a + b t ) ⋅ Sech 2 [ ( a + b t ) B T P x ] } B T P { A x + B θ m }               − Γ ( B T P x ) 3 − ρ b B T P x ⋅ Sech 2 [ ( a + b t ) B T P x ] (A.41)

γ 1 = α B T P B + β ^ B T P B + ρ ( a + b t ) B T P B ⋅ Sech 2 [ ( a + b t ) B T P x ] (A.42)

Step 1:

ω ˜ ˙ m = ω ˙ m − ω ˙ m * = ω ˙ m − χ 2 ( x , θ ˜ m , β ) − γ 2 ( x , θ ˜ m , β ^ ) Sat ( θ m − N 0 θ l ) (A.43)

ω m * = − σ 0 θ ˜ − 2 B T P x − χ 1 ( x , β ^ ) − ρ l γ 1 ( x , β ^ ) Tanh [ ( a + b t ) γ 1 θ ˜ ] (A.44)

ω ˙ m * = ∂ ω m * ∂ x x ˙ + ∂ ω m * ∂ β ^ β ^ ˙ + ∂ ω m * ∂ t (A.45)

ω ˜ ˙ m = 1 J m { τ − c m ω m − ( θ m − N 0 θ l ) + Sat ( θ m − N 0 θ l ) } − χ 2 ( x , θ ˜ m , β ^ )               − γ 2 ( x , θ ˜ m , β ^ ) Sat ( θ m − N 0 θ l ) (A.46)

γ ˙ 1 = τ 1 + τ 2 ⋅ Sat ( θ m − N 0 θ l ) = ∂ γ 1 ∂ x x ˙ + ∂ γ 1 ∂ β ^ β ^ ˙ + ∂ γ 1 ∂ t (A.47)

∂ γ 1 ∂ x = 2 ρ ( a + b t ) 2 B T P B Tanh ( ( a + b t ) B T P x ) Sech 2 [ ( a + b t ) B T P x ] B T P (A.48)

∂ γ 1 ∂ β ^ = B T P B β ^ ˙ = Γ B T P B ‖ B T P x ‖ 2 (A.49)

∂ γ 1 ∂ t = ρ b B T P B ⋅ Sech 2 [ ( a + b t ) B T P x ] B + ρ ( a + b t ) B T P B     ⋅ Sech 2 [ ( a + b t ) B T P x ] B Tanh [ ( a + b t ) B T P x ] ⋅ b B T P x (A.50)

τ 1 = 2 ρ ( a + b t ) B T P Tanh ( ( a + b t ) B T P x ) Sech 2 ( ( a + b t ) B T P x )             × ( a + b t ) B T P ( A x + B θ m ) + ∂ γ 1 ∂ β ^ + ∂ γ 1 ∂ t (A.51)

τ 2 = 2 ρ ( a + b t ) B T P Tanh ( ( a + b t ) B T P x ) Sech ( ( a + b t ) B T P x ) ( a + b t ) B T P B (A.52)

χ ˙ 1 = ∂ χ 1 ∂ x x ˙ + ∂ χ 1 ∂ β ^ β ^ ˙ + ∂ χ 1 ∂ t + ∂ χ 1 ∂ θ m θ ˙ m (A.53)

∂ χ 1 ∂ x = { − 2 ρ l [ ( a + b t ) B T P ] 2 { A x + B θ m } − 2 ρ l b ( a + b t ) B T P A − ρ l b B T P }   × Sech 2 ( ( a + b t ) B T P x ) Tanh [ ( a + b t ) B T P x ]   − { ρ l ( a + b t ) } B T P A Sech 2 ( ( a + b t ) B T P x )   − { α + β ^ } B T P A − 3 Γ ( B T P x ) 2 B T P (A.54)

∂ χ 1 ∂ β ^ β ^ ˙ = − Γ B T P { A x + B θ m } ‖ B T P x ‖ 2 (A.55)

∂ χ 1 ∂ t = − ρ l b B T P { A x + B θ m } Sech 2 ( ( a + b t ) B T P x )     − 2 ρ l { ( a + b t ) B T P ( A x + B θ m ) + b 2 B T P x } Sech 2 ( ( a + b t ) B T P x )     × Tanh [ ( a + b t ) B T P x ] (A.56)

∂ χ 1 ∂ θ m θ ˙ m = B T P B ω m { − α − β ^ − ρ l ( a + b t ) Sech 2 ( ( a + b t ) B T P x ) } (A.57)

χ ˙ 1 = τ 3 + τ 4 ⋅ Sat ( θ m − N 0 θ l ) (A.58)

τ 3 = ∂ χ 1 ∂ x { A x + B m ( θ m − N 0 θ l ) } + ∂ χ 1 ∂ β ^ + ∂ χ 1 ∂ t + ∂ χ 1 ∂ θ m θ ˙ m (A.59)

τ 4 = − ∂ χ 1 ∂ x ⋅ B (A.60)

τ 4 = − 2 ρ l ( a + b t ) B T P B { ( a + b t ) B T P ( A x + B θ m ) + b B T P x }   + ρ l B T P B { ( a + b t ) A + b } Sech 2 [ ( a + b t ) B T P x ]   + ( α + β ^ ) B T P A B + 3 Γ ( B T P x ) 2 B T P B (A.61)

Step 2:

τ = − θ ˜ m − σ m ω ˜ m − 1 J d ( N 0 θ l − θ m ) − χ 2 + ρ m γ 2 tanh [ ( a + b t ) γ 2 ω ˜ m ]       − ρ l tanh [ ( a + b t ) ω ˜ m ] (A.62)

χ 2 = − σ 0 ω m + σ 0 χ 1 − 2 B T P { A x + B m ( θ m − N 0 θ l ) }       − ρ m + τ 3 − ρ l τ 1 Tanh [ ( a + b t ) γ 1 θ ˜ m ]       − ρ l γ l Sech 2 [ ( a + b t ) γ 1 θ ˜ m ] [ b γ 1 θ ˜ m + ( a + b t ) τ 1 θ ˜ m + ( a + b t ) γ 1 ( ω − χ 1 ) ] (A.63)

γ 2 = σ γ 1 + 2 B T P B + τ 4 − ρ l τ 2 Tanh [ ( a + b t ) θ ˜ ]       − ρ l γ 1 Sech 2 [ ( a + b t ) γ 1 θ m ] { ( a + b t ) τ 2 − ( a + b t ) γ 1 2 } (A.64)