Experimental Study on a Twin Rotor MIMO System

3.3 Adaptive Multiple Model Fault Tolerant Control (AMMFTC) for Uncertain Multi-

3.3.5 Experimental Study on a Twin Rotor MIMO System

uncertainties affecting system performance. Therefore, improvement of transient performance in terms of L∞ bounds at start up and post fault scenarios in both the outputs would definitely reflect better mitigation of coupling effects. The tracking error comparison in Figure 3.16 demonstrate superior transient and steady state performance as claimed in theory over single model adaptive FTC; besides achieving substantial alleviation of coupling effects. The output tracking error performance in both the outputs, that is, displacements of massesm₁ andm₂ denoted byy₁ andy₂is indeed promising. On the contrary, as evident from Figures 3.16-3.15, the single model adaptive FTC suffers from a fairly unsatisfactory output tracking while its multiple model counterpart shows remarkable performance without any considerable increase in the control usage. Furthermore, the output tracking performance and the extent of control usage or rather the input energy spent under the proposed control and its single model equivalent, are quantified in terms of relevant and suitable performance metrics in Table 3.3 . Thereafter, as an added testament to our claims, a tabular comparison of such calculated input and output performances are drawn. In fact, as anticipated, the comparison reverberates the prolific features of the proposed adaptive FTC methodology.

To physically describe the TRMS system, is indeed necessary to lend clear understanding of the

Figure 3.17: Schematic of cross-coupled twin rotor MIMO system or a 2-DOF helicopter for control development

system and its desired objective. Referring to Figure 3.16(b), the TRMS has two propellers driven by dc motors, place perpendicular to each other. These propellers are attached to both ends of a beam pivoted at its base in a manner that the assembly of beam and the rotors, can rotate freely in both vertical and horizontal planes. A counterbalance arm with a weight at its end is fixed to the horizontal beam at its pivot to stabilize the TRMS. The main rotor produces a vertical thrust allowing the beam to rise vertically. This vertical movement is called pitch motion. While, the tail rotor generates a horizontal thrust and provides yaw stabilization by appropriately moving along the horizontal plane either towards left or right. The set-up is so constructed taht the angle of attack is fixed. It is easy to observe that TRMS comprises of two subsystems, namely, the pitch and yaw subsystems controlled by varying the speed of the main rotor and tail rotor, respectively. Further, the speed of the propellers are controlled by varying the armature voltage of the corresponding DC motors. In fact, DC motors herein act as actuators producing the desired speed equivalent to the control input voltages. In addition, the inherent dynamic input and output interactions between each of the subsystems in the TRMS are shown in Figure 3.17, where θ_v and θ_h represent the pitch angle and the yaw angle; andu_v and u_h are control input voltages to the main and tail rotors, respectively.

The pitch and the yaw angle are measured using position encoders fitted at the pivot. Since, it has been obvious from various simulations and experiments that pitch and yaw interactions play a crucial role in determining the performance of a controlled system. Inputs to any of the rotors influence both the position angles. Hence, apart from other design objectives including fault tolerance, alleviating these strong coupling effects is indeed imperative for any adaptive control synthesis for TRMS.

To complete the description, the TRMS set-up is interfaced with a Windows 7 PC (4 GB RAM), installed with Matlab/Simulink R2011a using PCL-812 card. Two position encoders are available to communicate the pitch and yaw angle measurements to the PC. The control inputs are generated using these measurements and the reference signals along with their derivatives, in the data acquisition and control block in the form of digital signals. These digital signals are thereafter converted to analog

signals through an embedded D/A converter and sent to the two d.c. motors attached to the rotors to achieve the desired attitude control.

Deviating our attention to mathematical modeling, the differential equations describing the dynamics of TRMS are given below [123].

I_vθ¨_v =M₁−M_{F G}−M_Bθv−M_G (3.219)

I_hθ¨_h=M₂−M_Bθh−M_R (3.220)

Where M₁ andM₂denote the nonlinear static characteristics,M_{F G} represents the gravity momentum, M_Bθv andM_Bθh are momenta corresponding to friction forces, MG is the gyroscopic momentum and M_R defines the cross reaction momentum. Further, their mathematical definitions are as follows.

M1=a1L²_m+b1Lm;M2=a2L²_t+b2Lt (3.221) M_{F G}=M_gsinθ_v;M_G=k_gyM₁θ˙_hcosθ_v (3.222) M_Bθv =B_1θvθ˙_v−0.0326

2 θ˙_h²sin 2θ_v;M_Bθh=B_1θhθ˙_h (3.223) M_R=

k_c(T₀s+ 1) (T_ps+ 1)

M₁ (3.224)

The terms Lm and Lt define the momentum of the main rotor and the tail rotor, respectively. These momentum functions are related to their corresponding control input(armature) voltages u_v(t) and u_h(t) as

L_m=

k_m T₁₁s+T₁₀

u_v; L_t=

k_t T₂₁s+T₂₀

u_h (3.225)

All other remaining parameters in equations (3.219)-(3.225) and their respective values are defined and listed in Table 3.4. Equations (3.219) and (3.220) are derived using Newton’s laws of rotational motion. To obtain a state space model of the TRMS dynamics, the state variables are chosen as x := [x₁, x₂, x₃, x₄, x₅, x₆]^T := [θ_v,θ˙_v, θ_h,θ˙_h, L_m, L_t]^T. Herein, x₁ defines the pitch angle, x₂ is the pitch angular velocity, x₃ denotes the yaw angle, the variablex₄ is the yaw angular velocity. Further, x5 andx6 represent the momentum of the main and tail rotor, respectively. Using the above relations from (3.221)-(3.225) in (3.219) and (3.220) leads to the formulation of the following dynamics describing the motion of a TRMS.

˙ x1=x2

˙ x₃=x₄

˙

x₂=θ₁^∗x₅²+θ₂^∗x₅−θ₃^∗sinx₁−θ₄^∗x₂+θ^∗₅x₄²sin(2x₂)

−θ^∗₆x₄x₅²cos(x₁)−θ₇^∗x₄x₅cos(x₁)

˙

x₄=θ₈^∗x₆²+θ₉^∗x₆−θ₁₀^∗ x₄−θ^∗₁₁x₅²−θ₁₂^∗ x₅

˙

x₅=−θ₁₃^∗ x₅+θ^∗₁₄u_v

˙

x₆=−θ₁₅^∗ x₆+θ^∗₁₆u_h y₁=x₁, y₂=x₃

(3.226)

Table 3.4: Physical parameters of the TRMS Symb. Description, Value & Unit

Iv Moment of inertia of vertical rotor 0.068 kg-m² Ih Moment of inertia of horizontal rotor 0.02 kg-m² a1 Static characteristic parameter 0.0135 - b1 Static characteristic parameter 0.0294 - a2 Static characteristic parameter 0.02 - b2 Static characteristic parameter 0.09 -

Mg Gravity momentum 0.32 N-m

B1αv Friction momentum function parameter 0.006 N-m-s/rad B1α_h Friction momentum function parameter 0.1 N-m-s/rad

kgy Gyroscopic momentum parameter .0155 s/rad

km Main rotor gain 1.1 -

kt Tail rotor gain 0.8 -

T11 Main rotor denominator parameter 1.1 - T10 Main rotor denominator parameter 1 - T21 Tail rotor denominator parameter 1 - T20 Tail rotor denominator parameter 1 - Tp Cross reaction momentum parameter 2 - T0 Cross reaction momentum parameter 3.5 -

kc Cross reaction momentum gain -0.2 -

The outputs of the TRMS are pitch and yaw angles and are denoted by y₁ and y₂. The unknown parameter vector is defined as θ^∗ := [θ₁^∗, θ^∗₂, . . . , θ₁₄^∗ , θ^∗₁₅, θ₁₆^∗ ]^T. The definition of all the unknown parameters θ_i for i= 1, . . . ,14 are as follows.

θ₁^∗=a₁/I_v; θ^∗₂ =b₁/I_v; θ₃^∗=M_g/I_v; θ^∗₄ =B_1αv/I_v; θ^∗₅ = 0.0326/2I_v; θ^∗₆ =k_gya₁/I_v; θ₇^∗=k_gyb₁/I_v; θ₈^∗=a₂/I_h; θ₉^∗=b₂/I_h; θ^∗₁₀=B_1αh/I_h; θ^∗₁₁=−1.75k_ca₁/I_h;

θ₁₂^∗ =−1.75k_cb₁/I_h; θ₁₃^∗ =T₁₀/T₁₁; θ₁₄^∗ =k_m/T₁₁; θ₁₅^∗ =T₂₀/T₂₁; θ₁₆^∗ =k_t/T₂₁

(3.227)

The parametric model for unknown parameter estimation is written as,

˙ x=













˙ x₁

˙ x₃

˙ x2

˙ x₄

˙ x₅

˙ x6













=











 x₂ x₄ 0 0 0 0











 +













0_1×7 0_1×5 0_1×2 0_1×2 0_1×7 0_1×5 0_1×2 0_1×2 ϕ^T_1,2 01×5 01×2 01×2

0_1×7 ϕ^T_2,2 0_1×2 0_1×2 0_1×7 0_1×5 ϕ^T_1,3 0_1×2 0_1×7 0_1×5 0_1×2 ϕ^T_2,3













| {z }

Φ(x,u)











 θ^∗₁ θ^∗₂ ... θ^∗₁₄ θ^∗₁₅ θ^∗₁₆













| {z }

θ^∗

. (3.228)

The regressors are defined as ϕ_1,2 := [x²₅, x₅,−sinx₁,−x₂,−x²₄sin 2x₂,−x₄x²₅cosx₁,−x₄x₅cosx₁]^T, ϕ_2,2 := [x²₆, x₆,−x₄,−x²₅,−x₅]^T,ϕ_1,3 := [−x₅, u_v]^T andϕ_2,3:= [−x₆, u_h]^T. The multiple identification models for parameter estimation are designed using the above estimation model (3.228) consistent with equations (3.162)-(3.163). The parameter estimates from each of the identification models are thereafter utilized for second layer of adaptation to yield the final estimate of the parameter vector θ^∗. Following the estimation of unknown parameters, the proposed adaptive controller is developed

following the backstepping methodology detailed in Section 3.3.2.1.

For experimental implementation of the proposed adaptive controller using multiple models, all the states of the system should be available. However, among the six states in the state vector x(t), only x₁ (pitch angle) and x₃ (yaw angle) are available for measurement. Therefore, an Extended Kalman Filter (EKF, Kalman-Bucy filter) [124] is used to estimate the remaining states of the nonlinear system. The detailed design of the EKF is provided in Appendix A.3. The initial conditions of the system are x(0) = [0,0,0,0,0,0]^T. Similarly the EKF states and the states of the identification models are also initialized at zero. Since the number of unknown parameters is 16, the number of identification models required to satisfy the conditions of proposed design N=17. The relative degree of the pitch and yaw subsystems are found to be ℘₁ = ℘₂ = 3 and hence the controller gains are selected asc₁ = [{c_1,i}³i=1]=[15,2.5,2.5]andc₂ = [{c_2,i}³i=1]=[10,2.5,2]. The damping gains are chosen as¯c_1,2 = ¯c_2,2= ¯g_1,3 = ¯g_2,3 = 0.5. The gain matrices of the adaptive identification models are considered asA_µ0 = [diag{−14,−14,−16,−16,−14,−14}]for µ= 1,17 andγ = 2. The adaptive rate parameter Γ_µ is chosen as Γ_µ = 20I₁₆. The adaptive weight parameters wˆ_µ for second layer of adaptation are initialized at w(0) =ˆ ₁₇¹ 117×1. The control input voltages u_h and uv are maintained within their allowable range of[−2.5,2.5]V. The reference trajectoriesy_r,1andy_r,2to be tracked by the outputsy₁= θvandy2=θ_hare the outputs of the reference systemGr(s) =

"

diag

( 0.2

9s²+ 10s+ 4, 0.2 9s²+ 10s+ 4

)#

with y_d,1 and y_d,2 as the inputs. In the following, the system is subjected two scenarios of step signal and sinusoidal signal tracking to illustrate the tracking capability of the proposed multiple model adaptive controller featuring transient performance improvement without significantly increasing the control energy.

• Scenario I: This scenario demonstrates sine command tracking performance of the proposed adaptive controller. The inputs to the reference system are chosen as y_d,1 = 6 + 4 sin(0.05πt) and y_d,2 = 12 cos(0.05πt). Figure 3.18(b)-(c) shows satisfactory output tracking capability for both pitch and yaw angles θ_v = x₁ and θ_h = x₃ with the proposed control scheme. From a closer glance at the tracking performance in Figure 3.18(a), it is observed that the pitch and yaw error signals exhibit no substantial transients and ultimately oscillates in the range [−0.02,0.04]and [−0.02,0.15]at steady state. The maximum percentage of steady state error in tracking with reference to the peak value of y_r,1 and y_r,2 corresponding to each of the outputs y₁ and y₂ is approximately found to be 6% and 14%, respectively. The control input voltages u_v and u_h are bounded within their limits. To further our insights on the prolific features of the proposed adaptive control methodology, a tabulation of output and input performances are provided in Table 3.5. A low value of root mean square error (RMSE) and integral of absolute error (IAE) essentially hints at a good steady state performance. Whereas, the integral of time absolute error (ITAE) metric is meant to measure the transient performance and is desired be of a small magnitude for fast tracking and stabilization. The inferences drawn from quantifications of the output tracking performance in Table 3.5 are fairly consistent with our claims of acceptable transient and steady state performance. The input performance is measured using control energy(CE) defined as the second norm of the control signal and total variation

Table 3.5: Tabulation of input and output performances quantified using suitable performance measures Reference

Signal

Output Performance Input Performance

RMSE IAE ITAE Mp Mu CE TV

Sine tracking θv 0.027 2.364 117.42 - - uv 298.93 217.72

θh 0.062 5.389 276.63 - - uh 164.11 809.38

Step tracking θv 0.011 1.023 20.56 19% 14% uv 288.96 59.40

θh 0.032 3.336 84.08 - - uh 86.15 25.95

(TV) [113] which indicates how frequent the control is used. A small value of TV for a control scheme is beneficial for practical applications as mentioned in Remark 3.5. The CE and TV of the control inputs u_v and u_h corresponding to the proposed controller indicate a fairly good input performance without excessive chattering. Nevertheless, on a critical assessment of the error performances, the reader may find the steady state error bounds on a slightly higher side.

However, it is to be mentioned that the performance of the TRMS under the action of the proposed controller is indeed appreciable given the fact that real time experiments on the TRMS are subjected to noise, air gust, voltage fluctuations and other unforeseen external disturbances.

• Scenario II: In this case, we study the performance of the proposed control methodology under step command tracking. Considering the same reference system G_r(s) driven by inputs y_d,1 = 4U(t) + 4U(t−25)−4U(t−75) and y_d,2 = 6U(t), its outputsy_r,1 and y_r,2 serve as the desired trajectories for system output tracking. Here, U(t) is defined as the unit step input. From the results in Figure 3.19, it is evident that the proposed control scheme is capable of rendering a good tracking performance for step reference inputs with minimum chattering in the control input voltages. Further, the control inputs reside within their permissible ranges while realizing the desired control objective. The pitch tracking error show some transient behaviour with an overshoot of 19% and an undershoot of 14% with reference to yr,1 = 0.2 while the yaw tracking error exhibits no peaks beyond y_r,2 = 0.3. The maximum steady state error is approximately calculated to be 5% forθ_v and9%forθ_h. Tabulation of calculated ITAE, IAE, and RMSE values under step tracking in Table 3.5 reveals a satisfactory and appreciable output performance.

Further, the TV and CE values in Table 3.5 are reasonably low, indicating a promising input performance. The effect of cross-coupling The yaw tracking performance can be however further improved using an integral action in the proposed controller. Taking into consideration several issues in real time implementation as mentioned in the preceding scenario, the performance of the proposed adaptive control system is certainly acceptable. The authors do believe that the obtained results are indeed appreciable and encouraging.

Remark 3.5. At this juncture, it should be remarked upon that the proposed adaptive control method- ology contributes in two directions. Firstly, it offers a control solution to the tracking problem in unknown nonlinear uncertain systems featuring an improved output transient performance. Secondly,

Time [s]

0 10 20 30 40 50 60 70 80 90 100

Output tracking error (rad)

-0.2 -0.1 0 0.1

0.2 x₁−y_r,1 x₃−y_r,2

(a)

0 10 20 30 40 50 60 70 80 90 100

0 0.2 0.4 0.6 0.8

Time [s]

Pitch angle (rad)

(b)

0 10 20 30 40 50 60 70 80 90 100

−0.5 0 0.5 1

Time [s]

Yaw angle (rad)

(c)

Time [s]

0 10 20 30 40 50 60 70 80 90 100

Control input voltages (V) ^-2 -1 0 1 2 3

Main rotor voltage u_v Tail rotor voltageu_h

(d)

Figure 3.18: (a) pitch and yaw angle tracking error (rad) ; (b) pitch angleθv=x1(rad); (c) yaw angleθh=x3(rad); (d) control input voltagesuvanduh(V).

Time [s]

0 10 20 30 40 50 60 70 80 90 100

Tracking error (rad)

-0.1 -0.05 0 0.05 0.1

x₁−y_r,1 x₃−y_r,2

(a)

Time [s]

0 10 20 30 40 50 60 70 80 90 100

Pitchangleθv=x1(rad) 0 0.1 0.2 0.3 0.4 0.5

y_r,1 x₁

(b)

Time [s]

0 10 20 30 40 50 60 70 80 90 100

Yawangleθh=x3(rad) 0 0.1 0.2 0.3 0.4 0.5 0.6

y_r,2 x₃

(c)

Time [s]

0 10 20 30 40 50 60 70 80 90 100

ControlInputs(V)

-2 0 2 4

Main rotor voltageu_v Tail rotor voltageu_h

(d)

the obtained solution can be further utilized to solve FTC problems using adaptive control and cater to the requirements of enhanced post failure transient performance without substantial increase in control usage. Adhering to the aim of FTC, the performance of the proposed AMMFTC scheme was demon- strated for control of MIMO nonlinear uncertain systems affected by abrupt actuator faults through extensive simulations in Section 3.3.4. Further, the proposed control methodology was experimentally implemented to control the attitude of a twin rotor MIMO system (TRMS). Given the fact, an abrupt decrease in thrust forces generated by the rotors would emulate the occurrence of actuator faults, it is certainly not possible to practically realize actuator faults in such an experimental set-up. It should be clearly understood that scaling the parameters θ^∗₁₄ and θ^∗₁₆ associated with the control voltages u_v and u_h do not model partial loss of effectiveness of actuators. Hence, the experiments were conducted without considering the occurrence of actuator faults. Nevertheless, the parameters of the TRMS were assumed to be unknown and estimated using the adaptation algorithm proposed in this chapter. The results obtained are promising. Any loss of actuation effectiveness will lead to change in the values of constant unknown parameters associated with L_m andL_t. Therefore, under this scenario, if a satisfac- tory transient and steady state performance is achieved at start up, it is reasonable to assume that the proposed adaptive controller would exhibit similar attributes at post fault scenarios.