Adaptive Sliding Mode Controller for Multiple Input Multiple Output
4.3 The twin rotor MIMO System
4.3.11 Control parameters for the vertical subsystem
For the vertical subsystem, the PI sliding surface parameters cv = [c1v c2v] (4.67) are chosen as c1v = [11 8], c2v = 12 andc3v = [7 0.5] to satisfy the condition mentioned in Section 4.3.5. The values of the reaching mode coefficientk1v and the adaptive tuning parameterγv are selected as 1.3 and 1.25
respectively (4.71, 4.75). The value of κv is chosen as 2.0. The adaptive tuning law for the TRMS-VS is formed as T˙ˆv = 0.8|σv| and ˆTv(0) = 0.5. The positive definite matrices Pv and Rv satisfying the stability condition as explained in Section 4.3.8 are found as
Pv =
105.10 30.49 59.89 30.49 288.55 28.08 59.89 28.08 199.89
, Rv =
15.50 1.50 8.30 1.50 35.60 1.30 8.30 1.30 30.10
(4.88)
The TRMS controlled by using our proposed adaptive sliding mode controller is studied for the position as well as the tracking control problem. In the position control example, the initial condition of the TRMS is considered asx(0) = [0 −0.5 0 0 0 0]T. In order to study the performance of the proposed controller for both the cases of position and tracking control, the simulation is performed by applying three different reference signals: 1) Step input with 1 rad in the horizontal subsystem and step input with 0.2 rad in the vertical subsystem; 2) Sine wave having amplitude 0.5 rad and frequency 0.025 Hz for the horizontal subsystem and sine wave having amplitude 0.2 rad and frequency 0.025 Hz for the vertical subsystem; 3) Square wave having amplitude 0.5 rad and frequency 0.025 Hz for the horizontal subsystem and square wave having amplitude 0.2 rad and frequency 0.025 Hz for the vertical subsystem.
0 5 10 15 20 25 30 35 40 45 50
−0.5 0 0.5 1 1.5
Time(sec) Yaw angle (rad) control signal (volt)
system output reference control signal
0 5 10 15 20 25 30 35 40 45 50
−0.5 0 0.5 1
Time(sec) Pitch angle (rad) control signal (volt)
system output reference control signal
Figure 4.2: Step response of the TRMS using the proposed adaptive SM controller
4.3 The twin rotor MIMO System
0 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 2
Time(sec) AdaptivegainˆTh
adaptive gain for yaw angle
0 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10
Time(sec) AdaptivegainˆTv
adaptive gain for pitch angle
Figure 4.3: Adaptive gain parameter of proposed adaptive SM controller
The desired and the actual step responses of the TRMS along with the control inputs for both the horizontal and the vertical subsystems are plotted in Fig.4.2. The adaptive gain parameters ˆTh and Tˆv for the step response analysis are shown in Fig.4.3.
Table 4.2: Transient performance of the TRMS for step input Time response parameters
Reference Rise time (sec) Settling time (sec) Peak time (sec) Peak overshoot %
Step H 3.45 4.60 5.50 0
V 2.45 2.50 2.95 2
Table 4.2 summarises the transient performance of the TRMS using the proposed adaptive sliding mode controller for the step input case. It can be observed from Table 4.2 that for both the hori- zontal and the vertical subsystems, the TRMS settles quickly to the desired position without much oscillation. In the tracking control example with sine wave, the TRMS settling time is less than 10 sec which was reported in [84]. The actual and the desired trajectory tracking (square and sine) responses as well as the control inputs for both the horizontal and the vertical subsystems are plotted in Figs.
4.4-4.5.
0 5 10 15 20 25 30 35 40 45 50
−1
−0.5 0 0.5 1
Time(sec) Yaw angle (rad) control signal (volt)
system output reference control signal
0 5 10 15 20 25 30 35 40 45 50
−0.5 0 0.5 1
Time(sec) Pitch angle (rad) control signal (volt)
system output reference control signal
Figure 4.4: Square wave response of the TRMS using the proposed adaptive sliding mode controller
0 5 10 15 20 25 30 35 40 45 50
−1
−0.5 0 0.5 1
Time(sec) Yaw angle (rad) control signal (volt)
system output reference control signal
0 5 10 15 20 25 30 35 40 45 50
−1
−0.5 0 0.5 1 1.5
Time(sec) Pitch angle (rad) control signal (volt)
system output reference control signal
Figure 4.5: Sine wave response of the TRMS using the proposed adaptive sliding mode controller
4.3 The twin rotor MIMO System
0 5 10 15 20 25 30 35 40 45 50
−0.5 0 0.5 1 1.5
Time(sec) Yaw angle (rad) control signal (volt)
system output reference control signal
0 5 10 15 20 25 30 35 40 45 50
−1
−0.5 0 0.5 1
Time(sec) Pitch angle (rad) control signal (volt)
system output reference control signal
Figure 4.6: Position tracking using the proposed controller subjected to an external disturbance
In order to study the robustness of the adaptive sliding mode controller, an external disturbance d(t) as given below is applied to the TRMS.
d(t) =
0.2, 20sec≤t≤25sec 0, otherwise
(4.89)
Fig. 4.6 illustrates that the TRMS controlled by using the adaptive sliding mode controller stays at the desired position even in the presence of disturbance and thereby proving its robustness.
Juang et al. in [84] illustrated that the proportional integral differential (PID) control with improved RGA (modified real-value-type genetic algorithm (M-RGA)) offered superior control performance than the conventional PID control and conventional realtime GA PID (C-RGA) control. In [84] the perfor- mances of the PID, C-RGA and M-RGA controllers are compared by computing the error and control indices which are defined as the sum of their absolute values [8]. In order to study the relative perfor- mance of our proposed adaptive sliding mode controller against these afore-mentioned control schemes, the same performance criteria are applied. In our comparison analysis, the error and control indices are calculated from 0 to 50 sec with a sampling period of 0.05 sec and are tabulated in Tables 4.3 and 4.4. The error index is defined as the sum of the absolute values (i.e., error index of HS TRMS=
∑n
k=1|x1(k)−xd1(k)|and error index of VS TRMS= ∑n
k=1|x2(k)−xd2(k)| wheren is the number of sampling data andx1(k),xd1(k) andx2(k),xd2(k) being the actual and desired states of the HS TRMS and VS TRMS respectively). The control index is defined as the sum of the absolute values of control actions (i.e., control index of HS TRMS =∑n
k=1|uh(k)|and control index of VS TRMS=∑n
k=1|uv(k)| where uh anduv are the control inputs for the HS TRMS and VS TRMS respectively.
Table 4.3: Comparison of Error Index among different controllers Error index
Reference PID [84] C-RGA [84] M-RGA [84] Adaptive sliding mode
Step H 81.2 69.09 54.52 45.23
V 40.11 34.92 27.46 23.08
Sine H 23.21 19.33 20.92 32.33
V 65.74 51.78 52.61 42.20
Square H 150.22 141.52 134.03 83.70
V 112.85 96.36 90.21 45.80
Table 4.4: Comparison of Control Index among different controllers Control Index
Reference PID [84] C-RGA [84] M-RGA [84] Adaptive sliding mode
Step H 76.71 51.34 40.47 12.45
V 812.36 701.23 617.10 645.98
Sine H 27.41 20.12 18.93 10.68
V 611.70 500.2 501.78 515.42
Square H 202 171.28 165.32 42.34
V 656.37 591.65 551.59 487.29
It is evident from these two tables that the proposed adaptive sliding mode controller exhibits lesser error and requires lesser control action in majority of the cases as compared to the other control methods [84].