4. Simulation and results

Consider the case that only the trolley motion is activated, we numerically simulate the distributed system dynamics (20)–(25) driven by either conventional Lyapunov-based input or barrier Lyapunov-based law. The finite difference method is applied for programing the control system in MATLAB environment. The system parameters used in simulation are composed of

The simulation results are depicted in Figures 3–6. Trolley and payload approach to destination qd = 2 m precisely and speedy without maximum overshoots. The payload swing stays in a small region during the transient state and absolutely suppressed at steady state (or payload destination). However, the longer length of cable is, the lager the payload swings. The system responses show the robustness in the face of parametric uncertainty. Despite the large variation of cable length, the system responses still kept consistency as shown in Figures 3–5. It can be seen from Figure 6 that with the application of the barrier Lyapunov function, payload fluctuation is controlled in an area defined by kb. Because the motion of the trolley in X and Y directions is forced to travel the same distance to reach the desired location, system responses

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in X and Y directions are similar.

Figure 4. System responses in the case of L = 6 m and m = 6 kg.

$$m = 5\,\text{kg};\ \ M = 1\,\text{kg};\ \ L = 3, 6, 9\,\text{m};\ \ K\_d = 200;\ \ K\_p = 5;\ \ K\_d = 42;$$

Figure 3. System responses in the case of L = 3 m and m = 3 kg.

Figure 4. System responses in the case of L = 6 m and m = 6 kg.

4. Simulation and results

328 Adaptive Robust Control Systems

Figure 3. System responses in the case of L = 3 m and m = 3 kg.

composed of

Consider the case that only the trolley motion is activated, we numerically simulate the distributed system dynamics (20)–(25) driven by either conventional Lyapunov-based input or barrier Lyapunov-based law. The finite difference method is applied for programing the control system in MATLAB environment. The system parameters used in simulation are

m ¼ 5 kg; M ¼ 1 kg; L ¼ 3, 6, 9m; Ka ¼ 200; Kp ¼ 5; Kd ¼ 42;

The simulation results are depicted in Figures 3–6. Trolley and payload approach to destination qd = 2 m precisely and speedy without maximum overshoots. The payload swing stays in a small region during the transient state and absolutely suppressed at steady state (or payload destination). However, the longer length of cable is, the lager the payload swings. The system responses show the robustness in the face of parametric uncertainty. Despite the large variation of cable length, the system responses still kept consistency as shown in Figures 3–5. It can be seen from Figure 6 that with the application of the barrier Lyapunov function, payload fluctuation is controlled in an area defined by kb. Because the motion of the trolley in X and Y directions is forced to travel the same distance to reach the desired location, system responses in X and Y directions are similar.

Figure 5. System responses in the case of L = 9 m and m = 9 kg with conventional Lyapunov function approach.

system responses despite the large variation of cable length and payload weight. Enhancing

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Department of Industrial Automation, School of Electrical Engineering, Hanoi University of

for 3D motion with carrying rope length will be proposed in the future studies.

Figure 6. System responses in the case of L = 9 m and m = 9 kg with barrier Lyapunov function approach.

Author details

Tung Lam Nguyen\* and Minh Duc Duong

Science and Technology, Hanoi, Vietnam

\*Address all correspondence to: lam.nguyentung@hust.edu.vn

#### 5. Conclusions

The dynamic model of overhead crane with distributed mass and elasticity of handling cable is formulated using the extended Hamilton's principle. Based on the model, we successfully analyzed and designed two nonlinear robust controllers using two versions of Lyapunov candidate functions. The first can steer the payload to the desired location, while the second can maintain payload fluctuation in a defined span. The proposed controllers well stabilize all

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Figure 6. System responses in the case of L = 9 m and m = 9 kg with barrier Lyapunov function approach.

system responses despite the large variation of cable length and payload weight. Enhancing for 3D motion with carrying rope length will be proposed in the future studies.
