Some Useful Inequations (updating...)
Zhao Zhao
  2022-05-11 23:23:06 2022-05-11 23:23    

Lemma 1： If then the following inequality holds:

Zhu, Z., Xia, Y., & Fu, M. (2011). Attitude stabilization of rigid spacecraft with finite‐time convergence. International Journal of Robust and Nonlinear Control, 21(6), 686-702.

Lemma 2： Let . If and , the inequality condition holds.

Lee, J., Jin, M., Kashiri, N., Caldwell, D. G., & Tsagarakis, N. G. (2019). Inversion-free force tracking control of piezoelectric actuators using fast finite-time integral terminal sliding-mode. Mechatronics, 57, 39-50.

Lemma 3：For any real numbers and , the following inequality holds:

Zhu, Z., Xia, Y., & Fu, M. (2011). Attitude stabilization of rigid spacecraft with finite‐time convergence. International Journal of Robust and Nonlinear Control, 21(6), 686-702.

Lemma 4：Suppose is a smooth positive-definite function, and is a negative semi-definite function for , then there exists an area such that any to in finite time. Moreover, if is the time needed to reach , then where is the initial value of .

Zhu, Z., Xia, Y., & Fu, M. (2011). Attitude stabilization of rigid spacecraft with finite‐time convergence. International Journal of Robust and Nonlinear Control, 21(6), 686-702.

Lemma 5：An extended Lyapunov description of finite-time stability can be given with the form of fast terminal sliding mode as and the settling time can be given by

Zhu, Z., Xia, Y., & Fu, M. (2011). Attitude stabilization of rigid spacecraft with finite‐time convergence. International Journal of Robust and Nonlinear Control, 21(6), 686-702.

Lemma 6：Consider the system . Suppose that there exist continuous function , scalars , and such that Then, the trajectory of system $x=f(x,u) is finite-time stable (PFS).

Proof:

There exists a scalar such that the above inequality can be expressed as: Clearly, if . Therefore, the decrease of in finite time drives the trajectories of the closed-loop system into . Therefore, the trajectories of the closed-loop system is bounded in finite time as where . And the time needed to reach the bound is bounded as where is the initial value of .

Lipschitz condition： satisfies the inequality for all and in some neighborhood of .

Definition: TSM and fast TSM can be defined as the following non-linear differential equation, respectively where are positive odd numbers and .

According to finite-time stability theory the equilibrium point of the above differential equation is globally finite-time stable, i.e. for any given initial condition , the system state x can converge to 0 in finite time

Zhao, D., Li, S., & Gao, F. (2009). A new terminal sliding mode control for robotic manipulators. International Journal of control, 82(10), 1804-1813.

Lemma 7：If function , scalar , and , we have .

Cheng, L., Hou, Z. G., & Tan, M. (2009). Adaptive neural network tracking control for manipulators with uncertain kinematics, dynamics and actuator model. Automatica, 45(10), 2312-2318.

Lemma 8： For any real numbers and , we can obtain the following inequality: Lemma 9：If , we have the following inequality: Lemma 10：(Cauchy–Schwarz inequality) If , we have Lemma 11：(Young’s inequality )For , we have Lemma 12: For any real variable x, the following inequality holds where is a position constant.

Lemma 13 : For , the following inequality holds Lemma 14 : For , then

Wang, F., & Lai, G. (2020). Fixed-time control design for nonlinear uncertain systems via adaptive method. Systems & Control Letters, 140, 104704.

Lemma 15 : For real variables and , and any positive constants , and , the following inequality is true Lemma 16 : For , the following inequality is satisfied:

Meng, F., Zhao, L., & Yu, J. (2020). Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash. Journal of the Franklin Institute, 357(16), 11281-11297.

Lemma 17: Let and a real valued function. Then

Lemma 18: Let . If , the inequality condition holds:

Qian, C., & Lin, W. (2001). Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization. Systems & Control Letters, 42(3), 185-200.

Lemma 19: For is an integer, the following inequalities hold: Aa a consequence, when is an odd integer,

Wang, Y., Song, Y., Krstic, M., & Wen, C. (2016). Fault-tolerant finite time consensus for multiple uncertain nonlinear mechanical systems under single-way directed communication interactions and actuation failures. Automatica, 63, 374-383.

Lemma 20: For , where are positive odd integer, then

Wang, Y., Song, Y., Krstic, M., & Wen, C. (2016). Fault-tolerant finite time consensus for multiple uncertain nonlinear mechanical systems under single-way directed communication interactions and actuation failures. Automatica, 63, 374-383.

Wang, Y., & Song, Y. (2016). Fraction dynamic-surface-based neuroadaptive finite-time containment control of multiagent systems in nonaffine pure-feedback form. IEEE transactions on neural networks and learning systems, 28(3), 678-689.

Lemma 21: , are odd integers,