| Time delay exits in engineering applications due to the system response laggingand the signal transmission speed limited. It is inevitable in real life. The stability ofthe time-delay systems and time-delay control is an important topic and has been thesubject of many studies. The stability study of linear time-invariant and periodicsystems with time delay is presented in this thesis. The five degree of freedom (5DOF)robotic manipulator is used as an experimental subject of this study. The theoreticaland experimental research of the delayed feedback is carried out.(1) Three Lyapunov-Krasovskii (L-K) functionals for stability studies oftime-delayed linear dynamic systems are compared. A benchmark second order linearsystem under delayed PD feedback controls is considered. The stability domains inthe feedback gain parameter space are computed from the LMIs of different L-Kfunctionals, and are compared with that calculated from the characteristic equation ofthe linear system. The results show that while most L-K functionals provide sufficientconditions for stability, which are conservative, Gu’s complete L-K functional is theleast conservative and the most accurate, and provides a necessary and sufficientcondition for stability. Gu’s complete L-K functional involves implicitly infinitenumber of matrices, hence, requires a huge computational effort. When the Lyapunovstability theory is adopted for control design, the conservative stability conditions maybe used, while Gu’s complete L-K functional is more favorable to the design ofcontroller.(2) The numerical methods of continuous time approximation (CTA),Chebyshev continuous time approximation (Chebyshev CTA) and lowpass filter basedcontinuous time approximation（LPCTA）are introduced. The delayed differentialequations are converted into differential equations without delay by using the threemethods. With the help of the three methods and Floquet theory, the stability study oflinear periodic systems with time delay is presented. The classical Mathieu equationwith delayed feedback is considered. The stability domains in the feedback gainparameter space are computed from the CTA methods, and are compared with thatcalculated from the semidiscretization (SD) method. The results show that theChebyshev CTA and LPCTA can compute geometrically complex stability boundariesin the parameter space with high accuracy. The stability domains by the ChebyshevCTA and LPCTA methods are nearly overlapped with the boundaries obtained by the SD method.(3) An approach to analyze kinematics of a five degree of freedom (5DOF)robotic manipulator is proposed. The effect of the time delay in control on the stabilityof the motor control system is considered. The delayed differential equations areconverted into differential equations without delay by using the lowpass filter basedcontinuous time approximation (LPCTA). The optimal control method is adopted todesign the controller. Experimental results show that the proposed kinematicsalgorithm is accurate with less computational cost, and is suitable for real time controlapplication of5DOF manipulator. The time delayed control system of5DOFmanipulator can accurately track the desired trajectory.(4) Based on dynamics of the five degree of freedom (5DOF) roboticmanipulator, the delay control is analyzed. The kinematic equation of5DOF roboticmanipulator is constructed with the Lagrange method. First, without considering thetime delay, the5DOF robotic manipulator as a strongly nonlinear system is controlledunder PD feedback controls. Second, considering the time delay, the delayed PDfeedback control is analyzed. The feedback gains of delayed PD feedback controlsystem are selected by solving a multi-objective optimization problem (MOP). TheSimple Cell Mapping (SCM) method is employed to search the Pareto optimalsolutions in cell space. Numerical results provide validations of the proposedalgorithm. It turns out the SCM algorithm is quite effective for high-dimensionmulti-objective optimization. |
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