A Comparative Study of Controller Designs for Linear Time Variant System withTime Delay Asif Mahmood Mughal 1 , Faisal Aftab 2 , Muhammad Sharjil 3 , Shakil Ahmad 4 National Engineering Scientific Commission, Islamabad, Pakistan Email: asifmahmoodmughal@gmail.com 1 ,m_faisalaftab@yahoo.com 2 , muhammad.sharjil@gmail.com 3 , shakilawan1@yahoo.com 4 Abstract-A linear time variant (LTV) system with high initial overshoots has different stability and control problems, due to its model dynamics, disturbances and nonlinearities. The problems are further enhanced due to time delay in the sensor model and input noise. In this research, we studied a 2 nd order LTV model of this particular system with 2 nd order sensor and 3 rd order actuator model with time delay in the loop. We used Pade approximation of 2 nd order for a time delay to yield a total 9 th order system. We further employed controller design techniques for a closed loop feedback scheme for the system. In this paper, we provided results with lead-lag, PID and H 2 (LQR/LQG) based regulators for a single input to control the system. Furthermore, we also provided a gain scheduling scheme based upon time varying eigenvalues of the system to maintain the bounds on the control effort. Our results indicate the merits and demerits of these linear techniques for this LTV system with time delay, and provide comparative understanding of the system for further analysis and design. Keywords: Time delays, linear time varying (LTV) systems, robust design, optimal controller I. INTRODUCTION Linear or nonlinear time varying systems poses difficult problem to controller design schemes. This problem enhances with time delays either in feedback loops, sensors, or actuation systems. Time delays cause system to slowdown and require robust design to get an appropriate response. Time delays are also sensitive to disturbances in the system, unmodeled dynamics and nonlinearities. In ref [1], discrete controller was first introduced with time delays in the control, and solved the linear tracking quadratic problem for two systems with and without time delays. Ref [2] discussed linear time invariant (LTI) systems with time delay controller with its stability conditions and convergence and thus making it as a part of repetitive control algorithm. They also described that time delay control scheme is an alternative approach for systems with unknown dynamics and unpredictable disturbances. Ref [3] further elaborated this approach by using time delay control for unknown disturbances in LTI model of 2 link robotic manipulator. They showed that this time delay control scheme is robust and effective for single input single output system with simplicity of control algorithm. A robust control scheme in [4] showed the time delay compensation in communication networks. This is 2DOF controller based upon compensation of the time delay from the system without delay, which shows the robustness in the design with variation in systems parameters and uncertainty in the model. Another research in [5] shows the optimal and discrete-time implementation of a controller in the bilateral teleoperation with a constant delay in the feedback loop. The linear quadratic Gaussian (LQG) controller implemented in master side with compensation of time delay from feedback channel from slave side showed experimentally the applicability of the scheme. Ref [6] discussed the time linear delay system with H 2 filtering with delay dependent criteria. Similarly ref [7] discussed the LMI with H∞ for neutral system with time delay in the states. Uncertain nonlinear system in ref [8] studied with neural or adaptive identification and development of control law in the presence of time delays. Other researchers in [9] discussed the controller law design in the presence of TD in the brushless servo control mechanism. They showed that the varying reference model according to the size of change in the reference input produces better results of overshoots and steady state responses. In this paper, we discuss a linear time varying system with high initial varying dynamics such a small scale air to ground vehicles with powerful thrust (e.g. power glider). A second order model for roll angle of the system with time varying eigenvalue and gains pose a slightly difficult control problem. This problem further increases with time delay in the sensor output (feedback loop). We compare three simple control law design using lead controllers, PID design and finally we introduce a higher order controller with H 2 design. These are linear time invariant controllers with time varying gain scheduling to the design. In this scheme, we also propose gain scheduling of LTI controller based upon time varying eigenvalue of the model. At the end we discuss the merits and demerits of proposed scheme and show the comparative understanding of the system. II. LINEAR TIME VARYING MODEL A second order nonlinear model with roll angle and roll angle rate is given as ൤ ݔሶሺݐሻ ݔሷሺݐሻ ൨ൌ൤ 0 1 0 ߣሺݐሻ ൨·൤ ݔሺݐሻ ݔሶሺݐሻ ൨൅൤ 0 ሺݐሻ ൨· ݑሺݐሻ (1) Proceedings of International Bhurban Conference on Applied Sciences & Technology Islamabad, Pakistan, 11 – 14 January, 2010 110