# On optimal tracking and sliding mode control with application to vibration quenching

Present day research in modeling and control of flexible structures has gained a renewed interest, particularly in aerospace and industry, where robot manipulators play a very significant role. In the present dissertation, the infinity-horizon or Steady State Linear Quadratic Tracking Problem (SS LQT) has been applied to suppress vibrations in flexible beams. It is well known that the SS LQT problem does not have a solution in the strict sense because in general the cost is unbounded. However, for applications where the reference signal is generated by an asymptotically stable system, the problem is well posed and enjoys a bounded cost. Computationally, one term in the solution is found by solving an Algebraic Riccati Equation; and the second term involves an auxiliary function v(t) found by solving a differential equation backward in time to determine v(0) which is then used in the actual control run. An important contribution of this dissertation is the development of a linear system of algebraic equations to determine v(0) that will allow implement the controller without having to solve off-line and backward in time for v( t). A second problem that is addressed in this dissertation is the simulation of the tracker. Once that we have v(0) is not always possible to simulate and implement the controller, this is due to round-off and truncation errors. To solve this problem a model follower technique is applied. In addition, we consider that the flexible beam is driven for a nonlinear actuator, and a technique based on Sliding Mode Control is presented to reintroduce the nonlinear model to our original design