Fundamentals of Control Theory

Control theory is the discipline that concerns itself with the response of dynamical systems to input. Control theory is a eld that spans both mathematics and engineering as it nds the vast majority of its applications while providing many complex problems as the topic of much mathematical research. Such problems often deal with questions such as stability, robustness (tolerance in variation of the parameters), tracking (small error for certain input) and others. One often deals with the concept of a plant. The plant represents the dynamic system and has input u(t),output y(t) and state x(t) signals which contain the information on the state of the system. This may be something simple like a scalar quantity like speed or it could be a vector that contains values for multiple parameters de ning the state of the system; for example an airplane may need to take into account pitch, yaw and roll. For systems with scalar signals the system is referred to as SISO (single input single output) whereas when the signals are vectors the system is referred to as MIMO (multiple input multiple output). The goal in control theory is generally to nd a control (often an input) to ensure that the state/output has the desired properties. There are two major types of control that need to be considered; open loop control and closed loop control. Open loop control is the simple situation where a choice of u(t) is made to ensure the desired behaviour whereas closed loop control is where the input is dependent on the output: u(t) = g(y(t), t) for some function g(.) known as the control law. This control law is what must be chosen to ensure desired behaviour in the feedback scenario. Often the dynamic system can be expressed as a di erential equation (or system of di erential equations). One approach to analyzing such a system is to transform from the time-domain to the frequency-domain by use of Laplace transforms. The system can then be expressed in the following way: