# EXERCISES AND PROBLEMS OF CLASSICAL CONTROL THEORY. INTEGRATION OF CLASSROOM AND LABORATORY

The aim of this work is how to integrate theoretical issues of control theory in the corresponding set of exercises and problems using computational procedures, for example, matlab and simulink to teach the classical control theory in the third course of Electrical Engineering.

Control systems is the engineering discipline that applies control theory to design systems with predictable behaviour. The practice uses sensors to measure the output performance of the device being controlled and those measurements can be used to give feedback to the input actuators that can make corrections to achieve the desired performance.

In particular the scope of classical control theory is limited to single-input and single-output (SISO) system design. The system analysis is carried out in time domain using differential equations, in complex-s domain with Laplace transform or in frequency domain by transforming from the complex-s domain. So, control theory is an interdisciplinary subfield of science, strongly related to physics, mathematics and dynamical systems.

It has been proposed a set of exercises to learn some concepts of classical control theory. Each exercise represents different dynamical system (electrical, hydraulic, mechanical and thermal) and is divided into two steps:

First, the system is introduced in open loop. So, in this step, students need to obtain the dynamical model in the state space representation, sometimes using Taylor approximation to linearise the nonlinear dynamics and the concept of steady state is also introduced. Then, they have to obtain the corresponding transfer function using Laplace transform and study the relationship between poles and zeros of the transfer function and the dynamical behaviour of the system.

Second, the control system is added to the previous system to study the closed loop problem. Now, students have to identify the control objective, the reference, the input disturbance, sensor, actuator and controller, and build the corresponding block diagram. Then, they have to learn how to impose control specifications, for example:

•Stability, the controller must ensure that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even worsen the stability of the open-loop system, which must normally be avoided.

•Rejection of step disturbance.

•Time-response of the closed-loop system: for example the rise time (the time needed by the control system to reach the desired value after a perturbation), peak overshoot (the highest value reached by the response before reaching the desired value).

Finally the root locus analysis is introduced for examining how the roots (poles) of a system change with variation of a certain system parameter, commonly the gain of a feedback system and in this way achieve the control specification proposed before.

In the framework for Qualifications of the European Higher Education Area, the academic and professional profiles of the Engineering degree are related to the identification and development of skills and competences and to decisions about how a student should achieve them, in particular, interpreting of results and critical reflection, which are the grounds of science knowledge. In these practices, each dynamical system and its control system are simulated in simulink to allow the students to check the behaviour of the systems and the effect of the considered control specifications in a visual way.