Control System Analysis

# Control systems are dynamic systems.

## Dynamic system definition.

The first thing you must understand in the analysis of control systems is that control systems are dynamic systems. According to Ogata (1987), a system is dynamic when its output in present time depends on its input in the past. If the system output in the present time depends just on an input in the present time, the system is called static. In a dynamic system, the output changes with the time if the system is not in its equilibrium state, while, in a static system, the output keeps constant if the input doesn’t change; i.e. the output changes only when the input changes. See an excellent introduction by Prof. Pedro Albertos from UPV: Systems and Signals Examples.

Figures 1 and 2 are examples of static systems and dynamic systems respectively. The first shows the balance relation of a lever supported over a fulcrum. The present value of y(t) depends on the present value of the input u(t). The second shows that the speed and the position of a vehicle depend on an input in the past.

Figura 2. An example of a dynamic system (Albertos, 2016).

The artificial systems such as the Off-Shore Platform of Figure 3 and the Aircraft cockpit of Figure 4, are also examples of high-complex dynamic systems made by the human beings:

## The transient response.

Regarding the control systems, the author Nise defines dynamic systems as follows:  “A control system is dynamic: It responds to an input by undergoing a transient response before reaching a steady-state response that generally resembles the input” (Nise, 2011, p. 10). Figure 5 shows a system to control the position of an antenna. Here, the output is the angular position  (Azimuth Angle), while the input is the signal  sent by the potentiometer. Figure 6 shows the output (blue line) of the system showed in Figure 5, in terms of the transient response and the steady-state response, both for high gain and low gain.

The goal of the control system of Figure 5 is to place the antenna into the position determined by the input. That´s why the output follows the input. Analyzing Figure 6 we can observe the main characteristic of dynamic systems, which is that, the response in any time after t=0 depends on the input in the past, i.e. the input in t=o determines the output in any time in the future. The transient response is the response of the system before reaching completely the steady-state response (the final value). Looking to the Figure 6, we can find two transient responses. The first one corresponds to a high gain. It generates a lot of fluctuations before the system reaches its steady-state, but it has the advantage of being faster in getting the final value. Here we can image the antenna getting its final position with a fast moving but zigzagging around it. The second one corresponds to a low gain, where there are no fluctuations and we get a very cushioned movement, but the system takes much more time in getting the final value. The selection of one or the other kind of transient response depends on the requirements of the operation and the limits of the system in order to maintain stability.

## LTI Systems.

Studying control system implies to obtain as a first step its model. Indeed, before analyzing a control system, we have to develop a mathematical model of a dynamic system.

A mathematical model is perceived as a set of equations representing the dynamic of the system in an exact or approximate way. Such a dynamic, being the system an electrical, a mechanical or a biological one, can be represented by mean of differential equations (Ogata, 2002). In general, resolving a problem requires getting a simple and simplified model in the first stage, in order to visualize the solution by mean of the most practical way. To obtain a simplified model, the control engineer must decide which variable are important and which factors can be ignored.

Once we have an approximate idea of the kind and scope of the solution, the natural or forced response of the system, the model can be optimized and be transformed in one of more complexity, which requires the application of specialized software to be simulated and analyzed, with the aim of obtaining hidden and valuable information.

Talking about making a model, the MIT professor John Sterman shares its philosophy about the efficacy of a model as follows:

“Every model is a representation of a system…But for a model to be useful, it must address a specific problem and must simplify rather than attempt to mirror an entire system in detail…the usefulness of models lies in the fact that they simplify reality, creating a representation of it, we can comprehend…Von Clausewitz famously cautioned that the map is not the territory. It´s a good thing it isn´t: A map as detailed as the territory would be of no use” (Sterman, 2000, p. 89) .

To obtain the equations which set up the models of dynamic systems, the engineers use the laws of the physics applied to the properties of the systems, always searching for the easier path when they are building the model. Among the most useful properties to accomplish this objective, we have the properties of linearity and invariance in time, basically for two main reasons. In the first place, a huge quantity of physical processes, overall those which concern to science, have both properties. In the second place, the linear and time-invariant systems (LTI Systems) are widely accessible in terms of available tools for their analysis. The science of signals and systems has reached a powerful development on these software tools for the systems analysis, allowing people from different academic fields to easily approach to the study of LTI systems (Oppenheim, 1996). That’s why the comprehension of LTI systems becomes the next task at the engineers’ training for the analysis of control systems.

## Analysis and Design definition.

Before getting deeper on the characteristics of the LTI Systems, the engineers need to strictly define the basic areas of their work as control engineers [1]: the analysis, the design and the synthesis of systems.

Analysis: it is the study of the functioning of a system at specific conditions, which mathematical model is known. Generally, they are varied, the values of the parameters involved in the mathematical models in order to observe the different responses and from there to get conclusions. As the analysis depends on the mathematical model, it is independent of the kind of the system studied, being this mechanical, electrical or hydraulic.

Design: given a specific task, it is the process whereby we can find the system which accomplishes that task.  Usually, it is not a direct process and requires essay and error. The design implies to make it clear the requirements of the system, typically given in qualitative and quantitative terms. Subsequently, the engineer uses the synthesis. Once he has a model, the engineer analyzes the system so foresee the compliance of the requirements by mean of computerized simulation. By applying essay and error, the engineer modifies the model until it approximately meets the desired result. If it is possible, the engineer builds a prototype and continues the analysis until it meets the final goal.

Synthesis: it is the use of a specific procedure to find a system which works in a specific way. In this case, the characteristics of the system are postulated at the beginning and afterward the engineer uses several mathematical techniques to come up with the right system.

There are therefore two methods for designing (Distefano et al, 1995):

1. Design from the analysis: it is made by mean of the modification of the characteristics of a system which already exists;
2. Design from the synthesis: it is the definition of a system starting from its specifications.

In relation to control systems, Nise defines the functions of an engineer as follows:“…we discuss three major objectives of systems analysis and design: producing the desired transient response, reducing steady-state error, and achieving stability” (Nise, 2011, p. 10).