Control of closed loop systems in industries is achieved through myriad control structures.
Few of them will be investigated further in this project
• Classical PID controller
• State space observer-based estimation 2.3.1 PID controller
This specific control structure has become almost universally used in industrial control as a result of its robustness and simplicity. However, simplicity of these controllers is also their downfall, since it limits the range of plants that they can control satisfactorily. Indeed, there exists a set of unstable plants which cannot even be stabilized with any member of the PID family. Nevertheless, the surprising versatility of PID control ensures continued relevance and popularity for this controller. (Ziegler, 1942)
Over time, various tuning methods have been invented to tune PID controller of a system to achieve stability and desired control. (Åström, 1995)
Two of the majorly used tuning methodologies are as follows:
1. Conventional Tuning Methods Ziegler and Nichols proposed in 1942 two different tuning strategies for PID controllers. These methods are successful but
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not extensively used due to its unpredictable execution thus leading to unintended disturbance in the process.
2. Tuning with Relay Method The relay method presented as an alternative to the conventional method of Ziegler-Nichols for closed loop, in the identification of model systems by Åström KJ, Hägglund T (Åström, 1984)in 1984, seems to be the solution. It has the advantage to generate and maintain controlled oscillations.
This method is highly popular due to the simplicity of identification and calibration of a system. Relay method is very efficient in determining the critical gain Rcrit and critical (Eigen) frequency ωcrit.
2.3.2 State-space controller
In a SISO system, a simple representation has sufficient effect on the desired output.
However, when we consider a more complex system with multiple inputs and multiple outputs, it is best to use special model representation. One of the most flexible and useful structures is the state space model. (Víteček, 2013) (C. Goodwin, 2000)
A state space representation is a valuable and frequently used tool for plant modeling. State variables form a set of inner variables which is a complete set, in the sense that, if these variables are known at some time, then any plant output, y(t), can be computed, at all future times, as a function of the state variables and the present and future values of the inputs.
For linear, time invariant systems the state space model is expressed in the following eigenvalues of the matrix A. (Luenberger, 1972)
A transfer function can always be derived from a state space model also the vice-versa is possible, i.e., a state space model can be built from a transfer function model. However, only the completely controllable and observable part of the system is described in that state space model. Thus, the transfer function model might be only a partial description of the system.
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The design of this controller is done under the assumption that all system variables are available for control (State space feedback controller) or designing an observer (state space observer) to estimate the values of the variables that are not available for measurement and control as easily. This is discussed in detail in the subsequent chapters of this project.
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3 Modelling and Simulation of Mathematical Models
Owing to increase in understanding of steady state and dynamic behavior of fluid power systems in recent years the creation of computer based mathematical model seems like an attractive possibility for various reasons. By creating a model of the dynamic system, we can describe the dynamic properties of the system. Modelling is not only used to understand the behavior of system but also gives the ability to simulate faults and observe changes in system that would not be possible to detect without investing in expensive components.
The description is brought about by using differential equations representing the properties of the system. Physical laws can be used to describe the dynamic properties and the equations. The model enables us to study system transients and steady state performance.
(Watton, 2007)
Let’s talk about complexity of the models:
• Model accuracy required for control system design is typically simpler than the model accuracy used for system simulation.
• Simpler models are modelled by using a few features such as:
o Overlooking some physical phenomena, o Approximate linearly nonlinear characteristics
• For the design of a control system:
o A preliminary simplified model is constructed for conceptual design of the process.
o A more accurate model is then used for controller design and parameter determination.
After modelling of the system to satisfaction the next step is to simulate it accordingly.
Simulation is usually done using specific numerical method from a plethora of numerical methods available in the market. The results are typically in form of graphs showing the response of the model generated. Nowadays there are simulation programs available in the market that employ these numerical methods, making the process of model solution much easier. The process is pretty simple and straight forward. A simulation model is created from the mathematical model in the simulation program, the condition of the solution are defined and the solution is on track. Subsequently the plots of system variables show the system response.
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