The Core Concepts of State-Space Control -  Part 1

When you talk about any form of control, you need a system. In engineering, a system is defined as anything that takes in input to generate a particular output. In the mathematical sense, a system is a function that links the input and the output. In the area of control design, the input is represented by "u," and the output by "y". 

You can find systems all around you, but some examples relevant to learning the introduction to state-space control are internal combustion (IC) engines, drones, and steering systems for vehicles.  

An Example

To illustrate the concept, consider an IC engine. The accelerator pedal's position in your car is used to control the throttle input to the engine. The engine takes in this input and the mixture of air and fuel and generates the torque, which is the output of the system. The result of the torque is the speed of the vehicle. 

Similarly, if you work with a vehicular dynamics system, you have another set of input parameters: the steering angle and the vehicle's velocity. The output for this system is the yaw rate (the lateral movement) that the vehicle produces. 

Note that this is a highly simplified scenario used to explain the concept. In reality, every system has multiple inputs and outputs. In the case of the combustion engine, you should ideally also consider the spark timing and the air-to-fuel ratio are inputs. 

The Goal of a System

The goal should be to operate a given system to get the desired output from it, which could be in the form of:

  • The desired torque or speed from an engine
  • The desired motion of a robotic arm (to reach a specific position or to pick up a particular object)
  • The desired path that a self-driving car should follow

To achieve this, you should find the necessary input parameters or quantities that ensure the system operates in the way you wish. Every system is primarily a function, as mentioned before. Using that function and knowing the output you want, you should be able to calculate the desired input. Mathematically, it means that you need to find the inverse of the function. 

This method of operating systems is called the open-loop way. The process of calculating the input considering only the desired output and not the actual output is called feedforward control. 

System Behavior

It's evident that the first thing you need to figure out is the system function, or, in technical terms, the system dynamics. However, no mathematical model is 100% accurate, and a close approximation is the best result you can get. 

Real-world systems always have some factors of uncertainty like noise, instability, and external disturbances. So, the feedforward control is never perfect and may not work in the real world all the time. 

Cruise Control System

The cruise control system in cars is a mechanism by which you set the desired speed, and the vehicle drives on its own at that speed without your intervention. Consider the system components here: the cruise control mechanism sets the accelerator pedal at the correct position (input) to achieve that speed (output).

Source

According to the feedforward control model, you first need to calibrate the input by assuming the desired output. For this purpose, you would consider a flat, smooth road. By doing so, you get a one-to-one mapping between the speed required and the position of the accelerator pedal. 

Having configured this model, you wish to test out the model in the real world. On a flat road, let's say that the mapping works out fine. However, if you encounter an inclined surface or a bumpy road, your model is no longer accurate. The limitation of this model is that it is valid only in ideal scenarios; in essence, the model is wrong in most real-world situations. 

Feedback Control Systems 

To get the calculations right, you need to consider the desired and the actual outputs. In other words, you want a model that is more sophisticated to also include system behavior like system stability, output tracking and regulation, and rejection of any internal or external disturbances. These models are called feedback control systems. 

The workflow of this model is as follows:

  • A controller generates the input (u) and feeds it into the system.
  • The system gives out the actual output (y).
  • The model compares the error (e) or difference between the desired output and the actual one.
  • This error parameter is fed into the controller, which takes it in and generates a new input parameter accordingly.
  • The process repeats until the actual and desired output are approximately equal.

For example, your initial model could predict that to achieve a speed of 25 km/h, the position of the accelerator is 25%. On a slope, at 25%, you get only 20 km/h. This error of 5 km/h is fed into the controller, which changes the input to, say, 30%, and the cycle repeats.  

What Is a Control System? 

A control system is one that manages, regulates, and stabilizes the behavior of the system of interest. In the domain of control design, there are primarily two types: classical control and modern control. 

Classical control involves control design in the frequency domain, i.e., using system transfer functions. In contrast, modern control entails control design in the state-space area using differential equations in the matrix form. Each type has a particular set of design and analysis techniques. 

Current Trends 

  • If you can represent the system behavior in the form of differential equations, you can then convert them to the form of a matrix. You can then apply the state-space control techniques to the matrix.
  • This model is fundamental to several areas of engineering, like linear systems, fault detection, and automotive control, which is why every engineer should know state-space control.
  • State-space control also finds applications in the industry, especially in IC engines, aerospace engines, etc.

Conclusion

Control systems are crucial to a variety of engineering verticals, and an engineer who is an expert in these is very likely to be in high demand anywhere in the world. 

If you are interested in taking a professional course on control design, you have come to the right place. Check out these master's programs offered by Skill-Lync, sign up, and start learning today!

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