# Analyzing and Controlling a MIMO System: MATLAB Case Study on a Robotic Arm

July 13, 2023 Anna Scott
Australia
MIMO System
She is a control systems engineer and MATLAB expert specializing in MIMO systems analysis and providing guidance to undergraduate students.
This blog post was written with undergraduate students in universities in mind who are currently enrolled in control systems engineering courses and are completing MATLAB assignment for class. Transfer functions are utilized throughout this extensive study of the analysis and control of Multi-Input Multi-Output (MIMO) systems. Students can gain valuable insights into MIMO system analysis and control by working through a practical case study centered on a robotic arm, while also sharpening their skills in MATLAB in the process. Students can improve their comprehension of MIMO systems and the ways in which they are applied in the real world by adhering to the guidance and looking at the examples that are provided in this blog post. The case study on the robotic arm serves as a tangible and relatable example that bridges the gap between theoretical concepts and their applications in the real world. Students will be able to confidently approach their MATLAB homework after reading this blog post because they will be equipped with the knowledge and tools necessary for analyzing and controlling complex MIMO systems. Make sure to successfully completing your transfer function assignment before the deadline to ensure you understand the concepts thoroughly. ## Understanding MIMO Systems

Let's get a basic understanding of MIMO systems under our belts before we delve into the case study. A Multiple Input Multiple Output (MIMO) system is a type of control system that has multiple inputs and outputs that are connected to one another. Inputs and outputs may take the form of physical quantities, signals, or variables respectively. In the case of a robotic arm, for example, the inputs could be the torque or current that is applied to each joint, while the outputs could be the joint angles or positions. Similarly, the inputs and outputs could also be swapped. MIMO systems are widely used in many different industries, such as robotics, aerospace, chemical processes, and telecommunications, to name a few. In comparison to Single-Input Single-Output (SISO) systems, they have a number of advantages, the most notable of which are improved performance, flexibility, and robustness.

## Transfer Functions and Their Role in MIMO Systems

Transfer functions are tools that are indispensable in the analysis and design of control systems. They offer a mathematical representation of the relationship that exists in the frequency domain between the signals that are input to a system and those that are output by that system. Transfer functions are typically represented as matrices in MIMO systems. These matrices are referred to as transfer function matrices. The transfer function matrix is the representation of the relationship between the inputs (torques or currents applied to the joints) and outputs (joint angles or positions) of the robotic arm in the context of our case study on a robotic arm. Every cell in the transfer function matrix contains a representation of the transfer function that is used between a particular input and output pair.

## Analyzing MIMO Systems Using Transfer Functions in MATLAB

The field of control systems engineering makes extensive use of the powerful MATLAB software. It offers a wide variety of tools and functions, all of which can efficiently analyze and control MIMO systems. Let's take a look at some of the most important steps involved in analyzing MIMO systems in MATLAB by using transfer functions:

1. System Identification The first step is to determine the transfer function matrix of the MIMO system. Experiments, mathematical modelling, and various other techniques for system identification can all be utilized to accomplish this goal.
2. Representation of the Transfer Function After the transfer function matrix has been acquired, it can be represented in MATLAB by making use of the tf function. It is possible to think of the transfer function matrix as an array of transfer functions, with each element standing in for the transfer function that exists between a particular input-output pair.
3. Analysis of the Frequency Response MATLAB provides functions such as bode, nyquist, and freqresp for analyzing the frequency response of MIMO systems. With the help of these functions, we are able to investigate the gain, phase, and stability characteristics of the system at a wide range of frequencies.
4. Stability is an extremely important component of control systems. MATLAB provides functions such as eig and pole that can be used to determine whether or not a MIMO system is stable by conducting an analysis of the eigenvalues or poles of the transfer function matrix associated with the system.
5. MATLAB provides several control design techniques, such as PID, LQR, and robust control, which can be used to design controllers for MIMO systems. Students are given the ability to put control algorithms into action in order to achieve the desired behavior of the system thanks to these techniques.

## Case Study: Analysis and Control of a Robotic Arm

Let's take a look at a case study involving a robotic arm to demonstrate how transfer functions and MATLAB can be applied in the context of analyzing and controlling MIMO systems. For the purpose of this case study, we will be concentrating on a three-degrees-of-freedom (DOF) robotic arm that has three joint inputs (in the form of torques) and three joint outputs (in the form of angles).

Using MATLAB, we will accomplish the following:

1. Determine the transfer function matrix of the robotic arm.
2. Make use of the tf function in MATLAB to create a representation of the transfer function matrix.
3. Utilizing functions such as bode and nyquist, conduct an investigation into the frequency response of the robotic arm.
4. Using the eig function, determine whether or not the system is stable.
5. Create a controller for the robotic arm that will achieve the desired trajectory tracking or disturbance rejection.

Undergraduate students will gain valuable insight into the real-world applications of control systems engineering by working through this case study and gaining hands-on experience in analyzing and controlling MIMO systems. This experience will be gained by working through the case study.

## Benefits and Challenges of MIMO Systems

In comparison to SISO systems, MIMO systems have a plethora of advantages, but they also come with their own unique set of difficulties. In the context of our case study on a robotic arm, let's take a look at the benefits and difficulties associated with MIMO systems:

• MIMO systems are able to achieve higher bandwidth, faster response times, and improved tracking accuracy compared to SISO systems. This is due to the fact that MIMO systems have more antennas.
• MIMO systems exhibit increased robustness to disturbances and parameter uncertainties, which makes them more reliable in real-world applications because of their increased resistance to these types of challenges.
• MIMO systems enable flexible system design and optimization, which enables engineers to meet a wider variety of performance requirements.
• Multi-Input Multiple-Output (MIMO) techniques can significantly improve the data throughput and spectral efficiency of communication systems.

### Challenges of MIMO Systems:

• Complexity: Due to the interconnections between multiple inputs and outputs, MIMO systems frequently involve complex mathematical modelling and analysis. This presents a challenge for designers.
• The Complexity of Computational Requirements: When compared to SISO systems, the computational requirements for analyzing and designing MIMO systems can be significantly higher. As a result, effective algorithms and computational resources are required.
• Coupling Effects: In MIMO systems, the interdependence between inputs and outputs can lead to coupling effects, which makes it difficult to isolate and control individual inputs and outputs. This is due to the fact that coupling effects can be caused by interdependence.
• Estimation and Control: Estimating the state variables and designing control strategies for MIMO systems require advanced techniques such as state observers and multivariable control methods. These techniques are necessary because MIMO systems have a high degree of complexity.

When beginning their studies in control systems engineering, it is essential for undergraduate students to have a solid understanding of both the benefits and the challenges presented by MIMO systems. Students are able to approach MIMO system analysis and control with a well-rounded perspective and develop effective solutions for problems that occur in the real world if they are aware of these aspects and apply them.

## Conclusion:

Control systems engineering involves a number of fundamental aspects, one of which is the analysis and control of MIMO systems. Undergraduate students can improve their knowledge and problem-solving skills in this field by becoming familiar with the concept of MIMO systems and making use of the transfer functions that are available in MATLAB. This blog post presents a case study on a robotic arm as an illustrative example of how transfer functions and MATLAB can be utilized to effectively analyze and control complex systems. The case study focuses on controlling the movement of the robotic arm. Students can confidently tackle MATLAB homework relating to MIMO systems and contribute to advancements in control systems engineering if they are armed with this knowledge.