# State Observer Design in MATLAB: A Theoretical Guide for University Students

September 30, 2023 Max Sterling
Australia
State Observer Design
Professor Max Sterling, a distinguished scholar with a wealth of experience in control theory and MATLAB is your go-to expert for conquering state observer design challenges. Holding a Ph.D. in Electrical Engineering from the University of Melbourne, Professor Sterling is committed to helping students excel in MATLAB assignments and control system design.

State observers play a crucial role in control theory and engineering, enabling us to estimate the unmeasurable states of a dynamic system based on available measurements. Understanding the process of state observer design is essential for students pursuing studies in control systems and related fields. In this comprehensive guide, we will delve into the theoretical aspects of state observer design using MATLAB, providing university students with a detailed understanding of this fundamental concept. If you need help with your state observer design assignment using MATLAB, this guide is here to assist you.

## Basics of State Observers

Before diving into the MATLAB-based design process, it's essential to grasp the fundamental concepts of state observers.

### What is a State Observer?

A state observer can be thought of as a mathematical tool or computational framework that serves a critical purpose in control theory. It's essentially a set of equations or algorithms designed to perform a specific task: estimating the internal states of a dynamic system. These internal states represent crucial information about how a system behaves over time. However, in many practical scenarios, these states are not directly measurable using available sensors or measurements. Imagine you're driving a car, and you want to know the current speed, engine temperature, and fuel level, but you only have access to the steering angle and the brake pedal position. A state observer is like a smart system that takes these limited inputs (steering angle and brake pedal position) and computes an educated guess of the hidden or unobservable states (speed, engine temperature, fuel level) based on the known dynamics of the car.

### Why Do We Need State Observers?

State observers are indispensable tools in control systems engineering for several compelling reasons:

1. Fault Detection: In many critical systems like aircraft, power plants, or autonomous vehicles, it's vital to detect faults or abnormalities in realtime. State observers can help identify when the actual system's behaviour deviates from its expected behaviour, indicating a potential problem.
2. Control Synthesis: Control systems often rely on feedback control, where the system's state is used to adjust control inputs in realtime. When not all states are directly measurable, observers provide estimates that enable effective control synthesis.
3. Monitoring Complex Systems: Some systems, particularly large and complex ones, have numerous internal states that are difficult or expensive to measure directly. State observers allow us to gain insight into the system's internal dynamics without the need for an excessive number of sensors.

In essence, state observers bridge the gap between what we can directly measure and what we need to know about a system's behaviour for effective control, monitoring, and diagnosis. They are essential tools in the control engineer's toolbox, enabling the design of robust and efficient control systems for a wide range of applications.

## Mathematical Background

To design state observers effectively, one must have a strong grasp of the mathematical foundations. This section provides a theoretical foundation.

### System Dynamics

Understanding the dynamic behaviour of a system is crucial. We'll discuss linear and time-invariant systems, which are often the focus of state observer design.

### State-Space Representation

In control systems engineering, the state-space representation is a foundational concept that provides a comprehensive and unified framework for describing the behaviour of dynamic systems. It's an essential component for understanding and designing state observers.

At its core, the state-space representation represents a dynamic system by defining two primary sets of variables:

• State Variables: These variables, often denoted as x(t), represent the internal states of the system. They encapsulate all the information needed to predict the future behavior of the system given its current state and inputs. For example, in a mechanical system, state variables might include position, velocity, and acceleration.
• Input Variables: These variables, typically denoted as u(t), represent the inputs or control signals applied to the system. Inputs can be forces, voltages, or any external influences that affect the system's behaviour.

Furthermore, the state-space representation includes a set of differential equations that describe how the system's state variables change over time in response to inputs. These equations are usually written in the form:

dx(t)/dt = f(x(t), u(t))

This equation specifies how the rates of change of the state variables (dx(t)/dt) depend on the current values of the state variables (x(t)) and the inputs (u(t)).

The state-space representation is a powerful tool because it allows complex systems to be described concisely and systematically. It also provides a natural framework for analysis and control design.

### Observability

Observability is a critical property of dynamic systems, and it directly relates to the feasibility of estimating the system's internal states based on available measurements. In other words, it answers the question: Can we reconstruct the hidden states of a system accurately using the data we can observe?

The concept of observability is closely tied to the state-space representation. In the context of state observers, here's what you need to know:

• Observability Matrix: This is a key tool in assessing observability. It's a matrix that combines the system's dynamics (described by the state-space equations) with the measurement equations (how states relate to measurements). The observability matrix, often denoted as O, is used to determine whether a system is observable or not.
• Observable Systems: If the observability matrix is full rank (i.e., its determinant is non-zero), the system is said to be observable. In such systems, it is theoretically possible to estimate all of the internal states accurately using measurements.
• Unobservable Systems: If the observability matrix is not full rank (i.e., its determinant is zero), the system is unobservable. In such cases, there exist internal states that cannot be estimated solely from the available measurements.

## State Observer Design Process

Now that we have the necessary theoretical background, we can move on to the state observer design process using MATLAB.

### Luenberger Observer

We'll introduce the Luenberger observer, a widely used observer design technique, and explain the theory behind it.

### Observer Design Objectives

Discuss the primary objectives of observer design, including stability, accuracy, and robustness.

### MATLAB Implementation

Step-by-step guide on implementing a Luenberger observer in MATLAB. This includes defining system matrices, calculating observer gains, and simulating the observer.

## Extended Topics in State Observer Design

To provide students with a comprehensive understanding, we'll touch on advanced topics related to state observer design.

### Nonlinear State Observers

While the discussions thus far have focused on linear systems, the real world often involves nonlinear systems with complex behaviours. Nonlinear state observers extend the concepts covered to these more intricate systems. In such cases:

• Nonlinear Observability: The concept of observability still applies but becomes more complex in nonlinear systems. Observability analysis may involve tools like observability Grampians or nonlinear observability matrices.
• Nonlinear State Estimation: Instead of simple linear equations, nonlinear observers rely on techniques like extended Kalman filters (EKF) or unscented Kalman filters (UKF) to estimate states. These methods use nonlinear models to predict and correct state estimates, accommodating the inherent nonlinearities in the system.

Nonlinear state observers are vital for applications where linear approximations fall short, such as in robotics, aerospace, and biochemical systems.

### Optimal State Observers

Optimal state observers, like the famous Kalman filter, offer a more sophisticated approach to state estimation. These observers aim to minimize estimation errors and are widely used in various fields:

• Kalman Filter: The Kalman filter is a recursive algorithm that optimally estimates the state of a linear dynamic system while considering the uncertainties in measurements and system dynamics. It's highly efficient and widely applied in navigation, robotics, and finance.
• Extended Kalman Filter (EKF): For nonlinear systems, the EKF extends the principles of the Kalman filter by linearizing the system dynamics at each time step. It's a powerful tool for nonlinear state estimation.
• Unscented Kalman Filter (UKF): The UKF avoids linearization and directly captures the uncertainty propagation through a set of carefully chosen sample points. It's more accurate for strongly nonlinear systems.

Optimal state observers excel in scenarios where precise state estimation and the handling of noisy measurements are critical, such as in autonomous vehicles, GPS navigation, and aerospace.

### Practical Considerations

When designing state observers in real-world applications, several practical considerations come into play:

• Measurement Noise: Real-world measurements are often noisy. Effective state observer design should account for measurement noise to provide accurate state estimates. This involves incorporating noise models into the observer equations.
• Discrete-Time Systems: Many control systems operate in discrete time rather than continuous time. The transition to discrete-time observer design requires careful consideration of sampling rates and discrete dynamics.
• Real-World Applications: Different industries and applications have specific requirements and constraints. State observers need to be tailored to these needs, whether it's for aircraft control, medical devices, or industrial automation.
• Robustness: Robust observer design ensures that the observer can still provide reliable state estimates even in the presence of uncertainties or modelling errors.

## Case Study

To reinforce the theoretical concepts discussed, you’ll need to present a practical case study where state observer design is applied to a real-world problem using MATLAB.

## Conclusion

This theoretical guide serves as a valuable resource for university students seeking to understand the process of state observer design using MATLAB. By comprehensively covering the theoretical foundations, mathematical concepts, and practical implementation steps, students will be well-equipped to solve their MATLAB assignments and projects related to control systems and observer design. Mastery of this topic is not only essential for academic success but also for a successful career in engineering and control systems.