State Observer for State Space Model

This article summarizes state estimators (state observers) for systems represented by state equations. A video explaining the state observer for systems expressed by state equations is provided at the bottom. Observers are also called software sensors because they observe information that cannot be directly measured through computation.

For an overview of state feedback control, please refer to the following article:

Summary of State Feedback Control and Control Based on State Equations

Overview of State Observers
Basic Concept of State Observers
- Formulation of Observer
Observer-Based System
Related Videos on State Estimation
Research on State Observers
Related Pages on the Control Engineering Channel (State Observers)

Overview of State Observers

Let’s explain state observers. In control systems, it is not always possible to sense all signals of a system, so it is necessary to estimate the state. In the state equation, variables such as (velocity and position) are generally considered states, distinct from the observed output. In implementing state feedback using state values, it is necessary to estimate unknown states. For example, if the states are [position] and [velocity], but only [position] is observed, it is necessary to estimate the velocity.

Consider a control object given as shown in the figure, where state feedback control is applied with u=-Kx. State feedback control is one of the most fundamental control methods in the field of control engineering. In state feedback control, it is assumed that all states are used. If estimates are obtained through a state observer, state feedback control can be realized by using these estimates instead.

Diagram showing the relationship between state quantities and state feedback control — State Feedback Control and State Quantities

Generally, there is a relationship between the number of states and the observable outputs, as shown in the figure, where the number of states is greater than the number of observable outputs.

Diagram showing that the number of sensors is less than the number of states — Relationship Between Number of Sensors and States

Basic Concept of State Observers

Here we explain the basic concept of observers.

First, when the dynamics of the control object are given as shown in the upper part of the figure, a copy of the control object with the same A, B, C matrices is created on a computer. This is called a mathematical model. When control input is applied to the control object, an output appears, and if the same input is applied to the copy of the control object, the state of the mathematical model behaves similarly to the state of the control object.

Diagram representing the behavior of states — Relationship Between States of Control Object and Model (Stable)

If the poles of the A matrix of the system are stable and the initial values of the state of the control object and the mathematical model are the same, the states of the control object and the mathematical model will completely match at all times in a disturbance-free environment.

Even if the initial values of the states do not match, if A is stable, the influence of these initial values decreases over time, so after some time, the state of the control object and the state of the mathematical model, which is a copy of the control object, become close.

Therefore, the state of the mathematical model can be considered the state of the control object and used in state feedback control.

Furthermore, by using the deviation between the output of the control object and the output of the copy of the control object in feedback, the accuracy of convergence can be improved (see the figure below).

Diagram showing the basic structure of an observer — Basic Structure of Observer

This is the concept of the observer.

Formulation of Observer

Specifically, the basic state observer is given by the following equation:

$\dot {\tilde x} (t) = A\tilde x (t) + B u(t) - L(C\tilde x(t) - y(t))$

Here, $y(t)$ is the observed output.

By constructing in this manner, the difference from $\dot x(t) = Ax(t)+Bu(t)$ is considered, and the dynamics concerning the state estimation error $e(t) = x(t) -\tilde x(t)$ are given by the following dynamics:

$\dot e(t) = (A-LC) e(t)$

By setting L so that the term $A - LC$ is stable, e asymptotically approaches zero. This observer can be implemented even if $A$ is unstable as long as the pair $(C,A)$ is observable.

This is the mechanism of the state observer, and since the deviation between the state and the estimated state becomes zero, the estimated state is considered the state and used in control, ensuring the entire control system works effectively.

Diagram representing the equations of a state observer — Dynamics of Estimation Error in State Observer

So far, the basic structure and formulation of the state observer have been explained. This concludes the explanation of the state observer.

Observer-Based System

A system using both a state observer and state feedback is called an observer-based system (observer-based state feedback control).

Observer-Based System

The system's order is $2n$ , and the closed-loop poles obtained by the observer structure ( $poles of A-LC$ ) and the closed-loop poles by state feedback ( $poles of A-BK$ ) determine the characteristics of the control system. This feature, where the observer and controller can be independently set, is called the separation principle.

In an observer-based system, the control input is $u = -K\tilde x$ , using the estimate $\tilde x$ instead of $u = -Kx$ . Therefore, the state deviation $e(t)$ should converge quickly, so the observer poles are placed more to the left on the complex plane.

Relationship Between Observer and Feedback Control Poles

blog.control-theory.com

The following video explains the state observer for a system expressed by state equations.

youtu.be

Research on State Observers

When state estimation is performed with sensor signals containing outliers, the estimation results can be significantly degraded by those outliers. The research results on observers that prevent such degradation are summarized below.

research.control-theory.com

>>Control Engineering Channel: A Portal Site with Over 500 Control Videos Control Engineering Channel - State-Estimation (control-theory.com)

>>Control Engineering Channel: A Portal Site with Over 500 Control Videos Control Engineering Channel - 01-Prof.Okajima-03 (control-theory.com)