# Chapter seven

State space methods

## Section four: STATE SPACE FEEDBACK CONTROL AND OBSERVERS

The focus of this section is on state feedback design. It starts with basic pole-placement approaches and then continues onto a brief mention of optimal control.

However, state space control laws are often based on state information and this information may not be readily available. Consequently and an equally important topic is observer design, where the role of the observer is to estimate the state information for use in the feedback.

## State space feedback

### 1. Introduction

This resource introduces the concept of state feedback and demonstrates how this has an impact on the poles, behaviour and steady-state. It demonstrates the need for systematic design methods as a simplistic approach is not manageable in general.

A talk through video is on YouTube. Read the notes (PDF, 347KB).

### 2. Pole placement with canonical forms

This resource introduces the concept of pole placement using control canonical forms. It explains how one can easily choose the values of a state feedback gain to achieve precisely the desired closed-loop poles.

It demonstrates that with SISO examples in controllable form, the selection of the gain parameters is straightforward.

A talk through video is on YouTube. Read the notes (PDF, 423KB).

### 3. Transformation to a canonical form

The previous video showed that when a system is in control canonical form and has full state observability, it is straightforward to design a state feedback to place the closed-loop poles.

This video considers the issue for a more general system structure. It shows that, assuming controllability, there always exists a similarity transformation that will convert a system into control canonical form.

Using this transformation, one can do placements using the canonical form and transformation to find the implied state feedback. A step by step algorithm is defined and demonstrated.

A talk through video is on YouTube. Read the notes (PDF, 753KB).

### 4. Ackermann's approach to pole placement

Ackermann's method for pole placement requires far fewer steps than the transformation approach of the last video. It can be defined with a simpler algorithm and therefore is easier to implement. Although, the required computations are not pen/paper ones in general.

However, the derivation of Ackermann's approach is more involved. This video first gives the derivation (which viewers could skip if they wish) and then demonstrates application of the algorithm on a few examples.

A talk through video is on YouTube. Read the notes (PDF, 560KB).

### 5. Tutorial examples and use of MATLAB

This resource gives a few worked examples (2 state, 3 state and 4 state systems).

It demonstrates the use of the three alternative design methods of

canonical forms

state transformation

Ackermann

Moreover, it emphasises the numerical demands of these approaches and shows MATLAB code for doing the computations (matrix algebra). It finishes with a summary of MATLAB shortcuts.

A talk through video is on YouTube. Read the notes (PDF, 506KB).

### 6. Challenges of pole placement

This resource illustrates how system behaviour varies significantly depending on where the user decides to place the closed-loop poles. This demonstrates a key point that being able to place poles arbitrarily is the same as knowing where to place them.

Future work must look at systematic design methods which suggest good pole locations, as well as considering how to include tracking/integral action.

A talk through video is on YouTube. Read the notes (PDF, 468KB).

### 7. Optimal control

This resource gives a brief introduction to optimal control as a mechanism for designing a feedback which gives reasonable closed-loop pole positions, in combination with managing input activity.

It does not derive the underlying formulae in depth but gives some background and insight to the derivation. MATLAB is used for the numerical computations and illustrations which are not paper and pen exercises in general.

It gives a brief discussion of equivalent results for discrete time systems.

A talk through video is on YouTube. Read the notes (PDF, 527KB).

### 8. Dead-beat control

The previous resources focussed on the continuous time state space case, although were implicitly applicable to discrete time systems as well. One key exception however is dead-beat design which is applicable only to the discrete case.

In this resource, dead-beat control is defined and illustrated alongside a brief discussion of its potential role.

A talk through video is on YouTube. Read the notes (PDF, 448KB).

## State space observers

### 1. Introduction

This resource introduces the concept of an observer using layman's terms. It discusses how there is a need to use context information and other knowledge to extract the maximum information from any measurements available.

A talk through video is on YouTube. Read the notes (PDF, 439KB).

### 2. Basic structures

This resource introduces the concept of an internal model, and the classical structure of an observer used to estimate the values of the system states. It demonstrates the concept of observer gain and how the choice of this affects the convergence rates of the state estimates.

A talk through video is on YouTube. Read the notes (PDF, 450KB).

### 3. Observer design by pole placement

This resource introduces the concept of duality by demonstrating the analogies between an observer design and a feedback design. It uses the insights to propose formal design methods for a classical observer. There are a number of numerical examples using MATLAB.

A talk through video is on YouTube. Read the notes (PDF, 563KB).

### 4. System stability

This resource analyses overall system behaviour where a state feedback is combined with an observer. It introduces the separation principle but also illustrates the potential dangers that arise when an observer is needed. There are a number of numerical examples.

A talk through video is on YouTube. Read the notes (PDF, 436KB).