Modelling and control by Anthony Rossiter

# STATE SPACE FEEDBACK CONTROL AND OBSERVERS

This is the 4th section in the chapter on state space models, behaviours and control. The first focus here is on state feedback design; starting with basic pole-placement approaches and then 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 equally important topic is observer design, where the role of the observer is to estimate the state information for use in the feedback.

 It is implicit in several of these chapters that students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials and Laplace. More information on these can be found in the Mathematics theme on the left hand toobar.

This section contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and occasional tutorial resources for users to test themselves.

## State space feedback 1 - introduction

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

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space feedback 2 - pole placement with canonical forms

Introduces the concept of pole placement using control canonical forms whereby one can easily chose the values of a state feedback gain to achieve precisely the desired closed-loop poles. Demonstrates that with SISO examples in controllable form, the selection of the gain parameters is straightforward.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space feedback 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 placement using the canonical form and transformation to find the implied state feedback. A step by step algorithm is defined and demonstrated.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space feedback 4 - Ackermann's approach to pole placement.

Ackermann's method for pole placement requires far fewer steps than the transformation approach of video 3 and can be defined with a simpler algorithm and thus is easier to implement, although the required computations and 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 summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space feedback 5 - tutorial examples and use of MATLAB.

Gives a few worked examples (2 state, 3 state and 4 state systems). Demonstrates the use of the 3 alternative design methods of: (i) canonical forms; (ii) state transformation and (iii) Ackermann. Moreover, emphasises the numerical demands of these approaches and shows MATLAB code for doing the computations (matrix algebra). Finishes with a summary of MATLAB shortcuts.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space feedback 6 - challenges of pole placement.

Illustrates how system behaviour varies significantly dependent on where the user decides to place the closed-loop poles. This demonstrates a key point which is being able to place poles arbitrarily is the same as knowing where to place them. Hence future work must look at systematic design methods which also suggest good pole locations, as well as considering how to include tracking/integral action.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space feedback 7 - optimal control.

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. 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. Gives a brief discussion of equivalent results for discrete time systems.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space feedback 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. Here dead-beat control is defined and illustrated alongside a brief discussion of its potential role.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space observers 1 – introduction.

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

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space observers 2 – basic structures.

Introduces the concept of an internal model and the classical structure of an observer used to estimate the values of the system states. Demonstrates the concept of observer gain and how the choice of this affects the convergence rates of the state estimates.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space observer 3 – observer design by pole placement.

Introduces the concept of duality by demonstrating the analogies between an observer design and a feedback design. Uses the insights to propose formal design methods for a classical observer. Gives a number of numerical examples using MATLAB.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## State space observer 4 – system stability.

Analyses overall system behaviour where a state feedback is combined with an observer. Introduces the separation principle but also illustrates the potential dangers that arise when an observer is needed. Gives a number of numerical examples.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.