Loop-shaping Robust Control

By Philippe Feyel

The loop-shaping approach consists of obtaining a specification in relation to the open loop of the control from specifications regarding various closed loop transfers, because it is easier to work on a single transfer (in addition to the open loop) than on a multitude of transfers (various loopings such as set point/error, disturbance/error, disturbance/control, etc.). The simplicity and flexibility of the approach make it very well adapted to the industrial context.
This book presents the loop-shaping approach in its entirety, starting with the declension of high-level specifications into a loop-shaping specification. It then shows how it is possible to fully integrate this approach for the calculation of robust and efficient correctors with the help of existing techniques, which have already been industrially tried and tested, such as H-infinity synthesis. The concept of a gap metric (or distance between models) is also presented along with its connection with the prime factors of a set of systems shaping a ball of models, as well as its connections with robust synthesis by loop-shaping, in order to calculate efficient and robust correctors. As H-infinity loop-shaping is often demanding in terms of the order of correctors, the author also looks at loop-shaping synthesis under an ordering constraint. Two further promising lines of research are presented, one using stochastic optimization, and the other non-smooth optimization. Finally, the book introduces the concept of correction with two degrees of freedom via the formalism of prime factorization.
Avenues for future work are also opened up by the author as he discusses the main drawbacks to loop-shaping synthesis, and how these issues can be solved using modern optimization techniques in an increasingly competitive industrial context, in accordance with ever more complex sets of functional specifications, associated with increasingly broad conditions of usage.

Contents

Introduction
1. The Loop-shaping Approach
2. Loop-shaping H-infinity Synthesis
3. Two Degrees-of-Freedom Controllers
4. Extensions and Optimizations
Appendix 1. Demonstrative Elements on the Optimization of Robust Stabilization with Order Constraint
Appendix 2. Establishment of Real LMIs for the Quasi-Convex Problem of Optimization of the Weighting Functions

About the Authors

Philippe Feyel is an R&D Engineer for the high-tech company Sagem Défense Sécurité, part of the defence and security business of the SAFRAN group, in Paris, France.

• Language: English
• Publisher: Wiley
• Release date: Jun 12, 2013
• ISBN: 9781118574751

Book preview

Introduction

I.1 Presentation of the book

In an increasingly competitive industrial context, an automation engineer has to apply servo-loops in accordance with ever more complex sets of functional specifications, associated with increasingly broad conditions of usage. In addition to this, the product is often destined for large-scale production. Thus, the engineer has to be able to implement a robust servo-loop on a so-called prototype, whilst taking account of this broad spectrum in its entirety, at the very earliest stage of design.

An example of such a system, upon which most of the examples given in this book are based, is a mass-produced viewfinder, for which the automation engineer has to inertially stabilize the line of sight, whose usage conditions may be extremely varied – indeed there are often as many potential applications as there are types of carriers (aircraft, ships, etc.). In addition, the viewfinder is required to deliver increasingly high-end functionalities – e.g. target tracking, guidance, etc. In order to moderate and reduce development costs, there is a growing tendency to carry out so-called generic stabilizations. This is possible only if the servo-loop designed has a certain degree of robustness, which needs to be taken into account as an a priori constraint on synthesis.

In the 1990s, automation engineering made a great leap forward, with the emergence of H∞-based controller synthesis techniques:

– Firstly, it became possible to obey a complex set of frequency specifications by using frequency weighting functions on exogenous inputs and on monitored signals, and then minimizing the H∞ transfer norm between those signals by using a stabilizing controller whose state-space representation was explicitly formulated in [DOY 89], inspired by a dichotomy in the solution of Riccati equations (the so-called γ-iteration) and based on the following standard form:

Figure I.1. Standard form for control

Introduction-image001

where e represents the exogenous inputs (reference points, disturbances, etc.), z represents the signals being monitored (error signals, commands, etc.) and y represents the measurements used by the controller to calculate the command u.

– Secondly, the small-gain theorem gives us a necessary and sufficient condition for the stability of the loop obtained for any uncertainty Δ(s) such that ||Δ(s)||∞ < γ−1 . This is stable if and only if (iff) ||Tez (s)||∞ < γ, and in this knowledge, we can take account of objectives of robustness during the synthesis process.

Figure I.2. Standard form for robustness analysis

Introduction-image002

Thus, with the standard approach to robust control, the complexity of controller calculus – hitherto usually based on examination of the open loop – is now reflected in the complexity of determining the set of relevant frequency weights, which make a crucially important contribution to the performances of the final controller. Owing to the difficulty in calculating these weights, the know-how that this operation requires and the conceptual difference from conventional frequency automation engineering, certain engineers are deterred from using the standard approach to robust control, preferring to employ more conventional open-loop concepts.

However, at the same time, the world witnessed the publication of the explicit solution to the robust stabilization of normalized coprime factor plant descriptions [MCF 90], based on the following form.

Figure I.3. Robust coprime factor plant description stabilization

Introduction-image003

– This method, which is highly attractive because of its simplicity, consists of solving two LQG-type Riccati equations. In its 4-blocks equivalent representation, it is a particular case of the standard H∞ approach to robust control. Noting that we can model the direct and complementary sensitivity functions by modeling the open-loop response, and seeing that any loop transfer is proportional to those sensitivity functions, it is therefore possible to model any loop transfer by working on a single transfer – the open-loop response. This is the principle upon which loop-shaping synthesis is founded. Drawing inspiration from frequency-shaped LQG synthesis, we shape the singular values of the open-loop response using weighting functions on the input and output of the system, thereby creating a loop-shape for which a stabilizing controller can be calculated. This is the definition of H∞ loop-shaping synthesis.

– However, thanks to the notion of the gap metric (which expresses a distance between two systems in mathematical terms) as well as the small-gain theorem, the stability of the loop can be evaluated even before the controller has been explicitly formulated.

There is a growing interest in H∞ loop-shaping synthesis. Obviously, it is less general than the standard H∞ approach, because the number of degrees of freedom is constrained by the dimensions of the system. However, the adjustment of the input and output weighting functions on the basis of the concepts of conventional frequency automation makes the loop-shaping technique extremely attractive and easy to access – all the more so as it has the qualities of robustness which are inherent to H∞ techniques.

In Chapter 1, we introduce the loop-shaping approach by showing how to obtain a specification on the open-loop response of the servo-loop from a complex frequency specification on multiple loop transfers. Chapter 2 introduces the robust stabilization of a normalized coprime factor plant description. Along with the notion of the gap metric which we then introduce, it constitutes the basis for robust H∞ loop-shaping synthesis. Chapter 3 relates to two-degrees-of-freedom controllers (2 d.o.f controllers), and two techniques that are closely linked to H∞ loop-shaping synthesis are presented, thus greatly extending the possibilities for the use of the method. Finally, Chapter 4 opens up avenues for future work: it discusses the main drawbacks to loop-shaping synthesis, and how to solve these issues using modern optimization techniques.

I.2. Notations and definitions

Below, we review a number of fundamental notions and notations that are frequently employed in the various chapters of this book.

I.2.1. Linear Time-Invariant Systems (LTISs)

I.2.1.1. Representation of LTISs

An n-order linear time-invariant system with m inputs and p outputs is described by a state-space representation defined by the following system of differential equations:

Introduction-image004

where¹:

– x(t) ∈ Rn is the state of the system;

– x(t0 ) is the initial condition;

– u(t) ∈ Rm is the system input;

– y(t) ∈ Rp is the system output;

– A ∈ Rn×n is the state matrix;

– B ∈ Rn×m is the control matrix;

– C ∈ Rp×n is the observation matrix;

– D ∈ R p×m is the direct transfer matrix.

For a given initial condition x(t0), the evolution of the system’s state and its output is given by:

Introduction-image005

The system is stable (in the sense that it has bounded input/bounded output) if the eigenvalues of A all have a strictly negative real part, i.e. if:

Introduction-image006

where λi (A) is the ith eigenvalue of A.

For a zero initial condition, the input/output transfer matrix of the system is defined in Laplace form by:

Introduction-image007

For the sake of convenience, we represent this as:

Introduction-image008

or:

Introduction-image009

When H(∞) is bounded, H is said to be proper². When H(∞)=0, then the system is said to be strictly proper, and D = 0.

Finally, for the same transfer matrix, there are an infinite number of possible state-space representations. Indeed, consider the linear transformation T ∈ Rn×n, where T is invertible, such that:

Introduction-image010

In this case, the initial state-space representation becomes:

Introduction-image011

The corresponding transfer function is:

Introduction-image012

I.2.1.2. Controllability and observability of LTISs

The system H or the pair (A,B) is said to be controllable if, for any initial condition x(t0) = x0, for any t1 > 0 and for any final state x1, there is a piecewise continuous command u(.) which can change the state of the system to x(t1) = x1.

We determine controllability by checking that for any value of t > t0, the controllability Gramian Wc(t) is positive definite:

Introduction-image013

An equivalent condition is that the matrix (B AB A² B … An−1B) must be full row rank, i.e n.

The system H or the pair (C, A) is observable if, for any value of t1 > 0, the initial state x(t0) = x0 can be determined by the past values of the control signal u(t) and of the output y(t) in the interval [t0, t1].

We determine observability by checking that, for any value of t > t0, the observability Gramian Wo(t) is positive definite:

Introduction-image014

An equivalent condition is that the matrix:

Introduction-image015

must be full column rank, i.e. n.

I.2.1.3. Elementary operations on LTISs

Consider H, the transfer system:

Introduction-image016

The transpose of H is defined by the system:

Introduction-image017

The conjugate of H is defined by the system:

Introduction-image018

If D is invertible, the inverse of H is defined by the system:

Introduction-image019

Now consider two systems H1 and H2, whose respective state representations are:

Introduction-image020

The serial connection of H1 with H2 (or the product of H1 by H2) gives us the system:

Introduction-image021

The parallel connection (or addition) of H1 to H2 gives us the following system:

Introduction-image022

The looping of H2 with feedback from H1 gives us the system:

Introduction-image023Introduction-image024

where R12 = I + D1D2 and R21 = I + D2 D1.

Many notions about linear time invariant systems are explained in [ZHO 96].

I.2.2. Singular values

I.2.2.1. Definition

The singular values of a transfer matrix H(s) of dimensions p×m are defined as the square roots of the eigenvalues of the product of its frequency response H(jω) by its conjugate:

Introduction-image025

The singular values are positive or null real numbers and can be classified. The largest singular value, also called the maximum singular value, is denoted as xvii (H), and the smallest, also called the minimum singular value, is denoted as xviia (H).

Introduction-image025a

In the case of a monovariable system (i.e. m=p=1), the unique singular value is equal to the gain of the frequency response:

Introduction-image026

Hence, the singular values extend the notion of gain established with monovariable systems to multivariable systems. We say that H is high-gain if xviia (H) is large and is low-gain if xvii (H) is small.

I.2.2.2. Properties

In this book, we make abundant use of the following properties:

Introduction-image027

In the case of two parallel systems, we use the following properties:

Introduction-image028

In the case of two serial systems, an important property is:

Introduction-image029

In particular, we shall use the following specific cases:

Introduction-image030

In the case of the sum of two systems, an important property is:

Introduction-image031

In particular, we shall use the following two specific cases:

Introduction-image032

which lead us to:

Introduction-image033

or indeed:

Introduction-image034

Finally, we use the following property:

Introduction-image035

The interested reader