Faculty of Engineering
PhD in Aerospace Engineering XX Cycle PhD ThesisH
∞Loop-Shaping Control
of Aerospace Systems
Supervisor CandidateProf. Giovanni Mengali Renato Panesi
Director of PhD Course Prof. Maria Vittoria Salvetti
>AeÐ ti kainän mèra paideÔetai
(Sempre ogni giorno fa imparare qualcosa di nuovo) Euripide
Loop-Shaping design with H∞ controller synthesis plays a role of primary
importance for the synthesis of robust multi-variable controllers. However, a typical difficulty is that the designer has to develop considerable expertise in this matter before obtaining good results. In fact, the choice of the weighing matrices, that are at the heart of the loop-shaping design, is not an easy issue.
The aim of the work described in this dissertation is to improve the well known H∞loop-shaping approach, by simplifying the shaping procedure and
by extending its range of effectiveness, through some useful additions helping the designer in the controller synthesis.
The first major improvement consists in the definition of a scaling method aiming to minimize the condition number of the plant. This simplifies the choice of weights and increases the closed-loop robustness.
A second important addition is the development of a design procedure help-ing the designer in the selection of non-diagonal weights, which seems the best solution to overcome difficulties related to plants with highly coupled dynamics.
A Matlab software package has been created, which implements the proposed design procedures and assists the designer by simplifying the shaping activity and reducing his/her amount of work.
The exposed design methodologies are validated through some case stud-ies, where typical problems of flight control system design are dealt with. These applications are illustrated with the aid of a Matlab/Simulink lin-earized model of an aircraft.
La tecnica del loop-shaping con stabilizzazione robusta H∞ gioca un ruolo di
fondamentale importanza nel campo delle metodologie di sintesi di sistemi di controllo robusti multi-variabile. Tuttavia, essa presenta una difficolt`a, con-sistente nel fatto che il progettista deve accumulare una elevata esperienza in merito per poter ottenere risultati soddisfacenti. Infatti, la scelta delle matrici peso, che costituiscono il cuore della tecnica del loop-shaping, non `e un compito facile.
L’obiettivo del lavoro descritto in questa tesi `e proprio quello di migliorare la ben nota metodologia di sintesi di sistemi di controllo tramite H∞
loop-shaping, rendendo pi`u semplice la procedura di definizione delle funzioni peso ed aumentandone la versatilit`a: ci`o `e reso possibile grazie ad alcune aggiunte, apportate alla procedura, che assistono l’utente nelle varie fasi del progetto. Un primo notevole miglioramento consiste nella definizione di una procedura di scaling del sistema in esame, volta a minimizzare il numero di condiziona-mento del sistema stesso. Il raggiungicondiziona-mento di questo obiettivo, a sua volta, semplifica la scelta delle matrici peso e aumenta la robustezza del sistema in anello chiuso.
Un’altra importante modifica `e rappresentata dalla creazione di una metodolo-gia di progetto che assista il progettista nella costruzione di matrici peso non diagonali: queste ultime rappresentano di fatto l’unica soluzione efficace nel caso di sistemi dalle dinamiche altamente accoppiate.
Le procedure esposte sono state implementate in un pacchetto software, sviluppato in ambiente Matlab e concepito per assistere il progettista, sempli-ficando l’attivit`a di selezione delle funzioni peso e quindi riducendo lo sforzo richiesto all’utente.
La validit`a delle metodologie di progetto proposte `e dimostrata tramite al-cuni esempi applicativi, in cui sono trattati tipici problemi di progetto di sistemi di controllo del volo.
Queste applicazioni sono illustrate con l’ausilio di un modello di simulazione linearizzato di velivolo, realizzato in Matlab/Simulink.
Desidero approfittare di questo spazio per salutare e ringraziare le persone che hanno in diversi modi contribuito alla stesura di questa tesi, o che mi hanno aiutato e sostenuto durante il mio dottorato.
Ho sempre pensato che il corso di dottorato di ricerca sia un’esperienza al-tamente formativa, perch´e insegna a muovere i primi passi nell’affascinante mondo della ricerca scientifica. Sono quindi convinto che rappresenti una grande opportunit`a per un giovane laureato. `E con questo spirito che ho prima affrontato il concorso di ammissione e poi vissuto questi anni di studi. All’inizio sembrava che le cose non andassero per il verso giusto, poi una serie di eventi pi`u o meno favorevoli ha fatto in modo che potessi iniziare la mia avventura. Se sono riuscito ad arrivare in fondo a questa esperienza, lo devo alla mia tenacia, alla mia determinazione e al sostegno di alcune persone che mi hanno aiutato lungo tutto il mio percorso.
Ringrazio dunque, innanzitutto, la persona che pi`u di ogni altra ha reso pos-sibile la realizzazione di questo mio sogno: il Prof. Mengali. Ti ringrazio, Giovanni, per i tuoi saggi consigli, per i tuoi suggerimenti esperti e per il tempo che mi hai dedicato, spesso in orari scomodi, a correggere quello che scrivevo o a verificare i risultati che ottenevo con i miei studi. Ti ringrazio per la tua pazienza e per le tue parole di sostegno e conforto quando cre-devo di non farcela. Ma, soprattutto, ti ringrazio per aver capito fino in fondo le mie motivazioni, il mio entusiasmo: pi`u che per ogni altro motivo, ti ringrazio profondamente perch´e hai compreso la passione che mi spingeva, l’hai assecondata e mi hai dato l’opportunit`a che cercavo. In questi quattro anni abbiamo spesso avuto modo di confrontarci, di conoscerci meglio e credo che sia nato un buon rapporto di amicizia. Spero che in futuro ci sia modo di proseguire lungo questa strada e che il mio dottorato rappresenti solo l’inizio di una collaborazione che possa durare nel tempo.
Un saluto va poi a tutti coloro che, al dipartimento, mi hanno dato una mano nei modi pi`u disparati. Un ringraziamento particolare ad Alessandro, per aver ascoltato sempre con pazienza i miei problemi e per i suoi preziosi consigli.
Anche i miei genitori hanno avuto un ruolo importante. Ringrazio mia madre per i suoi consigli nella stesura della tesi in lingua inglese, ed entrambi per avermi sempre sostenuto e confortato. Grazie, soprattutto, per aver sempre appoggiato ogni mia scelta.
Infine, un forte e caloroso abbraccio va a Martina. Inizio ringraziandoti per il tuo sostegno ”logistico”, visto che stando al dipartimento mi hai aiutato a sbrigare diverse pratiche amministrative e mi hai sempre messo al corrente delle novit`a e delle notizie importanti che potevano essere di mio interesse.
Ti ringrazio poi per il tuo apporto tecnico, per avermi aiutato ad affrontare alcuni problemi e per averne discusso con me chiarendomi le idee. Ma tu sai benissimo che non sono questi i motivi principali per cui ti devo tutta la mia gratitudine. Con te mi sono spesso sfogato, mi sono lamentato, ho tirato fuori tutto quello che avevo dentro e tu sei sempre stata l`ı ad ascoltarmi. Con te ho discusso dei miei dubbi, ho affrontato le mie paure, ho costruito le mie certezze. In ogni caso, tu c’eri. Grazie, per tutto quello che hai fatto e fai per me.
List of Symbols 1
I Design Methodologies 4
1 Introduction 5
1.1 Subject . . . 5
1.2 Highlights of the Thesis . . . 6
1.3 Historical Background . . . 8
2 Aerospace H∞ Robust Control Systems 10 2.1 Introduction . . . 10
2.2 Design Specifications . . . 10
2.3 H∞ Norm and its Importance . . . 13
2.4 H∞ Mixed Sensitivity Controller Design . . . 14
2.4.1 General Control Problem Formulation . . . 14
2.4.2 Linear Fractional Transformations . . . 15
2.4.3 Controller Design . . . 16
2.4.4 Solution of the H∞ control problem . . . 21
2.5 H∞ Loop-Shaping Controller Design . . . 22
2.5.1 Design Specifications for H∞ Loop-Shaping . . . 22
2.5.2 Coprime Factorization and Model Uncertainties . . . 25
2.5.3 Derivation of H∞ Loop-Shaping Methodology . . . 27
2.5.4 Controller Design Methodology . . . 30
2.5.5 Advantages of H∞ Loop-Shaping . . . 32
3 H∞ Loop-Shaping: Design Procedures 33 3.1 The Classical Design Procedure . . . 33
3.1.1 Introduction . . . 33
3.1.2 The Procedure . . . 34
3.1.3 Comments . . . 37
3.2 Optimum Scaling Procedure . . . 38
3.2.1 The Importance of Scaling . . . 38
3.2.2 Condition Number . . . 39
3.2.3 Relative Gain Array . . . 40
3.2.4 The Procedure . . . 41
3.2.5 Remarks . . . 44
3.3 Non-Diagonal Weights Design . . . 45
3.3.1 Introduction . . . 45
3.3.2 Design Procedure . . . 46
3.3.3 Co-Spectral Factorization of Ψ . . . 50
3.3.4 Remarks . . . 52
II Applications 54 4 Aircraft Mathematical Model 55 4.1 ADMIRE . . . 55
4.1.1 Introduction . . . 55
4.1.2 Model Geometric, Inertial and Aerodynamic Data . . . . 56
4.1.3 Aircraft Dynamic Model . . . 57
4.2 Trimming and Linearization . . . 59
5 Case Studies 61 5.1 Design of a V and γ Longitudinal Controller . . . 61
5.2 Design of a Pitch Pointing Control System . . . 72
5.3 Design of a Lateral Control System for Coordinated Turns . . . 86
6 Conclusions 99 A ADMIRE - Mathematic Model 103 A.1 Aircraft Dynamic model . . . 103
A.2 Flight envelope . . . 104
A.3 Software Description . . . 106
A.4 Remarks . . . 106
B Loop-Shaping Design Tool: Software Description 107
List of Tables 120
A = state matrix
B = input matrix
C = output matrix
D = feedforward matrix
DI = input diagonal scaling matrix
DO = output diagonal scaling matrix
Fl = lower linear fractional transformation
Fu = upper linear fractional transformation
G = plant transfer function
GS = shaped plant transfer function
G? = scaled plant transfer function
H = Hamiltonian matrix
I = identity matrix
K = controller
K∞ = feedback H∞ controller
M = Mach number
Ml = transfer matrix, factor of a coprime factorization of a plant
Nl = transfer matrix, factor of a coprime factorization of a plant
P = generalized plant transfer function SI = input sensitivity function
SO = output sensitivity function
TI = input complementary sensitivity function
TO = input complementary sensitivity function
Tss = throttle stick setting
U = matrix of left singular vectors of G V = matrix of right singular vectors of G
or flight velocity (m/s), according to the context W1 = pre-compensator
W1r = reduced order pre-compensator W2 = post-compensator
Wa = align matrix
Wg = matrix that provides control over actuator usage
Wp = matrix that realizes the dynamic shaping of G
X = matrix, solution of Riccati equation dI = input disturbance
dO = output disturbance
h = flight altitude (m)
j = imaginary unit
m = measurement noise
p = roll rate (deg/s)
p = uncertainty parameter vector q = pitch rate (deg/s)
r = reference input
r = yaw rate (deg/s)
t = time (s)
u = control signal
u = vector of control variables (in the generalized plant description) or input vector (in the state-space) according to the context v = vector of sensed outputs (in the generalized plant description) w = vector of exogenous inputs
xv = longitudinal position in local vertical reference frame (m)
x = state vector
y = plant output
y = output vector
yv = lateral position in local vertical reference frame (m)
zv = vertical position in local vertical reference frame (m)
z = vector of exogenous outputs
Γ = transfer matrix that shapes Σ reflecting the conditioning of W1
Λ = relative gain array
Π = transfer matrix that shapes Σ reflecting the conditioning of W2
Σ = matrix containing singular values of G ΣS = matrix containing shaped singular values
Ψ = auxiliary matrix, see equation (3.22) α = incidence angle (deg)
β = sideslip angle (deg) γ = flight path angle (deg)
γ = H∞ cost function
δe = elevon deflection
δr = rudder deflection
= robust stability margin θ = pitch attitude angle (deg)
κ = condition number
κ? = minimized condition number σi = i-th singular value
σ = maximum singular value σ = minimum singular value φ = roll angle (deg)
ψ = yaw attitude angle (deg)
ω = frequency (rad/s)
ωH = upper limit of frequency range (rad/s)
ωL = lower limit of frequency range (rad/s)
ωd = design frequency (rad/s)
Subscripts
c = commanded actuator deflection
i, j = indexes (row,column) of a matrix entry − = spectral factor of a transfer matrix
Superscripts
T = transpose
∗ = conjugate transpose
∼ = pertransposed system, A∼(s) = A(−s)T
? = scaled variable
Chapter
1
Introduction
1.1
Subject
In the preface of his book [1] Zhou wisely observes that ”robustness of control systems to disturbances and uncertainties has always been the central issue in feedback control. Feedback would not be needed for most control systems if there were no disturbances and uncertainties. Developing multi-variable robust control methods has been the focal point in the last two decades in the control community. The state-of-the-art H∞robust control theory is the
result of this effort”. These few remarks describe in a very effective way the importance of H∞robust control in the field of control engineering and allow
one to perceive how the studies and the use of controller design methods, based on H∞ approach, are extremely diffused.
As it is reminded in section 1.3, H∞ control has got many different
appli-cations, in both the academic and the industrial communities, and in the aerospace field too. As a matter of fact, an aircraft must work in a wide vari-ety of flight conditions inside its flight envelope and the presence of external disturbances is continuous. Therefore, it is very unlikely for a linear model, created to reproduce mathematically the real physical plant, to be suitable to adequately simulate the aircraft dynamics at each point of the flight en-velope. Moreover, available multi-variable control design methods are useful especially in the aerospace field, since it is often required to simultaneously control two or more variables, ensuring in the meanwhile a good decoupling between them.
Among the available design methodologies, the so-called H∞ loop-shaping
plays a role of primary importance. This technique, born in the early nineties, became popular and attractive, especially in the aerospace field, because of its properties and advantages; basically, as described in Chapter 2, the method
consists in a shaping, through weighing functions, of the system open-loop transfer matrix L instead of a simultaneous shaping (as usually done in H∞
control) of closed-loop transfer functions S, KS and T which is much more difficult and imposes to manage trade-offs between conflicting requirements. Robustness to unstable perturbations and uncertainty is maximized with re-spect to normalized coprime-factor uncertainties, rather than multiplicative or additive ones. This uncertainty description is very useful, since it is quite general and has some remarkable advantages (fully described in section 2.5.2) with respect to the multiplicative or additive models. Furthermore, H∞
loop-shaping permits to avoid iterative design processes, very expensive from both the computational and the temporal point of view.
However, a limitation in its use is due to the fact that, even if one can find in literature many guidelines that help him/her to design a control system, the designer must develop considerable expertise in this matter before good results are achieved.
In such a context the work illustrated in this dissertation finds its place. The goal of the work consists in an improvement of the existing control systems design procedures that make use of the H∞loop-shaping concepts; the main
task is the achievement of better results, in terms of performance of the de-signed control system, reducing in the meanwhile both the effort required to the designer and the time needed to carry out the design procedure.
1.2
Highlights of the Thesis
Beginning from the state-of-the-art H∞loop-shaping multi-variable controller
design procedure, the result of the work described in this dissertation is the development of some additions to be applied to such a procedure, in order to make its use easier and improve its performance. The final outcome consists in a straightforward step-by-step controller design methodology that allows one to design an H∞ loop-shaping based control system in a short time and
with a small work load. Many steps of the procedure are automatized, and the activities that are left to the designer are simplified, thus allowing to achieve satisfactory results even without a considerable expertise.
The main advantages introduced by the proposed design procedure are sum-marized below.
• The open-loop plant is scaled so that its conditioning is reduced as much as possible for frequencies close to the desired crossover frequency. Such an optimum scaling is obtained by means of a dedicated minimization process. The designer can optionally choose adequate constraints re-garding the plant inputs and outputs (depending on sensors and actu-ators capabilities and project requirements).
• The possibility to easily design non-diagonal weighting matrices has been introduced. This is particularly helpful in the case of plants with highly coupled dynamics, where diagonal weighting functions do not guarantee the desired loop shape to be achieved. Non-diagonal pre-and post-compensators are computed automatically, beginning from diagonal matrices that the designer is asked to define to adequately shape the matrix of singular values of the open-loop system. Therefore, the work to be carried out by the designer results extremely simplified, since each singular value of the nominal plant can be shaped separately.
The controller design procedure has been implemented through a set of ded-icated Matlab functions, that have been created to carry out the various steps. A main Matlab script, that makes use of such functions, has been de-veloped to assist one in the complete design of the controller. The execution of such script allows the designer to achieve a solution with a high level of automation. Once the controller is found, its order can be reduced, if desired, through several functions already provided in Matlab.
The dissertation consists of two parts. The first one is dedicated to the ex-position of the theoretical features and is organized as follows:
Chapter 2 provides an introduction to H∞control and a description of the
main existing methodologies for control systems design.
Chapter 3 begins with a description of the state-of-the-art H∞loop-shaping
controller design procedure, that represents the starting point of this work, then the proposed additions are discussed.
Afterwards, section 3.2 deals with the first addition to the design procedure, that is the optimum scaling methodology: a full description is provided and the advantages of this strategy are emphasized.
Finally, section 3.3 discusses how non-diagonal weighting matrices can be selected to overcome the difficulties related to a possible cross-coupling be-tween the controlled variables. This is done by presenting the complete and revised controller design procedure. Further minor improvements contained in the procedure, as the reduction of the order of the controller model, are discussed.
The second part of the dissertation is dedicated to the applications; a mathe-matical model of an aircraft has been purposely chosen and, once linearized, the developed control systems design procedures have been verified with the above mentioned Matlab script.
Chapter 4 summarizes ADMIRE (Aero Data Model in a Research Environ-ment), the utilized mathematical model corresponding to a single seat fighter aircraft in delta-canard configuration.
procedure is verified in the ambit of the design of a multi-variable longitu-dinal control system. Then, the non-diagonal weights design procedure has been used in the development of both a pitch pointing control system and a lateral controller for coordinated turns.
Chapter 6, finally, states the conclusions, summarizing the results of this work, and outlines the possible future developments.
Two Appendices complete this dissertation: Appendix A gives a more de-tailed description of the used aircraft mathematical model, whereas Ap-pendix B describes the software package that has been developed and that implements the full controller design procedure.
1.3
Historical Background
Even if an earlier application of H∞ optimization can be found in 1976 in
a work of Helton [2], the main studies about its use for robust control be-gan in the 1980’s, originating from a first important contribution of Zames (1981, [3]), and gave remarkable results at the end of that decade: see, for example, the works by Glover and Doyle (1988, [4] and 1989, [5]), that treat the derivation of the H∞ solution for the general case. Subsequently, many
other important contributions were brought: as an example, see the works by Stoorvogel (1992, [6]), Packard (1994, [7]), Chen and Zhou (1996, [8]). During the development of the H∞ theory, it was understood that strong
relations exist between the two different approaches of H∞ and H2 control
(as observed by Doyle et al.,1989, [9]), so that nowadays the two theories are often introduced together in the literature (as, for example, in Refs. [1], [10]). The H∞ loop-shaping design methodology treated in this dissertation
com-bines the H∞ robust stabilization with the classical loop-shaping. This
ap-proach is possible by adopting a description of the H∞ robust stabilization
problem that uses normalized coprime factorization, as presented by Glover and McFarlane in 1989, [11]. The H∞loop-shaping strategy using normalized
coprime factorization was proposed by McFarlane and Glover (1990, [12] and 1992, [13]). A step-by-step procedure for H∞ loop-shaping design was
devel-oped in 1991 by Hyde [14], and the method was successfully applied to the design of scheduled control systems for a VSTOL (vertical and short take-off and landing) aircraft by Hyde and Glover in 1993 [15]. Subsequently, the method was applied to a wide variety of aerospace applications: see for ex-ample Refs. [16] and [17] (helicopters), Ref. [18] (VSTOL aircrafts), Ref. [19] (satellites), Ref. [20] (missiles) and Ref. [21] (wind tunnel model).
Further contributions helped to extend the range of applicability of Hyde’s procedure. Among these, a paper of Limebeer et al. (1993, [22]) showed how the design procedure can be extended by introducing a second
degree-of-freedom in the controller, while a work of Papageorgiou and Glover (1997, [23]) introduced a methodology for designing non-diagonal weights, to facilitate the shaping of transfer matrices with coupled dynamics (the paper represents the starting point for the development of the non-diagonal weights design proce-dure described in this dissertation).
A good description of H∞ loop-shaping design methodology peculiarities,
emphasizing the advantages of such a design strategy in the ambit of robust flight control, can be found in another work of Papageorgiou and Glover [24]: this paper is particularly significant, not only because it describes how the H∞ loop-shaping design methodology is effective to design robust flight
con-trollers, but also because the industrial viewpoint is discussed and the ease with which controllers can be implemented is pointed out.
Chapter
2
Aerospace H
∞
Robust Control Systems
2.1
Introduction
An introduction about existing multi-variable robust control design tech-niques that are used in the aerospace field is necessary to better understand how H∞ loop-shaping works, to appreciate its strong and weak points and
to compare it with other design techniques. Therefore, this Chapter is de-voted to a general description of the most important topics regarding robust control: first, a description of typical design specifications is given, then the most diffused control systems design techniques are introduced. After that, we will go through a description of the main concepts of H∞ loop-shaping,
showing how design requirements can be reviewed in a way that makes their management easier, and emphasizing the advantages of such a design tech-nique. The sections below follow the treatment of Refs. [10] and [25].
2.2
Design Specifications
Any aerospace control system is subject to the following main design speci-fications:
• Robust stability: the linear controller is required to provide a suf-ficient stability against both structured and unstructured uncertainty. A set of linear models of the plant must be built, outlined by a well defined structured uncertainty model, and an adequate stability mar-gin must be guaranteed for all these models, even against structured uncertainty.
noises, together with a good reference tracking and a sufficient decou-pling between commands are required to be provided by the linear controller. These requirements must be satisfied for the nominal lin-earized model of the plant (nominal performance) and for the other linear models in the set defined by the structured uncertainty model (robust performance).
For multi-variable systems, if we limit ourselves to consider only the nominal linearized model of the plant, these design requirements can be expressed in terms of singular values. As a matter of fact, the design of a MIMO con-troller involves the shaping of appropriate multi-variable transfer functions, that is the shaping of singular values of such transfer functions. It follows that specifications on robustness can be formulated in terms of singular val-ues, more in detail by defining adequate bounds on the minimum and/or maximum singular values of various closed-loop transfer function matrices. This implies that design requirements are defined in the frequency domain, taking into account the corresponding specifications in the time domain. To express our design objectives, we consider a generic one degree of freedom multi-variable feedback control system, like the one shown in Figure 2.1. Be-ing TI = K G/(I + K G)−1, TO= G K/(I + G K)−1, SI = I/(I + K G)−1and
SO = I/(I + G K)−1, the following fundamental equations can be written for
such a system: G K dO dI m u r y
-Figure 2.1: Multi-variable feedback control system
y = TOr + SOG dI+ SOdO− TOm (2.1)
u = K SOr − K SOdO+ SIdI − K SOm (2.2)
These equations allow one to define the nominal specifications by considering input and output sensitivity and complementary sensitivity functions, SI,
SO, TI, TO. In particular, the maximum singular values of such closed-loop
transfer function are considered. The following specifications can be set:
1. For an adequate reference tracking both σ(TO) and σ(TO) are required
to be close to 1.
2. Attenuation of disturbances:
• For attenuation of output disturbances at the plant output σ(SO) should be small.
• For attenuation of output disturbances at the plant input σ(KSO)
should be small.
• For attenuation of input disturbances at the plant output σ(SOG)
should be small.
• For attenuation of input disturbances at the plant input σ(SI)
should be small.
3. Attenuation of noises:
• For attenuation of measurement noise signals at the plant output σ(TO) should be small.
• For attenuation of measurement noise signals at the plant input σ(KSO) should be small.
4. A reduction of control energy (avoidance of large control signals) can be obtained with σ(KSO) small.
The presence of unstructured uncertainties imposes to add to the nominal specifications the following objectives, to satisfy the requirements for robust stability:
1. For robust stability to input multiplicative uncertainty σ(TI) is required
to be small.
2. For robust stability to output multiplicative uncertainty σ(TO) is
re-quired to be small.
3. For robust stability to additive uncertainty σ(KSO) is required to be
small.
Most of these requirements are conflicting, therefore they cannot be satisfied simultaneously. This is because
SI+ TI = I (2.3)
SO+ TO = I (2.4)
It follows that it is necessary a trade-off that will depend on the objectives of the design under consideration. However, one must consider that differ-ent design requiremdiffer-ents are important over differdiffer-ent frequency ranges (as an example, disturbances attenuation is typically required at low frequencies, whereas noise attenuation is important at high frequencies) and this makes the analysis easier.
In addition to the above mentioned specifications, it is important to take into account that the achievable performance of a feedback control system (in presence as well as in absence of system uncertainties) can be limited by other factors, depending on actuators capabilities (limitations in control signal rates and amplitudes may occur) and on the presence of right half-plane-zeros in the plant transfer function (as described in detail in Ref. [10]). Moreover, it must be noticed that, as assumed before, the above mentioned design requirements reflect only design specifications related to the nominal linearized plant, thus providing robustness guarantees (in terms of stability and performance) only to the nominal model. This limitation arises because it is impossible to give design specifications, in terms of singular values, about robust stability and performance of all available linear models of the aircraft that describe the structured uncertainty model. Furthermore, the knowledge of such a structured uncertainty affecting the model (that is the information about the difference between the design model and the real aircraft consid-ered a the design point) is usually poor and this makes very difficult to define design objectives for the uncertain system.
In spite of that, H∞ design methodologies permit in most cases to achieve
good results in terms of robustness against both structured and unstructured uncertainty with a relatively simple synthesis methods and structures of the feedback controller. A very powerful method is the mixed sensitivity H∞
controller design technique, described in section 2.4.
2.3
H
∞Norm and its Importance
The H∞ norm of a stable multi-variable Linear Time Invariant (LTI) system
G(s) is defined as the peak value over frequency of the largest singular value of the frequency response G(jω):
|G(s) |∞= max
The singular values Bode magnitude plot of a stable LTI system represent an effective way to display its H∞ norm.
Many different time domain interpretations and implications can be argued about the H∞ norm, all discussed in detail in Refs. [1], [10]. It is here
important to remember that H∞ norm represents a measure of the largest
possible steady-state gain of a system for sinusoidal inputs of any frequency. Moreover, for an LTI system G(s) with w and z respectively as input and output, the H∞ norm is equal to the induced worst-case 2-norm in the time
domain. In practice, the following relationship holds:
|G(S) |∞= max w6=0 |z(t) |2 |w(t) |2 (2.6) where |z(t) |2 = q R∞ 0 P
i|zi(t)|2dt is the 2-norm of the vector signal.
Therefore, minimizing the H∞ norm of a system amounts to minimizing the
energy related to the worst-case output signal vector: the output energy is then minimized over all the non-zero finite energy input signals.
Equation (2.5) shows that the H∞ norm is the peak of a transfer function
magnitude. Thus, in a control design perspective, if we introduce suitable weights we can consider the H∞norm as the magnitude of a specified
closed-loop transfer function relative to a defined upper bound. This last consider-ation leads to define the system performance in terms of weighted sensitivity or mixed sensitivity, as described in the next section.
2.4
H
∞Mixed Sensitivity Controller Design
2.4.1 General Control Problem Formulation
The scheme of Figure 2.2 shows an alternative configuration to describe a generic closed-loop control system. This formulation of the general control problem has been introduced by Doyle [26], [27]. P (s) and K(s) are re-spectively the generalized plant and controller, whereas w is the vector of exogenous inputs (pilot commands, disturbances, noises), z is the vector of exogenous outputs, v is the vector of measured outputs and u represents the vector of control variables. Positive feedback is used.
The purpose is to minimize a norm (as an example, the H∞ norm) of the
transfer function from w to z. This result is obtained by finding a controller K that, on the base of the measures fed back from the plant by means of the vector v, generates a control signal u able to counteract the influence of w on z, thus minimizing the closed-loop norm from w to z.
P
K
u
v
w
z
Figure 2.2: Robust multi-variable controller: general configuration
in section 2.4.3, whereas a full description can be found in Ref. [10], pp. 99. What is important to outline here is that almost any linear closed-loop con-trol system can be described by use of the block diagram of Figure 2.2.
2.4.2 Linear Fractional Transformations
Consider the general configuration of Figure 2.2. The transfer matrix P can be partitioned in the form
P = P11 P12 P12 P22 (2.7)
in such a way that
z v = P11 P12 P12 P22 w u (2.8) u = k v
After some algebra, it is possible to obtain
z = [P12K (I − P22K)−1P21+ P11] w (2.9)
If we define
Fl(P, K) , P12K (I − P22K)−1P21+ P11 (2.10)
one has
Fl(P, K) is called lower linear fractional transformation (LFT) of P and K.
A system in its state-space form can be represented as an LFT; in fact, if we let P = D C B A , K = 1 s
by substituting such expressions in equations (2.8), we can easily get
Fl(P, K) = C (s I − A)−1B + D (2.12)
The use of the LFT to express the transfer function from w to z is funda-mental in the standard H∞optimal control problem. In fact, this is based on
the minimization of the H∞ norm of Fl(P, K) over all stabilizing controllers.
2.4.3 Controller Design
Consider the control configuration depicted in Figure 2.2. The H∞ control
consists in finding a controller K that minimizes the H∞ norm of Fl(P, K)
over all possible controllers that stabilize the plant in the closed-loop. There-fore, K has to be determined so that
inf
K |Fl(P, K)(jω) |∞= infK maxω σ(Fl(P, K)) = infK maxw6=0
|z(t) |2 |w(t) |2
(2.13)
The computation of such an optimal H∞controller is a hard task, but usually
it is not necessary, whereas it is sufficient and much more simple to compute a sub-optimal controller. This can be done by use of an iterative procedure. Let γminbe the minimum value of |Fl(P, K)(jω) |∞over all stabilizing controllers
K. In practice γ is assigned, so that γ > γmin, and the sub-optimal H∞
control problem consists in finding all stabilizing controllers K that satisfy the inequality
|Fl(P, K)(jω) |∞< γ (2.14)
The problem can be solved by using an algorithm developed by Doyle et al. [9]. The core is that, by reducing iteratively the value of γ, the designer comes near to an optimal solution. However, this step will be discussed in more detail in section 2.4.4 and we now concentrate on the first problem for the designer, that is how P can be determined, so that Fl(P, K) is equivalent to
the assigned configuration. The following example explains in detail how the problem can be treated, beginning from the given closed-loop configuration. Assume the control set-up of Figure 2.3.
G
K
d
r
y
-Figure 2.3: Given control configuration
P
G
K
w=d
z=y
-u
v
Figure 2.4: Equivalent control configuration
Such a configuration can be easily converted in the form of Figure 2.2; as a matter of fact, it is equivalent to the configuration of Figure 2.4, that is in the required form.
The following relationships hold:
z = I w + G u (2.15)
v = −I w − G u (2.16)
from which we obtain the expression of P
P = I G −I −G (2.17)
It is important to observe that P does not depend on K.
It is now possible to determine Fl(P, K) by substituting the obtained entries
of P in equation (2.10), then the controller K can be synthesized by solving equation (2.14). This last step can be conducted by following the already mentioned algorithm of Doyle et al., described in section 2.4.4.
However, before the controller synthesis, in order to satisfy the design re-quirements, it is often useful to apply weighting matrices to the generalized plant P . This is better clarified by the following example.
Suppose that a multi-variable system G is required to track a reference sig-nal r. Moreover, keeping in mind the design specifications of section 2.2, bounds must be set on σ(S) to guarantee a good performance in terms of disturbances rejection, on σ(T ) to improve robustness against multiplicative uncertainties and to make the system less sensible to measurements noises, on σ(KS) to contain the control energy and to improve robustness against additive uncertainties. Each one of these goals can be achieved by apply-ing adequate weightapply-ing functions WS, WT and Wu. Referring to the typical
closed-loop control system set-up, the configuration to be used is shown in Figure 2.5.
G
K
u
r
y
-W
uW
TW
Sv
z
1z
2z
3Figure 2.5: Application of weighting matrices
How can we select the weighting matrices to satisfy the design objectives? And how a controller K can be synthesized taking into account the presence of such weighting matrices?
The answer to these questions is that it is possible to achieve an expression of Fl(P, K) that directly correlates WS, WT and Wu to the closed-loop transfer
the iterative procedure to determine the sub-optimal controller K takes into account that WS, WT and Wu appear in the expression of Fl(P, K).
Referring to the block diagram in Figure 2.5, it is possible to state that
v = r − G u (2.18) z1 = WSr − WSG u (2.19) z2 = Wuu (2.20) z3 = WTG u (2.21) so, if we set z , z1 z2 z3 , v = v , w = r , u = u
we get the expression of the generalized plant:
z v = P w u (2.22) where P11= WSI 0 0 , P12= −WSG WuI WT G , P21= I , P22= −G.
The corresponding block diagram is depicted in Figure 2.6.
In practice, the weights appear inside the entries of P . By substituting these entries in equation (2.10), we obtain
Fl(P, K) = −WSG WuI WTG K (I + G K) −1 + WSI 0 0 (2.23) = −WSG K (I + G K)−1+ WSI WuK (I + G K)−1 WTG K (I + G K)−1
If we now remember that G K (I + G K)−1 = T , (I + G K)−1 = S and I − T = S, we finally get Fl(P, K) = −WSS WuK S WT T (2.24)
P G K r=w z -u v WS z1 Wu z2 WT z3
Figure 2.6: Equivalent control configuration
In conclusion, the problem of the H∞ control for the given configuration
consists in finding the controller K that minimizes the ∞-norm of a matrix ( |Fl(P, K) |∞) that weights σ(S), σ(KS) and σ(T ). For this reason, the
de-scribed control problem is called H∞ mixed sensitivity control.
It is here important to observe that weights Wi must be proper and stable.
This is a requirement of the algorithm to solve the H∞ problem of
equa-tion (2.14). The selecequa-tion of weighting matrices is left to the designer; how-ever, suggestions are available in literature helping in the choice of weights (as an example, some solutions are discussed in Refs. [10] and [25]).
For instance, a typical choice for WS is a diagonal matrix, whose entries WS i
are of the form
WS i=
s/M + ωB∗
s + ωB∗ A (2.25)
2.4.4 Solution of the H∞ control problem
As mentioned in section 2.4.3, the first step of the H∞ control problem
con-sists in finding a solution to the following inequality:
|Fl(P, K)(jω) |∞< γ (2.26)
with γ assigned. The problem can be solved using the algorithm developed in Ref. [9], which is summarized below. Consider the control configuration of Figure 2.2, with the generalized plant P known in its state-space form:
P = A B1 B2 C1 D11 D12 C2 D21 D22 (2.27)
Several assumptions about the entries of P are typically made in H∞
prob-lems, to give a simple form to the general algorithm. We consider valid such assumptions, that are fully described in Ref. [10], pp. 363.
Then, there exists a stabilizing controller K such that equation (2.26) is satisfied, if and only if the following conditions are verified:
1. X∞≥ 0 is a solution to the algebraic Riccati equation:
ATX
∞+ X∞A + C1TC1+ X∞(γ−2B1B1T− B2B2T) X∞= 0 (2.28)
such that Re λi[A + (γ−2B1B1T− B2B2T) X∞] < 0, ∀i;
2. Y∞≥ 0 is a solution to the algebraic Riccati equation:
A Y∞+ Y∞AT+ B1B1T+ Y∞(γ−2C1TC1− C2TC2) Y∞= 0 (2.29)
such that Re λi[A + Y∞(γ−2C1TC1− C2TC2)] < 0, ∀i;
3. ρ(X∞Y∞) < γ2
All existing controllers are then given by the following lower LFT:
K = Fl(Kc, Q) (2.30) where Kc= A∞ −Z∞L∞ Z∞B2 F∞ 0 I −C2 I 0 (2.31) with F∞= −B2TX∞ , L∞= −Y∞C2T , Z∞= (I − γ−2Y∞X∞)−1 (2.32)
A∞= A + γ−2B1B1TX∞+ B2F∞+ Z∞L∞C2 (2.33)
and Q is a stable and proper transfer function such that |Q |∞< γ. If we let
Q = 0, we obtain
K = −Z∞L∞(s I − A∞)−1F∞ (2.34)
that is called central controller, having the same number of states as the gen-eralized plant P . In order to achieve a controller that is close to the optimal solution, with a specified tolerance, the so-called γ-iteration is necessary. In practice, it is possible to perform a bisection of γ until its value is sufficiently close to γmin.
This algorithm is widely used and is implemented in existing software. For instance, Matlab function hinfsyn computes K and automatically performs the bisection.
For a more general case of H∞ algorithm, with assumptions relaxed with
respect to those in Ref. [10], pp. 363, the reader is referred to Ref. [4].
2.5
H
∞Loop-Shaping Controller Design
2.5.1 Design Specifications for H∞ Loop-Shaping
We now derive some relationships that enable us to define a set of specification to be used for H∞Loop-Shaping control systems design. To do this, consider
that the singular values of a generic matrix J have following properties (proofs are given in Ref. [10], pp. 505):
σ(J−1) = 1
σ(J ) (2.35)
σ(J ) − 1 ≤ 1
σ(I + J )−1 ≤ σ(J ) + 1 (2.36) If both σ(KG) > 1 and σ(GK) > 1, recalling that SI = I+K GI and SO =
I
I+G K, from the above equations we obtain:
1 σ(KG) + 1 ≤ σ(SI) ≤ 1 σ(KG) − 1 (2.37) 1 σ(GK) + 1 ≤ σ(SO) ≤ 1 σ(GK) − 1 (2.38)
therefore, the following relationships can be achieved:
σ(SI) 1 ⇔ σ(K G) 1 (2.39)
Moreover, being TI = I+K GK G and TO = I+G KG K , the following equivalencies
hold:
σ(TI) ≈ σ(KG) for frequencies at which σ(KG) 1 (2.41)
σ(TO) ≈ σ(GK) for frequencies at which σ(GK) 1 (2.42)
σ(SOG) ≈ 1/σ(K) for frequencies at which σ(GK) 1 (2.43)
σ(KSO) ≈ 1/σ(G) for frequencies at which σ(GK) 1 (2.44)
By use of equations (2.39) to (2.44), design objectives stated in section 2.2 can be rewritten in terms of input and output open-loop transfer matrices KG and GK. This helps to overcome the conflicting requirements expressed in terms of closed-loop transfer matrices, because it defines the frequency ranges for which every objective is valid. The new set of objectives is as follows:
1. For a good reference tracking σ(G K) is required to be large; valid for frequencies at which σ(G K) 1
2. Attenuation of disturbances:
• For attenuation of output disturbances at the plant output σ(G K) should be large; valid, for frequencies at which σ(G K) 1. • For attenuation of output disturbances at the plant input σ(K)
should be small; valid for frequencies at which σ(G K) 1. • For attenuation of input disturbances at the plant output σ(K)
should be large; valid for frequencies at which σ(G K) 1. • For attenuation of input disturbances at the plant input σ(K G)
should be large; valid for frequencies at which σ(K G) 1.
3. Attenuation of noises:
• For attenuation of measurement noise signals at the plant output σ(G K) should be small; valid for frequencies at which σ(G K) 1.
• For attenuation of measurement noise signals at the plant input σ(K) should be small; valid for frequencies at which σ(G K) 1.
4. A reduction of control energy (avoidance of large control signals) can be obtained with σ(K) small; valid for frequencies at which σ(G K) 1 To satisfy the requirements for robust stability, the following additional ob-jective are imposed:
1. For robust stability to input multiplicative uncertainty σ(K G) is re-quired to be small; valid for frequencies at which σ(K G) 1.
2. For robust stability to output multiplicative uncertainty σ(G K) is re-quired to be small; valid for frequencies at which σ(G K) 1.
3. For robust stability to additive uncertainty σ(K) is required to be small; valid for frequencies at which σ(G K) 1.
Figure 2.7: Typical design specification for singular values of the open-loop transfer matrix L = G K
Usually, reference tracking and disturbances attenuation are objectives typi-cal at low frequencies, whereas attenuation of noises and reduction of control energy are peculiar in high frequencies. Thus, the choice of K to shape the singular values of both G K and KG seems not to be a difficult task: Figure 2.7 shows how G K is usually shaped to satisfy the above described specifica-tions. As a matter of fact, closed-loop stability must be guaranteed too, and the fulfilment of this latter condition cannot be verified from the behavior of the singular values of an open-loop transfer matrix.
and phase at frequencies close to the crossover. Thus, the constraints im-posed by the closed-loop stability can be managed by obtaining an adequate roll-off rate of GK in the crossover region. An analogous relationship be-tween gain and phase in the crossover region is expressed for MIMO systems in terms of eigenvalues of GK, rather than singular values. Therefore, sta-bility constraints are much more difficult to manage.
In practice, what is done is to split the design activity in two separate tasks. First, the singular values of the plant are shaped according to the design spec-ifications, without verifying if the closed-loop stability is guaranteed. This last condition is met in the second task, when the H∞ robust stabilization
problem, applied to the shaped plant, is solved. Section 2.5.4 describes in detail this approach.
2.5.2 Coprime Factorization and Model Uncertainties
A plant transfer matrix G(s) can always be factored as in equation (2.45):
G = Ml−1Nl (2.45)
The above expression represents a left-coprime factorization of G if Ml(s)
and Nl(s) are both stable and coprime. The fact that Ml and Nl are stable
means that Mlmust contain all the unstable poles of G, whereas the fact that
they are coprime implies that there should be no common right-half plane zeros in Ml and Nl, which induce pole-zero cancellations when factorization
is built. Mathematically, this means that a couple of stable transfer matrices Ul(s) and Vl(s) exists satisfying the following Bezout’s identity:
MlVl+ NlUl= I (2.46)
Coprime Factorization of a system is not unique. However, a particular case occurs if Ul(s) and Vl(s) correspond to the pertransposed systems Ml∼(s) and
Nl∼(s) respectively:
Ul(s) = Nl∼(s) = NlT(−s) (2.47)
Vl(s) = Ml∼(s) = MlT(−s) (2.48)
In this case
MlMl∼+ NlNl∼= I (2.49)
and the coprime factorization is said to be normalized.
The normalized coprime factorization is widely used to describe systems un-structured uncertainties models and is useful to understand the concepts of H∞ loop-shaping. Assume a nominal plant G described by equation (2.45),
with Mland Nl satisfying equation (2.49), and let ∆M and ∆N be stable
un-known transfer matrices quantifying the whole of the uncertain parameters, defined so that the ∞-norm |
∆N ∆M |∞is smaller than some quantity
and allowing to describe the perturbed plant Gp as
Gp = (Ml+ ∆M)−1(Nl+ ∆N) (2.50)
If u and y are respectively the input and the output of Gp, that is
y = (Ml+ ∆M)−1(Nl+ ∆N) u (2.51)
after some algebra one can achieve the following relationship, describing the uncertain model: Mly = Nlu + w (2.52) where w = ∆N −∆M u y (2.53)
The corresponding block diagram is depicted in Figure 2.8.
M
l-1u
-y
D
Nw
N
lD
M + + +Figure 2.8: System uncertainty description based on normalized coprime fac-torization
The physical meaning of the normalized coprime factor uncertainty descrip-tion is less intuitive than that of the classical uncertainty models, additive and multiplicative. Nevertheless, this description reveals some advantages particularly important for the description of aerospace systems.
• Both additive and multiplicative uncertainty models cannot be used to describe situations where variations of some parameters cause a stable system to become unstable. This happens because the corresponding uncertainty matrices are required to be stable, meaning that all the models in each set of uncertain models must have the same number of right-half plane poles as the nominal plant. The above situations,
easily occurring in aerospace applications, can easily be represented using normalized coprime factor uncertainty models. A demonstration is given in Ref. [25], pp. 53.
• Normalized coprime factor uncertainty description allows to wisely de-scribe uncertainty in the locations of lightly damped resonant poles.
This form of unstructured uncertainty description is quite general, both sim-ple and flexible, and is very useful in applications. In detail, this uncertainty description is particularly suitable to be used in a controller design procedure where one of the objectives is to maximize the magnitude of the uncertainty, so that robust stability is guaranteed (this means to maximize , which there-fore represents the stability margin). Such a controller design methodology is the H∞ Loop-Shaping.
2.5.3 Derivation of H∞ Loop-Shaping Methodology
Assuming a zero reference signal, the closed-loop control configuration asso-ciated to the system of Figure 2.8 is shown in Figure 2.9. From such a block
M
l-1u
-y
D
Nw
N
lD
M + + +K
Figure 2.9: Closed-loop control configuration
diagram, it is possible to deduce that
y = Ml−1Nlu + Ml−1w (2.54)
Since Ml−1Nl= G and u = K y, we obtain
and finally we get
y = (I − G K)−1Ml−1w (2.56)
u = k (I − G K)−1Ml−1w (2.57)
The two latest relationships can be rewritten in the form u y = K I (I − G K)−1Ml−1w (2.58)
with w defined in equation (2.53). If we put
M = K I (I − G K)−1Ml−1 (2.59) ∆ = ∆N −∆M (2.60)
it is possible to conclude that u y = M w (2.61) w = ∆ u y (2.62)
These two latest equations can be represented in a block diagram by the M − ∆ structure shown in Figure 2.10, that is typical in robustness analysis.
M
w
D
⎡ ⎤ ⎢ ⎥ ⎣ ⎦u
y
Figure 2.10: Typical M − ∆ structure
The M − ∆ structure is widely used in studying the problem of robustness; M represent the known part of the system, that is the nominal plant and the controller, whereas ∆ represents the whole uncertainty elements present in the system.
It is important to observe that the M −∆ structure of Figure 2.10 is analogous to the P − K one, depicted in Figure 2.2. In fact, while the P − K structure
is related to the lower LFT Fl(P, K), the M − ∆ structure is associated to
an upper LFT. In particular, if we consider a vector of exogenous inputs r and a vector of exogenous outputs z, we can say that
z = Fu(M, ∆) r (2.63)
where Fu(M, ∆) is the upper LFT of M and ∆:
Fu(M, ∆) , M22+ M21∆ (I − M11∆)−1M12 (2.64)
Ref. [10], pp. 304, gives a full description of the problem of robust stability of the M − ∆ structure; it comes out a very important result for our purposes. Provided that the nominal system M is stable and the perturbations ∆ are stable, for all perturbations ∆ satisfying (see in section 2.5.2)
|
∆N ∆M |∞≤ (2.65)
the M − ∆ structure is robustly stable, with stability margin , if
|M |∞≤
1
(2.66)
Since the M − ∆ structure has been derived from the closed-loop system of Figure 2.9, we can deduce that the perturbed plant Gp is closed-loop stable
only if the nominal plant G is stable and the following inequality holds:
γ , | K I (I − G K)−1Ml−1|∞≤ 1 (2.67)
where is the stability margin compatible with the controller K. The pre-vious inequality can be easily obtained by substituting the value of M from equation (2.59) in equation (2.66).
It is important to observe that γ is the H∞norm from w to
u y
. The minimum value of γ satisfying inequality (2.67) is
γmin= inf K γ =
1 max
(2.68)
therefore γmin defines the maximum stability margin max of the closed-loop
system. Glover and McFarlane showed in Ref. [11] how γmin can be obtained
from a minimal state-space realization of the nominal plant G, by simply solving two algebraic Riccati equations. No iterations, usually needed to solve the general H∞ problem, are required.
K that, from now on, we will indicate with K∞. K∞ robustly stabilizes the
closed-loop system with a stability margin . In Ref. [12], formulae to easily determine a controller K∞ (the so called central controller), for a specified
γ > γmin, are available. These formulae are already implemented in existing
software, therefore the computation of K∞ does not represent a problem. In
practice, in the classical loop-shaping design procedure (described in section 3.1), this sub-optimal controller is synthesized, instead of the optimal one. Indeed, the optimal controller is more difficult to be computed and there is no practical advantage in using it. When max> > 25%, that is γmin< γ < 4,
the design is usually successful. This concept is cleared in the next section. Obviously, in a controller design perspective, the closed-loop robust stabiliza-tion provided by K∞is not sufficient. In fact, the controller should also allow
the system to track a reference signal by following the assigned specifications. The consolidated controller design methodology, that allows to get both the results, is fully described in the next section.
2.5.4 Controller Design Methodology
The loop-shaping design methodology here described is based on a combi-nation of the H∞ robust stabilization with classical loop-shaping. It was
proposed in the early nineties by McFarlane and Glover [12], [13] and, sub-sequently, has been subject of considerable developments and applications in the aerospace field. In essence, the methodology consists in a two-stage design process, as already mentioned in section 2.5.1. First, a pre- and a post-compensator W1 and W2 are designed and applied to the open-loop
plant to give the desired shape to the singular values of the open-loop fre-quency response. The resulting shaped plant is then robustly stabilized with respect to normalized coprime-factor uncertainties (the most general type of unstructured uncertainty) through H∞optimization (equation (2.67)), which
provides the controller K∞.
A commonly used and recommended set-up of the loop-shaping controller is depicted in Figure 2.11. The constant prefilter K∞(0) W2(0) is added to
ensure a zero steady-state tracking error.
It is very important to underline that the second step does not require any weight selection or development of an uncertainty model and that the two stages can be handled separately. In fact, McFarlane and Glover [12] gave a theoretical proof of the fact that if γ is small (which means a stability margin large), the shape of the singular values of the robustly stabilized plant is similar to that of the open-loop shaped plant: in other words the robust stabilization does not modify in a significant way the singular values of the shaped plant open-loop frequency response. In practice, as mentioned
G
W
1W
2K
∞K
∞(0)W
2(0)
r
−u
y
+Figure 2.11: Typical implementation of the H∞loop-shaping controller
in section 2.5.3, when γmin< 4 (max>0.25) the design is usually successful.
On the contrary, if γmin > 4, this means that the shape that W1 and W2
assign to the singular values of the augmented plant is not compatible with the closed-loop system robust stability: in this case the pre- and/or post-compensator must be modified.
Once K∞ is found, the obtained overall controller is given by
K = W1K∞W2 (2.69)
K∞ is a dynamic compensator of order equal to that of the shaped plant
(which means the order of G plus the order of the weighting matrices W1
and W2), therefore the order of the overall H∞ loop-shaping controller K is
equal to the one of G plus twice the order of the pre- and post-compensator. The following guidelines are usually taken into account in the selection of the entries of the weighting matrices:
• both W1 and W2 can usually be chosen as diagonal;
• high gains are typically required at low frequency, for a good reference tracking: this can be obtained if W1 contains integral action;
• a roll-off rate of about −20dB/decade is desirable in the crossover re-gion. Phase-advance can be placed in W1 to reduce the slope of the
frequency response near crossover;
• for a good disturbances rejection, high roll-off rates are required at high frequency: this can be obtained by adding phase-lag to W1;
• W2 is often set as the identity matrix. Alternatively, it is diagonal of constant entries reflecting the importance of the outputs fed back to the controller.
2.5.5 Advantages of H∞ Loop-Shaping
H∞loop-shaping design technique has many characteristics that make it very
appealing, among robust control systems design methodologies, especially in aerospace applications. Above all, the main advantages provided by H∞
loop-shaping are:
• as already discussed in section 2.5.1, the possibility to handle project specifications that are defined in terms of an open loop transfer func-tion avoids the possibility of conflicting requirements that arise when the objectives are expressed in terms of sensitivity and complementary sensitivity functions, as it usually occurs in H∞ control problems;
• the system robustness is maximized against normalized coprime-factor uncertainty, that is the most general type of unstructured uncertainty and can describe situations where a stable system becomes unstable because of variations of some parameters, as stated in section 2.5.2. As already mentioned, no problem-dependent uncertainty modeling is required to synthesize the robust controller;
• no iterations are required to synthesize the controller. In fact, the optimal γmin can be found without iterations, which are usual in H∞
control problems.
Many other peculiarities and advantages are extensively discussed in Ref. [25]. Among these, the most important to be recalled are:
• possibility to achieve an exact observer implementation; the resulting overall controller can be expressed as an exact plant observer plus state feedback;
• no pole-zero cancellations are provoked by the controller, as it usually happens in mixed sensitivity H∞ control problems;
• robustness properties and performance at the plant input and output result to be balanced.
Chapter
3
H
∞
Loop-Shaping: Design Procedures
3.1
The Classical Design Procedure
3.1.1 Introduction
As discussed in the previous Chapter, the H∞ loop-shaping design
method-ology is constituted by two independent phases. The second one, consisting in the robust stabilization, does not present any conceptual difficulty, but only computational one that can be easily handled with standard software. On the contrary, skill is required to carry out the first phase of the design, that is the selection of the pre- and post-compensators. Even if the basic simple guidelines described in section 2.5.4 are followed, a certain expertise is necessary to obtain satisfactory results in terms of both performance and robust stability margin.
However, a well consolidated systematic design procedure is available, help-ing largely to overcome the difficulties arishelp-ing in weights selection.
The origin of such a classical design procedure is dated back in 1991, in the PhD thesis of Hyde [14], that developed it by working with Glover on the multi-variable robust control of a VSTOL (vertical and short take-off and landing) aircraft. Their work terminated with the development of a gain scheduled H∞controller [15] and was crowned with a successful flight test of
the H∞loop-shaping control laws implemented on a Harrier research aircraft
in 1993.
Afterwards, many other aerospace applications followed this design procedure (as an example, the works cited in section 1.3) and many studies brought im-provements as well as more or less significant additions or modifications. The result is the state-of-the-art multi-variable H∞loop-shaping controller design
procedure. The latter represents the starting point of this work, aimed to improve some of its aspects in order to make it easier to use, to extend its
range of applicability and, in the meanwhile, to achieve better results. The procedure is fully described below and its understanding is fundamental to have a better insight of the work presented in the following sections.
3.1.2 The Procedure
Consider a linear time-invariant plant model G, with m inputs and n outputs. The systematic step-by-step classical H∞ loop-shaping design procedure, as
recommended in literature, is as follows:
1. Scale all the plant inputs and outputs. This step is fundamental to improve the conditioning of the system and can strongly simplify the choice of the pre- and post-compensator. Many scaling methods are available, sharing the common approach of making variables less than one in magnitude, by dividing them by their maximum expected or allowed changes. In practice, the following guidelines are useful:
• In order to consider cross-coupling between all outputs of equal importance, outputs have to be scaled to obtain equal magnitudes of cross-coupling between all channels.
• Inputs should be scaled to reflect the relative actuators capabili-ties, therefore the actuator signals should be normalized by their maximum allowable values.
A convenient scaling method is available in Ref. [10], pp.6.
In practice, for MIMO system scaling is performed by use of two diag-onal matrices DI and DO that are respectively the input and output
scaling matrices. Thus, the scaled plant G? is given by
G? = DOG DI (3.1)
2. Reorder the inputs and the outputs, which means rearrange both rows and columns of G?to make the plant as diagonally dominant as possible. This step aims to make easier the selection of the weighting matrices, that will be chosen to be diagonal.
3. The elements of the diagonal pre- and post-compensators W1 and W2
must be selected, to obtain the shaped plant GS
GS = W2G W1 (3.2)
For this purpose, consider W1 as split in three matrices:
The first task is to choose both Wp and W2 so that the singular values
of W2G Wp reach the desired shape. The selection of the entries of Wp
is usually made by taking into account the guidelines already stated in section 2.5.4, that we here recall for convenience:
• integral action is required to achieve high gains at low frequency; • phase-advance is required for keeping the desired roll-off rates in the bandwidth region: a gentle slope of about 20dB/decade is usually adequate;
• phase-lag is required to increase the roll-off rates at high frequency, for a good disturbances rejection.
Therefore, Wp contains the dynamic shaping.
W2, instead, is usually chosen as a diagonal matrix of constant terms,
reflecting the relative importance of the outputs to be controlled and, more generally speaking, of all the measures fed back to the controller; often W2 can be set as the identity matrix.
Seldom W2 is chosen to contain some dynamic shaping; however, to
avoid a too small stability margin when the robust stabilization is per-formed, a lot of trial and error is required even for an expert designer. In any case, the weights W p and W2 should be chosen so that no
un-stable hidden modes generate in GS.
4. Align the singular values at the desired bandwidth by means of the diagonal align matrix Wa, of constant elements, cascaded with Wp. Wa
plays the role of a decoupler. Aligning the singular values at some frequency improves the conditioning of the system, therefore reducing the sensitivity to model uncertainties.
This step has to be carried out only if the plant is not ill-conditioned (see section 3.2.2). In fact, aligning the singular values at one point in the loop may deteriorate robustness and performance properties at another point. While this is acceptable for well conditioned systems, for which the reduction of performance is very limited, an ill-conditioned plant will give rise to an align matrix with high condition number, thus leading often to poor robustness.
5. Add an additional diagonal matrix Wg of constant elements, cascaded
to Wa, to provide control over actuator usage. In practice, the entries of
Wgare tuned so that actuators rate limits are not exceeded for reference
demands plus output disturbances. The plant which we refer to, is the scaled plant.
At the end of this step the shaped plant is obtained as GS= W2G W1,
with W1 = WpWaWg.
6. Robustly stabilize the shaped plant and determine K∞. First of all,
cal-culate the maximum stability margin max= 1/γmin. As pointed out in
section 2.5.3, this can be done as shown in Ref. [11], by solving two al-gebraic Riccati equations and no iteration is needed. Existing software allows one to manage this task very easily. The resulting max
repre-sents the maximum robust stability margin compatible with the plant augmented by the selected pre- and post-compensators. If the margin is too small, that is max< 0.25 (γmin> 4), the weighting matrices need to
be modified: in fact, a too small value of max(a too high value of γmin)
indicates that the shaped plant singular values loop-shapes are incom-patible with the robust stability requirements (as justified in Ref. [12] and shown in section 2.5.4). On the contrary, if max> 0.25 (γmin< 4),
at least 25% coprime-factor uncertainty is allowed and the shape of the singular values of the robustly stabilized plant is similar to that of the open-loop shaped plant. Select a value of γ > γmin (typically by
about 10%) and synthesize the corresponding sub-optimal controller. As mentioned in section 2.5.3, there is no practical advantage in using the optimal controller and the sub-optimal controller can be computed in an easier way. In particular, this can be done by use of existing software, that implements the formulae of Ref. [12]; as an example, in Matlab K∞can be computed with the function ncfsyn. The resulting
overall controller is K = W1K∞W2. The number of states of K∞ is
equal to that of GS, therefore the order of the overall controller K is
equal to that of G plus twice that of the weighting functions.
7. Create the closed loop. The recommended configuration is that of Fig-ure 3.1. As mentioned in section 2.5.4, the pre-filter K∞(0) W2(0) must
be applied to the reference signal to ensure a zero steady-state tracking error.
However, the robust stabilization through H∞ optimization is already
implemented in existing software and the designer should not worry too much about the set-up to be used. For instance, in Matlab the robust stabilization is performed automatically through the already cited func-tion ncfsyn, that synthesizes a controller K∞ by adopting the
closed-loop configuration of Figure 3.2.
Once the closed-loop is created, check the system performances in terms of time responses, actuator use and robustness.