Advanced control for constrained processes and systems a unified and practical approach

The second part is devoted to multivariable constrained control problems: improving system decoupling under different plant or controller constraints, and reducing the undesired effects

Advanced Control

for Constrained Processes and

Fabricio Garelli, Ricardo J Mantz

and Hernán De Battista

The Institution of Engineering and Technology www.theiet.org

978-1-84919-261-3

Advanced Control for Constrained Processes and Systems

Fabricio Garelli is currently Associate Professor

at the National University of La Plata (UNLP) and Official Member of the National Research Council

of Argentina (CONICET) He is the author of an awarded Ph.D Thesis and more than 30 journal and conference papers His research interests include multivariable systems and constrained control.

Ricardo J Mantz serves as Full Professor at UNLP and is an Official Member of the Scientific Research Commission (CICpBA) He is the author of a book and more than 150 papers in scientific journals and conferences His primary area of interest is nonlinear control

Hernán De Battista is Senior Professor at UNLP and Official Member of CONICET He has published

a book and more than 70 journal and conference papers His research interests are in the field of nonlinear control applications and renewable energy

The three authors are with LEICI, EE Dept., UNLP, Argentina.

This book provides a unified, practically-oriented treatment to many constrained control paradigms Recently proposed control strategies are unified in a generalised framework to deal with different kinds of constraints The book’s solutions are based on reference conditioning ideas implemented by means of supervisory loops, and they are complementary to any other control technique used for the main control loop Although design simplicity is a book priority, the use of well established sliding mode concepts for theoretical analysis make it also rigorous and self-contained.

The first part of the book focuses on providing a simple description

of the method to deal with system constraints in SISO systems It also illustrates the design and implementation of the developed techniques through several case studies The second part is devoted to multivariable constrained control problems: improving system decoupling under different plant or controller constraints, and reducing the undesired effects caused by manual-automatic or controller switching.

The key aim of this book is to reduce the gap between the available constrained control literature and industrial applications.

Advanced Control

for Constrained Processes and

Systems

Volume 2 Elevator traffic analysis, design and control, 2nd edition G.C Barney and

S.M dos Santos

Volume 8 A history of control engineering, 1800–1930 S Bennett

Volume 14 Optimal relay and saturating control system synthesis E.P Ryan

Volume 18 Applied control theory, 2nd edition J.R Leigh

Volume 20 Design of modern control systems D.J Bell, P.A Cook and N Munro (Editors)

Volume 28 Robots and automated manufacture J Billingsley (Editor)

Volume 32 Multivariable control for industrial applications J O’Reilly (Editor)

Volume 33 Temperature measurement and control J.R Leigh

Volume 34 Singular perturbation methodology in control systems D.S Naidu

Volume 35 Implementation of self-tuning controllers K Warwick (Editor)

Volume 37 Industrial digital control systems, 2nd edition K Warwick and D Rees (Editors)

Volume 39 Continuous time controller design R Balasubramanian

Volume 40 Deterministic control of uncertain systems A.S.I Zinober (Editor)

Volume 41 Computer control of real-time processes S Bennett and G.S Virk (Editors)

Volume 42 Digital signal processing: principles, devices and applications N.B Jones

and J.D.McK Watson (Editors)

Volume 44 Knowledge-based systems for industrial control J McGhee, M.J Grimble

and A Mowforth (Editors)

Volume 47 A history of control engineering, 1930–1956 S Bennett

Volume 49 Polynomial methods in optimal control and filtering K.J Hunt (Editor)

Volume 50 Programming industrial control systems using IEC 1131-3 R.W Lewis

Volume 51 Advanced robotics and intelligent machines J.O Gray and D.G Caldwell

(Editors)

Volume 52 Adaptive prediction and predictive control P.P Kanjilal

Volume 53 Neural network applications in control G.W Irwin, K Warwick and K.J Hunt

(Editors)

Volume 54 Control engineering solutions: a practical approach P Albertos, R Strietzel

and N Mort (Editors)

Volume 55 Genetic algorithms in engineering systems A.M.S Zalzala and P.J Fleming

(Editors)

Volume 56 Symbolic methods in control system analysis and design N Munro (Editor)

Volume 57 Flight control systems R.W Pratt (Editor)

Volume 58 Power-plant control and instrumentation D Lindsley

Volume 59 Modelling control systems using IEC 61499 R Lewis

Volume 60 People in control: human factors in control room design J Noyes and

M Bransby (Editors)

Volume 61 Nonlinear predictive control: theory and practice B Kouvaritakis and

M Cannon (Editors)

Volume 62 Active sound and vibration control M.O Tokhi and S.M Veres

Volume 63 Stepping motors: a guide to theory and practice, 4th edition P.P Acarnley

Volume 64 Control theory, 2nd edition J.R Leigh

Volume 65 Modelling and parameter estimation of dynamic systems J.R Raol, G Girija

and J Singh

Volume 66 Variable structure systems: from principles to implementation

A Sabanovic, L Fridman and S Spurgeon (Editors)

Volume 67 Motion vision: design of compact motion sensing solution for

autonomous systems J Kolodko and L Vlacic

Volume 68 Flexible robot manipulators: modelling, simulation and control M.O Tokhi

and A.K.M Azad (Editors)

Volume 69 Advances in unmanned marine vehicles G Roberts and R Sutton (Editors)

Volume 70 Intelligent control systems using computational intelligence techniques

A Ruano (Editor)

Volume 71 Advances in cognitive systems S Nefti and J Gray (Editors)

Volume 73 Adaptive Sampling with Mobile WSN K Sreenath, M.F Mysorewala,

D.O Popa and F.L Lewis

Volume 74 Eigenstructure Control Algorithms: applications to aircraft/rotorcraft

Advanced Control

for Constrained Processes and

Systems

Fabricio Garelli, Ricardo J Mantz

and Herna´n De Battista

The Institution of Engineering and Technology

The Institution of Engineering and Technology is registered as a Charity in England & Wales (no 211014) and Scotland (no SC038698).

† 2011 The Institution of Engineering and Technology

by the Copyright Licensing Agency Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address:

The Institution of Engineering and Technology

Michael Faraday House

Six Hills Way, Stevenage

Herts, SG1 2AY, United Kingdom

www.theiet.org

While the author and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them Neither the author nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause Any and all such liability is disclaimed.

The moral rights of the author to be identified as author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data

A catalogue record for this product is available from the British Library

ISBN 978-1-84919-261-3 (hardback)

ISBN 978-1-84919-262-0 (PDF)

Typeset in India by MPS Ltd, a Macmillan Company

Printed in the UK by CPI Antony Rowe, Chippenham, Wiltshire

1 An introduction to constrained control 1

1.2.2 Structural constraints 41.2.3 Dynamic restrictions 41.3 Some typical effects of constraints 5

1.3.3 Control directionality problem 91.4 Other constraint implications 101.5 Different approaches to constrained control 12

1.7 Short outline of the main problems to be addressed 14

2 A practical method to deal with constraints 17

2.2 Preliminary definitions 182.3 Sliding mode reference conditioning 182.3.1 Basic idea for biproper systems 182.3.2 Illustrative example 212.4 Biproper SMRC: features and analysis 23

2.4.2 SMRC operation analysis 282.4.3 Implementation issues 31

2.5.2 Method reformulation 342.5.3 Illustrative examples 362.6 SMRC and non-linear systems 422.6.1 Geometrical interpretation of SM 422.6.2 Geometric invariance via SMRC 442.6.3 SMRC in strictly proper non-linear systems 47

2.7.1 SM existence domain 49

3 Some practical case studies 53

3.1 Pitch control in wind turbines 533.1.1 Brief introduction to the problem 533.1.2 Pitch actuator and control 553.1.3 SMRC compensation for actuator constraints in the pitch

3.1.4 Application to a wind energy system for water pumping 573.2 Clean hydrogen production plant 593.2.1 Brief introduction to the problem 613.2.2 System description 623.2.3 SMRC algorithm to deal with electrolyser constraints 663.2.4 Simulation results 67

3.3.1 Brief introduction to the problem 703.3.2 Classical control scheme for robotic path tracking 713.3.3 Tracking speed autoregulation technique 733.3.4 Application to a 2R manipulator 753.4 Control of a fed-batch bioreactor 803.4.1 Brief introduction to the problem 80

4.4 Performance limitations in non-minimum phase systems 107

5.2 Control directionality changes 1125.3 Dynamic decoupling preservation by means of SMRC 115

5.3.1 Method formulation 1155.3.2 Sliding surfaces design 117

5.4 Minimum-phase example 1225.5 Non-minimum phase examples 1245.5.1 Revisiting Example 1.3 1245.5.2 Sugar cane crushing station 125

6 Interaction limits in decentralised control architectures 131

6.1 Introduction to decentralised control 1316.1.1 Architecture description 1316.1.2 Interaction measure 1326.1.3 Control structure selection: the TITO case 1346.1.4 Decentralised integral controllability 1376.2 Interaction effects on multiloop strategies 1396.3 Limiting interactions in decentralised control via SMRC 143

6.3.3 Output dynamics during conditioning 1476.3.4 Behaviour in presence of output disturbances 1486.4 Two-degrees of freedom PID controller with adaptive set-point

6.5 Case study: Quadruple tank 1506.5.1 Plant model analysis 1516.5.2 Interactions limits in non-minimum phase setting 1556.6 Delay example: catalytic reactor 161

7 Partial decoupling and non-minimum phase systems 163

7.1 Some introductory comments 1637.2 Right-half plane zeros directionality and partial decoupling 1647.2.1 Algebraic interpolation constraint 1647.2.2 Inverse response on a particular output 1677.3 Interpolating diagonal and partial decoupling 1707.4 Partial decoupling with bounded interactions via SMRC 170

7.6 Case study: quadruple tank 175

8.2 Switching at the plant input 1828.3 A simple SMRC solution for SISO systems 1838.4 MIMO bumpless transfer 1858.4.1 Some concepts on collective sliding modes 1858.4.2 A MIMO bumpless algorithm 188

8.5 Application to the quadruple tank process 1918.5.1 Manual–automatic switching 1918.5.2 Automatic–automatic commutation 193

An introduction to constrained control

In every real control loop, there exist physical limits that affect the achievableclosed-loop performance Particularly, it is well known that mechanical stops ortechnological actuator limitations give rise to unavoidable constraints at the input

to the plant, which must be taken into account to meet performance specifications

or safety operation modes

In single-input single-output (SISO) systems, these physical limits at the plantinput are the principal cause of highly studied problems like controller and plantwindup However, there are other types of constraints that have been dealt with to amuch lesser extent but that also have an effect on the achievable performance Forinstance, the few degrees of freedom of industrial controllers (typically PID) orplants having non-minimum phase features will generally restrict the evolution ofthe controlled variables What is more, some well-known problems such as bumpytransfers can be attributed to the fact of constraining the controller types and theirswitch scheduling Thus, there also exist structural or dynamic constraints thattogether with performance specifications, environmental regulations or safety rulesusually require system states or outputs to be bounded

Although constrained control problems have been studied primarily in SISO

systems, the majority of the real-world processes have more than one variable to be

controlled and possess more than one control action for this objective These tems are called multivariable systems or multiple-input multiple-output (MIMO)systems Actually, SISO systems frequently are a given subsystem of an overallMIMO system

sys-Multivariable systems can be found almost everywhere In the bathroom of ahouse, the water temperature and flow rate are important variables for a pleasantshower In chemical processes, it is commonly required to simultaneously controlpressure and temperature at several points of a reactor An automated greenhouseshould ensure that the lighting, relative humidity and temperature are adequate for agiven cultivation A robot manipulator needs six degrees of freedom to have a fullpositioning rank, whereas in a plane or a satellite there are dozens of variables to becontrolled

There are some phenomena that are present only in MIMO systems and do not

occur in SISO systems For example, the presence of directions associated to

input/output vectors is exclusive to MIMO systems On this account, a variable system may have a pole and a zero at the same location that do not canceleach other, or in a minimum-phase MIMO system the individual elements of thetransfer matrix might have their zeros in the right-half plane (RHP), or vice versa.Nevertheless, the most distinctive property of a multivariable system is probably

multi-the crossed coupling or interactions between its variables In effect, in a MIMO

system each input variable affects not only its corresponding output but also all theremaining controlled variables of the system This makes controller design a dif-ficult task, and in most applications precludes one from doing it as if the systemconsisted of multiple mono-variable loops, since the gains of a single-loop con-troller will have impact on the other loops and may even cause instability This isthe reason why crossed interactions are generally considered the main difficulty ofmultivariable control systems [144] Therefore, in a multivariable process theeffects and demands of the aforementioned system constraints are worsened

because of the directionality and interactions present in this kind of plants The

search for solutions to such problems has motivated several research works in theprevious years [46,63,123–125,127]

Despite the large number of existing methods to address constrained controlproblems, a common practice of engineers is to design control systems usingconventional methods in such a way that they avoid reaching the system limits and,

at the same time, achieve a reasonable performance for a given operating region.However, this conservative approach is seldom feasible in complex systems control

or high-performance applications Moreover, even in relatively simple industrialproblems, the resulting closed-loop performance can be significantly improved ifsystem constraints are taken into account

Contrary to what some practising engineers believe, this does not necessarilyrequire a complete redesign of the control system and abandoning their valuableexperience on nominal control design Indeed, the basic ideas behind the simplestanti-windup (AW) schemes highly accepted in industry can be further exploited togain robustness, closed-loop performance and design simplicity in more complexcontrol problems with different kinds of constraints, while preserving conventionalcontrol methods for the nominal controller design This will be a major topic in thebook: the first part (Chapters 1–3) mainly devoted to SISO constrained systems andthe second part (Chapters 4–8) devoted to some relevant multivariable controlproblems

Let us start by giving a classification of system constraints affecting closed-loopperformance As could be noted from the introductory comments, constraints shall

be understood in a ‘wide sense’ That is, we will refer to system constraints not only

to mention physical limits of the control loop components but also to mention anyother structural constraint or dynamic restriction that affects closed-loop perfor-mance and can be tackled by delimiting a given signal in the loop

Basically, we broadly classify the constraints from their source type into threecategories:

conse-PERFORMANCE REQUIREMENTS

Input signal bound

Internal signal bound

Output signal bound

Some examples, although obvious, seem to be opportune:

● Every electrical engine has a voltage limit and a maximum speed that shouldnot be exceeded

● A valve can be opened neither more than 100% nor less than 0%

● The slew rate of a hydraulic actuator will always be limited

In closed-loop systems, performance requirements frequently lead the controlaction to hit these physical limitations at the plant input, i.e actuator saturation isreached Saturation can occur in either amplitude, rate of change or higher-ordersignal derivatives, such as acceleration If the control action exceeds these limitsfor any reason, severely detrimental behaviours may occur Therefore, the controlsystem design must somehow account for the unavoidable actuator limits by con-fining the signal at the plant input to adequate values

However, although physical limits generally appear at the plant input, they donot exclusively require delimiting the commanded signal to the plant In effect, wewill see later that this kind of constraint may also demand restricting internal oroutput system variables to avoid performance degradation

In the next section, we briefly present some of the most common effects duced by this kind of constraints To emphasise its significance, it is interesting to

pro-recall that physical limits, frequently present in simple industrial control problems,have also been involved in extremely serious accidents, like various aircraft crashesand environmental disasters [128].

● Linear controllers coping with non-linear systems

● Multiloop or decentralised control structures in multivariable applications.Although we will deal with many other examples of structural constraintsthroughout the book, PID controllers are probably the most illustrative case Notonly are they highly accepted in industry applications, but the use of any other type

of controller is often turned down Despite their well-known advantages, it is alsotrue that they are not able to deal with every type of process and specification.Thus, if the controller type cannot be changed, it will result in an additional con-straint for the control designer when trying to achieve quite demanding controlobjectives

With this term, we refer to those dynamic characteristics of the process to becontrolled (or the controller to be employed) that directly affect the achievableclosed-loop performance and the evolution of the signals in the control loop Forinstance,

● Plants with RHP zeros and poles

● Systems or controllers with particular combinations of pole and zero locations

● Non-linear processes in which detrimental dynamic behaviours are excited bythe growth of a given internal variable

It is well known that RHP zeros produce undesired inverse responses in thecontrolled variable Also, systems with a stable zero closer to the origin than thedominant poles may give rise to large overshoots in the step response In complexprocesses with non-linear behaviours, the regulation of a variable of interest canlead auxiliary variables to dangerous or undesirable regions (see the last case study

in Chapter 2)

Consequently, this group of constraints may also translate into boundingrequirements on the loop signals because of performance specifications or safetyoperation This will be pointed out in Section 1.4 and further addressed fromChapter 3

1.3 Some typical effects of constraints

This section discusses some classic examples of the undesired effects that straints may have on the closed-loop responses We focus now on the effects related

con-to physical limits at the plant input Other problems arising from the different types

of constraints previously mentioned are briefly enounced in this chapter (see thenext section), but they will be described and illustrated afterwards throughout theremainder of the book

As mentioned earlier, the closed-loop bandwidth requirements often lead the trol signal to reach the physical actuator limits at the plant input When this occurs,unless an action is taken to avoid it, the feedback loop is broken and the systemoperates in an open-loop fashion Certainly, once the limit is reached, the actuatorcontinues giving its maximal (or minimal) output independently of the controlleroutput, until it returns to its ‘allowed region’ This brings about an overreaction ofthe controller, which in turn produces a significant degradation of the closed-loopresponse with respect to the one expected for actuator linear operation Theresulting response typically presents large overshoots and/or long settling times.Since the effect was originally attributed to the fact that the controller inte-

con-grator winds up due to input saturation, it was called intecon-grator windup However,

it was then observed that the same effects could actually be caused by any otherunstable or sufficiently slow controller dynamics, and so the problem is generally

known as controller windup There exists a vast bibliography devoted to the study

of controller windup and its possible solutions, among which some of the mostcited works are References 5, 55, 72 and 96

To illustrate this phenomenon, the following example is presented:

Example 1.1: Consider a simple model corresponding to a distillation column with

a PI controller and a Smith predictor as shown in Figure 1.2 The plant model isgiven by

P ðsÞ : G p ðsÞe Lps ¼ 0:57

ð8:60s þ 1Þ2e 18:70s ð1:1Þwhereas the Smith predictor and the PI controller, designed by means of relay auto-tuning techniques in Reference 53, are

PI (s) G p (s)

G m (s)

ˆu y

– –

Figure 1.2 PI control and Smith predictor for plant (see (1.1)) with input saturation

The objective here is not to evaluate the behaviour of the set PI–Smith predictor,since any other controller could have been considered for actuator linear operation.Instead, we aim at showing how the closed-loop performance is deteriorated because

of input non-linearities, even when modern design techniques are employed.The simulation results for the closed-loop system without actuator constraintsare plotted with dotted line in Figure 1.3 The controlled variable response with anovershoot of approximately 15% and a settling time (5%) of 50 [time units] is thenconsidered the desired response for this control system In the lower box, the

controller output evolution (u, also in dotted line) can be observed, which coincides

with the plant input (^u) for each time instant.

The solid curves in Figure 1.3 correspond to the closed-loop response whenactuator constraints^u  u are considered, with u ¼ 2:2, u ¼ 2 and u ¼ 1:9 As the

lower part of the figure makes evident, the controller output (dashed lines) exceeds

the available control amplitude during long periods of time This leads the output y

to evolve slower than in the unconstrained case, and therefore the error e poorly

decays For this reason, the PI integral term winds up, taking a value much greater

than when the actuator operates in its linear region Hence, even when the output y gets close to the reference r, the control action u continues to be saturated because

of the integral term contribution This is a typical controller windup effect, whoseprincipal consequence is in this case the long settling time Figure 1.3 shows howthe settling time increases as saturation becomes harder (24% of increment for

u ¼ 2:2, 73% for u ¼ 2 and 250% for u ¼ 1:9).

As we will see in Chapter 2, the undesired behaviour caused by controllerwindup can be avoided by properly bounding the commanded signal to the plantinput, so as to prevent actuator saturation Then, controller windup provides anexample of physical limits requiring input signal delimitation to attain performancedemands (see Figure 1.1)

Not only slow/unstable compensator modes or integral actions may produceundesired transients when combined with physical input constraints, but a similareffect to controller windup can also be observed in closed-loop systems with staticcontrollers such as P controllers or state feedback configurations This is obviouslynot attributable to the controller dynamics, but it is related to the dynamics of theprocess under control, since plant states cannot be brought to their steady-statevalues fast enough due to input saturation Consequently, although less studied and

formally recognised, this effect is known as plant windup [60].

We next present an example to demonstrate the problem of plant windup whenusing a proportional controller

Example 1.2: Let the third-order system

5x þ

001

26

37

5u

y ¼ ½ 60 15 1  x

ð1:6Þ

we can obtain the output and state evolution when a unitary step reference isapplied The results are plotted as dotted lines in Figure 1.4 As can be appreciated,the response exhibits a linear behaviour with a vanishing tracking error.

If we now consider a physical limit (saturation) such that the plant input isconstrained to j^uj u ¼ 1, the transient response is seriously deteriorated, pre-

senting both long-settling times and large overshoots (solid lines in Figure 1.4).These effects cannot be associated with a controller windup phenomenon since thecontroller has no dynamic modes Here, input saturation winds up the system states

to values from where they are not able to come back fast enough due to the strained input signal, thus affecting the output signal See particularly how the

con-decay rate of the first state x3is limited by input saturation

These undesired effects can even lead to oscillatory responses for greaterreference changes or harder physical limits In general, whenever the closed-loopdynamics is excessively fast with respect to the input limits, there is risk of plantwindup

State x3(t)

Time

Figure 1.4 Plant windup effect on output and state responses

Differing from controller windup, to overcome plant windup it may benecessary to delimit an internal plant state, as will be seen in the next chapter Then,remembering Figure 1.1, plant windup provides an example of performancedegradation caused by physical limits, which can be tackled by delimiting aninternal signal

1.3.3 Control directionality problem

As was claimed in Section 1.1, when designing multivariable systems one has todeal with interactions among the loops To reduce or avoid such cross-coupling, theplant input vector must have a given direction, determined by a centralised MIMOcontroller However, if individual constraints affect each of the controller outputsindependently, the plant input direction required for closed-loop decoupling can bedrastically altered, for instance, because of the saturation of only one vector com-

ponent This effect, also caused by physical input constraints, is known as control

directionality problem and usually produces strong closed-loop couplings apart

from other performance deterioration

We briefly introduce this problem here with a simple example, skipping thedetailed study of multivariable system decoupling and control directionality, since

it will be one of the main subjects of Chapters 4 and 5

Example 1.3: Consider a two-input two-output plant with the following nominal

37

which can be synthesised following the ideas described in Chapter 4

Assuming ideal actuators, a linear decoupled response is obtained when thesystem is excited with a negative reference change in the second channel (shownwith dotted lines in the left graphic in Figure 1.5)

Introducing now two identical saturating actuators whose linear operations areconfined to the range½u i , u i  ¼ ½2, 2, i ¼ 1, 2, the achieved performance is sig-

nificantly worsened (solid lines in Figure 1.5) In effect, the closed-loop decouplingfor which the controller had been designed is completely lost The causes of thisinteraction can be found at the bottom left of the figure, where it is observed how

the controller outputs u1and u2(dashed lines) differ from the plant inputs^u1and^u2

(solid lines) as a consequence of saturation As the right box in Figure 1.5 shows,this leads to a transient directionality change of the plant input with respect to the

controller output (between t0and t3, t0< t1< t2< t3) Particularly, it can be seen

how the controller output direction changes from t1 ¼ 0.61 [time units] to t2 ¼0.74 [time units] (between the two longest arrows), while the plant input directionremains unchanged during this period (shortest arrow), when both^u1and^u2aresimultaneously saturating

Apart from the undesired effects of physical limits described beforehand, there aremany other troubles in control system design that may require delimiting theexcursion of signals in the loop (recall Figure 1.1) Although most of these pro-blems are not usually addressed by conventional constrained control literature, theyare closely related to the remaining types of restrictions defined in Section 1.2,namely structural or dynamic constraints As we have already mentioned, we willhere limit to enounce some of the main problems to be coped with, whereas theywill be discussed and described in greater detail in Chapter 3 and at the beginning

of Chapters 5–8

For instance, the output overshoot of a second-order system controlled by a

P controller will increase as the desired closed-loop bandwidth does Thus, anoutput constraint may be broken if fast responses are demanded and the controller

−2 0 2 4 6 8 10

dashed lines) and without (dotted lines) input saturation

structure cannot be modified More generally, a similar situation can be found forevery linear controller (typically PI or PID industrial controllers) designed for agiven operating point but manipulating a real plant, which is actually non-linearand uncertain, as closed-loop requirements are tightened up.

Another typical phenomenon that can be associated with a structural constraint

is the bumpy transfer, i.e the jump at the plant input caused by manual–automatic

or controller switching, which deteriorates the control system response In effect,such mode switching can be associated with the fact of using a single linear con-troller for each operating point of a non-linear process and a trivial schedulingstrategy If no other switching policy (like linear parameter-varying control) iswelcomed and non-linear controllers are not accepted, the suppression of the jumps

at the plant inputs and their associated effects must be performed by limiting the

controller states or input signals This is usually referred to as bumpless transfer

and will be discussed in Chapter 8

Some other signal-bound requirements arising from structural constraints that

we will cover in the book are as follows:

● Confining the controlled variable to a given range of values under existing control structures, which cannot be altered, in presence of externaldisturbances

pre-● Matching the output of a given plant with the input of a serial connected cess or device

pro-● Limiting the crossed coupling in MIMO systems when the controller is posed of individual SISO controllers for each loop (multiloop control).Furthermore, the following constraints caused by dynamic restrictions of theplant will be tackled to meet performance specifications:

com-● To maintain an internal or alternative output variable, which affects thecontrolled variable of a non-linear system, running along its operationallimits

● To avoid inverse responses in the main controlled variable of non-minimumphase MIMO processes while delimiting the interaction on the remainingvariables

Regarding the group of constraints listed just above, it is important to remarkthat they are strongly related to the so-called fundamental closed-loop design lim-itations, for which there are in the literature several analytical measures and tools,e.g the Bode and Poisson integral constraints.1However, as was mentioned earlier,

we will focus on fulfilling the signal bounds that result from these unavoidableperformance constraints That is, an engineering approach will be adopted in whichexplicit and direct signal constraints can be determined by the control designer toaccomplish a given goal (see Section 1.6)

1

Even though this subject will be revisited, for further details on fundamental design limitations the reader is referred to References 67 and 110 and references therein.

1.5 Different approaches to constrained control

This book is not intended to give a complete overview of all the existing controlmethods to deal with constraints Interesting books related to constrained controlare References 46, 51, 60, 62, 69, 107, 121 and 156 Basically, the existing tech-niques to address constrained control problems can be divided into two categories:

Within the first group, the main avenues are based on either optimal controltheory or numerical optimisation This latter approach includes model predictivecontrol (MPC), a very popular discrete technique for the control of chemical pro-cesses At each sampling time, the implicit form of this algorithm solves an onlineoptimisation problem to compute the control inputs over a future time horizon,making use of a process model to predict the future response Then, only the firstcomputed control value over the horizon is actually applied to the process At thenext sampling time, the optimisation problem is updated by shifting the horizonforward by one time step (the so-called receding horizon) and solved using the newmeasures from the plant

MPC is a one-step procedure because the online optimisation can incorporate

various constraints, including control or quality objectives, ‘from scratch’ Indeed,the main advantage of MPC is probably its ability to deal with either input or outputconstraints so that nominal performance specifications are met It is also arguedthat this is the reason why a significant portion of the literature is devoted to it.However, MPC has often been criticised for being applicable only to relativelyslow processes due to the online program running time Moreover, although it isnumerically very powerful, its analytical treatment is quite involved Since thisapproach will not be followed here, the reader is referred to Reference 102 for atutorial about MPC and to Reference 51 for a more recent analytical treatment ofthese techniques

The two-step approaches (also called evolutionary approaches) are the

approaches in which the nominal controller is designed without explicitly sidering constraints (first step), and then a compensation loop is added to reduce theadverse effects of constraints as much as possible The principal motivation for thedevelopment of this type of approaches is that they can be intuitively understood,designed and tuned In fact, any desired control strategy, including those conven-tional techniques for which operators are usually more trained, can be utilised at thefirst step for the design of the main control loop This is probably the main reasonwhy two-step techniques constitute the most widely spread approach to constraints

con-in con-industrial control

All the so-called AW techniques rely on this design procedure Among them,

we can distinguish two groups of methods The first group includes the tional anti-(reset) windup methods, which modify the controller (usually its integralterm) as a function of the plant input or its difference with the controller output

conven-See, for instance, the methods described in References 4, 5 and 16 The secondgroup includes the algorithms conditioning the reference signal so that the ‘realis-able reference’ is sought, which is, by definition, the reference that if applied fromthe beginning would just avoid the differences between the controller output andthe plant input, i.e saturation This highly successful approach was originally for-mulated by Hanus and co-workers [54,55], and was afterwards generalised by

Walgama et al [141] Despite the vast effort to unify and formalise AW techniques [60,72,96], most of the two-step algorithms are basically ad hoc methods without

the background of a well-established theory Also, since they are mainly conceived

as AW schemes, two-step approaches generally cope with input constraints only

This book is aimed at providing a practical and unified approach to deal withseveral control problems caused by either physical, structural or dynamic con-straints To this end, we adopt a two-step approach in an attempt to take advantage

of its intuitive features, while at the same time we explore the ways to furtherextend its applicability and to systematise its design

Particularly, we introduce a two-step algorithm, which can handle constraints

on both the manipulated (input) and the controlled (output) variables, as well asinternal state limitations The resulting control strategies are based on reference-conditioning ideas, implemented by means of an auxiliary or supervisory loop,which employs a discontinuous action to generate the maximum reference signalcompatible with the system constraints Although design simplicity is a bookpriority, well-established Variable Structure Systems (VSS) theory and slidingmode (SM) related concepts are used for the methodology analysis This theoreticalapproach not only gives the proposals a rigorous though conceptually simplemathematical support, but also provides them with distinctive robustness featuresand reduced dynamic behaviours To encourage the method implementation, SISOtechniques are exploited as much as possible, even when dealing with MIMOcontrol problems Then, a multivariable sliding mode methodology is also intro-duced to allow further developments

The approach taken is unified because the same basic ideas allow tackling many

constrained control problems (those with different types of constraints, involving

either linear or non-linear models, SISO or MIMO processes, etc.), and practical,

since the resulting techniques are very easy to be implemented and complementary toany other control strategy that has been used in the main control loop

As was mentioned in the previous section, the book does not intend to cover allthe existing control methods to deal with constraints Instead, the book focuses itsattention on the aforementioned unified approach conceived to cope with a widevariety of control problems It should not be expected either a purely mathematicaltreatment of the topic, as is quite usual in constrained control literature On thecontrary, an engineering perspective combining rigorous analysis with focus onconcrete applications is the main approach

1.7 Short outline of the main problems to be addressed

The first part of the book (Chapters 1–3) is devoted to SISO constrained controlsystems After introducing in this chapter the book motivations together with a briefdiscussion of the topic, the main approach of the book is presented in Chapter 2.The basic idea is first described and analysed in light of VSS theory Then, themethodology is extended to deal with signal bounds at any part of the controlsystems, arising from either physical, structural or dynamic constraints and per-formance requirements The proposed technique is finally cast within the frame-work of non-linear systems and geometric invariance concepts, while its distinctiveand robustness features are also shown

To illustrate the practical potentials of the approach, Chapter 3 presents severalcase studies with different constraints to be fulfilled: (1) the pitch control of windturbines with both amplitude and rate actuator saturation; (2) a clean-hydrogenproduction plant in which the electrolyser specifications require output bounds;(3) the tracking speed autoregulation of robotic manipulators to avoid path devia-tions; and (4) the regulation of ethanol concentration below a given threshold in thefed-batch fermentation of an industrial strain for overflow metabolism avoidance.The second part of the book (Chapters 4–8) revises some important tools ofmultivariable control theory and deals with relevant problems of MIMO processcontrol It aims at improving the closed-loop decoupling degree in the presence ofeither physical (actuator saturations), structural (decentralised controllers) ordynamic (non-minimum phase characteristics) constraints There is also room tocope with bumpy transfers in MIMO processes

The first MIMO problem to be treated is the dynamic decoupling preservation

of multivariable processes in the presence of plant input constraints (Chapter 5) Asalready seen, input saturation changes the amplitude and the direction of the controlsignal that is necessary to achieve dynamic decoupling Hence, in addition to theknown problem of windup, the control directionality problem appears, bringingabout the loss of the decoupling obtained for the ideal unconstrained case Most ofthe existing methods to deal with this problem successfully avoid the change ofcontrol directionality by conditioning the whole reference vector However, wehere address the problem from a different perspective: to preserve the closed-loopdecoupling as long as possible Therefore, the presented approach does not affectthose variables whose set points remain constant, avoiding in this way the gen-eration of undesired transients in these channels

As a second multivariable problem of major interest for industrial engineers,the reduction of cross-interactions in multiloop or decentralised control is con-sidered in Chapter 6 The great majority of industrial process control loops forMIMO systems are still based on this architecture However, in spite of theirpractical benefits, multiloop controllers are not able to suppress interactions Anillustrative example is given in the chapter to reveal the effects of crossed inter-actions on multiloop control When the process coupling is significant, the pairingproblem of choosing which available plant input is to be used to control each of the

plant outputs must first be properly solved Nonetheless, neither appropriate controlstructure selection nor controller tuning is sufficient to guarantee amplitude deli-mitation of the input–output coupling The conditioning technique developed inChapter 2 is shown in this chapter as a powerful tool to impose user-definedboundaries for the loop interactions in decentralised control systems, i.e to robustlyrespect output signal bounds All the chapter content is then applied to a benchmarkquadruple tank process, whose dynamics simply represents the behaviour of severalchemical and industrial processes.

A half-way strategy between diagonal decoupling and decentralised control isaddressed in Chapter 7 for non-minimum phase processes, regarding that for thesesystems the former strategy spreads RHP zeros (thus imposing additional closed-loop performance constraints), while the latter does not generally give satisfactoryresults for relatively demanding closed-loop requirements Hence, partial decou-pling is considered and studied throughout the chapter In a partially decoupledcontrol system, non-zero off-diagonal elements in the closed-loop transfer matrixhelp to relax the bounds on sensitivity functions imposed by the RHP zeros, andalso permit pushing the effects of these zeros to a particular output However, theoff-diagonal elements also give rise to interactions in a structured form, whichstrongly depend on the RHP zero directions The conditioning algorithm is herepropounded, analysed and designed to limit the remaining interactions in partialdecoupled systems while avoiding undershoots in the (decoupled) variable ofinterest as much as possible The same non-linear quadruple tank process ofChapter 6 is considered as a case study, but subjected to much more demandingcontrol specifications

Finally, Chapter 8 presents an algorithm for the reduction of the undesiredeffects caused by manual–automatic or controller switching in multivariable pro-cess control Of course, the method is also applicable to SISO systems as a parti-cular case It simply uses a switching device and a first-order filter to avoidinconsistencies between the off line controller outputs and the plant inputs As aconsequence, jumps at the plant inputs are prevented (i.e bumpless transfer isachieved) and undesired transients on controlled variables are significantlyreduced Some advantages of this method are its straightforward implementation,its robustness properties and that, unlike other bumpless proposals, it does not need

a model of the plant

A practical method to deal with constraints

This chapter presents the main ideas of the book to deal with constraints in back control systems The proposed method aims at preserving the simplicity ofconventional anti-windup (AW) algorithms, but at the same time it is supported by

feed-a solid theoreticfeed-al bfeed-ackground feed-and exhibits distinctive robustness fefeed-atures Amongthem, the method does not require an exact model of the plant (it only needsknowledge of the model structure), it can be employed to address constraints innon-linear processes and it can transparently deal with a wide variety of non-linearities apart from amplitude saturation, such as rate limiter, asymmetry, deadzone, and time dependency For the sake of simplicity, only single-input single-output (SISO) systems are considered in this chapter

In this chapter, we describe and analyse a unified and practically oriented method

to address several constrained control problems The methodology to be presented

is the key topic behind the book’s control approaches, since from the basic andsimplest idea the algorithm is then extended to deal with different types of con-straints in both SISO and multiple-input multiple-output (MIMO) systems.The method recovers the idea of shaping the reference signal, which, asmentioned in Section 1.5, is one of the possible schemes of two-step algorithms fordealing with constraints Here, we combine reference-conditioning ideas withsliding regime properties to improve robustness with respect to both the model ofthe constrained system and output disturbances This greatly extends the applic-ability of the method as well The implementation and design is simple enough forbeing applied to real industrial problems, whereas the corresponding analysis givesrigorous conditions for the proper operation of the method

With the aim of keeping the presentation as simple as possible, we firstdescribe conceptually how the method operates when it is applied to controllers orsubsystems described by biproper transfer functions (see Section 2.2) After asimple example, some basic concepts on variable structure systems (VSS), neces-sary for the theoretical study of the algorithm, are presented in Section 2.4 Thetechnique is then extended to deal with strictly proper controllers or subsystems inSection 2.5 From Section 2.6, the chapter generalises the method application tonon-linear systems and discusses some implementation and robustness issues

2.2 Preliminary definitions

First of all, we need to recall some simple but relevant definitions related todynamical systems

Definition 2.1 (Proper and improper systems): A linear dynamical system

lims!þ1jPðsÞj > 0 All systems that are not proper are improper.

From the above definition, a system P(s) is proper if the order of the numerator polynomial n Nis smaller than or equal to the order of the denominator polynomial

n D, otherwise it is improper Note that improper transfer functions cannot be sically realised

phy-Definition 2.2 (Relative degree): It is said that a proper system P(s) has relative

In this way, the relative degree of a given system indicates the number of timesthat the system output must be differentiated for the system input to explicitly appear

Definition 2.3 (Minimum phase): A system P(s) is non-minimum phase (NMP) if

its transfer function contains zeros in the right-half plane (RHP) or time delays Otherwise, the system is minimum phase (MP).

Figure 2.1 presents a block diagram of a simple scheme to deal with constraints indynamical systems Therein, one can distinguish a constrained dynamical system

S c (s) to which an auxiliary loop was added to avoid breaking the limits v and v on its constrained variable v.

It is important to remark that the system S c (s) generically represents a strained subsystem of the whole control system, and therefore the variable v may

con-r f r fp

+ –

correspond to any system variable subjected to constraints For instance, S c (s) may stand for a biproper controller facing input saturation (i.e v : u, u is the controller

output), or a strictly proper feedback control system subjected to output signal bounds

(v : y) In the latter case, the states of the constrained subsystem will be necessary

for the conditioning loop, as we will see further in Section 2.5 (Figure 2.9)

Now, we focus on the scheme of Figure 2.1 and assume S c (s) is a linear,

biproper, stable and MP system (e.g a PI controller) Therefore, its dynamicbehaviour can be represented by

where d is a disturbance signal.

A signal^v, which always accomplishes the specified constraints on v, can be generated by means of an artificial limiting element (depicted with double box in

other typical non-linearity could have been considered

The following commutation law is then implemented in the switching block of the auxiliary loop to fulfil the bounds v and v:

The resulting discontinuous signal w is employed to shape the system input r f

through a first-order low-pass filter represented by the block

Natu-the constrained dynamical system in such a way that Natu-the system response is not

deteriorated when the output v is within its allowed range.

Observe that (2.3) and (2.5) determine a linear region

R ¼ fx s 2 X s : s ¼ ^v  v ¼ 0g ð2:7Þand two limiting surfaces

S ¼ fx s 2 X s : s ¼ v  v ¼ 0g

S ¼ fx s 2 X s : s ¼ v  v ¼ 0g ð2:8Þ

When the system operates within its allowed regionRðv  v  vÞ, the signal w is

zero and no reference correction is performed, i.e the conditioning loop is inactive

However, when v tries to exceed its upper boundS, which makes s < 0, the signal

s > 0, w switches to wþ This situation is depicted in Figure 2.2 for amplitude

limits on v.

As can be noticed from (2.1), (2.2) and (2.6), the switching signal w directly affects the first time derivative of the constrained signal v and the switching

functions Thus, for a sufficiently large discontinuous signal w, the switching logic

((2.4) and (2.5)) ensures that

s ¼ 0) Consequently, the conditioned signal r f will be continuously adjusted in

such a way that the output v never exceeds the predefined limits.

Remark 2.1: The method does only employ a switching device and a first-order

filter, and even then it has many attractive properties that will be shown and strated throughout the book Among them, it does not depend on the system model (only on its relative degree), it is robust against uncertainty and external

illu-s illu-s < 0 v

disturbances, it is confined to the low-power side of the system and thus easy to implement, etc.

Remark 2.2: Because of the switching nature of the algorithm, well-established

VSS theory can be used to analyse most of the method features, which is the subject

of Section 2.4 It will also allow generalising the method to deal with strictly proper linear systems (Section 2.5) and non-linear systems as well (from Section 2.6).

Remark 2.3: The high-frequency switching of w, which results from the

commu-tation law (2.4) and (2.5), can be interpreted as a transient sliding regime on the

this methodology as sliding mode reference conditioning (SMRC).

Consider again the Example 1.1 given in Chapter 1 To avoid controller windup, weapply the SMRC scheme just described, which for the case of biproper controllers

is as simple as shown in Figure 2.3

Note that in this case the constrained variable v coincides with the main control action u Also, the subsystem S c (s) is here given by the closed-loop transfer function between r f þ d and u, with the disturbance d given by the perturbed component of y.

For greater details on this output decomposition, see Section 4.2.2 in Chapter 4

Observe that S c (s) is biproper because the PI controller in (1.3) is also biproper.

The eigenvalue of the filter (2.6) is taken aslf ¼ 1, which makes the filter

much faster than the closed loop The switching signal w commutes according to

(2.4) and (2.5) Note that the switching function s ¼ ^u  u has unitary relative degree with respect to w, since the error e (whose first derivative depends on w) is implicit in u through the proportional term of the PI controller.

The response of the closed-loop system with the SMRC technique as AWmethod is presented in Figure 2.4 for the same amplitude limits as the ones con-

sidered in Example 1.1 On the one hand, the evolution of the control action u and

the plant input^u reveals that u  ^u all the time, i.e the actuator never saturates On the other hand, the response and settling time of the controlled variable y greatly

improve with respect to the non-compensated case (see Figure 1.3) The dottedlines depict once more the ideal unconstrained responses for comparative purposes

r f r

s

+ –

Figure 2.5 shows the conditioned reference signals r f and the discontinuous

signals w corresponding to Figure 2.4 As could be expected, as the bound on the control action u increases, the conditioned reference tends to the original reference

integral state of the PI controller In the absence of windup compensation, the

control signal u enters the forbidden saturated region at instant t1, re-entering

linear operation only after instant t3 During the interval t1t3, an important

growth of the integral state x i, typical of the windup effect, can be observed

This gives rise to a long settling time in the controlled variable y The trajectory

of the compensated system coincides with the previous one until instant t1 Fromthen on, the SMRC loop produces consecutive abrupt changes in the trajectory

direction by means of the discontinuous signal w, which forces the control action

to re-enter the linear region This situation is repeated at high frequency, and aso-called sliding regime is established on the surface s ¼ 0 In t2, the systemtrajectory stops pointing outside the linear region Thus, the conditioning loopbecomes inactive and the system evolves with its own linear dynamics to theequilibrium point

As was already mentioned, VSS-related concepts can be used to formalise andextend the methodology described in the previous section To this end, some basicsabout VSS and sliding regimes are presented first Then, they are employed for theanalysis of the SMRC algorithm when dealing with biproper constrained sub-systems and to gain insight into the method properties

u

t3

u u

compensation (solid) Dotted line: ideal unconstrained case

2.4.1 VSS essentials

A VSS is a dynamical system composed of various continuous subsystems with aswitching logic In a VSS, the system structure is intentionally changed with,among others, the following objectives:

● To improve closed-loop performance

● To solve control problems that continuous control is not able to work out

● To attain robustness against model uncertainties and external disturbances.The discontinuous control action resulting from the switching logic of a VSSsystem achieves a particular operation when switching occurs at a very high fre-quency, constraining the system state to a surface on the state space This kind ofoperation is called sliding mode and has many attractive properties Among them, it

is robust to parameter uncertainties and external disturbances, the closed-loopsystem is an order-reduced one and its dynamics depends on the designer-chosensliding surface [114,135] Because of its interesting features, a large number ofworks presenting practical applications of SM have been reported during the pastdecades in the main journals on control systems

From the beginning of VSS studies [29,133], great attention has been paid tothe development of a theoretical framework to deal with these kinds of switchingsystems, which are typically described by differential equations with discontinuousright-hand side Nowadays, VSS are considered to be theoretically well founded.Let us now review some fundamental concepts of VSS and sliding regimes Theinterested reader is referred to books [26,136] that are completely devoted to SMcontrol and its applications

Consider a continuous linear system given by

If a switching functions(x) is defined as a smooth function s : X ! R, whose

gradient!s is non-zero in X, the set

S ¼ fx 2 X : sðxÞ ¼ 0g ð2:11Þ

defines a regular manifold in X of dimension n 1, which is called sliding fold or surface

mani-Such a surface can be reached by properly defining a variable structure control

law, which makes the control action u to take one of two possible values, depending

on the sign of the switching functions(x):

u¼ u if sðxÞ > 0

u if sðxÞ < 0 u 6¼ u



ð2:12Þ

Whenever the switching law (2.12) enforces the system to reach the surfaceS and

to remain locally around it, it is said that there exists a sliding regime on thesurface S To this end, the vector fields controlled by the two continuous sub-

We now look for necessary conditions so that the situation depicted by Figure 2.7holds Mathematically, it can be described by (note the similarities with (2.9))

s_ðxÞ < 0 if sðxÞ > 0

s_ðxÞ > 0 if sðxÞ < 0



ð2:13Þ

Equation (2.13) implies that the rate of change of the scalar functions(x) always

opposes the sign ofs(x), which guarantees crossing S from both sides of the

sur-face The same condition aroundS can be written in compact form as

must hold locally on S Condition (2.16) is a necessary condition for SM

estab-lishment, and it is generally known as transversality condition [26].

As a particular case, consider the following switching law

where the constant k ris chosen so that the steady-state value of the output equals

the set point r, and the feedback gains kTdetermine the linear dynamics during SM(see Figure 2.8)

Figure 2.8 Conventional SM control scheme

Clearly, the transversality condition for the switching function (2.17) is given

by kTb 6¼ 0 Moreover, the transfer function from input u to s results

H SM ðsÞ ¼ kTðsI  AÞ1b ð2:18Þwhich can be written as

H SM ðsÞ ¼ ðkTÞbs1þ ðkTÞAbs2þ    þ ðkTÞA m bs ðmþ1Þþ    ð2:19Þ

by applying the Taylor’s series decomposition Therefore, the transversality dition imposes that the first term of the Taylor’s series decomposition must be non-

con-zero (kTb is the first Markov parameter of the corresponding linear system) This

means that the transfer function from the discontinuous control action to the slidingfunctions must have unitary relative degree

From a theoretical point of view, during SM operation the system switches atinfinite frequency It is then discontinuous at every time instant, and its stateequation does not admit an analytical solution in the classical sense Alternatively,

a simple and useful way of computing the system dynamics during SM consists offinding an equivalent continuous control signal

With this aim, the ideal sliding mode is defined as an ideal operating mode for

which the manifold S is a system invariant Thus, once the system trajectoryreaches the surface, it slides exactly alongS The invariance condition of manifold

Equation (2.21) states that the trajectory remains on the surface, whereas u eq (x) represents a continuous equivalent control action for whichS is a local invariant

manifold of system (2.10) The function u eq (x) can be easily derived from (2.21),

As can be seen, the transversality condition (2.16) is a necessary and sufficient

condition for the equivalent control to be well defined Note also that u eqmakessense only on the surfaces(x) ¼ 0.

For the particular case of sliding function (2.17), the equivalent control isobviously given by

u eq ¼ ðkTbÞ1kTAx ð2:23Þ

We have already seen that the transversality condition is necessary to accomplish

condition (2.13), and therefore for sliding regime to exist Now, from the u eq

definition we can derive a necessary and sufficient condition to guarantee SMestablishment In effect, by subtracting (2.21) from the first line of (2.15) we have

@s

@x

 T

b ðuþ u eqÞ < 0 ð2:24Þwhile

@s

@x

 T

b ðu u eqÞ > 0 ð2:25Þresults from taking away (2.21) from the second line of (2.15) Then, from (2.24)

and (2.25) and assuming without loss of generality that uþ> u, the following

necessary and sufficient condition yields:

The resultant reduced dynamics during ideal SM can be obtained by substituting in

the state equation of system (2.10) the input u with the equivalent control action

(2.22), which gives

The state equations (2.28) and (2.29) redundantly describe the system dynamics

during SM In fact, one of their rows is linearly dependent on the remaining n 1

equations because the system state x satisfies the algebraic constraint s(x) ¼ 0 Then, the matrix A SMhas an eigenvalue at the origin, which has to be attributed tothis redundant description and does not imply sliding regime instability

Given the earlier review of VSS fundamentals, we are now interested in analysingthe SMRC method described in Section 2.3 from a VSS point of view This willallow us to show some important features of the algorithm

v to come to one of its limits In fact, since the switching law (2.4) makes the

discontinuous signal zero at one side of each surface, SMRC could be thought of asproducing a ‘one-side SM’, in which the system slides on the surface only if itcontinues trying to exceed the bounds by itself In this manner, SM is only atransient mode of operation to avoid the system surpassing its constraints (assum-ing sustainable references, see subsection 2.4.2.6)

As we have seen, for SM to be established the transversality condition (2.16) musthold, i.e the first time derivative of the switching function must explicitly depend

on the discontinuous control action Differentiating the sliding function (2.5) of the

SMRC approach, it is straightforward from (2.1) and (2.6) to derive that thisnecessary condition is satisfied provided

Thus, although the simple algorithm of Section 2.3.1 is useful to illustrate the mainidea of the method operation, it is only valid for systems having a direct path from

the input to the constrained variable v (i.e biproper systems with d s6¼ 0) In the

case of strictly proper constrained systems S c (s), additional system states should be

included in the switching function so that the transversality condition holds That is,other sliding function rather than the trivial one of (2.5) should be defined whose

first derivative explicitly depends on the discontinuous action w This case is

addressed in Section 2.5

The continuous equivalent signal to the high-frequency switching signal producedwhen the system reaches one of its limits can be obtained from the invarianceconditions, generically given by (2.20) and (2.21) For the SMRC algorithm, thiscondition corresponds tos~ ¼ 0 and s~_ ¼ 0, where the tilde is used to denote eitherthe upper or the lower bar (?~ ¼ ? or ~? ¼ ?) Using the system and filter repre-sentations (2.1) and (2.6), it can be easily shown that

It is evident that the continuous equivalent control is only well defined provided

d s6¼ 0, which confirms the arguments given in the previous paragraph

From the general condition (2.26) and the particular switching logic of the SMRCalgorithm (2.4), the following conditions can be derived for SM establishment oneach of the limiting surfaces:

where w< 0 < wþwas assumed and both w

eq and w eqare given by (2.31)

It is important to remark that the inequalities with respect to zero in (2.32) and(2.33) are verified whenever the dynamical system attempts, by its own, to cross

the bounds v or v When this occurs, the discontinuous signal amplitudes wor wþ

should be large enough to satisfy the remaining inequalities, so that SM is anteed and the system constraints are fulfilled According to (2.31), this can always

guar-be achieved for appropriate bounds on _d , r, _ ~v and _x s

It is also worthy of mention in this regard that the selection of wþand wcan

be made in a conservative manner because the SM loop is restricted to the