Anti-windup compensator design using Riccati inequality based approach⁎

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Anti-windup compensator design Anti-windup compensator design Anti-windup compensator design Anti-windup compensator design using Riccati inequality based using Riccati inequality based using inequality based  using Riccati Riccati inequality based approach approach  approach  Triparna Ghosh, ∗∗ approach Parijat Bhowmick, ∗∗ Sourav Patra ∗∗

Triparna Ghosh, ∗∗ Parijat Bhowmick, ∗∗ Sourav Patra ∗∗ Triparna Ghosh, ∗∗ Parijat Bhowmick, ∗∗ Sourav Patra ∗∗ Ghosh, Parijat Sourav ∗ Triparna Ghosh, Technology Parijat Bhowmick, Bhowmick, WB, Sourav Patra Patra Indian ∗ Triparna Indian Institute Institute of of Technology Kharagpur, Kharagpur, WB, India India -- 721302 721302 ∗ ∗ (e-mail: [email protected]; Indian Institute of Technology Kharagpur, WB, India - 721302 ∗ (e-mail: [email protected]; ∗ Indian Institute Institute of Technology Technology Kharagpur, WB, India India -- 721302 721302 Indian of Kharagpur, WB, (e-mail: [email protected]; [email protected]; [email protected]). [email protected]; [email protected]). (e-mail: (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). [email protected]; [email protected]; [email protected]). [email protected]). Abstract: This Abstract: This paper paper presents presents a a full-order full-order dynamic dynamic anti-windup anti-windup compensation compensation design design scheme scheme Abstract: Thissystems paper presents aRiccati full-order dynamicbased anti-windup compensation design scheme for stable LTI using a inequality approach. The scheme utilizes the for stable LTI systems using a Riccati inequality based approach. The scheme utilizes the Abstract: This paper presents a full-order dynamic anti-windup compensation design scheme Abstract: Thissystems paper presents aRiccati full-order dynamic anti-windup compensation design scheme Weston-Postlethwaite’s generalized anti-windup configuration and applies right coprime facfor stable LTI using a inequality based approach. The scheme utilizes the Weston-Postlethwaite’s generalized anti-windup configuration and applies right coprime facfor stable LTI LTI systems systems transfer using a a function Riccati inequality based approach. approach. The scheme scheme utilizes The the for stable using Riccati inequality based The utilizes the torization matrix to conditioning Weston-Postlethwaite’s generalized anti-windup and applies right scheme. coprime factorization of of the the plant plant transfer function matrix configuration to obtain obtain aa linear linear conditioning scheme. The Weston-Postlethwaite’s generalized anti-windup configuration and applies rightcontrol coprime facWeston-Postlethwaite’s generalized anti-windup configuration and applies right coprime factorization of the plant transfer function matrix to obtain a linear conditioning scheme. The proposed anti-windup scheme ensures the internal stability of the closed-loop system proposed anti-windup scheme ensures the internal stability the closed-loop control system torization of the transfer function to obtain aa of linear conditioning scheme. The torization of of the plant plantscheme transfer function matrix to stability obtain linear conditioning scheme. The in saturation and also reduces the performance degradation caused proposed anti-windup ensures thematrix internal the closed-loop control system in presence presence of actuator actuator saturation and also reduces the of performance degradation caused proposed anti-windup scheme ensures the internal stability of the closed-loop control system proposed anti-windup scheme ensures the also internal stability of the closed-loop control system by the saturation phenomena. The Riccati inequality based approach offers a simple and in presence of actuator saturation and reduces the performance degradation caused by presence the saturation phenomena. The and Riccati inequality based approach offers a simple and in of actuator actuator saturation also reduces the performance degradation caused in presence of saturation and also reduces the performance degradation caused straightforward anti-windup compensator design method which is independent of the design by the saturation phenomena. The Riccati inequality based approach offers a simple and straightforward anti-windup compensator design method which is independent of the design by the saturation saturation phenomena. The Riccati Riccati inequality based approach offers aaof simple simple anda by the phenomena. The inequality offers and straightforward anti-windup compensator methodbased whichapproach is is the design of The of the technique demonstrated through of nominal nominal controller. controller. The effectiveness effectiveness of design the proposed proposed technique isindependent demonstrated through a straightforward anti-windup compensator design method which of design straightforward anti-windup compensator design methodtechnique which is is isindependent independent of the the designa MIMO example. of nominal controller. The effectiveness of the proposed demonstrated through MIMO example. of nominal controller. of nominal controller. The The effectiveness effectiveness of of the the proposed proposed technique technique is is demonstrated demonstrated through through aa MIMO example. MIMO example. © 2018, example. IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. MIMO Keywords: Keywords: Anti-windup Anti-windup compensation, compensation, Riccati Riccati inequality, inequality, L L22 -gain, -gain, Lyapunov Lyapunov stability, stability, LMI. LMI. Keywords: Anti-windup compensation, Riccati inequality, L22 -gain, Lyapunov stability, LMI. stability, Keywords: Anti-windup L -gain, Lyapunov Lyapunov design stability, LMI. LMI. Keywords: Anti-windup compensation, compensation, Riccati Riccati inequality, inequality, L22 -gain, 1. other 1. INTRODUCTION INTRODUCTION other approach, approach, aa two-step two-step design method method is is followed followed – – 1. INTRODUCTION other approach, a two-step design method is followed – where a stabilizing controller is synthesized first without where a stabilizing controller is synthesized first without 1. INTRODUCTION other approach, a two-step design method is followed – 1. capacity INTRODUCTION other approach, a two-step design method is followed – considering the saturation constraint and then, a ‘linwhere a stabilizing controller is synthesized first without Limitation in the of actuators is a common the saturation constraint and then, a ‘linLimitation in the capacity of actuators is a common considering where a stabilizing controller is synthesized first without where a stabilizing controller is synthesized firstthe without ear conditioning’ scheme is used to take care effect Limitation in the capacity of actuators is a common considering the saturation constraint and then, a ‘linproblem in physical systems [Hippe (2006)]. It is mainly conditioning’ scheme is constraint used to take theaeffect problem in physical systems [Hippe (2006)].is Ita iscommon mainly ear considering the saturation andcare then, ‘linLimitation in the of actuators the saturation constraint and then, a ‘linLimitation in safety the capacity capacity of technological actuators a iscommon of saturation [Weston et (2000)]. Such kind AW ear conditioning’ scheme isal. used to take care theof because of issues and constraints. problem in the physical systems [Hippe (2006)].is It mainly considering of saturation [Weston et al. (2000)]. Such kind ofeffect AW because of the safety issues and technological constraints. ear conditioning’ scheme is used to take care the effect problem in physical systems [Hippe (2006)]. It is mainly ear conditioning’ scheme is used to take care the effect of saturation [Weston et al. (2000)]. Such kind of AW problem in physical systems [Hippe (2006)]. It is mainly compensators are in operation only during the saturation Due to the actuator constraints, when a large control because of the safety issues and technological constraints. compensators are in operation only during the saturation Due to the actuator constraints, when a large control of saturation [Weston et al. (2000)]. Such kind of because of theactuator safety issues and technological constraints. of saturation [Weston et al.actions (2000)]. Such kind of AW AW because of the safety issues and technological constraints. period and take necessary to preserve internal Due to the constraints, when a large control compensators are in operation only during the saturation input demand is required to counteract unexpected large and take necessary actions to preserve internal inputto demand is required to counteract unexpected large period compensators are in only during the Due the actuator constraints, when a control compensators are in operation operation only during the saturation saturation Due to the inputs actuator constraints, when a large large control stability of the closed-loop system and also counteracts the period and take necessary actions to preserve internal disturbance or abrupt major set-point changes, the input demand is required to counteract unexpected large of the closed-loop system andtoalso counteracts the disturbance inputs or abrupt major set-point changes,large the stability period and take necessary actions preserve internal input demand is required to counteract unexpected period and take necessary actions to preserve internal input demand isrespond required tothe counteract unexpected large performance degradation caused by actuator saturation. changes, the stability of the closed-loop system and also counteracts the actuator cannot to full control input demand. disturbance inputs or abrupt major set-point performance degradation caused by actuator saturation. actuator cannot respond to the full control input demand. stability of the closed-loop system and also counteracts the disturbance inputs orfails abrupt major set-point changes, the stability of thestage closed-loop system and also counteracts the performance degradation caused by actuator saturation. disturbance inputs or abrupt major set-point changes, the At an earlier of anti-windup research, it was usually Thus, the controller to take necessary corrective acactuator cannot respond to the full control input demand. At an earlier stage of anti-windup research, it was usually Thus, the controller fails to take necessary corrective acperformance degradation caused by actuator saturation. actuator cannot respond to the full control input demand. performance degradation caused by actuator saturation. actuator cannot respond to the full control input demand. thought the that windup phenomena occurs only because Thus, the controller fails to take necessary corrective acAt an earlier stage of anti-windup research, it was usually tion and as a consequence, the error between the plant thought the that windup phenomena occursitonly because tion and as a consequence, the error between the plant At an earlier earlier stage of anti-windup anti-windup research, was usuallya Thus, the controller fails take necessary corrective acAt an stage of research, itonly was usually Thus, the controller fails to toinput take necessary corrective acof integral of the controller. In [Fertik et al. (1967)], thought thepart that windup phenomena occurs because output and the reference keeps on increasing. In tion and as a consequence, the error between the plant of integral part of the controller. In [Fertik et al. (1967)], a output and the reference input keeps on increasing. In thought the that windup phenomena occurs only because tion and as a consequence, the error between the plant thought the that windup phenomena occurs only because of integral part of the controller. In [Fertik et al. (1967)], a tion andand as the a consequence, thegets error between the plant termed as the ‘anti-reset windup’ was introduced reference input keeps on increasing. In scheme this circumstance, the actuator saturated and aa severe output scheme termed as the ‘anti-reset windup’ was introduced this circumstance, the actuator gets saturated and severe of integral part of the controller. In [Fertik et al. (1967)], a output and the reference input keeps on increasing. In of integral part of the controller. In [Fertik et al. (1967)], a scheme termed as the ‘anti-reset windup’ was introduced output and the reference input keeps on increasing. In where the integral part of the controller was modified to performance degradation in closed-loop response, such this circumstance, the actuator saturated and a severe the integral part‘anti-reset of the controller was modified to performance degradation in the the gets closed-loop response, such where scheme termed as the windup’ was introduced this circumstance, the actuator gets saturated and a severe scheme termed asactuator the ‘anti-reset windup’ wasmodified introduced this circumstance, the actuator gets saturated and aetc., severe escape from the saturation. A similar procedure performance degradation in the closed-loop response, such where the integral part of the controller was to as large overshoot/undershoot, long settling time, is fromintegral the actuator A similar procedure as large overshoot/undershoot, long settlingresponse, time, etc., is escape where the partknown ofsaturation. the as controller was modified modified to performance degradation in such the part of the controller was to performance degradation in the the closed-loop closed-loop such of anti-reset windup, the antiescape fromintegral the actuator saturation. Aconventional similar procedure observed. The effect of actuator saturation becomes the as large overshoot/undershoot, long settlingresponse, time, etc., is where of anti-reset windup, known as the conventional antiobserved. The effect of actuator saturation becomes the escape from the actuator saturation. A similar procedure as large overshoot/undershoot, long settling time, etc., is escape from the actuator saturation. A similar procedure of anti-reset windup, known as the conventional antias large overshoot/undershoot, long settlingsometimes, time, etc.,the is windup technique, was reported in et worst when integral is used it observed. The effect controller of actuator becomes windup technique, was known reported in [Doyle [Doyle et al. al. (1987)], (1987)], worst when integral is saturation used and and sometimes, it of anti-reset windup, as conventional antiobserved. The effect controller of actuator actuator saturation becomes the of anti-reset windup, as inthe the conventional antiwindup technique, was known reported [Doyle etof al. (1987)], observed. The effect of saturation becomes the where the control input is modified instead modifying may lead to instability too. In order to reduce the adworst when integral controller is used and sometimes, it where the control input is modified instead of modifying may lead tointegral instability too. Inisorder to reduce the adwindup technique, was reported in [Doyle et al. (1987)], worst when controller used and sometimes, it technique, was reported in [Doyle et al. (1987)], worst whentointegral controller isorder used to and sometimes, it windup the integral part of the controller. In the sequel, a may lead instability too. In reduce the adwhere the control input is modified instead of modifying verse effects of the actuator saturation, a proper action the integral part of the iscontroller. In the sequel, a linlinverse effects of the actuator saturation, a properthe action where the control control input modified instead of modifying may lead to instability too. In order to adwhere the input iscontroller. modified instead of modifying may lead toThe instability too.of saturation, In order to aisreduce reduce the adear conditioning technique named as ‘Hanus conditioning the integral part of the In the sequel, a linis required. objective this paper to propose a verse effects of the actuator proper action ear conditioning technique named as ‘Hanus conditioning is required. The objective of this paper is to propose a the conditioning integral parttechnique of the the controller. In (1987)] the sequel, sequel, a linlinverse effects ofinequality the actuator saturation, ais proper proper actiona the integral part of controller. In the a ear namedet conditioning verse effects of the actuator a action scheme’ was in al. which linear matrix (LMI) to is required. objective of saturation, thisbased papertechnique to propose scheme’ was proposed proposed in [Hanus [Hanus etas al.‘Hanus (1987)] which was was linear matrixThe inequality (LMI) based technique to dedeear conditioning technique named as ‘Hanus conditioning is required. The objective of this paper is to propose a ear conditioning technique named as ‘Hanus conditioning is required. The objective of this paper is to propose a an extension of the ‘back-calculation strategy’ published in linear matrix inequality (LMI) based technique to descheme’ was proposed in [Hanus et al. (1987)] which was sign linear scheme required to an extension of the ‘back-calculation strategy’ published in sign a a matrix linear conditioning conditioning scheme required to form form an an scheme’ was proposed in [Hanus et (1987)] which was linear inequality (LMI) based technique to descheme’ was proposed inthis [Hanus et al. al. (1987)] whichinput was linear matrix inequality (LMI) based technique to the de[Fertik et al. (1967)]. In technique, the reference sign a linear conditioning scheme required to form an an extension of the ‘back-calculation strategy’ published in anti-windup (AW) compensation strategy following [Fertik et al. (1967)]. In this technique, the reference input anti-windup (AW) compensation strategy following the an extension ofshaped the ‘back-calculation ‘back-calculation strategy’ published in sign a conditioning scheme required to an extension of the strategy’ published in sign a linear linear(AW) conditioning scheme requiredfollowing toal.form form an an itself has been by providing an extra feedback loop [Fertik et al. (1967)]. In this technique, the reference input Weston-Postlethwaite’s AW scheme [Weston et (2000)] anti-windup compensation strategy the itself has been shapedInbythis providing an extra feedbackinput loop Weston-Postlethwaite’s AW schemestrategy [Weston following et al. (2000)] [Fertik et al. (1967)]. technique, the reference anti-windup (AW) compensation the [Fertik et al. (1967)]. In this technique, the reference input itself has been shaped by providing an extra feedback loop anti-windup (AW) compensation strategy following the activated at the instant of actuator saturation. A more for stable Weston-Postlethwaite’s scheme [Weston et al. (2000)] activated at the instant of actuator saturation. A more for stable LTI LTI plants. plants. AW has been shaped by an extra feedback loop Weston-Postlethwaite’s AW scheme scheme [Weston [Weston et et al. al. (2000)] (2000)] itself itself hasversion been shaped by providing providing an saturation. extra feedback loop Weston-Postlethwaite’s general of ‘Hanus conditioning scheme’ et for stable LTI plants. AW activated at the actuator A more general version ofinstant ‘Hanus of conditioning scheme’ [Hanus [Hanus et The actuator saturation problem is handled in two ways: activated at the instant of actuator saturation. A more for stable LTI plants. activated at was theofgiven instant of actuator et saturation. A which more for stable LTI saturation plants. al. (1987)] in [Walgama al. (1992)] The actuator problem is handled in two ways: general version ‘Hanus conditioning scheme’ [Hanus et al. (1987)] was given in [Walgama et al. (1992)] which in one approach, the controller is executed general version ofgiven ‘Hanus conditioning scheme’ [Hanus et The actuator saturation problemdesign is handled in twotaking ways: general version of ‘Hanus conditioning scheme’ [Hanus et is called ‘generalized conditioning technique’. A state obin one approach, the controller design is executed taking al. (1987)] was in [Walgama et al. (1992)] which The actuator saturation problemdesign is handled in and twotaking ways: is called ‘generalized conditioning technique’. A state obThe actuator saturation problem is handled in two ways: in one approach, the controller is executed the actuator saturation constraint into account; in the al. (1987)] was given in [Walgama et al. (1992)] which al. (1987)] was given in [Walgama et al. (1992)] which is called ‘generalized conditioning technique’. A state observer based AW compensator design process for a class theone actuator saturation constraint into account; andtaking in the server based AW compensator design process for a class of in approach, the design is of in one approach, the controller controller design is executed executed called ‘generalized conditioning technique’. A obthe actuator saturation constraint into account; andtaking in the is is called ‘generalized conditioning technique’. Aal.astate state obLTI plants has been introduced in [Walgama et (1990)].  server based AW compensator design process for class of The first and third authors gratefully acknowledge the financial the actuator saturation constraint into account; and in the LTI plants has been introduced in [Walgama et al. (1990)].  the actuator constraint account;the andfinancial in the server based AW compensator design process for a class of The first andsaturation third authors gratefullyinto acknowledge server based AW compensator design process for a class of Modern anti-windup techniques started flourishing in mid LTI plants has been introduced in [Walgama et al. (1990)].  support of the and Engineering Research Board Modern anti-windup techniquesinstarted flourishing in mid The first and Science third authors gratefully acknowledge the (SERB), financial support of the Science and Engineering Research Board (SERB), LTI plants has been introduced [Walgama et al. (1990)].  LTI plants has been introduced in [Walgama et al. (1990)].  The 90s. Static and low-order AW compensator synthesis were The first first and third authors gratefully acknowledge the financial Modern anti-windup techniques started flourishing in mid Department of Science and Technology, India, under the research support of the Engineering Research Board (SERB), and third authors gratefully acknowledge the financial Science and and Technology, India, under the research 90s. Static and low-order AW compensator synthesis were Department of Science Modern anti-windup techniques started flourishing in mid Modern anti-windup techniques started flourishing inwere mid support of Science and and Technology, Engineering Research Board 90s. Static and low-order AW(2004)] compensator synthesis discussed in [Turner et al. and the anticipatory grant SB/FTP/ETA-263/2012. support of the the Engineering India, Research Board (SERB), Department of Science under the (SERB), research discussed in [Turner et al. (2004)] and the anticipatory grant SB/FTP/ETA-263/2012. 90s. Static and low-order AW compensator synthesis were Department of Science Science and and Technology, Technology, India, India, under under the the research research 90s. Staticinand low-order AW(2004)] compensator were Department of discussed [Turner et al. and thesynthesis anticipatory grant SB/FTP/ETA-263/2012. discussed in in [Turner [Turner et et al. al. (2004)] (2004)] and and the the anticipatory anticipatory grant discussed grant SB/FTP/ETA-263/2012. SB/FTP/ETA-263/2012.

Copyright © 2018 IFAC 647 Copyright 647 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright © under 2018 IFAC 647 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 647 Copyright © 2018 IFAC 647 10.1016/j.ifacol.2018.05.103

5th International Conference on Advances in Control and 616 Triparna Ghosh et al. / IFAC PapersOnLine 51-1 (2018) 615–620 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

anti-windup scheme was proposed in [Wu et al. (2014)]. A plenty of LMI based design techniques are now available in the literature to develop AW compensators in a systematic way considering satisfactory performance, robustness and closed-loop stability [Tarbouriech et al. (2007); Mulder et al. (2001); Teel et al. (2011)]. An excellent survey of the modern AW compensation methods is published in [Galeani et al. (2009)]. The Recent advancements can be found in the literatures [Sofrony et al. (2015); Ofodile et al. (2015); Ofodile and Turner (2016); Turner (2017); Torrico et al. (2016); Ofodile et al. (2016); Donnarumma et al. (2016)] and in the references cited therein.

for a diagonal matrix with diagonal elements βi , i = A B 1, 2, · · · , m. R(s) = represents the state-space C D realization of a real-rational, proper transfer function matrix R(s). L2 [0, ∞) indicates the Lebesgue space which consists of all measurable functions f (·) : [0, ∞) → R ∞ such that f  (t)f (t) dt < ∞. L2e denotes the extended

In the present work, we use the Weston-Postlethwaite’s anti-windup compensation scheme [Weston et al. (2000)] shown in Figure 2 and focus on the design of a linear conditioning scheme governed by a single transfer function matrix M (s). A right coprime factorization of the plant transfer function matrix, i.e., G(s) = N (s)M (s)−1 has been adopted to design the ‘disturbance filter’ N (s) = G(s)M (s), shown in Figure 3, so that the poles of G(s) do not appear in N (s). The main advantage of using the coprime factorization method is that a single static stabilizing state feedback gain matrix F is required to form both M (s) and N (s) which constitute the anti-windup compensator. Apart from that, the present technique also decouples the nonlinear loop and the intended linear system as illustrated in Figure 3, and due to this internal separation, the AW design process becomes independent of the design of the nominal controller K(s). The coprime factorization based full-order AW compensator design technique facilitates easy implementation of the scheme widening its applicability for stable, LTI plants. The Riccati inequality based approach has been resorted to render a simple AW compensator design method which ensures both stability and satisfactory performance of the closed-loop system in the event of actuator saturation. In [Sofrony et al. (2007)], a similar Riccati equation based approach was pursued to develop a full-order anti-windup compensator.

3. DEFINITIONS AND SOME USEFUL RESULTS

The remainder of the paper proceeds as follows: in Section II, useful notations are given. Section III illustrates some fundamental concepts, definitions and lemmas which provide a background for proving the main results of this paper. Section IV contains the main contribution of this paper, i.e., the Riccati inequality based full-order antiwindup compensator design method. Section V presents a numerical example to show the usefulness of the proposed anti-windup compensation scheme. Section VI concludes the paper mentioning the future research directions. 2. NOTATIONS The notations and acronyms are standard throughout. AT and A denote the transpose and the complex-conjugate transpose of a matrix A. For a square and non-singular ma trix A, A− represents the shorthand for A−1 . (·)T also denotes the vector transpose. RH m×m is the space of all ∞ real-rational, proper, stable transfer function matrices of , then the infinity dimension m × m. Let R(s) ∈ RH m×m ∞ norm of R(s) is defined as R(s)∞ = sup σ ¯ [R(jω)] where ω∈R

σ ¯ [·] denotes the maximum singular value of a matrix. The notation diag{β1 , β2 , · · · , βm } is used as a shorthand 648

0

Lebesgue space. fT := (f (t))T indicates the truncation of the signal f ∈ L2e ∀T ∈ [0, ∞). Γ L2 represents the L2 -gain of an operator Γ : u → y.

In this section, some useful technical results and lemmas are presented which provide a background to prove the main results. 3.1 Coprime factorization [Boyd et al. (1994)] In a SISO setting, two polynomials m(s) and n(s) are coprime if there exist polynomials x(s) and y(s) such that x(s)m(s) + y(s)n(s) = 1, known as Bezout identity. In general, let P (s) be a real-rational, proper transfer function matrix. A right coprime factorization of P (s) is a factorization P (s) = N (s)M (s)−1 where  N (s) and A B is a M (s) are right-coprime over RH ∞ . Suppose C D stabilizable and detectable realization of P (s). Let F be a matrix such that A + BF is Hurwitz, then a right coprime factorization of P (s) = N (s)M (s)−1 is given by     A + BF B M F I . (1) = N C + DF D

3.2 Saturation and dead-zone nonlinearities

Below we represent the input-output characteristics of the saturation φ(·) and dead-zone Ψ(·) nonlinearities. (u)

 (u ) 1

1

1

0

1

u

0

u

1

1

Fig. 1. Saturation and dead-zone nonlinearities. Let us define the saturation nonlinearity:   1 u ≥ 1, u |u| < 1, φ(u) =  −1 u ≤ −1.

(2)

If there is no saturation present, φ(u) ≡ u. The dead-zone nonlinearity is defined as   u − 1 u ≥ 1, 0 |u| < 1, Ψ(u) = (3)  u + 1 u ≤ −1. It is also convenient to represent the dead-zone function Ψ(·) in terms of the saturation function as follows: Ψ(u) = u − φ(u). (4)

5th International Conference on Advances in Control and Triparna Ghosh et al. / IFAC PapersOnLine 51-1 (2018) 615–620 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

A characteristic of the dead-zone, used extensively in this paper, is that Ψ(·) ∈ sector[0, I], as defined below: Definition 1. (Sofrony et al. (2007)). A decentralized nonlinearity N = diag{η1 (·), · · · , ηm (·)} is said to belong to sector[0, I] if the following inequality holds: 0 ≤ ηi (ui )ui ≤ u2i

∀i ∈ [1, · · · , m].

(5)

3.3 Weston-Postlethwaite’s anti-windup scheme The Weston-Postlethwaite’s generalized anti-windup compensation scheme performs linear conditioning with M (s), a right coprime factor of the nominal plant G(s). In this framework, the anti-windup compensator design method is independent of the nominal controller (K(s)) design.

r

ulin 

K (s)



ylin

uˆ

u



ud

G( s)

R(x) > 0

M ( s)

G( s)

Mode II



 K (s)

M (s)  I



> 0,

R(x) > 0

and

G( s)

M ( s)

G( s)

ylin





(8)

Further (7) is equivalent to

Mode III

u

(7)

which is an LMI constraint [Boyd et al. (1994)]. In other way, (7) is satisfied if and only if (6) holds.

Q(x) > 0

ulin

Q(x) S(x) S(x)T R(x)

Q(x) − S(x)R(x)−1 S(x)T > 0,

ud u

Q(x) − S(x)R(x)−1 S(x)T > 0 (6)

and

can be expressed as 

Fig. 2. Weston-Postlethwaite’s generalized anti-windup compensation scheme [Weston et al. (2000)].

r

In Figure 3, when the signal ulin is within the interval [−usat , usat ], u ˜ remains zero and in turn, the disturbance signal yd also remains zero. Accordingly, the plant output y is equal to the nominal output ylin and this mode of operation is called Mode-I. When magnitude of ulin increases and exceeds the saturation level ±usat , Mode-II and Mode-III come into play. Thus, during the actuator saturation period, u ˆ is no longer equal to u in Figure 2 and as a result, it gives rise to nonzero u ˜ which produces the disturbance signal yd through the disturbance filter N (s) and yd deteriorates the output of the system as y = ylin − yd .

This lemma converts a nonlinear matrix inequality into a linear matrix inequality (LMI) constraint. If x is a variable and the matrices Q(x) = Q(x)T , R(x) = R(x)T and S(x) depend affinely on x, then the set of convex nonlinear inequalities



yd



as it produces the signal yd after filtering out the unwanted nonlinear signal u ˜ during the actuator saturation.

3.4 Schur complement lemma

y

u

M (s)  I

617

and

R(x) − S T (x)Q(x)−1 S(x) > 0 (9)

where Q(x) − S(x)R(x)−1 S(x)T and R(x) − S T (x)Q(x)−1 S(x) are called the Schur complements of R(x) and Q(x), respectively. 4. RICCATI INEQUALITY BASED ANTI-WINDUP COMPENSATOR SYNTHESIS

yd

y

Mode I

Fig. 3. Decoupled structure of the Weston-Postlethwaite’s anti-windup compensation scheme. The scheme shown in Figure 2 can be represented into the following three decoupled modules: [Weston et al. (2000)] 1. Intended Linear System (Mode-I), 2. Nonlinear Loop (Mode-II), 3. Disturbance Filter (Mode-III), as marked in Figure 3. The equivalent structure is used to analyze the stability and performance of the closedloop control system having the anti-windup compensator during the actuator saturation periods. In this scheme, yd is called the disturbance signal as it causes the plant output y to get deviated from the desired output ylin . The linear block N (s) = G(s)M (s) acts as a ‘disturbance filter’ 649

In this section, a Riccati inequality based full-order antiwindup compensator design process is introduced in which the Weston-Postlethwaite’s scheme (see Figure 2) is used as the underlying framework. A set of LMI conditions has been proposed which solves a minimization problem to yield the state feedback gain matrixF required to form A + BF B along anti-windup compensator M (s) = F I with minimizing the L2 gain of the operator Γ : ulin → yd as shown in Figure 3.   A B Theorem 1. Let be a minimal state-space realC 0 ization of an LTI plant G(s) ∈ RH p×m and suppose ∞ that the dead-zone nonlinearity in Figure 3 belongs to the sector [0, I]. If there exist a real matrix P = P T > 0, a positive scalar γ and a diagonal matrix W > 0 such that the following minimization problem yields a feasible solution: minimize γ subject to

5th International Conference on Advances in Control and 618 Triparna Ghosh et al. / IFAC PapersOnLine 51-1 (2018) 615–620 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India



 P A + AT P + C T C P B < 0, − 12 W BT P   2W W and > 0, W γ2I

(10) (11)

then the anti-windup compensator with F = W −1 B T P ensures that the L2 gain of Γ : ulin → yd is less than γ. Proof. The proof establishes that the nonlinear operator Γ : ulin → yd , as shown in Figure 3, is finite L2 -gain stable and also minimizes the L2 -gain of the operator Γ . Let us consider a quadratic Lyapunov function V (x) = xT P x with P = P T > 0. If the condition (12) V˙ (x) + ydT yd − γ 2 uTlin ulin < 0 is satisfied, then L2 -gain of the nonlinear operator Γ : ulin → yd is less than γ > 0. Further considering the sector-bounded property of the dead-zone nonlinearity Ψ(·), there exists a diagonal matrix W > 0 such that u ˜T W (u − u ˜) ≥ 0 holds. Now, exploiting this condition it is straightforward to assert that if uT W (u − u ˜) < 0 (13) V˙ (x) + ydT yd − γ 2 uTlin ulin + 2˜ holds, then (13) implies (12) and in turn Γ L2 < γ. The nonlinear operator Γ : ulin → yd is now represented in a linear fractional transformation (LFT) framework, as shown in Figure 4, which is governed by the following state-space equations:  x˙ = (A + BF )x + B u ˜,   u = u − u , lin d (14) Γ : u = F x and    d yd = Cx, where F is the stabilizing static state-feedback gain matrix to be determined such that A + BF is Hurwitz. Substituting the expression for V˙ (x) into the above in-

u

u



ulin li ud

yd

x

F Fig. 4. LFT representation of the Weston-Postlethwaite’s anti-windup scheme. equality (13) and using the state-space description (14) we have, xT (A + BF )T P x + u ˜T B T P x + xT P (A + BF )x + xT P B u ˜ + xT C T Cx − γ 2 uTlin ulin + u ˜T W ulin

−u ˜T W F x − 2˜ uT W u ˜ + uTlin W u ˜ − xT F T W u ˜<0

− (γ 2 uTlin ulin − u ˜T W ulin − uTlin W u ˜+

1 2 W + 2W )˜ u<0 γ2 ⇔xT (P A + AT P + P BF + F T B T P + C T C)x −u ˜T (−

+u ˜T (B T P − W F )x + xT (P B − F T W )˜ u 1 1 ˜)T (γulin − W u ˜) − (γulin − W u γ γ 1 −u ˜T (2W − 2 W 2 )˜ u < 0. γ

(15)

n The inequality (15) holds for any x ∈ L2e and u ˜, ulin ∈ m if the following set of sufficient conditions is satisfied: L2e

P A + AT P + P BF + F T B T P + C T C < 0,

(16)

PB = FTW and (17) 1 (18) 2W − 2 W 2 > 0, γ which in turn ensures V˙ (x) < 0 and Γ L2 < γ. Thus, the closed-loop stability is achieved in presence of the actuator saturation. Now, the inequality conditions (16) and (18) can easily be transformed into the LMI form as mentioned in (10) and (11) by applying the Schur complement Lemma given in Subsection 3.4 and using the equality constraint F = W −1 B T P from (17). Apart from checking the feasibility of the LMI constraints (10) and (11), we also minimize the L2 -gain of the operator Γ so that the effect of actuator saturation can be minimized. Hence, the proof is completed.  It may be noted here that (16) is the Riccati inequality with the optimal state feedback gain matrix F = W −1 B T P . In this respect, the proposed anti-windup compensator can be considered to be optimal. Remark 1. Apart from the set of LMI conditions already derived in Theorem 1, another sufficient-type inequality condition from the L2 -gain minimization problem (15) is likely to appear as given below:    P A + AT P + P BF T PB − F W     +F T B T P + C T C     < 0.   1 − 2W − 2 W 2 BT P − W F γ (19) Contrary to (10), the inequality (19) does not use the equality condition F = W −1 B T P and therefore, it may offer a broader solution space in some cases. The inequality (19) can equivalently be transformed into    AR + RAT + T T 0   BK + K T B T RC BV − K     CR −I 0 0  < 0 (20)     0 −2V I  V BT − K 0

⇔xT (P A + AT P + P BF + F T B T P + C T C)x

1 T 2 u ˜ W u ˜) γ2

0

I

−γ 2 I

by applying the Schur complement Lemma, two consecutive congruent transformations with diag{P −1 , I, I} and diag{I, W −1 , I} respectively, and finally denoting the following change of matrix variables: R := P −1 > 0, V := W −1 > 0 and K := F R. A feasible solution of (20) yields the desired state feedback gain matrix F = KR−1

+u ˜ T B T P x + xT P B u ˜ − γ 2 uTlin ulin + u ˜T W ulin

−u ˜T W F x − 2˜ uT W u ˜ + uTlin W u ˜ − xT F T W u ˜<0

⇔xT (P A + AT P + P BF + F T B T P + C T C)x +u ˜T (B T P − W F )x + xT (P B − F T W )˜ u

650

5th International Conference on Advances in Control and Triparna Ghosh et al. / IFAC PapersOnLine 51-1 (2018) 615–620 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

10

output 1

5

−10 0

The Figures 5, 6, 7, and 9 indicate the performance degradation such as extremely large and poorly decaying overshoot, undershoot and increased settling time of the uncompensated closed-loop system response (dotted line) in the event of actuator saturation. In Figures 5 and 6, the compensated outputs y1 and y2 (dash-dot line) are found to follow the unsaturated response (continuous line) closely and moreover, the return to nominal linear dynamics from the saturation is fast enough. It claims that the full-order AW compensator reduces the percentage overshoot and undershoot of the output response to a great extent and makes the response faster causing significant improvement of the closed-loop performance. Further, 651

50

100

150

200 time (s)

250

300

350

400

Output without saturation without anti−windup Output with saturation without anti−windup Output with saturation with anti−windup

Fig. 5. Output y1 of the closed-loop system. 10

5. NUMERICAL EXAMPLE

5

0

−5 0

50

100

150

200 time (s)

250

300

350

400

Fig. 6. Output y2 of the closed-loop system.

control input 1

0 −50 −100 −150 0

50

time (s)

100

150

Fig. 7. The required control input u ˆ1 for the closed-loop system. 20 10 control input 1

We consider a stable MIMOplant G(s) with  a minimal   −0.05 0 1 0 state-space realization A = , B = , 0 −0.05 0 1     −0.4 0.5 00 C = , D = and a PI controller K(s) 0.3 −0.4 00   00 , Bc = is designed for tracking given as Ac = 00       5.00 6.25 78.00 97.50 80 and Dc = − . , Cc = − 3.75 5.00 58.50 78.00 08 The saturation limits of the actuators are chosen as ±5 in both the channels. Now we check the feasibility of the minimization problem given by the LMI conditions (10) and (11) for this plant G(s)  using the Matlab  LMI 16.4142 2.5104 toolbox and we obtain P = > 0, 2.5104 15.1590   527.9590 0 γ = 16.2474 and W = which yield 0 527.9590   0.0311 0.0048 . The AW compenF = W −1 B T P = 0.0048 0.0287   A + BF B sator is designed by forming M (s) = and F I   A + BF B N (s) = . The step (simulation) responses C 0 are depicted in the Figures 5-10.

0 −5

output 2

required to develop the AW compensation scheme through M (s) and N (s).  Remark 2. A similar Riccati equation (ARE) based antiwindup synthesis technique has been reported in [Sofrony et al. (2007)] where a particular Riccati equation is given along with the conditions Z := 2W − DT D − γ12 W 2 > 0 and R = γ 2 I − DT D > 0. The desired state feedback gain matrix is found as F = γ12 (W −1 − γ12 I)R−1 (B T P − DT C) [(Sofrony et al., 2007, Theorem 1)]. It is straightforward to assert that, for strictly-proper case, the condition Z > 0 is the same as (11) proposed in this paper and F = (γ 2 I − W −1 )B T P , which is closely related to our result. But, in [Sofrony et al. (2007)], the diagonal matrix W and the L2 -gain γ are required to be selected ‘a priori’ contrary to the present result (Theorem 1), which finds the matrices W , P and the scalar γ by solving the LMI conditions (10) and (11) for a stable LTI plant. Moreover, Theorem 1 of this paper minimizes the γ value, whereas in [Sofrony et al. (2007)], based on an initial choice of γ > G(s)∞ , the Riccati equation is solved. 

619

0 −10 −20 −30 −40 −50 45

50

55 time (s)

60

65

Fig. 8. The magnified version of the u ˆ1 plot during the saturation period starting from t = 50s subject to a unit step reference input in both the channels. from the magnified version of the control inputs u ˆ1 and u ˆ2 shown in Figures 7 and 9 it is observed that inspite of the actuator saturation constraint control signals return to linear behaviour faster than the uncompensated system. It is also worth noting that the Riccati inequality based anti-windup compensator design scheme cannot minimize the γ-value below the H∞ norm of the plant G(s). In

5th International Conference on Advances in Control and 620 Triparna Ghosh et al. / IFAC PapersOnLine 51-1 (2018) 615–620 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

control input 2

0 −50 −100 −150 −200 0

50

time (s)

100

150

Fig. 9. The required control input u ˆ2 for the closed-loop system. 20

control input 2

10 0 −10 −20 −30 −40 −50 45

50

55 time (s)

60

65

Fig. 10. The magnified version of the u ˆ2 plot during the saturation period starting from t = 50s subject to a unit step reference input in both the channels. this example, G(s)∞ = 16.2462 and eventually the LMIs (10) and (11) yield the minimized γ = 16.2474, i.e., γmin > G(s)∞ . 6. CONCLUSION A Riccati inequality based full-order anti-windup (AW) compensator design method for stable LTI plants is proposed in this paper following the Weston-Postlethwaite’s scheme. The right coprime factorization of the plant has been resorted to design the disturbance filter N (s) = G(s)M (s) so that the poles of G(s) do not appear in N (s). Although the AW compensator design is independent of the design of the nominal controller K(s), but it is investigated that the AW compensation scheme does not work satisfactory when the actuator saturation continues for a long time. In fact, in case of constant input tracking, immense care must be taken to design a nominal PI controller so that the uncompensated plant response can be recovered once the saturation period is over. In future work, the disturbance filter N (s) may be designed to have a very low DC-gain so that, in a constant input tracking scheme, N (s) maintains yd at a very low value even when the actuator saturation persists for a long period of time. The present AW scheme can also be extended to incorporate uncertainty/parameter variations of the nominal plant. REFERENCES Boyd, S., Ghaoui, L. El., Feron, E., and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory Studies in Applied Mathematics, SIAM, volume 15, Philadelphia.

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Donnarumma, S., Zacearian, L., Alessandri, A., and Vignolo, S. (2016). Anti-windup synthesis of Heading and Speed Regulators for Ship Control with Actuator Saturation. In Proceedings of IEEE European Control Conference, 1284-1290, Denmark. Doyle, J. C., Smith, R. S., and Enns, D. F. (1987). Control of plants with input saturation nonlinearities. In Proceedings of IEEE American Control Conference, 2147-2152, Minneapolis. Fertik, H. A. and Ross, C. W. (1967). Direct digital control algorithm with anti-windup feature. ISA Transaction, volume 6, 317-328. Galeani, S., Tarbouriech, S., Turner, M., and Zaccarian, L. (2009). A Tutorial on Modern Anti-windup Design. European Journal of Control, volume 3-4, 418-440. Hanus, R., Kinnaert, M., and Henrotte, J. (1987). Conditioning technique, a general anti-windup and bumpless transfer method. Automatica, volume 23(6), 729-739. Hippe, P. (2006). Windup in Control: Its Effects and Their Prevention. Springer, London. Mulder, E. F., Kothare, M. V., and Morari, M. (2001). Multivariable anti-windup controller synthesis using linear matrix inequalities. Automatica, volume 37, 1407-1416. Ofodile, N. A. and Turner, M. C., (2016). Decentralized approaches to anti-windup design with application to quadrotor unmanned aerial vehicles. IEEE Transactions on Control Systems Technology, volume 24(6), 1980-1992. Ofodile, N. A., Turner, M. C., and Sofrony, J. (2016). Alternative approach to anti-windup synthesis for double integrator systems. In Proceedings of IEEE American Control Conference, 5473-5478, Boston, USA. Ofodile, N. A., Turner, M. C., and Ubadike, O. C. (2015). Channel by channel anti-windup design for a class of multivariable systems. In Proceedings of IEEE American Control Conference, 2045-2050, Boston, USA. Sofrony, J. and Turner, M. C. (2015). Anti-windup design for systems with input quantization. In Proceedings of IEEE Conference on Decision and Control, 7586-7591, Japan. Sofrony, J., Turner, M. C., and Postlethwaite, I. (2007). Antiwindup synthesis using Riccati equations. International Journal of Control, volume 80(1), 112-128. Tarbouriech, S., Gercia, G., and Glattfelder, A. H. (2007). Advanced Strategies in Control Systems with Input and Output Constraints. Springer, New York. Teel, A. R. and Zaccarian, L. (2011). Modern Anti-windup Synthesis. Princeton University Press, New Jersey. Torrico, B. C., Andrade, F. V., Pereira R. D. O., and Nogueira, F. G. (2016). Anti-windup dead-time compensation cased on generalized predictive control. In Proceedings of IEEE American Control Conference, 5449-5454, Boston, USA. Turner, M. C. (2017). Positive µ modification as an anti-windup mechanism. Systems & Control Letters, volume 102, 15-21. Turner, M. C. and Postlethwaite, I. (2004). A new perspective on static and low order anti-windup synthesis. International Journal of Control, volume 77(1), 27-44. Walgama, K., Ronnback, S., and Sternby, J. (1992). Generalization of the conditioning technique for anti-windup compensators. IEE Proceedings D (Control Theory and Applications), volume 139(2), 109-118. Walgama, K. and Sternby, J. (1990). Inherent observer property in a class of anti-windup compensators. International Journal of Control, volume 52(3), 705-724. Weston, P. and Postlethwaite, W. (2000). Linear conditioning for systems containing saturating actuators. Automatica, volume 36(9), 1347-1354. Wu, X. and Lin, Z. (2014). Dynamic anti-windup design in anticipation of actuator saturation. International Journal of Robust and Nonlinear Control, volume 24, 295-312.